SERIES ON SEMICONDUCTOR - University of Bathstaff.bath.ac.uk/pysdvs/FYP/microcav.pdf · Series on...

SERIES ON SEMICONDUCTORSCIENCE AND TECHNOLOGY

Series Editors

R.J. Nicholas University of OxfordH. Kamimura University of Tokyo

Series on Semiconductor Science and Technology

1. M. Jaros: Physics and applications of semiconductormicrostructures

2. V. N. Dobrovolsky and V. G. Litovchenko: Surface electronictransport phenomena in semiconductors

3. M. J. Kelly: Low-dimensional semiconductors4. P. K. Basu: Theory of optical processes in semiconductors5. N. Balkan: Hot electrons in semiconductors6. B. Gil: Group III nitride semiconductor compounds: physics and applications7. M. Sugawara: Plasma etching8. M. Balkanski and R. F. Wallis: Semiconductor physics and applications9. B. Gil: Low-dimensional nitride semiconductors

10. L. Challis: Electron–phonon interaction in low-dimensional structures11. V. Ustinov, A. Zhukov, A. Egorov, N. Maleev: Quantum dot lasers12. H. Spieler: Semiconductor detector systems13. S. Maekawa: Concepts in spin electronics14. S. D. Ganichev and W. Prettl: Intense terahertz excitation of semiconductors15. N. Miura: Physics of semiconductors in high magnetic fields16. A. Kavokin, J. J. Baumberg, G. Malpuech, F. P. Laussy: Microcavities

Microcavities

Alexey V. Kavokin and Jeremy J. BaumbergSouthampton University

Guillaume MalpuechUniversité Blaise Pascal

Fabrice P. LaussyUniversidat Autónoma de Madrid

1

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PREFACE

Effects originating from light–matter coupling have stimulated the development of op-tics for the last three centuries. Nowadays, the limits of classical optics can be reachedin a number of solid-state systems and quantum optics has become an important tool forunderstanding and interpreting modern optical experiments. Rapid progress of crystal-growth technology in the twentieth century allows the realization of crystal microstruc-tures that have unusual and extremely interesting optical properties. This book addressesthe large variety of optical phenomena taking place in confined solid-state structures:microcavities. Microcavities serve as building blocks for many opto-electronic devicessuch as light-emitting diodes and lasers. At the edge of research, the microcavity rep-resents a unique laboratory for quantum optics and photonics. The central object ofstudies in this laboratory is the exciton-polariton: a half-light, half-matter quasiparti-cle exhibiting very specific properties and playing a key role in a number of beautifuleffects including parametric scattering, Bose–Einstein condensation, superfluidity, su-perradiance, entanglement, etc. At present, hundreds of research groups throughout theworld work on fabrication, optical spectroscopy, theory and applications of microcav-ities. The progress in this interdisciplinary field at the interface between optics andsolid-state physics is extremely rapid. We expect the appearance of a new generation ofopto-electronic devices based on microcavities in the 2010s.

Both rich fundamental physics of microcavities and their intriguing potential appli-cations are addressed in this book, oriented to undergraduate and postgraduate studentsas well as to physicists and engineers. We describe the essential steps of developmentof the physics of microcavities in their chronological order. We show how differenttypes of structures combining optical and electronic confinement have come into playand were used to realize first weak and later strong light–matter coupling regimes. Wediscuss photonic crystals, microspheres, pillars and other types of artificial optical cavi-ties with embedded semiconductor quantum wells, wires and dots. We present the moststriking experimental findings of the recent two decades in the optics of semiconductorquantum structures.

The first chapter of this book contains an overview of microcavities. We present thevariety of semiconductor, metallic and dielectric structures used to make microcavitiesof different dimensions and briefly present a few characteristic optical effects observedin microcavities.

The next two chapters (2 and 3) are devoted to the fundamental principles of opticsessential for understanding optical phenomena in microcavities. We provide overviewsof both classical and quantum theory of light, discuss the coherence of light, its polar-ization, statistics of photons and other quantum characteristics. The reader will find herethe basic principles of the transfer matrix technique allowing for easy understanding oflinear optical properties of multilayer structures as well as the basics of the second quan-tization method. We consider planar, cylindrical and spherical optical cavities, introduce

v

vi PREFACE

the whispering-gallery modes and Mie resonances.In Chapters 4 and 5 we give the theoretical background for the most important light–

matter coupling effects in microcavities considered from the point of view of classical(Chapter 4) and quantum (Chapter 5) optics. We formulate the semiclassical non-localdielectric response theory and study the dispersion of exciton-polaritons in microcavi-ties. As an important toy model we consider a single exciton state coupled to a singlelight mode and study many variations of it. We describe important nonlinear effectsknown in atomic cavities (such as the Mollow triplet).

In Chapter 6 we discuss the physics of weak coupling, when interaction of the lightfield with the exciton acts as a perturbation on its state and energy. We discuss thePurcell effect that symbolizes this regime and lasing as its most important application.We also describe nonlinear effects such as bistability.

Chapter 7 addresses the resonant optical effects in the strong-coupling regime. Weoverview the most spectacular experimental discoveries in this area and present thequasimode model of parametric amplification of light. We also discuss the quantumproperties of optical parametric oscillators based on microcavities.

Chapters 8 and 9 discuss the future of microcavities. Chapter 8 is devoted to theBose–Einstein condensation of exciton-polaritons and polariton lasing. At the time ofwriting, polariton-lasers remain more a theoretical concept than commercial devices butwe believe that in a few years they will become a reality. Thus, for the first time, Bosecondensation would be observed at room temperature and used for the creation of a newgeneration of opto-electronic devices. The path toward this breathtaking perspectiveand the most serious obstacles in the way that are not yet overcome, are tackled in thisChapter.

The subject of Chapter 9 is “spin-optronics”: a new subfield of solid-state opticsthat emerged very recently due to the discoveries made in microcavities and other quan-tum confined semiconductor structures. How to manipulate the polarization of lighton a nanosecond and micrometre scale? What would be the polarization properties ofpolariton-lasers and which mechanisms govern spin-relaxation of exciton-polaritons?These questions are treated in this chapter.

The glossary is addressed to a non-specialist who is searching for the qualitativeunderstanding of the physics of microcavities or to any reader who has no time to gothrough the entire book but needs a simple and concise answer to one of the specificquestions related to microcavities. In the glossary, a number of important relevant issuesare treated without any equations on a simple and accessible level for the general reader.We pay special attention to explanation of terms frequently used in this field of physics,for example, “exciton-polariton”, “Rabi splitting”, “strong coupling”, “Bragg mirror”,“VCSEL”, “photonic crystal”, etc.

The book is intended as a working manual for advanced or graduate students andnew researchers in the field. It is written to a high standard of scientific and mathematicalaccuracy, but to allow an agreeable reading through the essential points unhampered bydetails, many sophistications or difficulties, as well as side issues or extensions, havebeen relegated to footnotes. These would be most profitably considered in a secondreading.

PREFACE vii

Exercises are sprinkled throughout the text and are an important part of it. Theyshould be read as a minimum, for otherwise the notions they introduce will be missingfor later development of the regular text. Starred exercises are straightforward or sys-tematic, those doubled starred are conceptually challenging or require involved compu-tations, those tripled starred are difficult and almost qualify as research problems. Weuse the international system of units, while in numerical examples the energies will begiven in electron-volts and the distances will be given in micrometres or nanometres.An extensive bibliography used throughout the text appears at the end of the text inHarvard format (identified by first author and date of publication).

Microcavities represent a young and rapidly developing field of physics. Our bookcovers the state-of-the-art in this field in the first half of 2006 observed from the prismof personal experience of four authors who have actively worked in the physics of mi-crocavities for a large part of their scientific lives. We wanted to give a personal touchto this book and we do not claim to be objective, which at this stage of the field is verydifficult. We shall be very grateful for any feedback, comments and critical remarksfrom our readers!

November 2006,The authors.

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ACKNOWLEDGEMENTS

This book owes much to our collaboration with active researchers in the field. It isour pleasure to express our gratitude to Yuri Rubo, Ivan Shelykh, Kirill Kavokin, Pav-los Lagoudakis, Mikhail Glazov, Maurice Skolnick, David Whittaker, Daniele Sanvitto,Pavlos Savvidis, David Lidzey, Elena del Valle and Carlos Tejedor.

ix

Alexey KavokinTo Sofia Kavokina,

Jeremy BaumbergTo Melisia Murry,

Guillaume MalpuechTo Anne Tournadre,

Fabrice LaussyTo my father, Raymond Laussy.

(1953–2006)

CONTENTS

1 Overview of Microcavities 11.1 Properties of microcavities 2

1.1.1 Q-factor and finesse 21.1.2 Intracavity field enhancement and field distribution 31.1.3 Tuneability and mode separation 31.1.4 Angular mode pattern 41.1.5 Low-threshold lasing 41.1.6 Purcell factor and lifetimes 51.1.7 Strong vs. weak coupling 5

1.2 Microcavity realizations 51.3 Planar microcavities 6

1.3.1 Metal microcavities 81.3.2 Dielectric Bragg mirrors 9

1.4 Spherical mirror microcavities 101.5 Pillar microcavities 121.6 Whispering-gallery modes 15

1.6.1 Two-dimensional whispering galleries 161.6.2 Three-dimensional whispering-galleries 18

1.7 Photonic-crystal cavities 191.7.1 Random lasers 20

1.8 Plasmonic cavities 201.9 Microcavity lasers 211.10 Conclusion 21

2 Classical description of light 232.1 Free space 24

2.1.1 Light-field dynamics in free space 242.2 Propagation in crystals 27

2.2.1 Plane waves in bulk crystals 272.2.2 Absorption of light 312.2.3 Kramers–Kronig relations 32

2.3 Coherence 322.3.1 Statistical properties of light 322.3.2 Spatial and temporal coherence 332.3.3 Wiener–Khinchin theorem 382.3.4 Hanbury Brown–Twiss effect 41

2.4 Polarization-dependent optical effects 432.4.1 Birefringence 432.4.2 Magneto-optical effects 44

2.5 Propagation of light in multilayer planar structures 45

xiii

xiv CONTENTS

2.6 Photonic eigenmodes of planar systems 492.6.1 Photonic bands of 1D periodic structures 52

2.7 Planar microcavities 592.8 Stripes, pillars, and spheres: photonic wires and dots 64

2.8.1 Cylinders and pillar cavities 662.8.2 Spheres 69

2.9 Further reading 73

3 Quantum description of light 753.1 Pictures of quantum mechanics 76

3.1.1 Historical background 763.1.2 Schrodinger picture 763.1.3 Antisymmetry of the wavefunction 853.1.4 Symmetry of the wavefunction 863.1.5 Heisenberg picture 883.1.6 Dirac (interaction) picture 93

3.2 Other formulations 953.2.1 Density matrix 953.2.2 Second quantization 973.2.3 Quantization of the light field 99

3.3 Quantum states 1003.3.1 Fock states 1003.3.2 Coherent states 1013.3.3 Glauber–Sudarshan representation 1023.3.4 Thermal states 1033.3.5 Mixture states 1053.3.6 Quantum correlations of quantum fields 1063.3.7 Statistics of the field 1103.3.8 Polarization 113

3.4 Outlook on quantum mechanics for microcavities 1153.5 Further reading 116

4 Semiclassical description of light–matter coupling 1174.1 Light–matter interaction 118

4.1.1 Classical limit 1184.1.2 Einstein coefficients 120

4.2 Optical transitions in semiconductors 1234.3 Excitons in semiconductors 127

4.3.1 Frenkel and Wannier–Mott excitons 1274.3.2 Excitons in confined systems 1314.3.3 Quantum wells 1324.3.4 Quantum wires and dots 135

4.4 Exciton–photon coupling 1374.4.1 Surface polaritons 1404.4.2 Exciton–photon coupling in quantum wells 142

CONTENTS xv

4.4.3 Exciton–photon coupling in quantum wires and dots 1474.4.4 Dispersion of polaritons in planar microcavities 1504.4.5 Motional narrowing of cavity polaritons 1604.4.6 Microcavities with quantum wires or dots 164

5 Quantum description of light–matter coupling in semiconductors 1695.1 Historical background 1705.2 Rabi dynamics 1705.3 Bloch equations 173

5.3.1 Full quantum picture 1765.3.2 Dressed bosons 179

5.4 Lindblad dissipation 1875.5 Jaynes–Cummings model 1925.6 Dicke model 1985.7 Excitons in semiconductors 199

5.7.1 Quantization of the exciton field 2005.7.2 Excitons as bosons 2025.7.3 Excitons in quantum dots 202

5.8 Exciton–photon coupling 2085.8.1 Dispersion of polaritons 2105.8.2 The polariton Hamiltonian 2115.8.3 Coupling in quantum dots 213

6 Weak-coupling microcavities 2156.1 Purcell effect 216

6.1.1 The physics of weak coupling 2166.1.2 Spontaneous emission 2176.1.3 The case of QDs, 2D excitons and 2D electron–hole pairs 2196.1.4 Fermi’s golden rule 2206.1.5 Dynamics of the Purcell effect 2236.1.6 Case of QDs and QWs 2256.1.7 Experimental realizations 226

6.2 Lasers 2286.2.1 The physics of lasers 2296.2.2 Semiconductors in laser physics 2336.2.3 Vertical-cavity surface-emitting lasers 2366.2.4 Resonant-cavity LEDs 2406.2.5 Quantum theory of the laser 241

6.3 Nonlinear optical properties of weak-coupling microcavities 2466.3.1 Bistability 2476.3.2 Phase matching 249

6.4 Conclusion 249

7 Strong coupling: resonant effects 2517.1 Optical properties background 252

xvi CONTENTS

7.1.1 Quantum well microcavities 2527.1.2 Variations on a theme 2547.1.3 Motional narrowing 2567.1.4 Polariton emission 256

7.2 Near-resonant-pumped optical nonlinearities 2587.2.1 Pulsed stimulated scattering 2587.2.2 Quasimode theory of parametric amplification 2637.2.3 Microcavity parametric oscillators 265

7.3 Resonant excitation case and parametric amplification 2687.3.1 Semiclassical description 2687.3.2 Stationary solution and threshold 2697.3.3 Theoretical approach: quantum model 2707.3.4 Three-level model 2717.3.5 Threshold 274

7.4 Two-beam experiment 2747.4.1 One-beam experiment and spontaneous symmetry breaking 2747.4.2 Dressing of the dispersion induced by polariton condensates 2767.4.3 Bistable behaviour 277

8 Strong coupling: polariton Bose condensation 2798.1 Introduction 2808.2 Basic ideas about Bose–Einstein condensation 280

8.2.1 Einstein proposal 2808.2.2 Experimental realization 2828.2.3 Modern definition of Bose–Einstein condensation 283

8.3 Specificities of excitons and polaritons 2848.3.1 Thermodynamic properties of cavity polaritons 2858.3.2 Interacting bosons and Bogoliubov model 2868.3.3 Polariton superfluidity 2898.3.4 Quasicondensation and local effects 292

8.4 High-power microcavity emission 2948.5 Thresholdless polariton lasing 2978.6 Kinetics of formation of polariton condensates: semiclassical picture 302

8.6.1 Qualitative features 3028.6.2 The semiclassical Boltzmann equation 3058.6.3 Numerical solution of Boltzmann equations, practical aspects 3078.6.4 Effective scattering rates 3078.6.5 Numerical simulations 308

8.7 Kinetics of formation of polariton condensates: quantum picture in theBorn–Markov approximation 310

8.7.1 Density matrix dynamics of the ground-state 3128.7.2 Discussion 3168.7.3 Coherence dynamics 317

CONTENTS xvii

8.8 Kinetics of formation of polariton condensates: quantum picture beyondthe Born–Markov approximation 319

8.8.1 Two-oscillator toy theory 3198.8.2 Coherence of polariton-laser emission 3298.8.3 Numerical simulations 3358.8.4 Order parameter and phase diffusion coefficient 336

8.9 Semiconductor luminescence equations 3388.10 Claims of exciton and polariton Bose–Einstein condensation 3418.11 Further reading 342

9 Spin and polarization 3459.1 Spin relaxation of electrons, holes and excitons in semiconductors 3469.2 Microcavities in the presence of a magnetic field 3519.3 Resonant Faraday rotation 3529.4 Spin relaxation of exciton-polaritons in microcavities: experiment 3559.5 Spin relaxation of exciton-polaritons in microcavities: theory 3609.6 Optical spin Hall effect 3649.7 Optically induced Faraday rotation 3669.8 Interplay between spin and energy relaxation of exciton-polaritons 3689.9 Polarization of Bose condensates and polariton superfluidity 3729.10 Magnetic-field effect and superfluidity 3749.11 Finite-temperature case 3789.12 Spin dynamics in parametric oscillators 3819.13 Classical nonlinear optics consideration 3819.14 Polarized OPO: quantum model 3839.15 Conclusions 3859.16 Further reading 386

Glossary 387

A Linear algebra 395

B Scattering rates of polariton relaxation 399B.1 Polariton–phonon interaction 399

B.1.1 Interaction with longitudinal optical phonons 400B.1.2 Interaction with acoustic phonons 401

B.2 Polariton–electron interaction 402B.3 Polariton–polariton interaction 404

B.3.1 Polariton decay 404B.4 Polariton–structural-disorder interaction 405

C Derivation of the Landau criterion of superfluidity and Landau formula 407

D Landau quantization and renormalization of Rabi splitting 409

References 413

1

OVERVIEW OF MICROCAVITIES

In this chapter we provide an overview of microcavities. We present thevariety of semiconductor, metallic and dielectric structures used to makemicrocavities of different dimensions and briefly present a fewcharacteristic optical effects observed in microcavities. Many importanteffects mentioned in this chapter are discussed in greater extent in thefollowing chapters.

2 OVERVIEW OF MICROCAVITIES

A microcavity is an optical resonator close to, or below the dimension of the wavelengthof light. Micrometre- and submicrometre-sized resonators use two different schemes toconfine light. In the first, reflection off a single interface is used, for instance froma metallic surface, or from total internal reflection at the boundary between two di-electrics. The second scheme is to use microstructures periodically patterned on thescale of the resonant optical wavelength, for instance a planar multilayer Bragg re-flector with high reflectivity, or a photonic crystal. Since confinement by reflection issometimes required in all three spatial directions, combinations of these approachescan be used within the same microcavity. In this chapter we will explore a number ofthe basic microcavity designs, and contrast their strengths and weaknesses. For practi-cal purposes in this book, the discussion of microcavities will be limited to cavities inwhich confining dimensions are below 100 µm.

Fig. 1.1: (a) single interface reflection and (b) interference from multiple interfaces.

1.1 Properties of microcavities

To help survey microcavity designs it is helpful to motivate a comparison of differentoptical properties of a microcavity. We assume in this section a microcavity with totalpower reflectivity R, and round-trip optical length L. The resonant optical modes withina microcavity have characteristic lineshapes, wavelength spacings and other propertiesthat control their use. A longitudinal resonant mode has an integral number of half-wavelengths that fit into the microcavity, while transverse modes have different spatialshape. However, in a microcavity this traditional distinction can lose its precision asmodes all exist on the same footing.

1.1.1 Q-factor and finesse

The quality-factor (or Q-factor) has the same role in an optical cavity as in an LCRelectrical circuit, in that it parametrizes the frequency width of the resonant enhance-ment. It is simply defined as the ratio of a resonant cavity frequency, ωc, to the linewidth(FWHM) of the cavity mode, δωc:

Q =ωc

δωc. (1.1)

The finesse of the cavity is defined as the ratio of free spectral range (the frequencyseparation between successive longitudinal cavity modes) to the linewidth (FWHM) ofa cavity mode (see Fig. 1.2):

PROPERTIES OF MICROCAVITIES 3

F =∆ωc

δωc=

π√

R

1−R. (1.2)

The Q-factor is a measure of the rate at which optical energy decays from within thecavity (from absorption, scattering or leakage through the imperfect mirrors) and whereQ−1 is the fraction of energy lost in a single round-trip around the cavity. Equivalently,the exponentially decaying photon number has a lifetime given by τ = Q/ωc.

Fig. 1.2: (a) longitudinal mode has integral wavelengths along the main cavity axis, giving (b) mode spectrum.

Because the mode frequency separation ∆ωc = 2πcL is similar to the cavity mode

frequency in a wavelength-scale microcavity, the finesse and the Q-factor are not verydifferent. This is not the situation for a large cavity, in which case the Q-factor becomesmuch greater than the finesse because of the long round-trip propagation time. Instead,the finesse parametrizes the resolving power or spectral resolution of the cavity.

1.1.2 Intracavity field enhancement and field distribution

The on-resonance optical intensity enhancement is given by

Iintracavity

Iincident 1

1−R=

F

π√

R, (1.3)

assuming the mirror losses dominate the finesse. In a travelling-wave cavity this willbe uniformly distributed. However in a standing-wave microcavity, this enhancementis found in the form of spatially localized interference peaks. Hence, it is not alwayssimple to couple an emitter directly to this enhanced optical field. The enhanced opticalfield inside the microcavity can be usefully harnessed for enhanced nonlinear opticalinteractions.

1.1.3 Tuneability and mode separation

The separation of longitudinal modes in a microcavity, ∆ωc, is inversely proportional tothe cavity length. However, cavities other than confocal cavities have transverse opticalmodes at different frequencies, and these also scale similarly with cavity length. Hence,microcavities have far fewer optical modes in each region of the spectrum than macro-scopic cavities. This can mean that specifically tuning the cavity mode to a particularemission wavelength becomes more important than in large cavities.

Various techniques for spectral tuning of modes have been advanced, however, noneis as yet ideal due to the difficulty of modifying the round-trip phase by π without intro-ducing extra loss. The simplest way is to scan the cavity length, however, in cases where


this can be altered, it becomes more difficult to then maintain consistently a fixed lengthonce the desired tuning has been reached. For many other monolithic systems, tuning ofthe cavity modes is extremely difficult, and is most advanced for semiconductor lasers(see Section 1.5).

1.1.4 Angular mode pattern

Microcavities are typically small in all three spatial directions, with aspect ratios closerto unity than macroscopic cavities. As a result, the angular mode emission patterns,which are Fourier related to the cavity mode spatial distribution, tend to have a widerangular acceptance. This means that microcavities emit into a large solid angle on reso-nance.

On the other hand, emission from a microcavity is still beamed into particular di-rections. For instance, by embedding an LED active region within a planar microcavity,the light is emitted in a forward-directed cone when the electroluminescence is resonantwith the cavity mode.

1.1.5 Low-threshold lasing

There are two reasons why microcavities can have lower lasing thresholds: their reducednumber of optical modes, and their reduced gain volume.

In a microcavity, an embedded emitter has a reduced range of optical states intowhich it is likely to emit. In free space it can emit into any solid angle and frequency, butthe microcavity acts to structure the optical density of states around the emitter. This isa particularly strong effect when the emitter has a large linewidth (e.g. an electron–holepair in a semiconductor) as the spectral overlap with different cavity modes is reducedby reducing the size of the cavity. Also in a microcavity, the angular acceptance of anyparticular microcavity mode is much larger. Because the lasing threshold occurs at thepoint at which a spontaneously emitted photon returns to the emitter and stimulates thenext photon emission, reducing the number of cavity modes has the effect of reducingthe laser threshold because spontaneously emitted photons are more likely to return tothe emitter. This effect is contained within a spontaneous emission coupling factor, β,which is defined as the fraction of the total spontaneous emission rate that is emitted intoa specific (laser) mode. This is typically below 10−5 in bulk lasers, but can be > 10%in microcavities. The laser threshold is given by

Pthr =ω2

c

2Qβ. (1.4)

A small optical cavity means that it contains a smaller volume that is pumped electri-cally or optically to provide gain. Because such systems have to be pumped so that oneof the energy levels is brought into inversion, the total energy needed to reach inversionscales with the volume of active material.

As a result of these two effects, microcavities have the smallest known thresholdsof any laser, having now reached the state of a single-photon intracavity field thresholdlevel—the first photon emitted turns the laser on.

MICROCAVITY REALIZATIONS 5

1.1.6 Purcell factor and lifetimes

Embedding an emitter inside a microcavity can lead to additional effects due to thechange in the optical density of states. When the emitter linewidth is smaller than that ofthe cavity mode (δλc), the emitter can be considered to couple to an optical continuumand the emission kinetics is given by Fermi’s golden rule. This describes the emissionlifetime, τ , modified from free space (τ0), in terms of the detuning between the emitter(λe) and cavity:

τ0

τ= FP

2

3

|E(r)|2|Emax|2

δλ2c

δλ2c + 4(λc − λe)2

+ f , (1.5)

which is controlled by the Purcell factor given by

FP =3

4π2

λ3c

n3

Q

Veff(1.6)

where n is the refractive index of the cavity, Veff is the effective volume of the mode,E(r) is the field amplitude in the cavity and |Emax|2 is the maximum of its intensity. Theconstant f in eqn (1.5) describes the losses into leaky modes, as discussed in Chapter 6.

The crucial ratio Q/Veff allows the emitter to emit much faster into the optical field(if both spectral and spatial overlaps are optimized). It also allows decay to be sup-pressed, though competition with non-radiative recombination generally means that thisis accompanied by a decrease in emission efficiency. Typically, the ratio Q/Veff is diffi-cult to enhance arbitratily since smaller cavities often have restrictions in the maximumQ-factor that is possible. The theory of Purcell effect is presented in Chapter 6.

1.1.7 Strong vs. weak coupling

If a resonant absorber is embedded inside a microcavity, then another new regime ofoptical physics can be reached when the absorption strength is large and narrow bandenough. If the total scattering rates of both the cavity photons and the excited absorberare less than the rate at which they couple with each other, new mixed light modes calledpolaritons will result. Spectrally tuning the absorber to the cavity resonance leads tomixing of the photon and absorber, resulting in new polariton states at higher and lowerenergies. This effect will be dealt with in detail in Chapters 4 and 5.

The condition for strong coupling is thus that the light–matter-induced splitting be-tween the new polariton modes (known as the Rabi splitting, Ω) is greater than thelinewidths of either cavity photon (γc, controlled by the finesse) or the exciton (γx, con-trolled by the inhomegeneous broadening of the excitons in the sample). On resonance,the new polaritons are half-light, half-matter, with wavefunctions that are superposi-tions:

ψ =1√2ψX ± ψC , (1.7)

where ψX , ψC are the exciton and photon wavefunctions, respectively.

1.2 Microcavity realizations

The most common microcavity is the planar microcavity in which two flat mirrors arebrought into close proximity so that only a few wavelengths of light can fit in between


Fig. 1.3: Strong coupling of excitons inside a QW microcavity, with repulsion of initial cavity photon (C)and exciton (X) states producing upper (UP) and lower (LP) polaritons.

them. To confine light laterally within these layers, a curved mirror or lens can be in-corporated to focus the light, or they can be patterned into mesas.

An alternative approach for microcavities uses total internal reflection within a highrefractive index convex body, to produce whispering-gallery modes that can exist withinspheres (3D modes) or disks (2D modes), or more complicated topological structures.

Finally, photonic crystals employ periodic patterning in 2 or 3 dimensions to confinelight to a small volume surrounding a defect of the structure.

The key issues that should be borne in mind when considering microcavities are:

• their optical losses or finesse

• coupling to incident light

• optical mode volume

• fabrication complexity and tolerance

• incorporation of active emitters, and

• practicality of electrical contacting.

1.3 Planar microcavities

The well-known Fabry–Perot cavity comprised of two plane mirrors can perform effec-tively when the mirror separation, L, is only a few wavelengths of light. The resultingcavity modes are equally spaced in frequency, apart from shifts caused by the varia-tion with wavelength in the phase change on reflection on each of the mirrors. TheseFabry–Perot modes have a characteristic dispersion in their frequency as the angle ofincidence, θ, is increased. Essentially, the condition for constructive interference afterone round-trip enforces a condition on the wavevector k⊥ = k cos θ perpendicular tothe mirror surfaces: at higher angles of incidence an additional wavevector parallel tothese surfaces means that the total k is larger than 2π/λ and hence the cavity frequencyincreases.

k⊥ × 2L = 2mπ , (1.8)

hence

ω =mπc/L√n2 − sin2 θ

, (1.9)

PLANAR MICROCAVITIES 7

Charles Fabry (1867–1945) and Alfred Perot (1863–1925).

Fabry developed the theory of multibeam interferences at the heart of the Fabry–Perot interferometer.He published 15 articles derived from his invention with Perot in the course of 1896–1902, applying itwith great success to spectroscopy, metrology and astronomy. Alone or with others, he derived from theinterferometer a system of spectroscopic standards, demonstrated Doppler broadening in the emission line ofrare gases and, in 1913, evidenced the ozone layer in Earth’s upper atmosphere. When he was elected to achair at the French “Academie des sciences” in 1927, he gathered 51 votes whereas all other candidates cameout with only one, including Paul Langevin. An enthusiastic teacher and popularizer of science, some of hislectures were so popular that the doors had to be closed for lack of space half an hour before the beginning.He is quoted as having said “My whole existence has been devoted to science and to teaching, and these twointense passions have brought me very great joy.”

Perot did not climb to the same fame as Fabry beside their joint naming of the interferometer, and the mostvalued source of information about his life is in fact the obituary written for him by Fabry, in “Alfred Perot”,Astrophys. J. 64, 208 (1926). While his colleague and friend—apart from the theory—would also carry outmost of the measurements and calculations, Perot was mainly involved in the design and construction of theapparatus where he deployed great skills that brought to the system immediate fame. He also initiated theproject by consulting Fabry on a problem of spark discharges of electrons passing between close metallicsurfaces. He later developed an interest in experimental testing of general relativity with some positiveoutcomes but a final failure to evidence a gravitational redshift.

Further informations can be found in sources compiled by J. F. Mulligan for the centenary anniversary of theinterferometers in “Who were Fabry and Perot?”, Am. J. Phys. 66, 9, (1998).

where n is the average refractive index of the microcavity and c is the speed of light.1

Planar microcavities illuminated with plane waves of infinite extent in the planeof the mirrors do not have any additional modes confined inplane. Such a situation ispractically unrealistic, and there is always some limit to the lateral extent either fromthe size of the mirrors, the width of the illuminating beam, or aperturing effects insidethe microcavity. The natural basis set for planar microcavities are wavelets of extentin both real and momentum space. One useful variety are Airy modes, because theseare anchored around a particular point on the mirror (which we normally impose byilluminating or detecting at a particular position). However, for most purposes plane

1The speed of light in vacuum is exactly 299 792 458 ms−1, as it is, since 1983, a definition rather than ameasurement. In turn, the unit of the SI system, the metre, is defined as the distance light travels in a vacuumin 1/299 792 458 of a second. The second itself is defined as the duration of 9 192 631 770 periods of theradiation in the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Itslabel c is for “constant”. We shall use this notation throughout the text without further mentioning it.


waves in the transverse direction are used to describe the field distribution.Microcavities in which the mirrors are not exactly parallel cause incident light to

slowly “walk” towards the region of larger cavity length through multiple reflections,as illustrated in Fig. 1.4. The accelaration of the confined light in this direction can bedirectly tracked in time and space. Thus, this lateral walk-off acts as an extra loss in thecavity modes, reducing the finesse.

Fig. 1.4: Wedged microcavity, showing walk-off of incident light in multiple reflections.

It is helpful to consider approaches to making these devices through comparisonbetween two sorts of reflectors: metals and distributed Bragg reflector (DBR) stacks:

1.3.1 Metal microcavities

The modes of a metal microcavity are limited by the fundamental material parametersof loss and reflectivity in metal films of varying thickness. For wavelengths further intothe infrared this situation improves and the finesse can be high, reaching Q-values of109 for superconducting cavities at microwave frequencies. However, for microcavitiesaround the optical region of the spectrum, the modes have Q < 500 (see Fig. 1.5).

100

80

60

40

20

0

Reflection/T

ransm

issio

n(%

)

200015001000500Wavelength (nm)

Au 25nm /SiO2 590nm / Au 25nm

expt theoryRT

Fig. 1.5: Reflection and transmission of a planar microcavity consisting of a gold-coated 590 nm glass spacer.


The boundary conditions for reflection of light at a metal imply that the optical fieldis nearly zero at the mirrors, while penetration of light into the metal mirrors is smallcompared to the wavelength. Note also that the phase change on reflection from themetal varies with wavelength depending on the dielectric constants (see eqn 2.133).

The ultimate Q-factor for a metal cavity is set by the trade-off between the real andimaginary parts of the dielectric constant, n = n + iκ, which control reflectivity andabsorption.

On the other hand, in a metal microcavity the modal extent can be relatively small,with penetration of light into the barriers limited to the exponential decay of the electricfield in the metal, ∝ exp(−δz) with δ = 2πκ/λ and z the distance along the normal tothe surface.

1.3.2 Dielectric Bragg mirrors

The situation is different when multilayers of many pairs of alternating refractive indexare used to make cavity mirrors (see Section 2.6.1). The complete structure has an extracavity spacer in between the Bragg mirrors, and the whole can be considered a 1Dphotonic crystal cavity with a central defect. A scanning electron micrograph of thecross-section of a typical semiconductor DBR microcavity is shown in Fig. 1.6.

Fig. 1.6: Scanning electron micrograph of GaAs/AlGaAs DBR microcavity on a GaAs substrate, from Sav-vidis et al. (2000).

The finesse of this cavity is set by the reflectivity of each mirror that depends onthe number of pair repeats and the refractive index constrast between the two materialsused. The key condition is that the optical path in each of the layers is a quarter of thedesired centre wavelength of reflection. In this case, the resonant field is maximum atthe dielectric interfaces and there is significant penetration of light into the surroundingmirror stacks (Fig. 1.7(c)). The reflectivity has a central flat maximum, which dropsoff in an oscillating fashion either side of the reflection- or stop-band. The spectralbandwidth of the mirror is set by the refractive index difference between the materials(Section 2.6.1).

The penetration into the mirrors limits the minimum modal length of the DBR mi-crocavity (see eqn 2.124), so the cavity mode volume is larger than for metal Fabry–Perot microcavities. For instance the cavity mode and the first mode of the Bragg mirrorsidebands have similar extents (Figs. 1.7(c) and (d)).

The finesse of these microcavities is ultimately limited by the number of multilayersthat can be conformally deposited, whilst ensuring that roughness or surface cracking,


Fig. 1.7: (a,b) Reflection of a planar DBR microcavity consisting of a top and bottom mirror with 15 and 21repeats of GaAs/AlAs. In (a) a 240 nm-thick bulk GaAs cavity is incorporated, while in (b) 3 InGaAs QWsare incorporated in the same region. (c,d) field distributions corresponding to (a) at wavelengths of (c) 841 nmand (d) 787 nm.

which creates scattering loss, does not increase unduly. Electron-beam evaporation ofdielectrics in general produces the smoothest results for oxides, while MBE of semicon-ductors produces very high quality epitaxial single-crystal Bragg mirrors. Other tech-niques have also been investigated, such as controlled etching of layer porosity in Si,polymer multilayers, and chiral liquid-crystalline phases. Finding suitable materials forBragg mirrors in the UV spectral region is in general more challenging, and the avail-able refractive contrast limits the bandwidth here. In practice, achieving transmissivityratios of 106 on and off the resonance wavelengths is achievable across wide spectralbandwidths.

Some of the earliest active planar microcavities were optically pumped dyes flowingbetween two DBR mirrors. These demonstrated many of the features resulting fromlateral confinement (in this case by the localized optical pump beam) of the microcavitymodes in the weak-coupling regime. By controlling the separation between the mirrors,the cavity mode can be tuned into resonance with the dye emission. The far-field patternshows how the lateral coherence length changes as the stimulated photon emission turnson. By optical pumping two neighbouring positions inside the dye-filled microcavity,the lateral coherence properties could be investigated as a function of their separation.

1.4 Spherical mirror microcavities

In order to fully control the photonic modes in the microcavity the light has to beconfined in the other two spatial directions. The way this is conventionally achievedin macroscopic cavities is to use mirrors with spherical curvature. This can also be

SPHERICAL MIRROR MICROCAVITIES 11

achieved in microcavities, however, new methods have to be utilized to produce spheri-cal optics with a radius of curvature below 100 µm.

Developing standard routes to spherical mirrors and polishing mirrors to ultralowroughness can give reflectivity in excess of R > 0.9999984. These have been success-fully used in 40 µm long cavities for atomic physics experiments, providing Q-factorsexceeding 108. For such extremely narrow linewidths, active stabilization of the cavitylengths is mandatory on the level of several pm using piezoelectric transducers, see thediscussion by Rempe et al. (1992).

Another route suggested has been to trap micrometre-sized air bubbles in coolingglass and then cut such frozen bubbles in two for subsequent coating. This has beensuccessfully used by Cui et al. (2006) to create microcavities with finesse up to severalhundred.

A separate technique has been to develop templating for micrometre-scale mirrors.In this strategy, latex or glass spheres of a selected size are attached to a conducting sur-face. This is followed by electrochemical growth of reflective metals around them. Sub-sequent etching of the templating spheres leaves spherical micro-mirrors with smoothsurfaces. Such mirrors down to 100 nm radius of curvature have been produced, forinstance by Prakash et al. (2004).

Fig. 1.8: Spherical gold mirrors (a,b) templating process, (c) SEM and optical micrograph, of 5 µm diameter,5 µm radius of curvature mirrors, from Prakash et al. (2004).

For all these spherical mirror microcavities, the optical mode spectrum of the micro-cavity becomes fully discretized. In the paraxial approximation these Laguerre–Gaussmodes are given by:

ωnpq =c

L

[n + εn + (2p + q) arctan

√L

R− L

], (1.10)

where n, p, q are integers for the longitudinal, axial, and azimuthal mode indices, R


is the radius of curvature of one mirror (the other mirror is plane) and they are sepa-rated by a distance L. The phase shifts from the mirror reflections are ε(λ). Typicallyin macroscopic-scale cavities the azimuthally symmetric mode symmetry is broken byslight imperfections in the mirror shape and the astigmatic alignment, and horizon-tal/vertical TEmn and TMmn modes result.2 However, for microcavities, the dominantsplitting is due to the breakdown of the paraxial approximation, which assumes all raysare nearly parallel to the optic axis so that sin θ θ. In microcavities, the cylindricalmode symmetry is maintained, while mode splittings are more pronounced for fieldswith different extents in the lateral direction (Fig. 1.9).

Fig. 1.9: (a) Spherical-planar microcavity, L = 10 µm, (b) Cavity mode transmission spectrum vs. radialdistance from centre of planar top mirror, (c) Optical near-field image of mode at ω=1.932 eV, courtesy ofB. Pennington.

Enhancing the finesse further requires deposition of DBR mirrors inside the spheri-cal micro-mirror. This becomes progressively more difficult as the radius of the spheri-cal mirror reduces, and the mode size shrinks laterally. Hence, there is a minimum limitto the mode volume of spherical microcavities.

1.5 Pillar microcavities

Another way to confine the lateral extent of the photonic modes inside planar micro-cavities is to etch them into discrete mesas. Total internal reflection is used to confinethe light laterally, while the confinement vertically is dependent on reflection from DBRmirrors.

For semiconductors with high refractive indices, like GaAs, the lateral confinementis quite strong. One way to think of these microcavities is as waveguides that have reflec-tors at each end. Light propagating at any angle less than 73 to the external wall surfaceis totally internally reflected. Once again the modes have discrete energies, which arefurther separated in frequency as the pillar area is reduced. From the waveguide pointof view, the modes can be labelled as TEMpq . A lowest-order description of the modesin square pillars assumes the electric field vanishes at the lateral surfaces, and hence

ωnpq =c

n

√k20 + k2

x + k2y , (1.11)

2TE and TM denote modes with transverse electric and magnetic field vectors, respectively.

PILLAR MICROCAVITIES 13

Fig. 1.10: Pillar microcavity from an etched planar DBR semiconductor microcavity (left), with emissionmode spectrum (right), from Gerard et al. (1996).

where k0 = 2πn/λ0 denotes the wavevector of the vertical cavity, and in the lateraldirections kx,y = (mp,q + 1)π/D, for a square pillar of side D with lateral photonquantum numbers mp,q ∈ N, labelling the transverse modes.

Typically, DBR planar microcavities are used as the basis of the pillar, and the etch-ing proceeds to just below the central defect spacer. In such structures, further splittingof degenerate modes is observed, due to the effects of pillar shape (square, elliptical,rectangular), strain (in non-centrosymmetric crystals like III-V semiconductors), andimperfections in the perimeter of the pillar.

The discrete modes of a hard apertured circular microcavity correspond to Airymodes, Al, in the angular dispersion,

Al(θ) =2Jl(kL cos θ)

kL cos θ, (1.12)

where Jl is the first-order Bessel function. The angle-dependent far-field coupling andenergies of these modes in larger microcavities falls on top of the inplane dispersionof the planar microcavity, showing the close connection between mode area and energysplitting. Depending on the geometry of the pillars, the modes may be mixed together,producing orthogonal TE/TM modes rather than axial symmetric modes as in the spheri-cal microcavities. More details on the optical modes in cylindrical and spherical cavitiesare given in Chapter 2.

A number of researchers have demonstrated that the growth of thin semiconduc-tor nanowires (such as GaN or ZnO) can produce a similar microcavity effect, withthe waveguide modes confined at each end by the refractive index contrast at air andsubstrate interfaces. In general, however, the smaller the pillar area, the more difficultit is to couple light efficiently into or out of the microcavities. In addition, such pillarmicrocavities are inherently solid, and emitters have to be integrated into the structure.Typically for semiconductors, the diffusion length of carriers is micrometres, and thus


once the pillar diameter is reduced below this, non-radiative relaxation on the surfacesof the pillar dominates the emission process reducing the emission intensities.

Typically, pillar DBR microcavities with diameters of 5 µm show Q factors in ex-cess of 104, and these can be straightforwardly measured by incorporating an emittinglayer such as InAs quantum dots in the centre of the cavity stack. These Q-values arelimited by imperfections in the DBR mirrors (mostly caused by the buildup of strainwithin the mirror stacks). The other main loss from the micropillar is the light that canleak out the sides of the pillar. To improve the finesse of such microcavities, they can becoated with metal around their vertical sides.

The most common active pillar-type microcavities are the so-called vertical-cavitysurface-emitting lasers (VCSELs), which form the basis of a huge and thriving industrydue to their ease of large-scale manufacturing and of in-situ testing of devices in waferform. These VCSELs normally use a combination of pillar etching and further lateralcontrol of the electrical current injection by progressively oxidizing an incorporatedAlAs layer that thus forms an insulating AlOx annulus. The oxide also provides extraphoton confinement (as the refractive index is much lower than the semiconductor inte-rior), improving the coupling of the optical mode with the electron–hole recombinationgain profile.

Tuning of VCSELs has been demonstrated using MEMs (movable external mirror)technologies, in which the upper mirror is suspended above the active cavity and lowerDBR mirror, and can be moved up and down to tune the main cavity mode wavelength.Other cavity tuning schemes use temperature or current through the mirrors to modifythe refractive index of the cavity. Wide bandwidth tuning across hundreds of nanometresin the visible and near-infrared has not yet been achieved in integrated structures, thoughthis would be extremely useful for many applications. More information on VCSELscan be found in Chapter 6.

Fig. 1.11: (a) VCSEL design incorporating oxide apertures, (b) MEMs VCSEL structure with cantileveredtop mirror, from Chang-Hasnain (2000).

WHISPERING-GALLERY MODES 15

1.6 Whispering-gallery modes

When light is incident at a planar interface from a high refractive index n1 to a lowrefractive index n2 medium, it can be completely reflected, provided the angle of inci-dence exceeds the critical angle, θc = arcsin(n2/n1). This total internal reflection canbe used to form extremely efficient reflectors in a microcavity, not dependent on themetal or multilayer properties. Because the light skims around the inside of such a highrefractive index bowl, it resembles the whispering-gallery acoustic modes first noticedby Lord Rayleigh in Saint Paul’s Cathedral in London.

However, in general it is not possible to make extremely small microcavities usingthis principle. The reason is that diffraction plays an increasing role at small scales, andany planar surfaces have to be connected by corners that act as leaks for the diffractinglight.

Whispering-gallery modes can be classified with integers corresponding to radial, n,and azimuthal, l, mode indices. High-Q whispering-gallery modes generally correspondto large numbers of bounces, l, (thus at glancing incidence to the walls) with n ≈ llocalizing the mode near the boundary walls (see Section 2.8.2).

The limit of these structures is a circle or in general a distorted or curved shape. Inthis regime of curved interfaces, the total internal reflection property is modified, andlight can tunnel out beyond a critical distance to escape. A simple way to see this is tomap the curved interface into a straight interface by varying the refractive index radially(in a well-defined way, see Fig. 1.12). This leads to an equation for the radial opticalpotential that for the perpendicular wavevector looks like tunnelling through a triangularbarrier. Solving this leads to the reflectivity per bounce, R:

ln(1−R) = −4πρ

λh[ln(h +

√h2 − 1)−

√1− h−2

], (1.13)

where h = n cos α, α is the angle of incidence on the curved boundary, and ρ is theradius of curvature.

Fig. 1.12: (a) Tunnelling through total internal reflection at curved interface, (b) conformal mapping and (c)optical potential.

This shows the limits of the total internal reflection even for structures that wouldgive high reflectivity at a planar interface. For instance, for micrometre-sized silica


spheres, the reflectivity per round trip drops below 50% when l < 5 bounces.The variants of whispering-gallery microcavities can be classified according to their

geometry, whether the multiple total internal reflections lie in a plane (2D) or circulatealso in the orthogonal direction (3D).

1.6.1 Two-dimensional whispering galleries

For any sufficiently high refractive index convex shape, light can be totally internallyreflected around the boundary. Providing n > 1/ arcsin 30 = 2, triangular whispering-gallery modes are the lowest-order number of bounces possible (Fig. 1.13).

Fig. 1.13: Microdisk triangle laser, with external coupling waveguide, from Lu et al. (2004).

In real structures, the third dimension remains important, and confinement in thisdirection is provided by a waveguide geometry, or DBR reflectors. Hence, the geomet-rical form of 2D whispering galleries is generally a thin disk of high refractive indexmaterial on a low effective refractive index substrate. An extreme form of this geometryis provided by the thumb-tack microcavity, in which a semiconductor disk is undercutand supported by a thin central pillar.

Fig. 1.14: Left: microdisk (lower ring) with upper electrical contact, from Frateschi et al. (1995) and right:microdisk intensity pattern inside disk.

The modes of a circular microcavity take the form of spherical Bessel functionsinside the disk (Section 2.8). As the microcavity becomes smaller, the photonic modeseparation increases, and can be estimated from the simple multibounce model basedon a ray treatment for constructive interference, kL = 2πm, after l bounces, hence

WHISPERING-GALLERY MODES 17

ωlm =m2πc

n2lR sin(π/l) mc

nR, (1.14)

with the disk radius R and refractive index n.As most of the field of the whispering-gallery modes is in the outer part of the disk,

the inside of the disk can be removed to form microring resonators with very similarcharacteristics. The lack of emission from the central part when optically pumped re-duces the background spontaneous emission, thus reducing the laser threshold as longas scattering losses from the whispering-gallery mode are not also increased.

Fig. 1.15: Microring resonator with integrated waveguide coupling, from Xu et al. (2005). On the photo, thegap between the disk and the waveguide is 200 nm, the width of the guide is 450 nm and the diameter of thering is 12 µm.

Since the leakage from a circular microdisk frustrates evanescent coupling to air, itproduces a structure that is extremely hard to selectively couple to and from. Favouredversions have included a waveguide in close proximity to provide selective evanescentcoupling in particular directions (Fig. 1.15). However, such coupling is exponentiallysensitive to the coupling gap between disk and waveguide, giving strict fabrication tol-erance on the nm scale.

To ameliorate this sensitivity, the basic shape of the microdisk can be altered, forinstance to a stadium shape that has bow-tie modes in a ray picture.

Fig. 1.16: Bow-tie resonator, from Nockel (2001).


Such shapes have classical chaotic ray orbits, and are related to a general class of“quantum billards” devices for ballistic electron transport. These have more sharplycurved sections, where most of the output coupling occurs, and hence the microcavityselectively emits in particular directions. A variety of other modified shapes have beenproduced to induce unidirectional, or spatially stabilized output (Fig. 1.17).

Since 2D whispering-gallery structures can be conveniently produced by conven-tional lithography processes, they are suited to dense integration applications. However,the main issue is that of obtaining efficient outcoupling without destroying the highQ-factor, and introducing huge fabrication sensitivity. By using high refractive indexcontrast GaAs on AlOx disks, Q-factors > 104 have been observed in 2 µm diameterwhispering-gallery resonators. Nevertheless, one of the remaining issues is the diffi-culty of tuning the wavelength of these cavity modes, and getting access to the internalelectric field.

Fig. 1.17: Cog microdisk laser, to pin the azimuthal standing wave field, from Fujita and Baba (2002).

1.6.2 Three-dimensional whispering-galleries

Very similar behaviour to the 2D disks is observed for 3D structures. The difference isthat the confinement in the third dimension is now provided not by a thin waveguide,but by additional total internal reflection.

The simplest example is the spherical microcavity, which can be simply formedby melting the end of a drawn optical fibre. The resulting sphere produced by surfacetension of the glass perches on top of the remaining fibre, and for light resonating aroundthe equator forms an excellent high-Q cavity.

Because of the increasing evanescent loss as the curvature increases, the highest Q(≈ 109) cavities are found in the largest spheres (> 100 µm). For micrometre-sizedspheres, Q-factors drop to several 1000, and the mode spectral spacing increases totens of nanometres. Typically, sets of azimuthal modes are seen (with the same l), with

PHOTONIC-CRYSTAL CAVITIES 19

Fig. 1.18: Spherical glass resonator atop optical fibre, with lasing circular whispering-gallery mode, fromVahala (2003).

different out-of-plane mode indices, m, provided by the extra degree of freedom in3D (see Section 2.8). These have also been spatially mapped using scanning near-fieldoptical microscopy showing the way the orbits converge around an equatorial orbit.

As in the 2D microdisks, deformations of the sphere provide another control vari-able for the emission direction and mode symmetry. Investigations of elongated liquiddroplets filled with dye have shown the relation between unstable classical ray orbits anddiffraction. Once again, the most difficult problem is that of efficiently coupling lightinto and out of the whispering gallery. As before, smaller spheres show lower Q-factorsdue to the curvature of the interface that permits light to evanescently escape.

1.7 Photonic-crystal cavities

Photonic crystals arise from multiple photon scattering within periodic dielectrics, andalso exist in 2D and 3D versions. The ideal 3D photonic crystal microcavity would bea defect in a perfect 3D photonic lattice with high enough refractive index contrast thatthere is a bandgap at a particular wavelength in all directions. In principle, this wouldprovide the highest optical intensity enhancement in any microcavity, however, cur-rently no fabrication route has yet demonstrated this. Currently, scattering determinesthe properties of most 3D photonic crystals.

Instead, 2D photonic crystals etched in thin high refractive index membranes haveshown the greatest promise, with the vertical confinement coming from the interfacesof the membrane.

These can show Q-factors exceeding 105 while producing extremely small modevolumes, which are advantageous for many applications. The main issues for such mi-crocavities are the difficulty of their fabrication, the large surface area in proximity tothe active region (which produces non-radiative recombination centres and traps diffus-ing electron–hole pairs), and the difficulty in tuning their cavity wavelengths. However,they currently offer the best intensity enhancement of any microcavity system.


Fig. 1.19: (a) Photonic crystal resonator (5-hole defect) close to a photonic crystal waveguide coupler, (b)emission spectrum from a number of microcavities of different shapes, from Akahane et al. (2005).

1.7.1 Random lasers

Related to regular 3D photonic crystals are media with wavelength-scale strong scat-terers that are randomly positioned. Such photonic structures support photonic modesthat can be localized, with light bouncing around loops entirely inside the medium. Byplacing gain materials such as dyes inside these media, and optically pumping them,so-called “random lasing” can result. While the optical feedback is not engineered inspecific orientations in these devices, it is spontaneously formed by the collection ofrandom scatterers.

1.8 Plasmonic cavities

Recently a new class of microcavity has emerged that is based on plasmons localized tosmall volumes close to metals. For noble metals particularly, a class of localized elec-tromagnetic modes exists at their interface with a dielectric, known as “surface-plasmonpolaritons”. If the metals are textured on or below the scale of the optical wavelength,these plasmons can be localized in all three directions, producing 0D plasmonic modes.Four implementations are of note: flat metals, metallic voids, spherical metal spheres,and coupled metal spheres, in order of increasingly confined optical fields (Fig. 1.20).While plasmons bound to flat metal surfaces are free to move along the surface, theplasmons on nanostructures can be tightly localized.

As in any microcavity, these confined modes can be coupled to other excitationssuch as excitons in semiconductors, and this has now become of interest as the plas-monic spatial extent can be significantly smaller than the wavelength of light, downto below 10 nm. On the other hand, the presence of absorbtion from the metal due to

MICROCAVITY LASERS 21

Fig. 1.20: Plasmon localization: on flat noble metals, metallic voids, metal spheres, and between metalspheres.

plasmon-induced excitation of single electrons places strict limits of the utility of theseplasmonic modes. Strong coupling has recently been observed in such plamonic micro-cavities using organic semiconductors.

1.9 Microcavity lasers

Microcavities can be used as laser resonators, providing the gain is large enough tomake up for the cavity losses. Because of the short round-trip length, the conditionson the reflectivity of the cavity walls are severe. However, these cavities are now thestructure of choice (in the form of VCSELs), due to their ease of integratable manufac-ture and their performance. Small cavity volumes are also advantageous for producinglow threshold lasers as the condition for inversion can be reached by pumping fewerelectronic states. On the other hand the total power produced from a microcavity is ingeneral restricted as eventually the high power density causes problems of thermal load-ing, extra electronic scattering, and saturation. One advantage of a microcavity laser isthe reduced number of optical modes into which spontaneous emission is directed, thusincreasing the probability of spontaneous emission in a particular mode and thus reduc-ing the lasing threshold. This is discussed in detail in Chapter 6. Polariton lasers areexpected to have a lower threshold than VCSELs as they do not require inversion ofpopulation (see Chapter 8). They represent one of the currently hottest subject in thephysics of semiconductors. Their characteristics are discussed in detail in the last twochapters.

1.10 Conclusion

This survey of microcavities shows the wide range of possible designs and blend ofoptical physics, microfabrication and semiconductor engineering that is required to un-derstand them. These will be taken apart and then put back together in the followingchapters.

2

CLASSICAL DESCRIPTION OF LIGHT

In this chapter we introduce the basic characteristics of light modes infree space and in different kinds of optically confined structuresincluding Bragg mirrors, planar microcavities, pillars and spheres. Wedescribe the powerful transfer matrix method that allows for solution ofMaxwell’s equations in multilayer structures. We discuss thepolarization of light and mention different ways it is modified includingthe Faraday and Kerr effects, optical birefringence, dichroism, andoptical activity.

24 CLASSICAL DESCRIPTION OF LIGHT

2.1 Free space

2.1.1 Light-field dynamics in free space

We start from Maxwell’s (1865) equations in a vacuum (setting the charge density dis-tribution ρ and electric current J identically to zero):3

∇∇∇·E(r, t) = 0 , ∇∇∇×E(r, t) = − ∂tB(r, t) , (2.2a)

∇∇∇·B(r, t) = 0 , ∇∇∇×B(r, t) =1

c2∂tE(r, t) , (2.2b)

where E and B are the electric and magnetic fields, respectively, both a function ofspatial vector r and time t. However, the equations are much simpler in reciprocal spacewhere fields are expressed as a function of wavevector instead of position, helping tolay down a simpler mathematical structure. The link between the field as we introducedit E(r, t) and its weight EEE (k, t) in the plane-wave basis eik · r is assured by Fouriertransforms:

E(r, t) =1

(2π)3/2

∫EEE (k, t)eik · r dk , (2.3a)

EEE (k, t) =1

(2π)3/2

∫E(r, t)e−ik · r dr . (2.3b)

James Clerk Maxwell (1831–1879) put together the knowl-edge describing the basic laws of electricity and magnetismto set up the consistent set of equations eqn (2.15) that formthe foundations of electromagnetism.

One of the giants of science, Einstein said of his work itwas the “most profound and the most fruitful that physicshas experienced since the time of Newton.”, Planck that“he achieved greatness unequalled” and Feynman that “themost significant event of the 19th century will be judgedas Maxwell’s discovery of the laws of electrodynamics”. Adeeply religious man, Maxwell composed prayers that werelater found in his notes. He once said that he was thanking“God’s grace helping me to get rid of myself, partially inscience, more completely in society.”

3Maxwell’s equations in the presence of charge and currents are, in MKS units:

∇∇∇·E(r, t) =1

ε0ρ(r, t) , ∇∇∇× E(r, t) = − ∂tB(r, t) , (2.1a)

∇∇∇·B(r, t) = 0 , ∇∇∇× B(r, t) =1

c2∂tE(r, t) +

1

ε0c2J(r, t) . (2.1b)

FREE SPACE 25

The properties of differential operators with respect to Fourier transformation turnMaxwell’s equations (2.2) into the set of local equations:

ik ·EEE (k, t) = 0 , ik× EEE (k, t) = − ∂tBBB(k, t) , (2.4a)

ik ·BBB(k, t) = 0 , ik×BBB(k, t) =1

c2∂tEEE (k, t) , (2.4b)

with obvious notations: the cursive letter and its straight counterpart are Fourier trans-form pairs related the one to the other like eqn (2.3). The locality is a first appealingfeature, as opposed to real space equations (2.2) where the value of a field at a pointdepends on other field values in an entire neighbourhood of this point. The picture clar-ifies further still by separating the transverse ⊥ and longitudinal ‖ components of thefield, EEE (k, t) = EEE ⊥(k, t) + EEE ‖(k, t) (with likewise definitions for other fields), thelongitudinal component at point k being the projection on the unit vector ek ≡ k/k,i.e.,

EEE ‖(k, t) =(ek ·EEE (k, t)

)ek , (2.5)

(with likewise definitions for other fields) and the perpendicular component at point k

being the projection on the plane normal to the unit vector ek, i.e., EEE ⊥(k, t) ≡ EEE (k, t)−EEE ‖(k, t).

Back in real space, E⊥(r, t) and E‖(r, t) are obtained, respectively, by Fouriertransform of EEE ‖(k, t) and EEE ⊥(k, t) as given above, which correspond to the divergence-free and the curl-free components of the field.4 The field is transverse in vacuum (themagnetic field always is) and these transverse components obey the set of coupled linearequations derived from eqn (2.4):

ik× EEE ⊥(k, t) = −∂tBBB(k, t)ic2k×BBB(k, t) = ∂tEEE ⊥(k, t) .

(2.7)

This linear system is diagonalized by introducing the new mode amplitudes a and b as:

a(k, t) ≡ − i

2C (k)(EEE ⊥(k, t)− cek ×BBB(k, t)) , (2.8a)

b(k, t) ≡ − i

2C (k)(EEE ⊥(k, t) + cek ×BBB(k, t)) , (2.8b)

4The benefits of decomposing the field into its transverse and longitudinal components are more com-pelling when charges are taken into account, as is the case in eqn (2.1). Then, the dynamics of the field arisingfrom the interplay between its electric and magnetic components (the transverse part) and the dynamics cre-ated by the sources (responsible for the static field if they are at rest) are consequently clearly separated. Forinstance, dotting eqn (2.5) with k/k2, one gets from the EEE ‖(k, t) = −i(k)k/(ε0k2), which, by Fouriertransformation of both sides, gives the electric field well known from electrostatic:

E‖(r, t) =1

4πε0

Zρ(r′, t)

r − r′

|r − r′|3 dr′ . (2.6)

The fact that eqn (2.6) is local in time is a mathematical artifice: only the whole field E(r, t) has a physicalsignificance and other instantaneous effects from the transverse field correct those of the longitudinal field.


with C a real constant as yet undefined (written as such for later convenience).5 Since E

and B are real, relation (2.3) implies that EEE (−k, t)∗ = EEE (k, t) (the same for BBB) whichallows us to keep only one variable, e.g., a, and write a(−k, t)∗ instead of b(k, t).Inverting, one obtains the expression of the physical fields (in reciprocal space) in termsof its fundamental modes:

EEE ⊥(k, t) = iC (k)(a(k, t)− a∗(−k, t)) (2.11a)

BBB(k, t) =iC (k)

c(ek × a(k, t) + ek × a∗(−k, t)) . (2.11b)

This mathematical formulation has therefore replaced the electric and magneticfields by a set of complex variables a(k, t) (which give the transverse components of thefields) and the phase space variables of sources (ri, ∂tri/m). All quantities of interestthat can be expressed in terms of the fields and the dynamical variables can naturally bewritten with the new set of variables. The energy of the field relevant for our descriptionof the vacuum comes with the Hamiltonian:

H⊥ = ε0

∫C (k)2[a∗(k) · a(k) + a(−k) · a∗(−k)] dk . (2.12)

As a function of this new variable for the field, eqn (2.7) becomes:5∂ta(k, t) = iωa(k, t)∂ta

∗(−k, t) = −iωa∗(−k, t), (2.13)

where ω = ck. This is the main result that asserts that the free electromagnetic fieldis equivalent to a set of harmonic oscillators.6 These oscillators—the complex-valuedvectors a of eqn (2.13)—result from the mathematical manipulations that we have de-tailed. The physical sense attached to such an oscillator is rather meager in its classicalformulation, being at best described as a modal amplitude for the field and thereforechiefly as a mathematical concept. Still, this affords a straightforward canonical quanti-zation scheme.

5The diagonalization of eqn (2.7) can be made by evaluating the crossproduct of k with both sidesof the first line, yielding on the l.h.s. ik × (k × EEE ⊥) = −ik2EEE ⊥ since EEE ⊥ is transverse, and on ther.h.s. −∂t(k ×BBB). Introducing ω = ck, eqn (2.7) then readsj

iωEEE ⊥(k, t) = ∂t(cek ×BBB(k, t))iωcek ×BBB(k, t) = ∂tEEE ⊥(k, t) .

(2.9)

Summing and subtracting both lines yields

∂t(EEE ⊥(k, t) ± cek ×BBB(k, t)) = ±iω(EEE ⊥(k, t) ± cek ×BBB(k, t)) (2.10)

which, when expressed in terms of a(k, t) and b(k, t) = a(−k, t)∗, is the main result (2.13).6Inserting back eqn (2.8) in Maxwell’s equations with source terms, eqns (2.1), the equations of motion

obtained following the same procedure become

∂ta(k, t) = −iωa(k, t) +i

2ε0C (k)JJJ ⊥(k, t) . (2.14)

Generally JJJ ⊥(k, t) depends non-locally on a. Interactions with sources therefore couple together the vari-ous modes of the fields that are otherwise independent.

PROPAGATION IN CRYSTALS 27

2.2 Propagation in crystals

2.2.1 Plane waves in bulk crystals

We now describe propagation in dielectric or semiconductor materials. Maxwell’s equa-tions get upgraded to the following closely related expressions:

∇∇∇·D =ρ

ε0, (2.15a)

∇∇∇·B = 0 , (2.15b)

∇∇∇×E = −1

c∂tB , (2.15c)

∇∇∇×B =1

ε0c2∂tJ +

1

c2∂tD , (2.15d)

where ρ is the free electric charge density7 and J is the free current density.8 In thefollowing, we only describe dielectric or semiconductor materials where ρ = 0 and J =0. The electric displacement field is defined as

D = ε0E + P = εεεE , (2.16)

where P is the dielectric polarization vector and εεε is the dielectric constant. In the mostgeneral case, εεε is a tensor. By a proper choice of the system of coordinates it can berepresented as a diagonal matrix:

εεε =

⎛⎝ε1 0 0

0 ε2 00 0 ε3

⎞⎠ . (2.17)

If the diagonal elements of this matrix are not equal to each other, the crystal is opticallyanisotropic. The effect of optical birefringence specific for optically anisotropic mediais briefly discussed at the end of this section. Throughout this book we always assumeε1 = ε2 = ε3 = ε (unless it is explicitly indicated that we consider an anisotropic case).We largely operate with the quantity n =

√ε/ε0 known as the refractive index of the

medium. In crystals having resonant optical transitions, eqns (2.15) have two types ofsolutions. One of them is given by the condition

∇·E = 0 . (2.18)

It corresponds to the transverse waves having electric and magnetic field vectors per-pendicular to the wavevector. The transverse waves in media are analogous to the lightwaves in vacuum. Another type of solution are the longitudinal waves. For them

ε = 0, ∇·E = 0, H = 0, ∇×E = 0 . (2.19)

In these modes the electric field is parallel to the wavevector, and the magnetic field isequal to zero everywhere. They have no analogy in vacuum (which has ε > 0 at all

7Free electric charges are those that do not include dipole charges bound in the material.8Free current densities are those that do not include polarization or magnetization currents bound in the

material.


Fig. 2.1: Elliptically polarized light. The curve shows the trajectory followedby the electric-field vector of the propagating wave together with its projec-tions on x- and y-axes and on the plane normal to the direction of motionwhere the “elliptical” character of the polarization becomes apparent.

frequencies) but play an important role in resonant dielectric media where the dielectricconstant can vanish at some frequencies. In any case, the solution of eqns (2.15) can berepresented as a linear combination of plane waves where the coordinate dependence ofthe electric field is given by

E(r, t) = E0 exp(i(k · r− ωt)

). (2.20)

k is the wavevector of light, and its modulus obeys

k = nω

c, (2.21)

and E0 is the amplitude of the plane wave. Its vector character is responsible for the po-larization of light. Note that the choice of the sign in the exponential factor of eqn (2.20)is a matter of convention. Note also that the vector can change with time even for afreely propagating plane wave. The longitudinal waves are linearly polarized along thewavevector, by definition. For the transverse waves it is convenient to take the curlof both parts of eqn (2.15c) and substitute the expression for ∇ × B from Maxwell’seqn (2.15d). This yields

∇×∇×E =n2

c2∂tE , (2.22)

therefore, from the identity ∇ × ∇ × A = ∇(∇·A) − ∇2A (for any field A) andsubstituting eqn (2.20), one obtains the wave equation:

∇2E = −k2E . (2.23)

The general form of the polarization for transverse waves can be seen as follows.Consider a plane wave propagating in the z-direction. The vector E0 can have x- andy-components in this case,


E =

(Ex

Ey

)=

(E0x cos(kz − ωt)

E0y cos(kz − ωt + δ)

), (2.24)

with9 δ ∈ [0, 2π[. This vector is the real part of the Jones vector of the polarized lightintroduced in 1941 by R. Clark Jones for concisely describing the light polarization.Since the phase constants can be picked arbitrarily for one component of the vector, weset it to 0 for Ex. After an elementary transformation

Ey

Ey0= cos(kz − ωt + δ) = cos(kz − ωt) cos δ − sin(kz − ωt) sin δ , (2.25)

which yields

Ex

Ex0cos δ − Ey

Ey0= sin(kz − ωt) sin δ =

√1−

(Ex

Ex0

)2

sin δ , (2.26)

squaring, we finally obtain

(Ex

Ex0

)2

+

(Ey

Ey0

)2

− 2Ex

E0x

Ey

E0ycos δ = sin2 δ , (2.27)

which is the equation for an ellipse in the (Ex, Ey) coordinate system, inclined at anangle φ to the x-axis given by

tan(2φ) =2E0xE0y cos δ

E20x − E2

0y

. (2.28)

The general polarization is therefore elliptical. If δ = 0 or π, light is linearly polar-ized, if δ = π

2 or 3π2 and E0x = E0y , it is circularly polarized. We call right circular

polarization denoted σ+ for the case where δ = π2 , and left-circular polarization, de-

noted σ−, for the case where δ = 3π2 .

The electric-field vector of the circularly polarized light rotates around the wavevec-tor in the clockwise direction or anticlockwise directions for σ+ and σ− polarization,respectively (if one looks in the direction of propagation of the wave).

In reality, light is usually composed of an ensemble of plane waves of the form (2.20)with their phases more or less randomly distributed. As a result, light can be partiallypolarized or unpolarized. In this case its intensity I0 > (|E0x|2 + |E0y|2)/2, whilethe equality holds for fully polarized light. It is convenient to characterize the partiallypolarized light by so-called Stokes parameters proposed by the English physicist and

9The notation [a, b[ means all the values between a included and b excluded. It is a common notationin countries such as France or Russia. Another widespread convention uses parenthesis for exclusion, whichwould read [a, b) in our case. Both are an occasional source for confusion but are recognized standards of theISO 31-11 that regulates mathematical signs and symbols for use in physical sciences and technology.


Henri Poincare (1854–1912) and George Gabriel Stokes (1819–1903).

Poincare was a French mathematician and philosopher. The number of significant contributions hemade in numerous fields is overwhelming. Among extraordinary achievements, he discovered the firstchaotic deterministic system and understood correctly ahead of his generation the implication of chaos.He formulated the “Poincare conjecture” one of the most difficult mathematical problems and one of themost important in topology only just recently solved by Grigori Perelman. The Poincare sphere arose inthe context of this conjecture. He was the first to unravel the complete mathematical structure of specialrelativity—christening Lorentz transformations that he showed form a group—though he missed the physicalinterpretation. He never acknowledged Einstein’s contribution, still referring to it by the end of his life asthe “mechanics of Lorentz”. He is also famous for the philosophical debates that opposed him to the Britishphilosopher Bertrand Russel. He liked to change the problem he was working on frequently as he thought thesubconcious would still work on the old problem as he would on the new one, a good match to his famoussaying “Thought is only a flash between two long nights, but this flash is everything.”

Sir George Gabriel Stokes was an Irish mathematician and physicist, also with exceptional productivity in awide arena of science. He is most renowned for his work on fluid dynamics (especially for the Navier–Stokesequation), mathematical physics (with the Stokes theorem) and optics (with his description of polarization).In 1852 he published a paper on frequency changes of light in fluorescence, explaining the “Stokes shift.”His production is even more remarkable given that he kept unpublished many of his first-rate discoveries,like Raman scattering in the aforementioned work, that Lord Kelvin begged him without success to bringto print, or the spectroscopic techniques (in the form of chemical identification by analysis of the emittedlight) which he merely taught to Lord Kelvin (then Sir William Thomson), predating Kirchoff by almost adecade. In a letter he humbly attributed the entire merit to the latter, saying that some of his friends had beenoverzealous in his cause. He had a tumultuous mixture of professional and sentimental feelings (not aidedby the historical context in Cambridge) and it is reported his bride-to-be almost called off the wedding uponreceiving a 55-page letter of the duties he felt obliged to remind her.

mathematician George Gabriel Stokes in 1852. The Stokes parameters S0,1,2,3 are de-fined as a function of the total intensity I0, the intensity of horizontal (x-linear) po-larization I1, the intensity of linear polarization at a 45 angle I2 and the intensity ofleft-handed circularly polarized light I3, and then defining

S0 = 2I0 , (2.29a)

S1 = 4I1 − 2I0 , (2.29b)

S2 = 4I2 − 2I0 , (2.29c)

S3 = 4I3 − 2I0 . (2.29d)

These are often normalized by dividing by S0. For the fully polarized light, they canalso be re-expressed in terms of the electric field as


S0 = E20x + E2

0y , (2.30a)

S1 = E20x − E2

0y , (2.30b)

S2 = 2E0xE0y cos δ , (2.30c)

S3 = 2E0xE0y sin δ . (2.30d)

In this form, it is clear that a relationship exists connecting these parameters:

S21 + S2

2 + S23 = S2

0 . (2.31)

For partially polarized light this equality is not satisfied.The components of the Stokes vector have a direct analogy with the components

of the quantum-mechanical pseudospin introduced for a two-level system, as will beshown in detail in Chapter 9. Condition (2.31) defines a sphere in the (S1, S2, S3) set ofcoordinates, called the Poincare sphere (as originally proposed in 1892 by the Frenchmathematician Henri Poincare).

Partially polarized signals can be represented by augmenting the Poincare spherewith another sphere whose radius is the total signal power, I . The ratio of the radii ofthe spheres is the degree of polarization p = Ip/I . The difference of the radii is theunpolarized power Iu = I − Ip. By normalizing the total power I to unity, the innerPoincare sphere has radius p and will shrink or grow in diameter as the degree of polar-ization changes. Modifications in the polarized part of the signal cause the polarizationstate to move on the surface of the Poincare sphere.

2.2.2 Absorption of light

Absorbing media are characterized by a complex refractive index with a positive imag-inary part in the convention adopted here, see eqn (2.20):

n = n + iκ . (2.32)

Fig. 2.2: Three-dimensional representation of thePoincare sphere. The Stokes parameters consti-tute the cartesian coordinates of the polarizationstate, which is represented by a point on the sur-face of the sphere. The radius of the Poincaresphere corresponds to the total intensity of thepolarized part of the signal.


An electromagnetic wave propagating in a given direction (say, z-direction) is evanes-cent in this case and can be represented as

E =

(E0x cos(nω

c z − ωt)E0y cos(nω

c z − ωt + δ)

)exp(−κ

ω

cz) . (2.33)

The parameter

α = κω

c(2.34)

is known as the absorption coefficient. α−1 is a typical penetration depth of light in anabsorbing medium. It can be as short as a few tens of nanometres in some metals.

2.2.3 Kramers–Kronig relations

The German physicist Ralph Kronig and the Dutch physicist Hendrik Anthony Kramershave established a useful relation between the real and imaginary parts of the dielectricconstant, known as the “Kramers–Kronig relations”:

ε(ω) = ε0 +2

πP

∞∫0

Ωε(Ω)Ω2 − ω2

dΩ , (2.35a)

ε(ω) =2ω

πP

∞∫0

ε(Ω) − ε0

Ω2 − ω2dΩ , (2.35b)

where the above integrals are to be understood in the sense of Cauchy and P denotestheir Cauchy principal value.10 In optics, especially nonlinear optics, these relationscan be used to calculate the refractive index of a material by the measurement of itsabsorbance, which is more accessible experimentally. The link between n and α followsas:

n(ω) = 1 +c

πP∫ ∞

0

α(Ω)

Ω2 − ω2dΩ . (2.36)

2.3 Coherence

2.3.1 Statistical properties of light

When it is mature enough, a physical theory needs to be extended to a statistical de-scription, where lack of knowledge of the system is distributed into probabilities for thevarious “ideal” situations to arise. In the case of light the ideal case would be typicallya sinusoidal wave of well-defined frequency ω, amplitude E0 and phase ϕ, propagatingalong some axis x with wavevector k = ω/c. Disregarding the polarization degree of

10The Cauchy principal value of an integral that presents a certain type of singularity is obtained byexcluding the singularity from the integration by approaching it asymptotically from both sides (on the realaxis).

COHERENCE 33

freedom (for which the argument extends in a straightforward and identical way) thesolution thus reads:

E(x, t) = E0ei(ωt−kx+ϕ) . (2.37)

However, such a perfect case is an ideal limit impossible to meet in practice. Forinstance, in the case where light is generated by an atom, obvious restrictions such as afinite time of emission (within the lifetime of the transition) already spoil the monochro-matic feature (light of a well-defined frequency): the wave has finite extension in timeand therefore a spread in frequency. The uncertainty in time emission results in uncer-tainty of the phase. Taking into account light from an ensemble of atoms brings furtherstatistics and more complications can be, and often must be, also added. Some systemscan by-pass such shortcomings, as is the case of the laser where many atoms emit pho-tons made identical by the stimulated process with the hope of overcoming the finitelifetime of a single atom and providing a continuously emitting medium, thereby re-ducing the spread of frequency. Very small linewidths are indeed achieved in this way,but whatever source of light one considers, there is ultimately always a complicationthat requires a statistical treatment. One universal cause calling for statistics is tempera-ture. In the case of the laser, the complication arises from spontaneous emission, that is,the fact that some atoms of the collective ensemble decay independently of the others,bringing some noise in the system. We shall see, however, that which property of lightis affected can be subject to some control. This suggests that this requirement for statis-tics is ultimately linked to Heisenberg’s uncertainty principle, although the argumenttranslates almost verbatim to classical concepts, as illustrated in Fig. 2.3.

2.3.2 Spatial and temporal coherence

Coherence measures the amount of perturbation—or noise—in the wave, that we shallcall the signal as the notion of coherence lies in information theory. Namely, coherencerefers to the ability of inferring the signal at remote locations (in our case, in space ortime) from its knowledge at a given point. In the upper case of Fig. 2.3, there is fullcoherence as the value at a given single point determines the state of the field entirelyand exactly. In the second case, this is limited in the window where the field is defined(where it is nonzero in this case). If the horizontal axis of the left panel of Fig. 2.3 istime, the corresponding interval defines its coherence time, if it is space, it defines itscoherence length. From the symmetry of time (t) and space (x) in eqn (2.37), the twonotions share not only similar definition and behaviour but are linked the one to the otherby the expression τc = lc/c, with τc and lc the coherence time and length, respectively.A general formula for the coherence time for a wavetrain of spectral width ∆λ centredabout λ is:

τc =λ2

c∆λ. (2.38)

The coherence length of a laser can be as high as hundreds of kilometres. For a dis-charge lamp it reduces to a few millimetres. It is very small given the speed at whichlight travels but it is high enough for experimental measurements by the time of Huy-gens and Fresnel.


0 1 2 3 4

Ele

ctric

fiel

d

ω0t

E=cos(ω0t)

0

0.5

1

-2 -1 0 1 2

Spe

ctra

l sha

pe

ω-ω0

0 1 2 3 4

Ele

ctric

fiel

d

ω0t

E=e-γtcos(ω0t), γ=1

0

0.5

1

-2 -1 0 1 2

Spe

ctra

l sha

pe

ω-ω0

0 1 2 3 4

Ele

ctric

fiel

d

ω0t

E=Σi cos(ωit), <ω>=ω0

0

0.5

1

-2 -1 0 1 2

Spe

ctra

l sha

pe

ω-ω0

Fig. 2.3: Statistical description of light. Even the ideal case of light in vacuum—which from Maxwell’sequations arise as sinusoidal functions (upper case) of well-defined frequency—need a statistical description.One simple illustration is the finite lifetime of the emitter, resulting in a Lorentzian spread in frequency(central case). Another more realistic case displays superpositions of different wavetrains

This definition or notion of coherence gives a good vivid picture of the concept, butit is mainly rooted in classical physics. As such, it describes perfectly well coherenceof classical waves like sound or water waves. In a modern understanding of optics, thisdescribes first-order coherence, still an important property of light, but definitely notencompassing the whole aspect of the problem.

The principle for measuring (first-order) coherence follows easily from the intuitivedescription that we have given. The notion that the knowledge of a signal at some pointinforms about its values at other points is mathematically described by correlation, andin this case, since it refers to the signal itself, by autocorrelation. The signal itself be-

COHERENCE 35

comes a random variable, here the field E, which means that when prompted for a value(when it is measured, for instance), the “variable” is sampled according to a probabilitydistribution. Physically, when the experimental setup performs a measurement, it drawsone possible realization of the experiment. From our description of coherence, we wantto know how much the knowledge of, e.g., the field E(t0) at a given time t0, tells usabout the values E(t) at other times t (here, E is still a random variable). A way toquantify this is to take the product E(t0)E(t) and to average over many possible re-alizations. We denote 〈E(t0)E(t)〉 this average. If the field is coherent, that is, if thevalue at t determines that at t0, each product will bear a fixed relationship that will sur-vive the average. If, however, the two values are not related to each other, the productis random and averages to zero. In intermediate cases, the degree of correlation definesthe degree of coherence. To retain some mathematical properties, the product is taken ashermitian, 〈E∗(t0)E(t)〉, and is normalized to make it independent of the field’s abso-lute amplitude, as well as writing t0 − t = τ to emphasize the importance of “delay” in“confronting” the two values of the field, we arrive at the first-order coherence degree:

g(1)(τ, t) =〈E∗(t)E(t + τ)〉

〈|E(t)|2〉 . (2.39)

This is a complex number in general satisfying the following properties:

g(1)(0, t) = 1, g(1)(τ, t)∗ = g(1)(−τ, t) . (2.40)

Equation (2.39) depends explicitly on time, as can be the case if the system is not inequilibrium or in a steady-state. Note that such a measure of coherence extends to otherdegrees of freedom of the fields.11 Often, however, one considers the time-independentcase, which in this context translates as stationary signals, i.e., systems that still varyin time “locally” but do not depend on time in an absolute way, as is the case in apulsed experiment for instance with clear difference before, during and after a pulse, asopposed to continuous pumping, where on the average the system does not evolve.

The distinction in terms of the notions we have just presented is that for a stationaryprocess the probability distribution is time independent, but the experimental quantityremains the random variable that fluctuates according to its distribution. However, itsmean and variance are also constant. An obvious example is when time variations ofthe fields are limited to fluctuations, though this is only a special case of a stationaryprocess. A periodic signal is also stationary. The ergodic theorem asserts that for suchprocesses time and ensemble averages are the same, so that in eqn (2.39), t is taken toassume values high enough so that steady-state is achieved.12 What kind of average is

11A more general formula quantifying simultaneously coherence both in time and space reads:

g(1)(r1, t1; r2, t2) = 〈E∗(r1, t1)E(r2, t2)〉‹q〈|E(r1, t1)|2〉〈|E(r2, t2)|2〉 . (2.41)

12The direct time dependence of g(1) disappears in the steady state. A rigorous definition reads:

g(1)(τ) = limt→∞

〈E∗(t)E(t + τ)〉〈|E(t)|2〉 . (2.42)


effectively performed is a rather moot point. There is ensemble averaging in most casesbecause many emitters combine to provide light, all of them independently emittingdifferent “wavetrains” that correspond to an ensemble averaging. On the other hand, thecoherence time of light is much too short to be detected directly by a detector that itselfperforms a time averaging. The importance of statistics and the relevance of correlationsespecially as defining coherence in optics has been realized in a large measure thanksto the guidance of Born and Wolf.

Max Born (1882–1970) and Emil Wolf (b. 1922) emphasized the importance of coherence in optics by de-veloping its statistical theory. They coauthored “Principles of Optics” (1959) that is the classic authoritativetext in the field.

Born was awarded the 1954 Nobel prize (half-prize) for “his fundamental research in quantum mechanics,especially for his statistical interpretation of the wavefunction.” Educated as a mathematician (with Hilbert),he identified the indices of Heisenberg’s notations for transitions rates between orbitals as matrix elements.Systematizing the approach with Pascual Jordan (then his student), they submitted a paper entitled “ZurQuantummechanik” (M. Born and P. Jordan, Zeitschrift fur Physik, 34, 858, (1924)) bearing the first pub-lished mention of the term “quantum mechanics.”

Wolf is the physicist of optics par excellence, bringing many advances in statistical optics, coherence, diffrac-tion and the theory of direct scattering and inverse scattering. He discovered the “Wolf effect” that is a redshiftmechanism to be distinguished from the Doppler effect. As a signature of his eponymous father, the effectfollows from partial coherence effects. It was experimentally confirmed the year following its prediction in1987.

The materialization of the mathematical procedure outline above is made in the lab-oratory with an interferometer, such as the Michelson interferometer or Mach–Zehnderinterferometer. In these devices, the field is superimposed with a delayed fraction ofitself and the time-averaged intensity of the light is collected at the output. Oscillationsin this intensity build up fringes with visibility defined as

V =Imax − Imin

Imax + Imin, (2.43)

with Imin/max the minimum, maximum intensity of the resulting interference pattern,respectively, and this equates to the modulus of coherence degree:

V = |g(1)(τ)| . (2.44)

COHERENCE 37

The visibility V varies with the time delay τ and of course also with position if relatedto the more general definition of eqn (2.41). It assumes values between zero, for anincoherent field, and one, for a fully coherent one. Intermediate values describe partialfirst-order coherence.

Albert Abraham Michelson (1852–1931) received the first American Nobel prize for physics in 1907 “forhis optical precision instruments and the spectroscopic and metrological investigations carried out with theiraid”. He designed the Michelson interferometer (shown on right) that superposes the light field onto itselfwith a delay (imparted by the moving mirror), allowing its first order coherence to be measured.

As a trivial example, the sine wave of eqn (2.37) gives by direct application of thedefinition of eqn (2.39)

g(1)(τ) = exp(−iωτ) , (2.45)

which corresponds to full first-order coherence (as its modulus, eqn (2.44), is one). Incomparison, a field that results from two sources, each perfectly coherent (in the sensethat they are both a sinusoidal wave of the type of eqn (2.37)) and with a dephasing ϕbetween them, give as the total scalar field:

E(x, t)/E0 = exp(i(k1z − ω1t)

)+ exp

(i(k2z − ω2t + ϕ)

). (2.46)

We assume they have common amplitude E0. If ϕ is kept fixed in the averaging, thesame results as for the purely sinusoidal field apply. If, however, ϕ varies randomlybetween measurements, the result of Exercise 2.1 is obtained.

Exercise 2.1 (∗) Show that the field given by eqn (2.46) with ϕ varying randomly yields:

V = |g(1)(τ)| = | cos(1

2(ω1 − ω2)τ)| . (2.47)

Another frequent expression for g(1) brings together in eqn (2.50) (derived in Exer-cise 2.2) the decoherence resulting from two typical atomic dephasing processes: colli-sion broadening and Doppler broadening.

Collision broadening is associated with the abrupt change of phase due to collisionsbetween emitters, which have probability γc per unit time, to see their phase randomizedduring emission (associated with the chance of colliding with a neighbour).


Doppler broadening results from the Doppler effect that shifts the frequency of emit-ters as a function of their velocity. From kinetic theory it can be shown that the distri-bution of frequencies is normal

f(ω) =1√2π∆

exp

(− (ω0 − ω)2

2∆2

), (2.48)

centred about the unshifted frequency ω0 and with root mean square:

∆ = ω0

√kBT/(mc2) , (2.49)

at temperature T for emitters of mass m. This effect is therefore important in a gas atnon-vanishing temperatures.

Exercise 2.2 (∗∗) Model the light field of sources subject to collision and Dopplerbroadening and show that their first-order coherence degree is given by:

g(1)(τ) = exp(−iω0τ − γc|τ | − 1

2∆2τ2) . (2.50)

Observe how in eqn (2.50) the dephasing that ultimately loses completely the phaseinformation results in a decay of |g(1)| rather than in oscillations, as was the case ineqn (2.47), as a result of the finite number of emitters contriving to dephase the system.This will become clear in the next section that links first-order coherence to the emittedspectra.

2.3.3 Wiener–Khinchin theorem

An important relationship was established by Wiener and Khinchin between the spectraldensity of a stochastic process and its autocorrelation function. Namely, they form aFourier transform pair.

The spectral density is, physically, the decomposition of a signal (or field) into itscomponents of given frequencies. The relation between frequency ω and energy E,13

E = ω , (2.51)

with = h/(2π) the reduced Planck constant,14 provides the experimental way torecord such a spectra: photons of a given energy are counted over a given time to buildup a signal. The name of “power spectra” is also commonly used, since an intensity perunit time is measured. We now discuss what physical quantities it relates to and how tomodel them mathematically.

13The relation (2.51) is Planck’s hypothesis that energy is emitted by quanta, the quantization being pro-vided by the box of the blackbody. It was fully developed by Einstein for his explanation of the photoelectriceffect.

14The reduced Planck constant is a shortcut for h/2π, where h = 4.135 667 43(35) × 10−15 eV s isthe Planck constant proper. It is the quantum of action and of angular momentum, or, following uncertaintyprinciple, the “size” of a cell in phase space.

COHERENCE 39

Norbert Wiener (1894–1964) and Aleksandr Yakovlevich Khinchin (1894–1959)

Wiener was an American mathematician who pioneered an important branch of study of stochastic processeswith his generalized harmonic analysis. He is also credited as the founder of “cybernetics”. His absent-mindedness and lively nature sparked many famous anecdotes. In his obituary, the Times reports how he“could offend publicly by snoring through a lecture and then ask an awkward question in the discussion”. Inhis essay “the quark and the jaguar”, Gell-Mann remembers how Wiener would hinder the circulation in theuniversity by sleeping in the stairs. The first tome of his autobiography is titled “Ex-Prodigy: My Childhoodand Youth”. Working with physicists, he remarked that “one of the chief duties of the mathematician in actingas an adviser to scientists is to discourage them from expecting too much from mathematics.”

Khinchin was one of the most prominent Russian mathematicians in the field of probabilities, largely domi-nated at that time by the Soviet school. He unravelled the definition of stationary processes and developed theirtheoretical foundations. He published “Mathematical Principles of Statistical Mechanics” in 1943, which heextended in 1951 to the highly respected text “Mathematical Foundations of Quantum Statistics”. He aimedat a complete mathematical rigour in his results as characterizes well the Wiener–Khinchin theorem, since—applying to signals without a Fourier transform—it is far more involved mathematically than physicists usu-ally appreciate.

Intuitively, given a time-varying signal E(t), one can quantify in the spirit of Fourierhow much the harmonic component eiωt exists in the signal by overlapping both andaveraging to get a number:

E(ω) =

∫ ∞

−∞E(t)eiωt dt . (2.52)

The “weight” amplitude E is now itself a function of ω. Its modulus square for a givenfrequency is related to the “strength” of this harmonic in the signal, and thus wouldseem to serve as a good spectral density, and sometimes does:

S(ω) = |E(ω)|2 . (2.53)

However, we already pointed out that an important class of physical signals are sta-tionary, which—being globally time invariant—are not square integrable15 and conse-quently do not admit a Fourier transform, which dooms the above approach. Wiener

15A function f is square integrable ifR |f |2 exists. It is a less stringent condition than integrability that is

required for many properties relative to integration to be meaningful.


showed in 1930 that, for a large class of functions z (encompassing all the cases rele-vant for us), the integral Γ(τ) = limT→∞ 1

2T

∫ T

−Tz∗(t)z(t + τ) dτ exists and also its

Fourier transform:

S(ω) =1

2π

∫ ∞

−∞Γ(τ)eiωτ dτ . (2.54)

Γ(τ) as defined above (and by Wiener) does not refer to an ensemble as there is noaveraging. It required Khinchin’s inputs on stationary random processes (that we willnot detail) and the ergodic theorem to arrive at the Wiener–Khinchin theorem such as itis known today, which is eqn (2.54) together with the definition already discussed Γ(t−t′) = 〈E∗(t)E(t′)〉 for the autocorrelation function (reverting notation to E for thefunction that has been z up to now). Hence:

S(ω) =1

2π

∫ ∞

−∞〈E∗(t)E(t + τ)〉eiωτ dτ . (2.55)

Normalising eqn (2.55) shows that

σ(ω) =1

2π

∫ ∞

−∞g(1)(τ)eiωτ dτ (2.56)

is the lineshape of the emission, that is, the normalized spectral shape.Applying formula (2.56) to eqn (2.50) shows that:

• A pure coherent state (without dephasing) emits a delta-function spectrum: thereis no spread in frequency, as expected.

• Homogeneous broadening, associated to exponential decay of g(1), results in aLorentzian lineshape:

σ(ω) =1

π

γc/2

(ω − ω0)2 + (γc/2)2. (2.57)

• Inhomogeneous broadening, associated to Gaussian decay of g(1), results in aGaussian lineshape:

σ(ω) =1

∆√

2πexp

(− (ω − ω0)

2

2∆2

). (2.58)

By convolution, the general case for g(1) given by eqn (2.50) combining homoge-neous and inhomogeneous broadenings yields the following expression of the lineshape(known as the Voigt lineshape):

σ(ω) =

∫ ∞

−∞

1

∆√

2πexp

(− x2

2∆2

)1

π

γc/2

(ω − x)2 + (γc/2)2dx . (2.59)

The Voigt lineshape interpolates between the other two cases, as shown in Fig. 2.4.

COHERENCE 41

0

0.05

0.1

0.15

0.2

0.25

0.3

-5 -4 -3 -2 -1 0 1 2 3 4 5

ω

γc/2=0 ∆=1.6γc/2=1 ∆=1γc/2=1.8 ∆=0

Fig. 2.4: Typical lineshapes (emission spectra) emitted by a source with two mechanisms of broadeningresulting in exponential and Gaussian decay of the coherence degree g(1). The limiting cases recover therespective Lorentzian and Gaussian lineshapes (lower and upper lines). The general case in between is knownas the Voigt lineshape.

2.3.4 Hanbury Brown–Twiss effect

Up to now, we have investigated “coherence” as a concept of correlations in the fieldbut in fact, of correlations in the amplitudes of the field, as those have been the relativevalues of E that were compared. Again, this corresponds experimentally to splittingthe beam in two and superimposing it onto itself with a delay, as schematized by theMichelson interferometer on page 37. The same approach extends to other quantitiesderiving from the field. The next step is to quantify correlations in the intensities ratherthan amplitudes, as has been done by Hanbury Brown and Twiss (1956) who correlatedthe signals of two photomultiplier (PM) tubes collecting light from the star Sirius assketched on the next page.

Hanbury Brown and Twiss evidenced a positive simultaneous correlation betweenthe two signals, meaning that the detection of a signal on any one of the PMs wasmatched by detection of a signal on the other PM more often than if the sources wereuncorrelated (if the PMs were aimed at two different stars, for instance). This is knownas bunching, as the detections appear to be grouped together. We shall see in the nextchapter that at the quantum level, this means that photons—the particles actually de-tected and amplified by the PMs—are lumped together in the light emitted by Sirius.This experiment, which in the above case is made at two different locations, can bemade with an extra delay in time. With a delay, one measures the likelihood that, givena first photon is measured at time t, a second one is measured at time t + τ . The HBTsetup then consists of a 50/50 beamsplitter directing the light on two photomultipliers(typically avalanche photodiodes). The time elapsed between the detection of two con-secutive photons is measured and the number n(τ) of photon pairs separated by a timeinterval τ is counted. This number gives the probability of joint detection P2(t, t + τ)


Robert Hanbury Brown (1916–2002) invented—with Richard Twiss—the intensity interferometer. Thesetup, now used in various “HBT experiments”, is schematized by Hanbury Brown and Twiss (1958) forapplications in radioastronomy to measure the radius of a star. Two stellar mirrors A spatially separated by adistance d collect light and focus it on photomultipliers P . B are amplifiers and C operates the coincidence(after delay τ has been imparted on one of the line). M integrates the signal.

The concept of photon interferences was initially met with great opposition, also from the highest authoritiesof the time such as Richard Feynman. In his autobiography, “BOFFIN : A Personal Story of the Early Days ofRadar, Radio Astronomy and Quantum Optics”, Hanbury Brown remembers “As an engineer my educationin physics had stopped far short of the quantum theory. Perhaps just as well, otherwise like most physicists Iwould have come to the conclusion that the thing would not work[. . . ]In fact to a surprising number of peoplethe idea that the arrival of photons at two separated detectors can ever be correlated was not only hereticalbut patently absurd, and they told us so in no uncertain terms, in person, by letter, in print, and by publishingthe results of laboratory experiments, which claimed to show that we were wrong. . . ”

at times t and t + τ . This probability can be linked to intensity correlations from pho-todetection theory as has been done by Mandel et al. (1964).16 They are proportional:

P2(t, t + τ) = α2〈I(t)I(t + τ)〉 , (2.60)

where I = |E|2 is the intensity, and α represents the quantum efficiency of the photo-electric detector.

Intensity correlations being the straightforward next-order extension of amplitudecorrelations, eqn (2.39), we introduce the notation g(2) and call “second-order coher-ence degree” the quantity

g(2)(t, τ) =〈E∗(t)E∗(t + τ)E(t)E(t + τ)〉

〈E∗(t)E(t)〉2 . (2.61)

We also introduce the centred, normalized correlation function:

λ(τ) =〈∆I(t)∆I(t + τ)〉〈I(t)I(t + τ)〉 , (2.62)

16The proportionality between g(2) and the joint detection probability is established in the full quantumcase in Exercise 3.20 on page 108.

POLARIZATION-DEPENDENT OPTICAL EFFECTS 43

where ∆I = I−〈I〉. Now eqn (2.60) reads P2(t, t+τ) = α2〈I(t)〉〈I(t+τ)〉[1+λ(τ)]and from the first result of Exercise 2.3, it follows that

P2(t, t) ≥ P2(t, t + τ) . (2.63)

Exercise 2.3 (∗) Show that λ(τ) (of eqn (2.62)) satisfies λ(τ) ≤ λ(0) and also thatlimt→∞ λ(t) = 0. Discuss the sign of this quantity.

We already noted that the detection process is at the quantum level, detecting singlephotons,17 whereas it was the field amplitude that was previously measured. For thisreason the HBT experiment is often regarded as the quantum optical measurement parexcellence, though it is merely correlating intensities and has been used by HanburyBrown and Twiss in a classical context (more of which is investigated in the problemat the end of this chapter). However, it is true that it also characterizes light bearing aquantum character, as shall be explained in the next chapter.

2.4 Polarization-dependent optical effects

We now give a brief account of the main optical effects dealing with the polarization oflight. Most of them have been observed in microcavities. Some of them allow measure-ment of the most important intrinsic characteristics of microcavities.

2.4.1 Birefringence

Birefringence is the division of light into two components (an ordinary and an extraor-dinary ray), found in materials that have different indices of refraction in different di-rections (no and ne for ordinary and extraordinary rays, respectively). Birefringence isalso known as double refraction.

The quantity referred to as birefringence is defined as

∆n = ne − no . (2.64)

Crystals possessing birefringence include hexagonal (such as calcite), tetragonal,and trigonal crystal classes and are known as uniaxial. Orthorhombic, monoclinic, tri-clinic crystal exhibit three indices of refraction. They are known as biaxial. Birefringentprisms include the Nicol prism, Glan–Foucault prism, Glan–Thompson prism, and Wol-laston prism. They can be used to separate different incident polarizations.

Dichroism is the selective absorption of one component of the electric field of a lightwave, resulting in polarization.

Optical activity arises when polarized light is passed through a substance containingchiral molecules (or non-chiral molecules arranged asymmetrically), and the directionof polarization can be changed. This phenomenon is also called optical rotation.

The Kerr effect discovered in 1875 by the English physicist John Kerr consists inthe development of birefringence when an isotropic transparent substance is placed in

17A single photon is detected in an avalanche photodiode by exciting a single electron–hole pair acrossthe bandgap. The high electric field across the diode accelerates the electron that acquires sufficient energy toexcite further electron–hole pairs—the avalanche—and producing a sizable current pulse.


Michael Faraday (1791–1867) and an illustration of the Faraday effect, similar to optical activity as in bothcases the polarization plane of light rotates as it propagates through the medium. The difference is that theFaraday rotation is independent of the propagation direction and can be accumulated if light makes severalround-trips through the medium, while the rotation caused by optical activity changes its sign for light propa-gating in the opposite direction. Thus, after any round-trip inside an optically active medium the polarizationof light remains the same.

Faraday was a self-taught genius who developed an interest for science by reading books he had to bind asa poor apprentice working in a bookshop. Through attendance and hard work he gained access to experi-mental laboratories where he excelled and developed experimental setups of an unprecedented standard. Hisevidence that magnetism could affect rays of light proved a relationship between the two, a finding preparingthe great first unification in physics of electricity and magnetism as two facets of electromagnetism. To thespectroscopist James Crookes, he once advised “Work. Finish. Publish”.

an electric field F. It is used in constructing Kerr cells, which function as variable waveplates with an extremely fast response time, and find use in high-speed camera shutters.Because the effect is quadratic with respect to F, it is sometimes known as the quadraticelectro-optical effect. The amount of birefringence (as characterized by the change inindex of refraction) due to the Kerr effect can be parametrized by

∆n = λ0KF 2 , (2.65)

where K is the Kerr constant (ranging between 10−12 and 10−15 mV−2 for differentmaterials) and λ0 is the vacuum wavelength. The phase change ∆φ introduced in a Kerrcell of thickness d under an applied voltage V is given by

∆φ =2πKλV 2

d2. (2.66)

2.4.2 Magneto-optical effects

2.4.2.1 Faraday effect The English physicist Michael Faraday experimentally dis-covered diamagnetism and observed what is now called the Faraday effect in 1845. Hedemonstrated that, given two rays of circularly polarized light, one with left-hand andthe other with right-hand polarization, the one with the polarization in the same direc-tion as the electricity of the magnetizing current travels with greater velocity. That iswhy the plane of linearly polarized light is rotated when a magnetic field is appliedparallel to the propagation direction (see Fig. 2.5).

PROPAGATION OF LIGHT IN MULTILAYER PLANAR STRUCTURES 45

Fig. 2.5: Orientation of electric and magnetic fields in TE- and TM-polarized incident on a planar boundary.

The empirical angle of rotation is given by

α = V Bd , (2.67)

where V is the Verdet constant (with units of rad/(Tm)), named after the French physi-cist Emile Verdet. The Faraday rotation angle is a measure of magnetization induced inthe medium by a magnetic field. In the quantum-mechanical description, it occurs be-cause imposition of a magnetic field B alters the energy levels of atoms or electrons(Zeeman effect).

2.4.2.2 Magneto-optical Kerr effect This effect has very much in common with theFaraday effect. It also consists in rotation of the polarization plane of light in the mediahaving a nonzero magnetization in the direction of light propagation. The differencebetween the two effects consists in the experimental configuration used to detect thepolarization rotation. For the Faraday effect the polarization of the transmitted light isanalysed, while for the magneto-optical Kerr effect the polarization of reflected lightis compared with polarization of the incident light. As a large part of reflected signalcomes from the surface reflection usually, the Kerr rotation is very sensitive to the sur-face magnetization.

Exercise 2.4 (∗∗) Consider a dielectric slab of thickness d subjected to a magneticfield B oriented normally to the surface of the slab. Let t+ and t− be amplitude trans-mission coefficients of the slab for σ+ and σ− polarized light, respectively, and |t+| =|t−|. Find the Verdet constant of the material.

2.5 Propagation of light in multilayer planar structures

In this section we present the transfer matrix method, which solves Maxwell equationsin multilayer dielectric structures. We consider the example of a periodical structure (aso-called Bragg mirror) and derive general equations for photonic eigenmodes in planarstructures. In the beginning we consider propagation of light normal to the layer planes


direction. Then we generalize to the oblique incidence case. We formulate the transfermatrix approach for TE and TM linear polarizations. By definition, TE-polarized (alsoreferred to as s-polarized) light has the electric-field vector parallel to the layer planes,TM-polarized light (also referred to as p-polarized) has the magnetic-field vector paral-lel to the planes (see Fig. 2.5).

The behaviour of the electromagnetic field at the planar interface between two di-electric media with different refractive indices is dictated by Maxwell’s equations (2.15).They can be solved independently in the two media and then matched for the electricand magnetic fields by the Maxwell boundary conditions at the interface. These con-ditions require continuity of the tangential components of both fields. They can be mi-croscopically justified for any abrupt interface in the absence of free charges and freecurrents.

Consider a transverse light wave propagating along the z-direction in a mediumcharacterized by a refractive index n that is homogeneous in the xy-plane but possiblyz dependent. The wave equation (2.23) in this case becomes for the field amplitude:

∂2zE = −k2

0n2E , (2.68)

where k0 is the wavevector of light in vacuum. The general form of the solution ofeqn (2.68) reads

E = A+ exp(ikz) + A− exp(−ikz) , (2.69)

where k = k0n, A+ and A− are coefficients. Using the Maxwell equation (2.15d) onecan easily obtain the general form of the magnetic field amplitude

B = nA+ exp(ikz)− nA− exp(−ikz) . (2.70)

If we consider reflection of light incident from the left side of the boundary (z = 0)between two semi-infinite media characterized by refractive indices n1 (left) and n2

(right), the matching of the tangential components of electric and magnetic fields gives

A+1 + A−

1 = A+2 , (2.71a)

(A+1 −A−

1 )n1 = A+2 n2 , (2.71b)

where A+1 , A−

1 and A+2 are the amplitudes of incident, reflected and transmitted light,

respectively. One can easily obtain the amplitude reflection coefficient

r =A−

1

A+1

=n1 − n2

n1 + n2, (2.72)

and the amplitude transmission coefficient

t =A+

2

A+1

=2n1

n1 + n2. (2.73)

The reflectivity, which is the ratio of reflected to incident energy flux, is given by

PROPAGATION OF LIGHT IN MULTILAYER PLANAR STRUCTURES 47

R = |r|2 , (2.74)

and the transmittance, or ratio of transmitted to incident energy flux, is

T =n2

n1|t|2 . (2.75)

In the last formula, the factor n2/n1 comes from the ratio of light velocities in thetwo media.

In multilayer structures, direct application of Maxwell’s boundary conditions at eachinterface requires solving a substantial number of algebraic equations (two per inter-face). A convenient method allows the number of equations to be solved to be strictlyminimized (four in the general case) and is the transfer matrix method, which we nowbriefly describe.

Let us introduce the vector

ΦΦΦ(z) =

(E(z)cB(z)

)=

(E(z)

− ik0

∂zE(z)

), (2.76)

where E(z) and B(z) are the amplitudes of the electric and magnetic field of any lightwave propagating in the z-direction in the structure. Note that ΦΦΦ(z) is continuous atany point in the structure due to Maxwell’s boundary conditions. In particular, it iscontinuous at all interfaces where n changes abruptly.

By definition, the transfer matrix Ta across the layer of width a is a matrix thatenforces

TaΦΦΦ|z=0 = ΦΦΦ|z=a . (2.77)

It is easy to verify by substitution of the electric and magnetic amplitudes (2.69) and(2.70) into eqn (2.77) that if n is homogeneous across the layer,

Ta =

(cos ka i

n sin kain sin ka cos ka

). (2.78)

The transfer matrix across a structure composed of m layers is found as

T =m∏

i=1

Ti , (2.79)

where Ti is the transfer matrix across the ith layer. The order of multiplication ineqn (2.79) is essential. The amplitude reflection and transmission coefficients (rs and ts)of a structure containing m layers and sandwiched between two semi-infinite mediawith refractive indices nleft and nright before and after the structure, respectively, canbe found from the relation between the fields ΦΦΦ on either side of the structure:

T

(1 + rs

nleft(1− rs)

)=

(ts

nrightts

). (2.80)

One can easily obtain


rs =nrightt11 + nleftnrightt12 − t21 − nleftt22t21 − nleftt22 − nrightt11 + nleftnrightt12

, (2.81a)

ts = 2nleftt12t21 − t11t22

t21 − nleftt22 − nrightt11 + nleftnrightt12. (2.81b)

The intensities of reflected and transmitted light normalized by the intensity of theincident light are given by

R = |rs|2, T = |ts|2 nright

nleft, (2.82)

respectively.Reciprocally, the transfer matrix across a layer can be expressed via reflection and

transmission coefficients of this layer. If the reflection and transmission coefficients forlight incident from the right-hand side and left-hand side of the layer are the same,and nleft = nright = n (the symmetric case of, in particular, a quantum well embeddedin a cavity), the Maxwell boundary conditions for light incident from the left and rightsides of the structure yield:

T

(1 + rs

n(1− rs)

)=

(tsnts

)and T

(ts−nts

)=

(1 + rs

−n(1− rs)

). (2.83)

This allows the matrix T to be expressed as

T =1

2ts

(t2s − r2

s + 1 − (1+rs)2−t2sn−n((rs − 1)2 − t2s) t2s − r2s + 1

). (2.84)

For a quantum well, as will be shown in the next section, ts = 1+ rs and eqn (2.84)becomes

TQW =

(1 0

−2nrs/ts 1

). (2.85)

In the oblique incidence case, in the TE-polarization, one can use the basis(Eτ (z), cBτ (z))T ( T means transposition), where Eτ (z) and Bτ (z) are the tangen-tial (inplane) components of the electric and magnetic fields of the light wave. In thiscase, the transfer matrix (2.78) keeps its form provided that the following substitutionsare made:

kz = k cos φ, n→ n cos φ , (2.86)

where φ is the propagation angle in the corresponding medium (φ = 0 at normal inci-dence).

In the TM-polarization, following Born and Wolf (1970), we now use the basis(cBτ (z), Eτ (z))T that still allows the transfer matrix (2.78) to be used, provided thatthe following substitutions are done:

kz = k cos φ, n→ cos φ

n. (2.87)

Note that the transfer matrices across the interfaces are still identity matrices, andeqn (2.79) for the transfer matrix across the entire structure is valid.

PHOTONIC EIGENMODES OF PLANAR SYSTEMS 49

Rene Descartes (1596–1650) and Willebrord Snellius (1580–1626).

Descartes is regarded as the father of modern mathematics and natural philosophy. His thinking instigated therevolution in western science that would break from eastern influence and mark the dominance of Europeanthinking. His famous saying “je pense donc je suis” (I think therefore I am) in “Discourse on method” ranksamong the most influential philophical statements. His name became synonymous with “rational” and “mean-ingful”, an eponymous tribute whose extent and glory was never equalled after him.

Snellius was a Dutch astronomer and mathematician. He discovered the law of refraction that is now namedafter him (although in a few countries, especially in France, this law is called Snell–Descartes). Arab as-tronomers apparently knew it long time before Snellius from the work of Ibn Sahl (984). Snellius also workedout a remarkably accurate (for his time) value of the radius of the earth and devised a new method to com-pute π, the first such improvement since ancient time.

In the formulas (2.81)–(2.83) for reflection and transmission coefficients, one shouldreplace, in the TE-polarization

nleft → nleft cos φleft, nright → nright cos φright , (2.88)

and in the TM-polarization

nleft → cos φleft

nleft, nright → cos φright

nright, (2.89)

where φleft, φright are the propagation angles in the first and last media, respectively.The same transformations would be applied to the transfer matrices (2.84) and (2.85).Note that any two propagation angles φi, φj in the layers with refractive indices ni, nj

are linked by the Snell–Descartes law:

ni sinφi = nj sinφj , (2.90)

which is also valid in the case of complex refractive indices, when the propagationangles formally become complex as well.

2.6 Photonic eigenmodes of planar systems

Consider a multilayer planar structure characterized by a transfer matrix T being aproduct of the transfer matrices across all the layers as given by eqn (2.79). Photonic


eigenmodes of the structure are the solutions of the Maxwell equations that decay out-side the structure and hence with the following boundary condition: no light is incidenton the structure either from the left (z → −∞) or from the right side (n → +∞). Thismeans that the electric field of the eigenmode at z →∞ can be represented as

E0 exp(ikx + iky + ikz) , (2.91)

with (kz) ≥ 0 and (kz) ≤ 0, while at z → ∞ the electric field can be representedin the form of eqn (2.91) with (kz) ≤ 0 and (kz) ≥ 0. In TE-polarization, let uschoose the system of coordinates in such a way that the electric and magnetic fields ofthe light mode are oriented as follows:

E =

⎛⎝ 0

Ey

0

⎞⎠ , B =

⎛⎝Bx

0Bz

⎞⎠ . (2.92)

The transfer matrix TTE links the vectors (Ey, cBx)T at the left and right bound-aries of the structure, so that

TTE

(Eleft

y

cBleftx

)= A

(Eright

y

cBrightx

), (2.93)

where A is a complex coefficient.Substitution of the electric field (2.69) into the first of Maxwell’s equations (2.15a)

yieldskz

k0Ey = cBx , (2.94)

where k0 = ω/c. This allows us to rewrite eqn (2.93) as

TTE

⎛⎝ 1

kleftz

k0

⎞⎠ = A

⎛⎝ 1

krightz

k0

⎞⎠ , (2.95)

where kleftz and kright

z are z-components of the wavevector of light on the left and rightsides of the structure, respectively. By elimination of A, eqn (2.95) can be easily reducedto a single transcendental equation for the eigenmodes of the structure:

tTE11

krightz

k0+ tTE

12

kleftz kright

z

k20

− tTE21 −

kleftz

k0tTE22 = 0 , (2.96)

where tij are the matrix elements of the transfer matrix TTE. Solutions of eqn (2.96) arecomplex frequencies, in general. Only those having a positive real part and negative (orzero) imaginary part have a physical sense. The imaginary part of the eigenfrequencyis inversely proportional to the lifetime of the eigenmode, i.e., a characteristic timespent by a photon going back and forth inside the structure before escaping from it tothe continuum of free light modes in the surrounding media. So-called waveguided or


Fig. 2.6: Propagation of guided modes in planar structures (a) is based on the total internal reflection effect (b).

guided modes are those that have an infinite lifetime (and consequently, zero imaginarypart of the eigenfrequency), see Fig. 2.6.

The equation for eigenfrequencies of TM-polarized modes can be obtained in asimilar way. One can choose the system of coordinates in such a way that the electricand magnetic fields of the light mode are oriented as follows:

E =

⎛⎝Ex

0Ez

⎞⎠ , B =

⎛⎝ 0

By

0

⎞⎠ . (2.97)

The transfer matrix T links the vectors (cBy,−Ex)T at the left and right boundariesof the structure, so that

TTM

(Bleft

y

−Eleftx

)=

(Bright

y

−Erightx

). (2.98)

The Maxwell equation (2.15a) yields in this case

− kz

n2k0By = Ex , (2.99)

where n is the refractive index of the media. This allows us to rewrite eqn (2.98) as

TTM

(1

kleftz

n2leftk0

)= A

(1

krightz

n2rightk0

), (2.100)

where nleft and nright are z-components of the wavevector of light on the left and rightsides of the structure, respectively. By elimination of A, eqn (2.100) can be easily re-duced to a single transcendental equation for the eigenmodes of the structure:

tTM11

krightz

n2rightk0

+ tTM12

kleftz kright

z

n2leftn

2rightk

20

− tTM21 − tTM

22

kleftz

n2leftk0

= 0 . (2.101)

At normal incidence, equations for the eigenmodes of light in TE- (eqn (2.96)) andTM-polarizations (eqn (2.101)), become formally identical. This is quite reasonable, as


at normal incidence there is no difference between TE- and TM-polarizations. One canshow using transformations (2.87) and (2.88) that they both reduce to

t11nright − t12nrightnleft − t21 + t22nleft = 0 , (2.102)

with tij being elements of the transfer matrix at normal incidence defined by eqns (2.78–2.79). Comparing condition (2.102) and expressions (2.71b) and (2.81b) one can seethat reflection and transmission coefficients rs and ts become infinite at the complexeigenfrequencies of the system. This result also holds for oblique incidence. It is notunphysical. We note that in optical measurements of reflectivity and transmission wealways detect signals at real frequencies and the condition

|rs|2 +nright

nleft|ts|2 ≤ 1 . (2.103)

holds (the equality holds in the case of no absorption and scattering).

2.6.1 Photonic bands of 1D periodic structures

Consider an infinite structure whose refractive index is homogeneous in the xy-planeand whose dependency on the coordinate z is a periodic function with period d. Theshape of this function is not essential, and we shall only assume that a transfer matrix Td

across the period of the structure can be written as a product of a finite number ofmatrices of the form of eqn (2.72). Let an electromagnetic wave propagate along thez-direction. For this wave

TdΦ∣∣z=0

= Φ∣∣z=d

, (2.104)

where Φ(z) is defined by eqn (2.76). According to the Bloch theorem, it can be repre-sented in the form:

Φ(z) = eiQz

(UE(z)UB(z)

), (2.105)

where UE,B(z) have the same periodicity as the structure and Q is a complex numberin the general case.

Note that the factor eiQz is the same for electric (E) and magnetic (B) fields in alight wave because in the normal incidence case they are linked by the relation

B(z) = − i

ck0

∂E(z)

∂z. (2.106)

Substitution of eqn (2.105) into eqn (2.104) yields

TdΦ∣∣z=0

= eiQdΦ∣∣z=0

, (2.107)

thus, eiQd is an eigenvalue of the matrix Td and therefore

det(Td − eiQd1) = 0 . (2.108)

Solving eqn (2.108), we use an important property of the matrix Td following fromeqns (2.78)–(2.79):


det(Td) = 1 . (2.109)

Thus, we reduce eqn (2.108) to

1− (T11 + T22)eiQd + e2iQd = 0 , (2.110)

where Tij are the matrix elements of Td. Multiplying each term by e−iQd we finallyobtain:

cos(Qd) =T11 + T22

2. (2.111)

The right-hand side of this equation is frequency dependent. The frequency bandsfor which ∣∣∣∣T11 + T22

2

∣∣∣∣ ≤ 1 (2.112)

are allowed photonic bands. In these bands, Q is purely real and the light wave canpropagate freely without attenuation. On the contrary, the bands for which∣∣∣∣T11 + T22

2

∣∣∣∣ > 1 (2.113)

are usually called stop-bands or optical gaps (see Fig. 2.7). In these bands, Q has anonzero imaginary part that determines the decay of propagating light waves. All this iscompletely analogous to electronic bands in conventional crystals. Equations (2.111)–(2.113) are also valid in the oblique incidence case, while the form of the matrix Td

is sensitive to the angle of incidence and band boundaries shift as one changes theincidence angle.

We recall that a Bragg mirror is a periodic structure composed of pairs of layers ofdielectric or semiconductor materials (see Fig. 2.8) characterized by different refractiveindices (say na and nb). The thicknesses of the layers (a and b, respectively) are chosenso that

naa = nbb =λ

4. (2.114)

Condition (2.114) is usually called the Bragg interference condition due to its sim-ilarity to the positive interference condition for X-rays propagating in crystals pro-posed in 1913 by English physicists Sir William Henry Bragg and his son Sir WilliamLawrence Bragg. The Bragg mirrors are also frequently called distributed Bragg re-flectors or DBRs. The wavelength of light λ marks the centre of the stop-band of themirror. For the wavelengths inside the stop-band the reflectivity of the mirror is close tounity. In the following we assume na < nb (na is the refractive index of the first layerfrom the surface, nb is the refractive index of the next layer). We describe the opticalproperties of the mirror within its stop-band using the transfer matrix approach.

At normal incidence, the transfer matrices across the layers that compose the mirrorare:

Ta =

(cos(kaa) i

nasin(kaa)

ina sin(kaa) cos(kaa)

), Tb =

(cos(kbb)

inb

sin(kbb)

inb sin(kbb) cos(kbb)

),

(2.115)


Fig. 2.7: Experimental and theoretical transmittance of a periodic structure composed by Si3N4 and SiO2

dielectric layers (a) compared to the calculated photonic dispersion for this structure (b), from Gerace et al.(2005). The stop-bands are shown in grey.

where ka = ωna/c and kb = ωnb/c. The transfer matrix T across the period of themirror is their product:

T = TbTa . (2.116)

An infinite Bragg mirror represents the simplest one-dimensional photonic crystal.Its band structure is given by eqn (2.111). Its solutions with real Q form allowed pho-tonic bands, while solutions with complex Q having a nonzero imaginary part formphotonic gaps or stop-bands.

At the central frequency of the stop-band, given by

ω =2πc

λ, (2.117)

the matrix T becomes:

T =

(−na

nb0

0 − nb

na

). (2.118)

Its eigenvalues are

exp[iQ(a + b)] = −na

nb, exp[−iQ(a + b)] = −nb

na. (2.119)

The reflection coefficient of a semi-infinite Bragg mirror at ω = ω can be foundfrom the condition:


Fig. 2.8: Electronic microscopy image of the GaN/AlGaN Bragg mirror grown by E. Calleja’s group inMadrid and reported by Fernandez et al. (2001).

William Henry Bragg (1862–1942) and William Lawrence Bragg (1890–1971).

Father and son, the two Braggs shared the 1915 Nobel Prize in Physics “for their services in theanalysis of crystal structure by means of X-rays.” When a young Bragg, aged 5, broke his arm by fallingoff his tricycle, he was radiographed by X-rays that his father had recently learned about from Rontgen’sexperiments. In 1912, aged 22 and a first-year university student, Bragg discussed with his father his ideasof diffraction by crystals that he would develop into Bragg’s law. His father developed the spectrometer. Hebecame the youngest Nobel prize. He is also credited as having played an important role in his support ofidentifying the DNA double helix, as then head of the Cavendish laboratory.

T

(1 + r

nleft(1− r)

)= −na

nb

(1 + r

nleft(1− r)

), (2.120)

which readily yields r = 1.In the vicinity of ω, one can derive a simple and useful expression for the reflection

coefficient, leaving in the matrix T only terms linear in


x = (ω − ω)λ

4c. (2.121)

The matrix is written in this approximation as:

T = −(

na

nbi(

1na

+ 1nb

)x

i(na + nb)xnb

na

). (2.122)

Equation (2.120) yields in this case

r =nleft

(na

nb− nb

na

)− i(na + nb)x

nleft

(na

nb− nb

na

)+ i(na + nb)x

(2.123a)

= exp

(i

nanbλ

2nleft(nb − na)c(ω − ω)

)= eiα(ω−ω) . (2.123b)

The coefficient

LDBR =nanbλ

2(nb − na)= αn0c (2.124)

is frequently called the effective length of a Bragg mirror. Note that it is close but notexactly equal to the penetration length L of the light field into the mirror at ω = ω. Thelength L can be easily obtained from the eigenvalues of the matrix (2.118):

L =a + b

ln nb

na

. (2.125)

One can see from eqn (2.123a) that at ω = ω the reflection coefficient of the Braggmirror is equal to 1, which means that the amplitudes of incident and reflected waveshave the same sign and absolute value at the surface of the mirror. This is why themaximum (antinode) of the electric field of light is at the surface. We note that this isonly true if na < nb. In the opposite case, the amplitude reflection coefficient at thecentre of a stop-band changes sign and the electric field has a node at the surface.

For a finite-size mirror, the reflection coefficient within the stop-band is differentfrom unity due to the nonzero transmission of light across the mirror. It can be foundfrom the matrix equation:

TN

(1 + r

n0(1− r)

)=

(t

nf t

), (2.126)

where r and t are the amplitude reflection and transmission coefficients of the mirror, Nis the number of periods in the mirror and nf is the refractive index behind the mirror.At the centre of the stop-band:

r =

(nb

na

)2N

− nf

n0(nb

na

)2N

+nf

n0

, t =

(− nb

na

)N

(nb

na

)2N

+nf

n0

. (2.127)

As follows from these formulas, the higher the contrast between na and nb, the betterthe reflectivity of the mirror.


A further important characteristic of a Bragg mirror is the width of its stop-band andthis can be found from eqn (2.111). The boundaries of the first stop-band are given bythe condition:

T11 + T22

2= −1 . (2.128)

Substituting the matrix elements of the product of matrices (2.115) one easily obtains

cos2 Ω− 1

2

(nb

na+

na

nb

)sin2 Ω = −1 , (2.129)

where Ω = kaa = kbb, therefore

cos Ω = ±nb − na

nb + na. (2.130)

This allows us to obtain the stop-band width (in frequency):

∆ =8c

λ

(π

2− arccos

nb − na

nb + na

)≈ 8c

λ

nb − na

nb + na. (2.131)

The stop-band width increases with an increase in the contrast between the two re-fractive indices. Figure 2.9 shows the calculated reflectivity of Bragg mirrors made ofdifferent semiconductor and dielectric materials, but all having λ = 1550 nm. One cansee that the stop-band width can achieve a few hundred nanometres for high contrast ofrefractive indices na and nb.

Finally, under oblique incidence the optical thickness layers composing a Braggmirror change. The phase gained by light crossing a layer of thickness a at an angle φa

is given by

θ =ω

cnaa cos φa , (2.132)

where na is the refractive index of this layer. It is evident that the frequency that fulfillsthe Bragg interference condition θ = π/2 is higher for oblique angles than for a normalangle. This is why, at oblique angles, stop-bands of any Bragg mirror shift towardshigher frequencies. More details on the phases of reflection coefficients of the mirrorsat oblique incidence can be found in the textbook by Kavokin and Malpuech (2003).

Metallic reflectivity is usually not so perfect as dielectric reflectivity. Metals reflectlight because they have a large imaginary component of the refractive index. Consideran interface between a dielectric having the real refractive index n1 and a metal havingthe complex refractive index n2 = n + iκ (we note that the absorption coefficient ofthe metal is proportional to the imaginary part of its refractive index: α = ωκ/c). Thereflection coefficient for light incident normally from the dielectric to the metal reads:

r =1− n− iκ

1 + n + iκ= exp

(−2i arctan

κ

1 + n

)

− 2n√(1 + n)2 + κ2

exp

(−i arctan

κ

1 + n

). (2.133)

Clearly, the reflectivity of a metal increases with increase of κ and decrease of n.In the limit of n → 0, κ → ∞, the reflection coefficient r → −1, which means that


Fig. 2.9: Reflectivity of different Bragg mirror structures. All stop-bands are centred at the same Bragg wave-length of 1550 nm (from the teaching materials of the Institut fur Hochfrequenztechnik, Technical Universityof Darmstand).

the incident and reflected waves compensate each other and the electric-field intensityis close to zero at the surface of the metal.

Fig. 2.10: The real and imaginary parts of the refractive index of gold, from Torok et al. (1998).

In everyday life one uses metallic mirrors. A method of backing a plate of flat glasswith a thin sheet of reflecting metal came into widespread production in Venice duringthe sixteenth century; an amalgam of tin and mercury was the metal used. The chemi-


cal process of coating a glass surface with metallic silver was discovered by Justus vonLiebig in 1835, and this advance inaugurated the modern techniques of mirror making.Present-day mirrors are made by sputtering a thin layer of molten aluminium or silveronto a plate of glass in a vacuum. The metal used determines the reflection characteris-tics of the mirror; aluminium is cheapest and yields a reflectivity of around 88%–92%over the visible wavelength range. More expensive is silver, which has a reflectivity of95%–99% even into the far infrared, but suffers from decreasing reflectivity (< 90%) inthe blue and ultraviolet spectral regions. Most expensive is gold, which gives excellent(98%–99%) reflectivity throughout the infrared, but limited reflectivity below 550 nmwavelength, resulting in the typical gold colour.

Exercise 2.5 (∗∗) Find the frequencies of the eigenmodes of an optical cavity composedby a homogeneous layer of width a and refractive index n sandwiched between twomirrors having the amplitude reflection coefficients r.

Exercise 2.6 (∗∗) If one of the layers in the infinite Bragg mirror has a different thick-ness from all other layers, it acts as a single defect or impurity in an ideal crystal.Localized photonic modes appear at such a defect. Find their eigenenergies.

2.7 Planar microcavities

In microcavities, the cavity layer can be considered as a “defect” layer within a regularBragg structure (see Exercise 2.6). The Fabry–Perot confined modes of light appearwithin the cavity layer under condition (from eqn (2.80)):

reikzLc = ±1 , (2.134)

where Lc is the cavity width and r is the reflection coefficient of the Bragg mirror.Alternatively, cavities with metallic mirrors can be used. In this case, the reflectioncoefficient r is given by eqn (2.133). In this section we only consider dielectric Braggmirrors characterized by the reflection coefficient r ≈ 1.

At normal incidence, for the ideal infinite Bragg mirror (r = 1) a linear equation forthe frequencies of the eigenmodes can be written:

α(ωc − ω) + kzLc = jπ, j ∈ N . (2.135)

The difference between microcavities and conventional cavities is in the value of Lc.In the case of microcavities, it is of the order of the wavelength of visible light dividedby the refractive index of the cavity material (i.e. typically 0.2–0.4 µm). The size ofconventional optical cavities is much larger. This is why the index j of the eigenmodesof microcavities is low and the spacing between their frequencies is so large that usuallyeach stop-band contains only one microcavity mode. On the contrary, in conventionalcavities, the spacing between eigenmodes is small and many of them are present withineach stop-band. Usually, the microcavity width is designed to be an integer number oftimes larger than one of the regular layers in Bragg mirrors, hence kzLc = jπ for ωc =ω. The electric-field profile in the eigenmode of a typical planar microcavity is shownin Fig. 2.11.


Fig. 2.11: Refractive-index profile and intensity of electric field of the eigenmode of a typical planar micro-cavity.

Transmission spectra of microcavities show peaks at the frequencies correspondingto the eigenmodes. Light is able to penetrate inside the cavity and be transmitted throughit at these frequencies. This is an optical interference effect that can be also understoodas the resonant tunnelling of photons: the photon from outside excites the eigenmode ofthe cavity and then jumps out, crossing the mirrors. Figure 2.12 shows the transmissionspectrum of a HfO2/SiO2 microcavity compared with the reflectivity of the single Braggmirror.

Fig. 2.12: Transmission spectrum of a HfO2/SiO2 microcavity and reflectivity of the single HfO2/SiO2

Bragg mirror containing 7 pairs of quarter-wave layers, from Song et al. (2004).

In the absence of absorption or scattering, the reflectivity R is linked to the trans-mission T by a simple relation:

R = 1− T , (2.136)

so that the reflection spectra of dielectric microcavities exhibit dips identical to the peaksof transmission spectra.


One can see that the transmission peak corresponding to the cavity mode is broad-ened. Broadening is inevitably present because of the finite thickness of the Bragg mir-rors and the resulting possibility for light to tunnel through the cavity even within thestop-bands of the mirrors. The quality factor of the cavity Q is defined as the ratioof the frequency of the cavity mode to the full-width at half-maximum of the peak intransmission corresponding to this mode. The quality factor can be also defined as:18

Q = ωcU

dU/dt, (2.137)

where U is the electromagnetic energy stored in the cavity, dU/dt is the rate of the en-ergy loss due to the tunnelling of light through the mirrors during a period of time dt,and ωc is the complex eigenfrequency of the cavity mode given by eqn (2.134). We notethat eqn (2.135) is not exact in real structures having finite Bragg mirrors. In particular,it yields purely real solutions while true eigenfrequencies of the cavity modes are com-plex. To determine the quality factor of a cavity it is important to know the imaginarypart of ωc, as we show below.

The probability that a photon goes outside is proportional to the number of photonsinside the cavity, which yields an exponential dependence of the energy U of the cavitymode on time:

U(t) = U0e−ωct/Q . (2.138)

τ = Q/ωc is the lifetime of the cavity mode. Having in mind the link between theenergy of an electromagnetic field and its complex amplitude E(t), namely U(t) ∝|E(t)|2, we obtain

E(t) = E0 exp(−(ωct/2Q)) exp(−iωct) , (2.139)

where E0 is the coordinate-dependent amplitude. Standard Fourier transformation givesus the frequency dependence of the field amplitude:

E(ω) =1√2π

∫ ∞

0

E0 exp(−(ωct/2Q)) exp(−iωct)dt , (2.140)

so that

U(ω) ∝ |E(ω)|2 ∝ 1

(ω −ωc)2 + (ωc/2Q)2. (2.141)

Expression (2.141) determines the transmission spectrum of the cavity. The resonanceshape has a full-width at half-maximum equal to ωc/Q, which shows the equalityof two definitions of the quality factor we have given. The denominator of expres-sion (2.141) vanishes if

ω = ωc + iωc/2Q . (2.142)

Having in mind that the complex eigenfrequency of the cavity is one that corre-sponds to the infinite transmission (see Section 2.6), we obtain from eqn (2.142)

18We note z and z the real and imaginary parts of the complex quantity z ∈ C, respectively.


Q =ωc

2ωc. (2.143)

For a specific normal mode of the cavity this quantity is independent of the mode ampli-tude. The imaginary part of the eigenfrequency of the cavity mode can be easily foundfrom eqn (2.134) as

ωc = − 1

αln |r| , (2.144)

where the absolute value of the reflection coefficient of the Bragg mirror with a finitenumber of layers is given by eqn (2.127). In high-quality microcavities, the qualityfactor can achieve a few thousands.

We note, that the quality factor is different for different eigenmodes of the samecavity, in general. In particular, Q →∞ (if there is no absorption) for the guided modesthat have purely real eigenfrequencies. From eqns (2.138) and (2.143), it follows thatthe lifetime of the cavity mode

τ =1

2ωc. (2.145)

This characterizes the average time spent by each photon inside the cavity beforegoing out by tunnelling through the mirrors. The lifetime of guided modes is infinite.Theoretically, light never goes out from the ideal infinite waveguide. In reality, eachphotonic mode of any structure has a finite lifetime. The photons escape from the eigen-modes due to scattering by defects, interaction with the crystal lattice, absorption, etc.The finesse of the cavity, F , (see Chapter 1), is linked to the lifetime of a cavity modeby the relation F = ∆ωcτ , where ∆ωc is the splitting between real parts of the eigen-frequencies of the neighbouring cavity modes.

In the following, we omit the prefix while speaking about the real part of thecavity eigenfrequency and will simply denote it as ωc for brevity. However, we shallremember that it also has an imaginary part,ωc = γc.

The deviation of the cavity mode frequency from the centre of the stop-band ofthe surrounding Bragg mirrors ω always takes place in realistic structures where thethicknesses of all layers change slightly across the sample. The detuning

∆ = ωc − ω (2.146)

is an important parameter, which governs the splitting between TE- and TM-polarizedcavity modes at oblique incidence.

At ∆ = 0 one can find the inplane dispersion of the cavity Fabry–Perot mode as thesolution to Exercise 2.5:

ω ≈ cπj

ncLc+

ck2xyLc

2ncπj, (2.147)

with nc being the cavity refractive index, which yields the effective mass of the photonicmode, from ω(k) ≈ ωc + 2k2/(2mph), where

mph =ncπj

cLc. (2.148)


To take into account the polarization dependence of the dispersion of microcavitymodes one should take into account the angle dependence of the reflection coefficient ofa Bragg mirror. At oblique incidence, one can conveniently define the centre of the stop-band as a frequency ω for which the phase of the reflection coefficient of the mirror iszero. The transfer matrices are modified in the case of oblique incidence and are differ-ent for TE- and TM-polarizations, as described in the previous section (see eqns (2.86)and (2.87)). Condition (2.123a) still holds at oblique incidence. It allows one to obtainthe reflection coefficient of the Bragg mirror in the form

rTE,TM = rTE,TM exp(iαTE,TM(ω − ωTE,TM)) (2.149a)

= rTE,TM exp(inc

cLTE,TM

DBR cos(φ0(ω − ωTE,TM))

, (2.149b)

where for TE-polarization:

rTE =

√1− 4

nf

n0

cos ϕf

cos ϕ0

(na cos ϕa

nb cos ϕb

)2N

, (2.150a)

ωTE =πc

2(a + b)

na cos ϕa + nb cos ϕb

nanb cos ϕa cos ϕb, (2.150b)

LTEDBR =

2n2an2

b(a + b) cos2 ϕa cos2 ϕb

n20(n

2b − n2

a) cos2 ϕ0, (2.150c)

where N is the number of periods in the mirror, ϕ0 is the incidence angle, ϕa,b are thepropagation angles in layers with refractive indices na, nb, respectively, and ϕf is thepropagation angle in the material behind the mirror, which has a refractive index nf .They are linked by the Snell–Descartes law:

n0 sin ϕ0 = na sinϕa = nb sinϕb = nf sinϕf . (2.151)

In TM-polarization:

rTM =

√1− 4

nf

n0

cos ϕ0

cos ϕf

(na cos ϕb

nb cos ϕa

)2N

, (2.152a)

ωTM =πc

2

na cos ϕb + nb cos ϕa

nanb(a cos2 ϕa + b cos2 ϕb), (2.152b)

LTMDBR =

2n2an2

b(a cos2 ϕa + b cos2 ϕb)

n20(n

2b cos2 ϕa − n2

a cos2 ϕb). (2.152c)

One can see that ωTM increases faster than ωTE with an increase of the incidence angle.LDBR increases with the angle in TM-polarization and decreases in TE-polarization.Finally, r increases with angle in TE-polarization and decreases in TM-polarizationif n0 = nf . One can see that the stop-bands move towards higher energies with increaseof the inplane component of the wavevector of light, both in TE and TM polarisations.This is why light from the cavity mode can be resonantly scattered to the so-called


leaky modes whose frequencies do not belong to the stop-bands. Via leaky modes, thephotons can escape from microcavities, provided that the scattering (e.g., on structuralimperfections) is efficient enough. The leaky modes limit quality factors of the planarcavities contributing to the broadening of the cavity modes.

Substituting into the equation for the cavity eigenmodes (2.134) the renormalizedstop-band frequencies (2.139), (2.143) and effective lengths (2.140) and (2.144), onecan obtain as shown by Panzarini et al. (1999a), the angle-dependent TE–TM splittingof the cavity modes:

ωTE(ϕ0)− ωTM(ϕ0) ≈ LcLDBR

(Lc + LDBR)22 cos ϕeff sin2 ϕeff

1− 2 sin2 ϕeff

∆ , (2.153)

where ϕeff ≈ arcsin n0

ncsin ϕ0 and LDBR is given by eqn (2.124). Obviously, the split-

ting is zero at ϕ0 = 0 as there is no difference between TE- and TM-modes at normalincidence. One can see that the sign of the TE–TM splitting is given by the sign of thedetuning between the cavity mode frequency at normal incidence and the centre of thestop-band. Changing the thickness of the cavity one can tune ∆ and change the TE–TMsplitting within large limits. Usually, the TE–TM splitting is much smaller than the shiftshown in eqn (2.136) (note the relation kxy = (ω/c) sin ϕ0).

Finally, note that in addition to the Fabry–Perot cavity modes described above, themicrocavities possess rich spectra of guided modes. Their spectrum can be found fromeqn (2.96) (for TE-polarization) and eqn (2.101) (for TM-polarization) at kxy > ω/c.

In summary to this section, the finite transmittivity of the Bragg mirrors leads tothe broadening of the peaks in transmission and dips in reflection corresponding to thecavity modes. This broadening is related to the imaginary part of the eigenfrequencyof the modes and is characterized by a quality factor of the cavity Q. The dispersion ofconfined light modes in microcavities is parabolic to a good accuracy, while the parabolacan have a different curvature in TE- and TM-polarizations. The splitting of TE and TMcavity modes can have a positive or negative sign depending on the difference betweenthe position of the mode at kxy = 0 and the centre of the stop-band of surroundingBragg mirrors.

Exercise 2.7 (∗∗) Find the quality factor of a GaAs microcavity (refractive index nc =3.5, thickness Lc = 244 nm) surrounded by AlAs/Ga0.8Al0.2As Bragg mirrors con-taining 10 pairs of layers each (refractive indices of AlAs, na = 3.0, of Ga0.8Al0.2As,nb = 3.4, layer thicknesses a = 71 nm, b = 63 nm, respectively).

2.8 Stripes, pillars, and spheres: photonic wires and dots

Progress in fabrication of so-called photonic structures, i.e., dielectric structures withintentionally modulated refractive indices, has made important the detailed understand-ing of the spectra, shape and polarization of the confined light modes in these structures.In general, this is not an easy task as the variety of photonic structures studied till nowis huge and the art of designing them (“photonic engineering”) is developing rapidly.

STRIPES, PILLARS, AND SPHERES: PHOTONIC WIRES AND DOTS 65

Lord Rayleigh (1842–1919) discovered and interpreted cor-rectly what is now known as Rayleigh scattering and surfacewaves, now called solitons.

He is by far much more renowned under his peerage than underhis real name, John Strutt. However, he acquired the title by histhirties. Other exceptional achievements include codiscovery ofargon for which he was awarded the Nobel prize in 1904. Withthe Rayleigh scattering, he was the first to explain why the skyis blue (this was in 1871). A gifted experimentalist despite tougheconomy resulting in basic equipment, he pushed teaching oflaboratory courses to undergraduates with fervour. His interestsalso touched less mundane topics such as “insects and the colourof flowers”, “the irregular flight of a tennis ball”, “the soaringof birds”, “the sailing flight of the albatross” and of course, theproblem of the Whispering Gallery. In a presidential British As-sociation address, he said: “The work may be hard, and the disci-pline severe; but the interest never fails, and great is the privilegeof achievement.”

The starting idea of the photonic crystal engineering formulated in 1986 by theAmerican physicist Eli Yablonovitch was to create the bandgap for light using the pe-riodic dielectric structures. The interference effects in planar structures can induce for-mation of the stop-bands or one-dimensional photonic gaps as we have discussed inthe previous section in relation to the Bragg mirrors. More complex structures wherethe refractive index is modulated along three cartesian axes allow for creation of three-dimensional photonic gaps. Theoretically, photonic crystals represent ideal non-absor-bing mirrors and can be efficiently used for the lossless guiding of light. They havea huge potentiality for applications in future integrated photonic circuits as Fig. 2.13shows.

Fig. 2.13: Future concept optical integrated circuits with use of photonic crystals, by Noda et al. (2000). Thecomplete three-dimensional photonic gap would allow lossless propagation of light in bent waveguides.


In reality, inevitable imperfections in photonic crystals lead to the losses becauseof the Rayleigh scattering of light. A detailed analysis of various photonic crystal sys-tems can be found in the textbooks by Yariv and Yeh (2002) and Joannopoulos et al.(1995). The description of photonic crystals is beyond the scope of this book. We shallmostly address the light–matter coupling in microcavities, i.e., cavities in the photonicstructures. The photonic engineering is indeed a powerful tool for the control of light–matter coupling strength: the density of states of the photon modes governs efficientlythe emission of light by the media (which is referred to as the Purcell effect, addressedin detail in Chapter 6). In planar structures considered in the previous section the pho-tonic modes have two degrees of freedom linked to the inplane motion (Fig. 2.14(a)).Additional confinement of light can be achieved in stripes where only the motion alongthe stripe axis remains free (Fig. 2.14(b)). Stripes as well as cylinders can be called pho-tonic wires. More radical enhancement of the photonic confinement can be achieved inpillar cavities (Fig. 2.14(c)). Here, inplane photonic confinement is not perfect, so thatthe leakage of light from the pillar is possible for a part of the eigenmodes, but thequality factor of the pillar can be high enough to strongly enhance the efficiency oflight–matter coupling with respect to the planar cavities. Finally, a realistic object al-lowing for a three-dimensional photonic confinement is a dielectric (or semiconductor)sphere (Fig. 2.14(d)). Both pillar cavities and spheres can be referred to as photonicdots. The light modes confined in such “dots” have a discrete spectrum and quite pe-culiar polarization properties. They can be coupled to the optical transitions inside the“dot”, which is potentially interesting for observation of the Purcell effect and variousnonlinear optical effects. In the rest of this section we give some basic formulae for thestructures having a cylindrical symmetry (cylinders and pillar cavities) and sphericalsymmetry (spheres).

2.8.1 Cylinders and pillar cavities

Let us solve the wave equation (2.23) in cylindrical coordinates. The Laplacian operatorreads in this case:

∇2 =1

r

∂

∂r

(r

∂

∂r

)+

1

r2

∂2

∂θ2+

∂2

∂z2, (2.154)

where r, θ, z are cylindrical coordinates (see Fig. 2.15). Let us consider an infinitecylinder of radius a and dielectric constant ε. For simplicity we only consider the modeswith cylindrical symmetry, which means that electric and magnetic fields are indepen-dent of θ. Solving the wave equation with the cylindrical Laplacian (2.154) one canrepresent the z-components of the field amplitudes in this case as

Fz(ρ) = J0(γρ), ρ ≤ a , (2.155a)

Fz(ρ) = AK0(βρ), ρ ≥ a , (2.155b)

where Fz is either an electric or a magnetic field, J0 is the Bessel function, K0 isthe modified Bessel function, A is a constant that can be determined from Maxwellboundary conditions (see Section 2.5), which require in our case conservation of the z-and θ-components of the fields, γ = ((ω2n2/c2)− k2

z)1/2, β = (k2z − (ω2/c2))1/2 and


Fig. 2.14: (a) Schematic representation of a planar microcavity; (b) a photonic stripe as seen by electronicmicroscopy by Patrini et al. (2002); (c) a pillar as seen by electronic microscopy by the group in Sheffield,where the pillars can be grown elliptically to split the polarization states; (d) schematic representation of asphere (photonic dot).

kz is the wavevector of light along the axis of the cylinder. Other components of thefields can be found using the Maxwell equations (2.15c) and (2.16). Inside the cylinder:

Bρ =ikz

γ2

∂Bz

∂ρ, Bφ =

in2kz

cγ2

∂Ez

∂ρ, (2.156a)

Eρ =c

n2Bφ , Eφ = −Bρ , (2.156b)

k0 = ω/c, and outside the cylinder

Bρ =−ikz

β2

∂Bz

∂ρ, Bφ =

−ikz

β2c

∂Ez

∂ρ, (2.157a)

Eρ = Bφc , Eφ = −Bρc . (2.157b)

The triplets (Bz, Bρ, Eφ) (TE-mode) and (Ez, Eρ, Bφ) (TM-mode) are indepen-dent of each other. Let us find the spectrum of TE-modes. From eqns (2.156) and (2.157)the field components inside the cylinder can be expressed as:

Bz =1

cJ0(γρ), Bρ = − ikz

cγJ1(γρ), Eφ =

ikz

γJ1(γρ) , (2.158)


Fig. 2.15: Cylindrical coordinates.

and outside the cylinder:

Bz =1

cAK0(βρ), Bρ = − ikzA

cβK1(βρ), Eφ =

ikzA

βK1(βρ) . (2.159)

Application of Maxwell boundary conditions at ρ = a yields

− J1(γa)

γJ0(γa)=

K1(βa)

βK0(βa). (2.160)

This is a transcendental equation for the eigenfrequencies of the cylinder that determinesthe dispersion of both guided modes (real eigenfrequency, infinite lifetime and qualityfactor, (ω/c) < kz ≤ n(ω/c)) and Fabry–Perot modes (complex ω having a finitenegative imaginary part, finite lifetime and quality factor, 0 ≤ kz ≤ (ω/c)). In a similarway one can obtain the spectrum of TM-modes:

−n2J1(γa)

γJ0(γa)=

K1(βa)

βK0(βa). (2.161)

In cylindrical waveguides, light can freely propagate along the z-axis. The electricand magnetic fields of the propagating modes can be found by multiplication of theamplitudes found above by an exponential factor exp

(i(kzz−ωt)

). In pillar microcav-

ities, light propagation is confined in the z-direction usually by the Bragg mirrors (seeFig. 2.14(c) for an electron microscopy image). Because of the photonic confinementin the z-direction, kz now takes discrete values approximately given by eqn (2.135),where Lc would be the distance between the two Bragg mirrors. The spectrum of eigen-frequencies of the pillar microcavity is also discrete, which allows it to qualify as aphotonic dot. As in the infinite cylinder, the eigenmodes of such a dot can be formallydivided in two categories: the “Fabry–Perot” modes having 0 ≤ kz ≤ (ω/c) and the


“guided” modes with (ω/c) ≤ kz . The profile of electric and magnetic fields in thez-direction can be obtained as for the planar cavities. If the Bragg mirrors confining thepillar cavity are infinite, the “guided” modes have an infinite lifetime, while “Fabry–Perot” modes have a finite lifetime and finite quality factor in all cases. In realisticstructures, the lifetime of all the modes is finite, while it can become very long in thecase of efficient photonic confinement. The quality factor record values, obtained intoroidal microresonators, exceed 108, as shown by Armani et al. (2003), which corre-sponds to a lifetime of the order of 10−7 s.

Pillar microcavities have attracted special attention very recently due to the experi-mental observation of the strong coupling of light with individual electron–hole statesin semiconductor quantum dots embedded inside the cavities. This is further discussedin Chapter 4.

2.8.2 Spheres

To describe the light modes in the dielectric or semiconductor microspheres it is conve-nient to rewrite the wave equation (2.23) in spherical coordinates.

Fig. 2.16: A dielectric sphere and the path of light in the whispering-gallery mode (left); localization of lightin a sphere due to multiple internal reflections (right).

The Laplacian in spherical coordinates reads:

∇2 =1

r2

∂

∂r

(r2 ∂

∂r

)+

1

r2 sin2 φ

∂2

∂θ2+

1

r2 sinφ

∂

∂φ

(sin φ

∂

∂φ

), (2.162)

where r, θ, φ are the radius, polar and azimuthal angle, respectively (see Fig. 2.16). Letus consider a sphere of radius a and dielectric constant ε surrounded by vacuum.

The solution for the amplitudes of the electric and magnetic fields inside the spherecan be represented for a given mode as:⎛

⎝F inr

F inθ

F inφ

⎞⎠ =

⎛⎝a1

a2

a3

⎞⎠ jl(kinr)Pm

l (cos θ)eimφ , (2.163)


where Fr, Fθ, Fφ are the components of either the electric or magnetic field, jl(x) isthe spherical Bessel function of the first kind, Pm

l (x) is the associated Legendre poly-nomial, l and m are angular and azimuthal mode numbers, a1, a2, a3 are coefficients(different for electric and magnetic fields, of course). An additional radial number ofthe mode n allows the radial wave number inside the cavity to be linked to the cavityradius a: kin ≈ (πn/a) with n ∈ N.

The fields outside the sphere are given by⎛⎝F out

r

F outθ

F outφ

⎞⎠ =

⎛⎝b1

b2

b3

⎞⎠hl(koutr)P

ml (cos θ)eimφ , (2.164)

where hl(x) is the spherical Hankel function of the first kind, kout is the wave num-ber outside the cavity, b1, b2, b3 are coefficients. The links between linear coefficientsfor the electric and magnetic field are always given by the Maxwell equations (2.15c)and (2.16). In general form they are rather complex. We address the interested reader tothe book by Chew (1995) containing a rigorous derivation of the spectra of the eigen-modes of dielectric spheres. Interestingly, there is no allowed optical modes having aspherical symmetry (i.e. having the angular number l = 0). Such a mode would havea diverging magnetic field in the centre of the cavity, which contradicts the Maxwellequation (2.15b). As in pillars, the sets of equations for TE- and TM-modes can be de-coupled. TE-modes in this case are defined as those having Ein

r = Eoutr = 0 and for

TM-modes Binr = Bout

r = 0.The spectrum of eigenmodes of the sphere is discrete. It can be obtained by matching

of the fields (2.163) and (2.164) by the Maxwell boundary conditions requiring

F inθ = F out

θ , F inφ = F out

φ . (2.165)

Substitution of the functions (2.163) and (2.164) into the conditions (2.165) givesthe equations for eigenmodes. In TE-polarization:

H ′l(kouta)Jl(kina) =

√εJ ′

l (kina)Hl(kouta), (2.166)

and in TM-polarization:

√εH ′

l(kouta)Jl(kina) = J ′l (kina)Hl(kouta), (2.167)

where Jl(x) = xjl(x) and Hl(x) = xhl(x), with ′ meaning derivative over the argu-ment of the function.

While an exact spectrum of the light modes in a sphere requires solution of the tran-scendental equations (2.166) and (2.167), for qualitative understanding of localizationof light in the sphere the images of ray propagation and arguments of geometrical op-tics are very helpful. Actually, light can be trapped by total internal reflection near thesphere’s surface in a resonant so-called whispering-gallery mode localized around theequator.


In the case of a dielectric sphere the whispering-gallery modes are the eigenmodeshaving high numbers l and m (high usually means higher than 10 in this context). Fig-ure 2.16 shows schematically how the whispering-gallery modes appear. Light is prop-agating along the surface of the sphere each time experiencing an almost total internalreflection (not exactly total because of the curvature of the surface). The cyclic boundaryconditions determine the energy spectrum of such modes.

These modes have a huge (but finite) quality factor and a long (but finite) lifetime.They can be qualified as “quasiwaveguided” modes. Propagation of whispers in thedome of Saint Paul’s cathedral is assured by such “quasiwaveguided” acoustical wavessubject to cyclic boundary conditions.

Whispering-gallery modes have been studied in micrometre-size liquid droplets andglass spheres from the early days of laser physics. Now, very high quality spheres areobtained by melting a pure silica fiber in vacuum, as done by Collot et al. (1993).The transverse dimensions of the modes can be reduced down to a few micrometres,the sphere’s diameter being about 100µm. The mode is strongly confined. Figure 2.17shows the calculated field intensity in a TE-polarized whispering-gallery mode in a sil-ica microsphere.

Fig. 2.17: TE whispering-gallery modes with mode numbers n = 1, m = l = 20 (from lectures notes byIkka Nitonen).

The value of the angular number l for such a mode is close to the number of wave-lengths of light on the optical length of the equator of the sphere. The value l −m + 1is equal to the number of the field maxima in the polar direction (i.e. perpendicular tothe equatorial plane). The radial number n is equal to the number of maxima in the di-rection along the radius of the sphere, and 2l is the number of maxima in the azimuthalvariation of the field along the equator. The resonant wavelength is determined by thenumbers n and l. The modes with lower indices l and m have a lower quality factor andshorter lifetime. They can be referred to as Fabry–Perot modes. These modes are bettersuited for coupling to the material of the sphere than whispering-gallery modes as theypenetrate deeper inside the sphere.

The German physicist Mie solved in 1908 the problem of scattering of a plane waveby a dielectric sphere and demonstrated the existence of resonances, now known as Mie


Eli Yablonovitch, produced the first artificial photonic crystals, the engineered counterparts of such struc-tures as on the right from Yablonovitch (2001): a butterfly wing that—with its incomplete bandgap—producesiridescent colours. Yablonovitch (1987) authored the second most highly cited Physical Review Letter.

resonances, linked to the eigenmodes of the sphere including the whispering-galleryand Fabry–Perot modes. Mie theory has allowed, in particular, to describe the scatter-ing of light of the Sun by droplets of water in the clouds. It explains the colour of thesky and the appearance of rainbows and glories. Figure 2.8.2 shows the results of calcu-lation of the colour of sky in the presence of two rainbows performed within Mie theoryassuming 500 micrometre-size water drops randomly distributed in the atmosphere. Thesimulation result is superimposed with a photograph to demonstrate the accuracy of themodel.

Gustav Mie (1869–1957). In background, a computer simulation (with MiePlot) for r = 500 µm waterdrops, superimposed on a photograph of a primary and secondary rainbows.

Mie is known essentially for the solutions he provided to the problem of light described by Maxwell’s equa-tions interacting with a spherical particle (commonly but incorrectly called “Mie theory”, when instead of a“theory” this is the analytical solution of the equation of an actual theory, namely, electromagnetism). Initiallya pure theorist, he indulged in experimental work toward the end of his career.

FURTHER READING 73

Exercise 2.8 (∗) The dome of Aya Sofya Mosque in Istambul has a radius of 31 metres.Find the wavelength of the whispering gallery-mode of this dome having an angularnumber l = 31.

Exercise 2.9 (∗∗∗)The classical Hanbury Brown–Twiss effectIn this chapter we have focused on the Hanbury Brown–Twiss effect in the temporal

domain and will again when we come back to it from the quantum point of view inthe next chapter. Historically, the method of intensity interferometry of light arose as ameans of measuring the angular diameter of stars, and is related to spatial correlations.

Compute the intensity correlation in space (at zero time delay) given by expression

C = 〈E∗(r1)E∗(r2)E(r1)E(r2)〉 (2.168)

for incident field E impinging on two points r1 and r2 (spatially separated detectors onEarth) and that is collected from the angular spread of the source seen from the Earthas plane waves with wavevectors k and k′, i.e., such that E(r) = Ek exp(ik · r) +Ek′ exp(ik′ · r) for both detectors. Compare with amplitude correlations such as mea-sured by a conventional optical interferometer and discuss how the apparent diameterof the star can be deduced from varying |r1 − r2|. What is the advantage of the HBTsetup over, e.g., the Michelson one?

2.9 Further reading

Many excellent books on light propagation in various photonic structures are availablethat will usefull supplement the content of this chapter. The interested reader will findfurther details on the subject of this Chapter in Born and Wolf (1970) and Jackson(1975) who give a general picture of classical optics, and the Yariv and Yeh (2002)and Joannopoulos et al. (1995) textbooks, which are devoted to optical properties ofdielectric structures including photonic crystals. More details on the transfer matrixmethod for description of the optical properties of microcavities can be found in thetextbook by Kavokin and Malpuech (2003). Rigorous derivation of the spectra of somephotonic structures including spheres is given in the monograph by Chew (1995).

3

QUANTUM DESCRIPTION OF LIGHT

In this chapter we present a selection of important issues, concepts andtools of quantum mechanics, which we investigate up to the level ofdetail required for the rest of the exposition, disregarding at the sametime other elementary and basic topics that have less relevance tomicrocavities. In the next chapter we will also need to quantize thematerial excitation, but for now we limit the discussion to light, whichallows us to lay down the general formalism for two special cases—theharmonic oscillator and the two-level system.

76 QUANTUM DESCRIPTION OF LIGHT

3.1 Pictures of quantum mechanics

3.1.1 Historical background

Historically, quantum mechanics assumed two seemingly different formulations: one byHeisenberg, called matrix mechanics, to which we return in Section 3.1.5, and anothershortly to follow by Schrodinger, based on wavefunctions. Although highly competitiveat the start, the two “theories” now framed in modern mathematical notations displayclearly their interconnection and unity.19 As the two pictures are useful both for physicalintuition and practical use, we study both of them. We start with the Schrodinger picturethat offers the best support for the postulates of quantum mechanics in the interpretationof the so-called Copenhagen school, which is nowadays the commonly agreed set ofworking rules to deal with practical issues, although as a worldview this interpretationis now largely discarded.20

3.1.2 Schrodinger picture

In the Schrodinger picture, one starts with the Schrodinger equation

i∂

∂t|ψ〉 = H |ψ〉 , (3.1)

written here with Dirac’s (1930) notation of bra and kets, an elegant convention cap-turing the essentials of the mathematical structures, as is discussed below. H is thequantum Hamiltonian of the system to be specified for each case under considerationand is the reduced Planck’s constant.

The first postulate of quantum mechanics: the quantum state

The postulates that govern quantum mechanics, essentially laid down by von Neumann(1932), provide the recipe to use the formalism and relate it to experiments:

I — A quantum system is described by a vector—called the state of the system—in a separable, complex Hilbert space H. This vector, in Dirac’s notations, isdenoted |ψ〉, where ψ is the set of variables needed to fully describe the system,but the notation used symbolically affords powerful abstract manipulations.21

19Quantum theory brought about many interesting developments in the history of science for all the con-troversies among its founding fathers and their personal views that often were the occasion for great drama.Beyond the famous opposition between Bohr and Einstein, there were also even animus feelings betweenSchrodinger and Heisenberg, and heated opposition amidst political tensions between Heisenberg and Bohrwho worked together on the Copenhagen interpretation. A theatrical unravelling on the birth of quantum me-chanics based on recently released documents provided the impetus for the recent play of M. Frayn, Copen-hagen.

20In the field of interpretation of quantum mechanics, although there is no consensus, the modern trendfavours the theory of decoherence and Everett interpretations of consistent realities (or parallel universes). Weshall briefly touch upon some of these aspects that intersect with the physics of microcavities, but otherwisewill remain oblivious and stick to the conventional Copenhagen interpretation. For further studies, cf., e.g.,Quantum Theory and Measurement, J. A. Wheeler and W. Zurek (Princeton Series in Physics), 1984.

21The main advantage of Dirac’s notation is the considerable simplification it brings when handling thedual space of H. Whereas a ket is a vector of some given nature, a bra is actually a linear application defined

PICTURES OF QUANTUM MECHANICS 77

Paul Dirac (1902–1984), Werner Heisenberg (1901–1976) and Erwin Schrodinger (1887–1961) in theStockholm train station, 1933, before the Nobel prize ceremony to award the 1932 prize to Heisenberg for thecreation of quantum mechanics and the shared 1933 prize to Schrodinger and Dirac for the discovery of newproductive forms of atomic theory. The delay in awarding the 1932 Nobel prize was due to the defiance of theNobel committee towards quantum mechanics.

In this chapter, the quantum system of ultimate interest is light, for which two ab-stract systems will eventually prove sufficient to describe it fully: a two-level systemwith associated vector space H2 = α |0〉 + β |1〉 , α, β ∈ C (which will describethe polarization state of light) and an harmonic oscillator, which in stark contrast tothe simple spaceH2 requires a functional space of square modulus integrable functionsHa = |ψ〉 , |〈ψ|ψ〉|2 <∞ (and that will describe the oscillations of the normal modeof the light field, cf. Section 2.1.1). These two specific cases will allow us to illustrate invery different cases the mechanism of the theory. We will describe the two-level systemin terms of a spin and the harmonic oscillator in terms of a mechanical oscillator, allow-ing us to recourse to widely used language and intuition. When it is time to return thesenotions to what we initially planned them for—the quantum description of light—we

on the ket space. With a little practice, one can almost forget entirely this underlying mathematical structure.Such shortcuts motivated Bourbaki mathematician Jean Dieudonne to state “It would appear that today’sphysicists are only at ease in the vague, the obscure, and the contradictory”. An interesting discussion ofthese conflicting approaches is given by Mermin in “What’s Wrong with This Elegance?” in the March 2000issue of Physics Today. We will, of course, make ample use of such simplifications. Such “rules of thumbs” areas follows: the ket |ψ〉 goes to 〈ψ|, coefficients are conjugated, α |ψ〉 → 〈ψ|α∗ and operators are transpose-conjugated, always written in reverse order, so that ΩΛ |ψ〉 → 〈ψ|Λ†Ω†. So for instance, the “dual” ofSchrodinger equation reads

−i∂

∂t〈ψ| = 〈ψ|H , (3.2)

since H is hermitian.


David Hilbert (1862–1943) and John von Neumann (1903–1957), two pure mathematicians as emblems ofthe inherent abstract nature of quantum theory. The quantum state is best described as a vector in a Hilbertspace, as has been axiomatized by von Neumann.

Hilbert’s interest in physics started in 1912 and became his main preoccupation. It provided impetus both toquantum mechanics and relativity. He invited Einstein to give lectures on general relativity in Gottingen atwhich occasion some believe he derived the Einstein field equations. He put forth 23 problems at the Inter-national Congress of Mathematicians in Paris in 1900 setting the edge of mathematical knowledge at the newcentury. The sixth one is “Axiomatize all of physics.” It is, as yet, unresolved.

Von Neumann was an extraordinary prodigy. At six, he could mentally divide two eight-digit numbers. Hewas famous for memorizing pages on sight and, as a child, he entertained guests by reciting the phone book.Beside axiomatization of quantum physics, which he connected to the Hilbert spaces—thereby solving thesixth problem in this particular case—he made crushing contributions—when he did not create the field—tofunctional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics,statistics, game theory and complexity theory. Many place him as among the greatest geniuses. Fellow math-ematican Stanislaw Ulam’s biography “Adventures of a Mathematician” is largely a tribute to his mentor withmany anecdotes of this peculiar character, famous for his hazardous driving, taste for parties and hypnotizationby women.

shall stick to the common practice of keeping the vocabulary of spin and analogies ofclassical mechanics, so these asides are not completely out of purpose. To later link withthe statistical interpretation, we further demand that

〈ψ|ψ〉 = 1 . (3.3)

Exercise 3.1 (∗) Show that the normalization condition, eqn (3.3), remains satisfied atall times through the dynamics of Schrodinger equation (3.1).

We have noted |0〉 and |1〉 two basis vectors of H2. Mathematically it is convenientto relate them to the canonical basis, i.e.,

|0〉 =

(1

0

)and |1〉 =

(0

1

). (3.4)

Physically, we could choose to represent the first state with right circular polarization ofthe light mode, |〉 = |0〉, and the second with left circular polarization, and |〉 = |1〉.We will refer to such states as spin-up and spin-down, respectively (for a true spin 1

2


particle we might prefer the depiction |↑〉 and |↓〉). The first postulate says that the mostgeneral state for a two-level system is |ψ〉 = (α, β)T /

√|α|2 + |β|2, with α, β ∈ C or,if the normalization has been properly ensured

|ψ〉 = α |0〉+ β |1〉 . (3.5)

With the description in terms of Jones vectors, such a general state describes an arbitrarypolarization. Basis (3.4) is not always the most convenient. In generic terms, anotherimportant basis reads

|+〉 =|0〉+ |1〉√

2, |−〉 =

|0〉 − |1〉√2

. (3.6)

Here, we have chosen the conventional notations for generic two-level systems, todayreferred to as qubits for their fundamental role in quantum computation, but of courseit transposes immediately with states of polarizations.

Exercise 3.2 (∗) Based on the definitions of the various possible polarization states,express the states of linear horizontal and vertical polarization |↔〉 , |〉 and lineardiagonal polarization |〉 , |〉 as a function of states |〉 and |〉. Typical exampleswould be the right and left circular polarization of light, given by, respectively:

|〉 =|↔〉+ i |〉√

2, |〉 =

|↔〉 − i |〉√2

. (3.7)

Obtain all other relations between any two bases. Write state (3.5) in each of thesebases. How are states of elliptical polarization described?

Exercise 3.3 (∗) Two bases are said to be conjugate when any vector of the first onehas equal projection on all vectors of the second. Study the conjugate character ofbases encountered so far.

The inner product is the vector scalar product, i.e.,

if |ψ〉 =

(α

β

)and |φ〉 =

(γ

δ

), 〈ψ|φ〉 = (α∗, β∗)

(γ

δ

)= α∗γ + β∗δ . (3.8)

This illustrates the richness of Dirac’s notation as the bra vector 〈ψ| is now tentativelyassociated to (α, β)∗ and the inner product reads as a product of bra and ket vectors,forming a “braket” (whence the names of each vector in isolation). Being of finite di-mension,H2 is trivially complete and separable.

A choice of basis for Ha first demands a choice of a space where to project thestates of the system. An oscillator could be characterized in real space by its position asit oscillates, or in momentum space by its velocity. A classical oscillator would requirespecification of both of these at a particular time to be fully specified. In quantum me-chanics, as the dynamics is ruled by a first-order differential equation (eqn (3.1)), thestate is fully characterized by one only of these pieces of informations. Later we shallsee that the simultaneous specification of both is in fact impossible.


If the state is projected in real space, a basis could consist of all the states |x〉 describ-ing an oscillator located at x on a 1D axis (without loss of generality). The first postulatein this case asserts that the most general state for an oscillator is |ψ〉 =

∫ψ(x) |x〉 dx

Now the linear superposition requires an integral as there is a continuous varying set ofbasis states.

The second and third postulates: observables and measurements

II — A physical observable is described by an hermitian operator Ω onH. The possiblevalues obtained from an observable are its eigenvalues. If the eigenstates of Ω arefound to be |ωi〉 with associated eigenvalues ωi, the result ωi0 is obtainedfor a system in state |ψ〉 with probability |〈ωi0 |ψ〉|2.

III — After measurement of an observable Ω that has returned the value ωi0 , the statehas collapsed to |ωi0〉, so that repeating the observation yields ωi0 , this time withcertainty.

An observable is a property of the state that one can determine through an appro-priate measurement on the system, or vividly “something that can be observed”. Suchaccuracy in defining basic notions has been made compulsory after the counterintuitiveimplications of quantum mechanics, of which we shall see a few in the following.

For a quantum system that has variables with classical counterparts, as is the casewith a quantum oscillator for which a position and momentum can still be measured,Bohr formulated the prototype of the second postulate in what came to be known asthe correspondence principle, which asserts that the classical variables x (for position,here in 1D) and p (momentum, also in 1D) are upgraded in quantum theory to hermitianoperators X and P defined, in the position basis, as

〈x|X |y〉 = xδ(x− y) and 〈x|P |y〉 = −iδ′(x− y) , (3.9)

where δ′ is the derivative of the delta function.Any dynamical variable function of these variables extends to the quantum realm

in this way, so for instance the kinetic energy 12mv2 is written 1

2p2/m and its quan-tum counterpart reads 1

2P 2/m. The classical Hamiltonian22 also extends in this wayto a quantum Hamiltonian, which appears in the Schrodinger equation (3.1). There-fore, one quantizes an harmonic (mechanical) oscillator of mass m and force constant κin phase space of position–momentum (x, p) starting from its (classical) HamiltonianHc = p2/(2m) + κx2/2 to read quantum mechanically

H =1

2ω(X2 + P 2) , (3.10)

through correspondence

22The Hamiltonian in classical mechanics is an analytic function that describes the state of a mechanicalsystem in terms of its phase space variables, typically position and momentum variables. It is a reformulationof Newton mechanics that is more suited to shift to quantum mechanics. For most practical use, the Hamilto-nian of a system can be understood as the energy of the system written in terms of specified coordinates. Formore detailed discussions, see for instance Classical Dynamics of Particles and Systems, S. T. Thornton andJ. B. Marion, Brooks Cole (2003).


x→ (κm)−1/4X and p→ (κm)1/4P , (3.11)

with ω ≡ √κ/m, where along with quantization of scalars (x, p) we have scaled the

operators in term of a dimensionless variable ω (as we eventually wish to describe os-cillations of light modes, without reminiscence of any mechanical embryos) .

For a quantum system that has no mechanical counterpart, as is the case with spin,23

the definition of mathematical observables to describe experimentally measurable prop-erties of a system, is the result of guesswork to adjust with experimental facts. This neednot concern us, however, as this procedure has been carried out a long time ago for allproperties that we will need to describe quantum mechanically in a microcavity.

The second postulate states that in H2, such an observable is described by a 2 × 2hermitian matrix. Any such matrix can be decomposed as a linear superposition, over C,of the identity 1 = (1 0

0 1) and the Pauli matrices:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

), (3.12)

written here, as will always be the case unless specified otherwise, in the canonicalbasis (3.4). If we consider, for instance, the observable Sz = σz , which provides aphysical dimension to the result obtained, the second postulate asserts that if Sz is mea-sured on a system in the state (3.5), the possible outputs are ± (the eigenvalues of Sz ,which are obtained straightforwardly as the operator is diagonal) and + is obtainedwith probability |α|2, while − is with probability 1 − |α|2. Postulate III states thatafter the measurement, |ψ〉, previously in the superposition (3.5), has collapsed to oneof the eigenstates |0〉 or |1〉, depending on which eigenvalue has been obtained.

So when a photon is absorbed, it transfers to the detecting material an angular mo-mentum of ±, depending on which state of circular polarization it is detected in. Theoutcome is deterministic if the photon was in one eigenstate |〉 or |〉. But accordingto postulate II and Exercise 3.2, which obtains the possible linear polarizations as a su-perposition of circular ones, then a linearly polarized photon still impinges one quantumof angular momentum, but now with a given probability. It is only when a beam madeup of many photons is considered that distinguishing features of linear polarization (likezero average angular momentum) appear. Still, all photons in a pure linearly polarizedbeams are identical.

Statistical interpretation

Quantum mechanics, according to postulate II, is a probabilistic theory: the outcome ofa given experiment is in general unknown, the theory can only account for the statisti-cal spread of repeated measurements. In this context, relevant quantities to compute areaverage and spread about this average, that is, the value obtained when an experiment isrepeated on different systems all in the same quantum state. There should be an ensem-ble of systems as once a measurement has been made on one of them, it has collapsed

23Although we contend that spin also describes polarization, which we have seen is a property of classicallight as well; this point will be clarified in Section 3.2.3.


on an eigenvalue of the observable, so the next measurement should not be made on thesame physical system but on another system in the same initial quantum state.

Since ωi0 is obtained with probability |〈ψ|ωi0〉|2 when Ω is measured on |ψ〉, theaverage value of this observable, written 〈Ω〉, is the weighting of all possible outcomes(that is, Ω eigenvalues) and so is

∑i ωi|〈ψ|ωi〉|2, so that through bra and ket algebra

〈Ω〉 =∑

i

ωi〈ψ|ωi〉〈ωi|ψ〉 , (3.13a)

= 〈ψ|(∑

i

ωi |ωi〉〈ωi|)|ψ〉 , (3.13b)

= 〈ψ|Ω |ψ〉 , (3.13c)

since eqn (3.13b) is Ω spelled out in its eigenstates basis, cf. eqn (A.3) in Appendix A.Note that 〈Ω〉 does not specify on which quantum state the average has been taken,which is usually clear from the context. In cases where the specification is important,Dirac’s notation once again provide a most convenient alternative, eqn (3.13c).

The dynamics of such an average follows from Schrodinger equation as:24

∂

∂t〈Ω〉 =

i

〈[H,Ω]〉 . (3.15)

Uncertainty principle

Coming back to a single experiment on a two-level system, assume that the value +

has been obtained as the result of measuring Sz on state (3.5), so that the system has nowcollapsed to state |0〉 according to the third postulate. If another measurement of Sz isperformed, the same value + will be obtained with probability one, and− with prob-ability |〈1|0〉|2 = 0. So the result is deterministic in this case. But what if the observableassociated to, say, Sx = σx is now measured? One can check that states (3.6) are theeigenstates of Sx with eigenvalue ± so that the system will collapse on one of themas the result of the measurement, with probability 1/2 (cf. Exercise 3.2). If, after this,one returns to Sz , the result has been randomized completely and the first measurementwill yield any possible value± with probability 1/2. This is a manifestation of the un-certainty principle that arises from the second and third postulate from non-commutingoperators.

24Spelling out the derivation of the equation of motion for a quantum average:

∂

∂t〈Ω〉 =

∂

∂t〈ψ|Ω |ψ〉 , (3.14a)

=“ ∂

∂t〈ψ|

”Ω |ψ〉 + 〈ψ|Ω

“ ∂

∂t|ψ〉

”, (3.14b)

=i

〈ψ|HΩ − ΩH |ψ〉 , (3.14c)

which, contracted, yields the result. In eqn (3.14b) we took advantage of the linearity of the differentialon Hilbert spaces and their dual remembering that in this case Ω is time independent. In eqn (3.14c) wesubstituted Schrodinger equation in the ket (3.1) and bra (3.2) spaces.


Generally, it is a necessary and sufficient condition for two operators Ω and Λ toshare a common basis of eigenstates on the one hand and to commute, [Ω,Λ] = 0,on the other hand. The uncertainty principle in its most general form reads as a lowerbound for the spread in the distributions of two observables:

Var(Ω)Var(Λ) ≥(

i

2〈[Ω,Λ]〉

)2

, (3.16)

where Var(Ω) = (Ω− 〈Ω〉)2.

Exercise 3.4 (∗∗) Prove eqn (3.16) with Schwarz inequality applied to vectors (Ω −〈Ω〉) |ψ〉 and (Λ− 〈Λ〉) |ψ〉.

From eqn (3.16) it is apparent that the commutation relations between operators arean important ingredient of quantum mechanics. One can check that

[X,P ] = i , (3.17)

by computing −i(x∂x − ∂xx) on a generic test function after applying the correspon-dence principle backward to get back to scalars from the operators. This is, when appliedto the general formula, the origin for the most famous form of the uncertainty relation:

∆x∆p ≥

2, (3.18)

Composite systems and symmetry

IV — The Hilbert space of a composite system is the Hilbert space tensor product ofthe state spaces associated with the component systems.

This postulate extends in the expected way the rules of quantum mechanics from aone-dimensional case to many: the dimensionality of the entire system’s Hilbert spacescales with the number of degrees of freedom to be described quantum mechanically.25

Observables also inherit this tensor product structure. The additional variable can per-tain to the same particle, e.g., be i) another spatial dimension or ii) a property of analtogether different character like spin, or iii) can be the same variable but for anotherparticle. Starting with |ψ1D〉 the state of a particle inH1 a single-particle Hilbert space,these three cases would lead to, respectively, i) |ψ2D〉 to be projected on 〈x| ⊗ 〈y| togive the function of two variables ψ2D(x, y), ii) |ψ1D〉 ⊗ σ and iii) |ψ1D〉 ⊗ |ψ′

1D〉.The general case of an observable being Ω1⊗Ω2, it is customary to drop the explicit

tensor sign and abbreviate it into a product, Ω1Ω2 |ψ1〉 |ψ2〉 or even and as commonly,simply |ψ1, ψ2〉 or |ψ1ψ2〉 for the state.

Composite systems, however, do not simply transport the quantum “weirdness” ofthe single-particle case to the higher-dimensional one. They bring one conceptual dif-ficulty of their own rooted in correlations and known as entanglement, which is at theheart of quantum information.

25Time is an example of a variable that remains classical in non-relativistic quantum mechanics, i.e., thatis not a quantum observable and therefore is not associated to a Hilbert space. This is to be contrasted withspecial relativity where, by contrast, time is shown to carry equivalent features with space variables.


The counterintuitive physics of entangled systems is best exemplified followingBohm and Aharonov (1957) who consider the singlet spin state of two particles:

|Ψ〉 =1√2

(|↑↓〉 − |↓↑〉) . (3.19)

We repeat that the notation |↑↓〉 is a shorthand for |↑〉 ⊗ |↓〉 and that first ket (in thiscase |↑〉) refers to one of the particles and the second ket to the other particle. These twoparticles are separated though remaining in the state (3.19) (the spatial wavefunctionpart of the system has not been written, which would change to reflect this spatial sepa-ration; the spin state can remain the same for separated particles). When the separationis significantly large, the spin of the “first” (or “left”) particle is measured. As a result,the wavefunction collapses on the measured eigenstate. This, however, has the effect ofalso collapsing simultaneously the state of the other particle. Such an experiment ex-hibits nonlocal quantum correlations, i.e., correlations with no classical counterpart ina sense that we now discuss in greater detail.

There is first the obvious correlation of the measurement that says that if the leftbranch has measured, say, spin-up, then the other is assured to measure spin-down. Thisis the correlation part just as it applies in a classical sense, and that ensures total spinconservation. However, correlations of state (3.19) are also non-classical because theyalso hold in all other bases, although in these cases the wavefunction does not specifythe outcome. Therefore, if one measures the first qubit value in the basis |±〉 of eqn (3.6)(with the same eigenvalues +1 and −1) and finds, say, spin-up again, then the other bitis also −1 in the new basis. Although the wavefunction does not specify the values ofall components, the correlations always match. These correlations are finally nonlocalbecause this agreement holds even for any separation with the bits possibly measuredsimultaneously. This has been confirmed experimentally by Weihs et al. (1998).

V — The wavefunction changes or retains its sign upon permutation of two identicalparticles.

This important postulate26 can be motivated by the insightful quantum-mechanicalproperty of indistinguishable identical particles, which asserts that two particles of thesame species bear no absolute or independent role to the wavefunction that describesthem both. To make this explicit, consider the wavefunction written as a function of thegeneralized coordinates qi for the ith particle out of N , that is, ψ(q1, · · · ,qN ). Thesystem would remain the same if the jth and kth particles were to be interchanged,provided that they are identical, i.e., refer to particles of the same species that cannot bedistinguished experimentally. The quantum state would therefore also remain the same,i.e.,

ψ(q1, · · · ,qj , · · · ,qk, · · · ,qN ) = αψ(q1, · · · ,qk, · · · ,qj , · · · ,qN ) , (3.20)

26It is little appreciated that the indistinguishable characters of the quanta is a postulate, motivated by ex-perimental evidence, but that in principle can be violated without undermining quantum mechanics. Messiahand Greenberg (1964) have emphasized this point for elementary particles.


with the phase factor α = eiθ, which does not change the observables (and thereforeyields the same quantum state). Doing this twice yields α2 = 1, i.e., α = ±1. Thisshows how postulate V derives from invariance of the quantum state upon interchangeof identical particles.

Pauli (1940) has shown in the context of relativistic quantum field theory how thisproperty of the wavefunction relates to the spin of the particles thus described and istherefore also an intrinsic property that is always and consistently satisfied. Particleswith integer spin are called bosons (after Bose) and those of half-integer spin are calledfermions (after Fermi). It is shown that wavefunctions for bosons keep the same sign asparticles are interchanged, while those for fermions change sign.27

This has considerable physical consequences of strikingly different characters de-pending on the sign, although the mathematical structure assumes a simple unifyingform:

ψ(qσ(1), · · · ,qσ(N)) = ζξψ(q1, · · · ,qN ) , (3.21)

with ζ = 1 for bosons and ζ = −1 for fermions,

where σ ∈ S is a permutation28 of [1, N ] and ξ the signature of σ.

3.1.3 Antisymmetry of the wavefunction

In the case of fermions, eqn (3.21) leads to Pauli blocking or Pauli exclusion that assertsthat two fermions cannot occupy the same quantum state. This is clear from eqn (3.20)with α = −1 since in this case the probability (amplitude) is zero when the qj = qk,which makes it impossible to find the system with two particles in the same projec-tions |qi〉. Pauli blocking can also be stated with overall quantum states in which casethe entire wavefunction vanishes. If the ith particle out of N is in state |φi〉, the wave-function of the whole system reads as a determinant since this is the mathematical ex-pression to associate sign swapping to function (column) or coordinates (row) permuta-tions:29

ψ(q1, · · · ,qN ) ∝ det1≤i,j≤N

(φi(qj)

). (3.22)

The constant of normalization of eqn (3.22) depends on the orthogonality of the |φi〉. Itdiverges as some of the |φi〉 overlap to unity, as a manifestation of Pauli exclusion.

27In addition to possible (small) deviations from Bose and Fermi statistics for elementary particles, whichwould be of a fundamental character, there are also deviations that result from cooperative or compositeeffects, or of reduced dimensionalities. Such emerging statistics are typical of solid-state physics and examplesare provided in the next chapter with excitons.

28A permutation σ of the set of integers [1, N ] = 1, 2, . . . , N is a one-to-one function from [1, N ]unto itself, e.g., σ(1) = 2, σ(2) = 1 and σ(3) = 3 is a permutation of [1, 3]. There are N ! permutationsof [1, N ]. The set of all permutations is written S. The signature ξ of a permutation, also known as itsparity, is ±1 according to whether an even or odd number of pairwise swappings is required to bring thesequence (1, . . . , N ) into (σ(1), . . . , σ(N)).

29With only two particles to simplify notations, in respective states |φi〉, i = 1, 2, the total fermion

wavefunction reads ψ(q1,q2) ∝ φ1(q1)φ2(q2) − φ1(q2)φ2(q1) =˛

φ1(q1) φ2(q2)φ1(q2) φ2(q1)

˛.


Exercise 3.5 (∗) Show that in the case where the N single-particle states are orthog-onal, 〈φi|φj〉 = δi,j , the constant of normalization of the symmetrized or antisym-metrized wavefunction is 1/

√N !.

3.1.4 Symmetry of the wavefunction

If the sign remains the same in eqn (3.21), there is no cancelling and the probabilitydoes not vanish for any given superpositions. Rather the opposite tendency holds thatthe probability is enhanced for two different particles to be found in the same quantumstate. The accurate and general formulation states that the probability that N bosonsbe found in the same quantum state is N ! times the probability that distinguishableparticles be found in the same state.30

A more familiar statement is that if N particles are in the same state, the probabilityfor the (N + 1)th to be found also in this same state is N + 1 times this probabilityfor distinguishable particles. It is proved with conditional probabilities, if “A” is thestatement “the (N +1)th particle is in some given state |ϕ〉”, while “B” is the statement“the N other bosons already are in state |ϕ〉”, thus the probability (with respect todistinguishable particles) that “A” is realized given that “B” is, is P(A ∩ B)/P(B),that is (N + 1)!/N !. This N + 1 coefficient characterises bosonic stimulation.31

Solving the Schrodinger equation

Now that the postulates and interpretation of quantum mechanics have been laid down,we can carry on with solving the equations.

30The proof is instructive and goes as follows: let |Ψ〉, the wavefunction of the state, be developed on abasis |φi〉 of H⊗N , first assuming distinguishable particles that do not require the symmetry postulate:

|Ψ〉C =X

i1,...,iN

αi1,...,iN|φi1 〉 · · ·

˛φiN

¸, (3.23)

(we subscripted the state with C for “classical”), then symmetrizing the state to ensure bosonic indistinguisha-bility (3.21):

|Ψ〉B =1√N !

Xi1,...,iN

Xσ∈S

αi1,...,iN

˛φσ(i1)

¸ ⊗ · · · ⊗ ˛φσ(iN )

¸. (3.24)

The probability amplitude that all distinguishable particles be in the same quantum state, say |φ1〉, is

〈φ⊗N1 |Ψ〉C = α1,··· ,1 , (3.25)

while for indistinguishable particles,

〈φ⊗N1 |Ψ〉B =

N !√N !

α1,··· ,1 . (3.26)

The ratio of these probabilities is N !. Note especially that it is independent of α (which should be nonzero,meaning that one cannot find all particles in the same state if they do not all have a projection in this state).This ratio is therefore independent of any linear combination of the α and thereby of any state of the system,thus showing that the probability to find all particles in the same state if they are indistinguishable bosonsis N ! the probability for distinguishable particles.

31As all results involving conditional probabilities—and in this case further complicated by the quantuminterpretation—the bosonic stimulation is more subtle than it appears. However, considered in first order ofperturbation theory, it becomes an exact and useful concept in the form of renormalization scattering rates ofemission or in rate equations, as shall be seen in greater detail in later chapters.


Exercise 3.6 (∗) Reduce by separation of variables the Schrodinger equation for a time-independent Hamiltonian to Schrodinger’s time-independent equation

H |φ〉 = E |φ〉 , (3.27)

with |φ〉 now time independent.

When the Hamiltonian is time independent, a formal solution is obtained as

|ψ(t)〉 = e−iHt/ |ψ(0)〉 . (3.28)

All the postulates of quantum mechanics apply on |ψ(t)〉, which now just happensto change with time.

Exercise 3.7 (∗∗) Solve the Schrodinger equation (3.1) for the quadratic potential (3.10).Technically this requires finding the eigenstates and eigenenergies of H .32

Exercise 3.8 (∗∗) Using the results of the previous exercise, study the time dynamicsof the initial conditions 〈x|ψ1〉 ∝ exp(−(x − x0)

2/L2) for various (relevant) sets ofparameters (x0, L).

Liouville–von Neumann equation

One can work with all operators by promoting the quantum state |ψ〉 to an operator

ρ = |ψ〉〈ψ| , (3.30)

which is the projector of state |ψ〉, and is known as the density operator. Its main interestarises when statistical physics is added to quantum mechanics, as we shall discuss inSection 3.2.1. For now it is just another representation for the state whose main effect33

is the introduction of new formulas to carry out computations, for instance the averageof an observable Ω in state (3.30) is now obtained through the formula:34

32The eigenstates |φn〉 of the Schrodinger equation for the harmonic potential are:

〈x|φn〉 =1√2nn!

“ mω

π

”1/4exp

„−mωx2

2

«Hn

„rmω

x

«. (3.29)

The associated energy spectrum is En = (n + 1/2)ω.33The density operator also makes it clear that the overall phase of a ket state is immaterial, as |ψ〉 =

α |0〉 + β |1〉 and eiφ |ψ〉 both have the same density operator independent of φ.34The trace of an operator Ω in some Hilbert space is defined as the sum of its diagonal elements (in

any base), i.e., Tr(Ω) =P

i 〈φi|Ω |φi〉, and thus is a number (a real number if the operator is hermitian).The partial trace, say over H2, of an operator Ω acting on a tensor Hilbert space H1 ⊗ H2, is the traceover diagonal elements of H2 in any of its basis, leaving unaffected the projection of Ω on H1. That is,decomposing Ω as Ω =

Pi

Pj ωijΩi ⊗ Ωj

TrH2Ω =

Xi

“ Xj

ωij

Xk

〈φk|Ωj |φk〉”Ωi =

Xi

ωiΩi , (3.31)

which is an operator on H1, with ωi the term in parentheses. The generalization to higher dimensions as wellas to any other selection of which spaces to trace out is obvious.


〈Ω〉 = Tr(ρA) , (3.32)

as 〈Ω〉 = 〈ψ|Ω |ψ〉 =∑

i 〈ψ|Ω |φi〉〈φi|ψ〉 by inserting the closure of identity (A.4),now 〈ψ|Ω |φi〉 and 〈φi|ψ〉 are two complex numbers, so they commute and 〈Ω〉 =∑

i〈φi|ψ〉〈ψ|Ω |φi〉 =∑

i 〈φi| ρΩ |φi〉 by definition of ρ, which is the result.

Exercise 3.9 (∗) Derive from the Schrodinger equation for |ψ〉 the equation for |ψ〉〈ψ|:

i∂

∂tρ = [H, ρ] . (3.33)

We shall refer to the Schrodinger equation (3.33) cast with the density operator asLiouville–von Neumman equation (which holds for a more general density operator thaneqn (3.30), as will be seen in Section 3.2.1).

3.1.5 Heisenberg picture

At this stage, we have presented the essential facts of quantum theory required for thequantum description of light that we shall soon undertake, of course omitting a lot ofmaterial not immediately or crucially needed for that purpose.

We now devote further considerations to alternative formulations along with in-clusions of other physics, like statistical physics or thermodynamics, because of theirimportance to microcavity physics.

The formal integration of the Schrodinger equation, eqn (3.28), shows that the timeevolution of the state is a rotation in Hilbert space. Taking advantage of this fact, one canuse a basis of rotating states and transfer the dynamics from the states to the operators,which were previously fixed, i.e., time independent. Therefore, considering the time-varying average35 〈Ω〉(t), which is a quantity physically measurable that should notdepend on which formalism is used, and starting from the definition we have given (inthe Schrodinger picture)

〈Ω〉(t) = 〈ψ(t)|Ω |ψ(t)〉 , (3.34a)

= 〈ψ(0)| eiHt/Ωe−iHt/ |ψ(0)〉 , (3.34b)

= 〈ψ(0)| Ω(t) |ψ(0)〉 , (3.34c)

we arrive at a time-varying operator Ω(t)

Ω = eiHt/Ωe−iHt/ , (3.35)

which acts on time-independent states (frozen to their initial condition |ψ(0)〉). Thisformulation of quantum mechanics where operators carry the time dynamics and thestates are fixed, is called the Heisenberg picture. We used the average as an intermediatebetween the two pictures, but there is an equation of motion for the operators directly,aptly called the Heisenberg equation, which reads

i∂

∂tΩ(t) = [Ω(t),H] . (3.36)

35Now, the time dependence is shown explicitly everywhere and is accurately attributed to which quantityis time dependent, e.g., 〈Ω(t)〉 is very different from 〈Ω〉(t).


Exercise 3.10 (∗) Derive the Heisenberg equation (3.36) from the Schrodinger equa-tion (3.1) and also the Schrodinger equation from the Heisenberg equation, therebyproving the complete equivalence of the two formulations. Note that in the commuta-tor of eqn (3.36), Ω is the time-dependent Heisenberg operator, while H is the time-independent Schrodinger Hamiltonian. In the course of your demonstration, show thatfor algebraic computation purposes, the Heisenberg equation can be read as:

i∂

∂tΩ(t) = [Ω,H] , (3.37)

where tilde means the operator has been transformed according to eqn (3.35), i.e.,

[Ω,H] = eiHt/[Ω,H]e−iHt/. Therefore, the algebra carried out in the commutatoris time independent throughout.36

Keeping in mind the functional analysis results of the quantum harmonic oscillator,we now consider the problem from an algebraic point of view, therefore closer in spiritto the Heisenberg approach (this solution is due to Dirac). It pre-figures the analysis weshall make on the more complicated system of the light field in Section 2.1.1.

Let us introduce the ladder operators

a =1√2

(X + iP ), a† =1√2

(X − iP ) , (3.38)

where the others quantities have been defined on page 80. It follows straightforwardlyfrom eqn (3.17) and the above definition that

[a, a†] = 1 , (3.39)

and also that the Hamiltonian (3.10) reads

36From the Schrodinger equation to the Heisenberg equation and back; we compute:

∂

∂tΩ =

∂

∂t

“eiHt/Ωe−iHt/

”

=

„∂

∂teiHt/

«Ωe−iHt/ + eiHt/

„∂

∂tΩ

«e−iHt/ + eiHt/Ω

„∂

∂te−iHt/

«

=

„∂

∂teiHt/

«Ωe−iHt/ + eiHt/Ω

„∂

∂te−iHt/

«,

since Ω is time independent

=

„iH

eiHt/

«Ωe−iHt/ + eiHt/Ω

„− iH

«e−iHt/

=i

HΩ − i

ΩH ,

since H commutes with e−iHt/

=i

[H, Ω] .


H = ω(a†a +

1

2

), (3.40)

so that the eigenvalue problem now consists in finding |φ〉 such that

a†a |φ〉 = φ |φ〉 . (3.41)

To do so, dot eqn (3.41) with 〈φ| (these states are assumed normalized) to get φ =〈φ| a†a |φ〉 = ‖a |φ〉 ‖2 that shows that the eigenvalues φ are positive. Using the alge-braic relations37

[a, a†a] = a , [a†, a†a] = −a† , (3.42)

we obtain the equality (a†a)a = a(a†a− 1), which leads to

(a†a)a |φ〉 = a(a†a− 1) |φ〉 = a(φ− 1) |φ〉 = (φ− 1)a |φ〉 , (3.43)

the two ends showing that a |φ〉 is an eigenstate of a†a, with eigenvalue φ − 1, at thepossible exception if a |φ〉 = 0 (the null vector, since it cannot be an eigenstate). Asfor a |φ〉, dotting it with its conjugate yields

‖a |φ〉 ‖2 = (〈φ| a†)(a |φ〉) = 〈φ| a†a |φ〉 . (3.44)

Iterating eqn (3.43) n times shows that an |φ〉 is also an eigenvector of a†a but thistime with eignvalue φ − n, which unless it becomes zero for some value of n (inter-rupting the iteration), will ultimately become negative, in contradiction to what pre-cedes. Therefore, the zero vector must be hit exactly, i.e., there exists n ∈ N∗ suchthat an |φ〉 = 0 but an−1 |φ〉 = 0. Consider the normalized eigenstate |φ− n〉 =an |φ〉 /‖an |φ〉 ‖ with eigenvalue φ− n. Applying a on this state yields zero by defini-tion of n, while eqn (3.44) gives its normed squared as φ−n. Equating the two providesthe structure of the solutions, φ = n, therefore the eigenstates of a†a are states |n〉,where n ∈ N. The “bottom” state |0〉 satisfies

a |0〉 = 0 . (3.45)

Note that |0〉 and 0 are two different entities. The latter is the mathematical zero, and inthis case the zero vector of the Hilbert space, while the former is more of a notation fora complicated mathematical object, in this case a Gaussian function once projected in acoordinate space.

Exercise 3.11 (∗) Repeat the previous analysis to prove the counterpart propertiesfor a†, i.e., a† |φ〉 is an eigenvector of a†a and ‖a† |φ〉 ‖ =

√φ + 1.

37Commutation relations between operators derived from the initial definition range from straightforwardto intractable with little difference in their shape. Equation (3.42) is easily obtained from eqn (3.39), forexample by brute expansion: [a, a†a] = aa†a − a†aa = (aa† − a†a)a = a.


As opposed to a, which eventually reaches the bottom rung |0〉 past which no otherstates can be obtained, a† can increase indefinitely the eigenstate label. These operatorsare quickly checked to satisfy:

a |n〉 =√

n |n− 1〉 , (3.46a)

a† |n〉 =√

n + 1 |n + 1〉 , (3.46b)

and also, applying them in succession:

a†a |n〉 = n |n〉 . (3.47)

Higher-order formulas are readily obtained, like eqn (3.48) below.

a†iaja†k |n〉 =(n + k)!

(n + k − j)!

√(n + i + k − j)!

n!|n + i + k − j〉 , (3.48)

with condition j ≤ n + k, the result being zero otherwise. Equation (3.48) can be usedto evaluate expressions often arising in the course of quantum algebra calculations,typically in connection with boson–boson interactions.38

38Applying repeatedly formulas (3.46), written here for convenience with their counterparts in the dualspace,

a |n〉 =√

n |n − 1〉 , 〈n| a = 〈n + 1| √n + 1 , (3.49a)

a† |n〉 =√

n + 1 |n + 1〉 , 〈n| a† = 〈n − 1| √n , (3.49b)

one obtains straightforwardly

ai |n〉 =

sn!

(n − i)!|n − i〉 , a†i |n〉 =

r(n + i)!

n!|n + i〉 , (3.50)

and therefore, still iterating:

(for i ≤ n + j) aia†j |n〉 =(n + j)!√

n!p

(n + j − i)!|n + j − i〉 , (3.51a)

(for i ≤ n) a†jai |n〉 =

√n!

p(n + j − i)!

(n − i)!|n + j − i〉 . (3.51b)

The result is zero if the condition on the left is not satisfied. Next step gives formula (3.48). With appropriatevalues for i, j and k, one finds for the following, of frequent use in many problems:

1. a2 |n〉 =p

n(n − 1) |n − 2〉 2. 〈n| a2 = 〈n + 2|p

(n + 1)(n + 2)

3. a†2 |n〉 =p

(n + 1)(n + 2) |n + 2〉 4. 〈n| a†2= 〈n − 2|

pn(n − 1)

5. a†a2 |n〉 = (n − 1)√

n |n − 1〉 6. 〈n| a†a2 = 〈n + 1|n√n + 1

7. aa†a |n〉 = n3/2 |n − 1〉 8. 〈n| aa†a = 〈n + 1| (n + 1)3/2

9. a2a† |n〉 =√

n(n + 1) |n − 1〉 10. 〈n| a2a† = 〈n + 1| √n + 1(n + 2)

11. aa†2 |n〉 =√

n + 1(n + 2) |n + 1〉 12. 〈n| aa†2= 〈n − 1| √n(n + 1)


The states |n〉 are sometimes called number states because they are states with adefinite number of particles. They are also known, from quantum field theory, as Fockstates (cf. Section 3.3.1). They are handy for calculations and offer a de facto canonicalbasis, i.e.,

|0〉 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

10000...

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, |1〉 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

01000...

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, |2〉 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

00100...

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, . . . |n〉 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...010...

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, . . . ,

with 1 as the (n + 1)th element of the vector. In this basis, the annihilation and creationoperators read:

a =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1

0√

2

0√

3. . .

. . .0√

n. . .

. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, a† =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

01 0√

2 0√3 0

. . .. . .√

n + 1 0. . .

. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where the nth column is being shown for the general case. In both cases, the diagonal iszero. All the algebra can be computed with this representation and expressions for op-erators obtained in this way. The number operator, for instance, is diagonal with values0, 1, 2, . . .

If we return to the Hamiltonian (3.40), the energy spectrum is seen to consist ofequally spaced energy levels with energy difference ω:

En =(n +

1

2

)ω . (3.53)

The state |0〉 of lowest energy is the ground-state of the system and is, by defini-tion, devoid of excitation. It is called the vacuum state (or a vacuum state in the case

13. a†aa† |n〉 = (n + 1)3/2 |n + 1〉 14. 〈n| a†aa† = 〈n − 1|n3/2

15. a†2a |n〉 = n

√n + 1 |n + 1〉 16. 〈n| a†2

a = 〈n − 1| (n − 1)√

n

17. a2a†2 |n〉 = (n + 1)(n + 2) |n〉 18. 〈n| a2a†2= 〈n| (n + 1)(n + 2)

19. a†2a2 |n〉 = (n − 1)n |n〉 20. 〈n| a†2

a2 = 〈n| (n − 1)n.

Also, note the useful relation obtaining |n〉 from the vacuum, following from eqn (3.51b):

|n〉 =a†n

√n!

|0〉 . (3.52)


of degeneracy). However, its energy is nonzero since 〈0|H |0〉 = ω/2. This term isassociated to quantum fluctuations that arise from the uncertainty principle, of whichwe shall see more later in a true field-theoretical setting. One more important point isthat this structure of equally spaced levels makes compelling the interpretation of |n〉 asa state with n quanta of excitations, each of energy ω, superimposed on the vacuumfluctuations, which cannot be eliminated.

The effects of the various operators just defined on the state |n〉 with n excita-tions entitle them to be called annihilation, cf. eqn (3.46a), and creation operators, cf.eqn (3.46b). Their net result is indeed to raise or lower the number of excitations.39 Theannihilation operating on the vacuum state, eqn (3.45), results in cancelling the term,which disappears from the process being computed.

The Heisenberg picture has many advantages, including computational or algebraicsimplicity, as we shall later appreciate. One further quality is the aid to physical intuitionafforded by this formalism as these are the observables that vary in time, in analogy withclassical physics.

3.1.6 Dirac (interaction) picture

The Dirac picture, also known as the interaction picture, is an intermediate case be-tween the Schrodinger picture (with fixed operators and time-varying states) and theHeisenberg picture (with time-varying operators and fixed states) where both operatorsand states are time dependent. Which fraction of the dynamics is attributed to each ofthem depends on the decomposition of the Hamiltonian.

First with the case of a total Hamiltonian H without time dependence, which isseparated as

H = H0 + HI , (3.54)

where H0 is typically a “simple” part of the dynamics, i.e., which can be solved exactlyand whose solutions will define the states of the “free” particle, or a “dominant” partthat will result in a fast dynamics, and the remaining HI , typically complicated or of alesser magnitude, which will be interpreted as an interaction terms between the particlesdefined by H0.

The underlying principle of the interaction picture is to concentrate on the compli-cation HI in the Hamiltonian by embedding H0 dynamics into the operator by defining:

Ω = eiH0t/Ωe−iH0t/ , (3.55)

which equation of motion is reminiscent of the Heisenberg equation (3.36) but withsome important redefinitions of the quantities involved:

i∂

∂tΩ(t) = [Ω(t), HI ] , (3.56)

where tilde means the operator has been transformed to the Heisenberg picture accord-ing to eqn (3.55). As opposed to the Heisenberg equation, this transformation applies to

39A more accurate terminology would call these operators “ladder operators” in the case of a genericquantum harmonic oscillator, comprising a raising and lowering operator, and to reserve the terms for anni-hilation and creation in a field-theoretic context.


what remains of the Hamiltonian as well, HI = eiH0t/HIe−iH0t/, which, however,

can remain time independent under this transformation, leading to HI = HI . In allcases, HI and not H—like in the Heisenberg or Schrodinger equations—appears in thecommutator. The similarities are strong enough, however, for eqn (3.56) to be called theHeisenberg equation in interaction picture.

Exercise 3.12 (∗) Derive eqn (3.56).

If the computation is difficult, one can compute the commutation first, as [Ω, H] =

[Ω,H]. An efficient approach to solve a general problem is to separate it, as we said, intoa “dominant” part, which can be solved exactly or approximated easily, and consider theminor parts left out initially as perturbations. What is meant by dominant depends onthe problem at hand, but a typical case would be the time-independent part of a generalHamiltonian H(t), which we could separate as

H(t) = H0 + H1(t) , (3.57)

where H0 is time independent, resulting in a uniform time rotation in Hilbert spacee−iH0t/ |ψ0〉, cf. eqn (3.28).

The Heisenberg and Dirac pictures often require a lot of algebra, which means thatcommutators have to be computed by expansion and simplification through gatheringand cancelling with relations such as eqn (3.39). To allow such an evaluation whenfaced with a general commutator of the type [

∏ni=1 Ai,

∏mj=1 Bj ], where Ai, Bj are

some operators, the most general expansion is a sum over all combinations of commuta-tors [Ai, Bj ] with other operators factored outside of the commutator. Their relative po-sition is of course important in the most general case where their relative commutationrules are unknown. Their placement is made as follows for the commutator [Ai, Bj ]: alloperators A1 · · ·Ai−1 placed before Ai and all operators B1 · · ·Bj−1 placed before Bj

are placed before the commutator [Ai, Bj ], in this order, and all operators Ai+1 · · ·An

and all operators Bj+1 · · ·Bm are placed after the commutator, in the opposite orderof A and B. This is illustrated below, for the case where the operators that remain in thecommutator are Ai and Bj :

[(A1 · · ·Ai−1)Ai(Ai+1 · · ·An), (B1 · · ·Bj−1)Bj(Bj+1 · · ·Bm)]

(A1 · · ·Ai−1)(B1 · · ·Bj−1)[Ai, Bj](Bj+1 · · ·Bm)(Ai+1 · · ·An)

Application of this rule on the six arbitrary operators A, · · · , F written as followsyields:

OTHER FORMULATIONS 95

[ABC,DEF ] =[A,D]EFBC + D[A,E]FBC + DE[A,F ]BC (3.58a)

+ A[B,D]EFC + AD[B,E]FC + ADE[B,F ]C (3.58b)

+ AB[C,D]EF + ABD[C,E]F + ABDE[C,F ] . (3.58c)

There are in this case nine terms as there are 3 × 3 combinations for commutators,with operators distributing as illustrated. Such evaluations usually simplify extensivelywhen they are carried over a family of boson operators ai, obeying the following alge-bras:

[ai, aj ] = 0, [ai, a†j ] = δij . (3.59)

For instance, here a useful relation obtained in this way is:

a†a†aa = a†a(a†a− 1) . (3.60)

Exercise 3.13 (∗) Given three Bose operators a1, a2 and a3 obeying commutation re-lations (3.59), evaluate the following expressions often encountered in the Heisenbergand interaction pictures: [a1, a

†1a1], [a1, a

†21 a2

1], [a1, a†1a1a

†2a2], [a1, a

†n1 am

1 ].

3.2 Other formulations

3.2.1 Density matrix

The formulation of the theory so far—in whatever representation—describe so-calledpure states, which for now we can regard as a synonym for ket states, that is, states forwhich there exists a “wavefunction” |Ψ〉. A first reason why there would be no suchstate is if one would attempt to describe only part of a composite system. Namely, ifthe system of our interest S is in contact with another system R (the notations are for“system” and “reservoir”), the (pure or ket) state to describe the whole system is, ingeneral

|ΨSR〉 =∑

i

∑j

cij |φi〉 |ϕj〉 , (3.61)

where the |φi〉 are basis states for S and |ϕj〉 are basis states for R. Note that in thesense that each state has its associated Hilbert space and set of observables, it is definedindependently of the other. If the cij are such that eqn (3.61) can be written as a (tensor)product

|ΨSR〉 =(∑

i

cSi |φi〉

)(∑j

cRj |ϕj〉

), (3.62)

(that is, if cij = cSi cR

j ), then |ΨSR〉 can be likewise decomposed as |ΦS〉 |ΦR〉 and thesystem S considered in isolation with its quantum state |ΨS〉.

If, however, such a decomposition (3.62) is not possible, in which case the systems Sand R are said to be entangled, then it is not possible to consider any one of themindependently, at least exactly. The density operator is the approximated state that ariseswhen the total wavefunction is averaged over unwanted degrees of freedom of the totalsystem.


Exercise 3.14 (∗∗) Show that the state that best describes S in isolation is

ρS = TrR(|ΨSR〉〈ΨSR|) , (3.63)

where the partial trace34 is taken over the Hilbert space of R.

Such an operator ρS—the so-called reduced density matrix—which cannot be de-scribed by a ket state, is called a mixed state—as opposed to the pure state—because ofthe “averaging” that mixes up quantum states.

Another reason to have recourse to a density operator is when statistical mechanicsis incorporated into quantum mechanics, i.e., when some indeterminacy is injected inthe system as the result of knowledge not available only in practice (in direct analogy tothe classical statistical theory and direct opposition to the quantum indeterminacy thatis intrinsic to the system). For instance, the state of a radiation mode that can take uppossible energies Ei in thermal equilibrium at temperature T requires such a statisti-cal description, as some excitations are randomly poured into or conversely removedfrom the system by thermal kicks issued by the reservoir. In accordance with thermo-dynamics, such a state would be described as being in the (pure) quantum state |Ei〉of energy Ei with probability pi = e−Ei/kbT /Z with Z =

∑i e−Ei/kbT the partition

function.The average 〈Ω〉 given by eqn (3.32) with density matrix

ρ =∑

i

pi |Ei〉〈Ei| , (3.64)

yields 〈Ω〉 =∑

i pi 〈Ei|Ω |Ei〉, from which it is seen that the average is now the quan-tum average (3.13c) weighted over the classical probabilities pi, so eqn (3.64) describesa system where a quantum state is realized with probability pi.

It is important, though sometimes subtle, to distinguish between the quantum and aclassical indeterminacy. For instance, the density matrix

1

2

(1 11 1

)=

1√2(|〉+ |〉)(〈|+ 〈|) 1√

2, (3.65)

which describes a pure state of linear polarization (cf. Exercise 3.2), is physically dif-ferent from the mixed state

1

2

(1 00 1

)=

|〉 with probability 1/2|〉 with probability 1/2

, (3.66)

which describe a single photon that has either circular polarization with probability 1/2all along. This is different from eqn (3.65) where the circular polarization becomesleft or right only as a result of a measurement in this basis and is the rest of the timeotherwise undetermined.

It therefore appears that off-diagonal elements of the density matrix are linked withthe pure or mixed character of the state. Indeed, as we shall have many occasions toappreciate in this text, these elements relate to quantum coherence and their contributionto the equations of motion discriminates between classical and quantum dynamics.


Mathematical properties

The density matrix is hermitian and its trace is unity, in agreement with its statisticalinterpretation. The trace of its square is also one in the case of a pure state since in thiscase ρ = ρ2 as there exists |ψ〉 such that ρ = |ψ〉〈ψ|, whence ρ2 = |ψ〉〈ψ|ψ〉〈ψ| = ρaccording to eqn (3.3). But this is not the case for a mixed state and this provides auseful criterion for identification of pure states:

Exercise 3.15 (∗) Show that for a mixed state,

Tr(ρ2) < 1 . (3.67)

The minimum for the trace of ρ2—to indicate how “mixed” a state is—depends on thesize of the Hilbert space. Show that for a space of size n it is given by 1/n.

A density matrix is also positive40 and reciprocally, any positive operator whose traceequals 1 is eligible to be a density matrix.

3.2.2 Second quantization

We have presented in the previous sections the basic concepts of quantum physics fromits “mechanical” point of view where the object of quantization is a mechanical attributeof the particle, like its motion or its spin. When the physical object is a field—as is thecase with light—quantum mechanics is upgraded to the status of a quantum field theory.There are various possible theoretical formulations but for the needs of this book weshall be content with a simple and vivid picture, known as second quantization. Oneconceptual benefit of this reformulation of the theory is the valuable concept it affordsof a particle as an excitation of the field, in the terms we are about to present. In theformulation given so far, we have already used repeatedly the term “particle” to describethe object to which to attach the wavefunction or one of its attributes (like the spin). Onedifficulty, however, arises when the number of particles is not conserved, as is the casein a statistical theory in the grand-canonical ensemble, or if particles are unstable andcan decay into other particles (calling for the necessity to remove and add particles inthe quantum system). Also, particles are generally considered in a collection, so that thesymmetry requirements are to be taken into account.

Second quantization starts with the occupation number formalism that provides anelegant and concise solution to all these desiderata. Let H1 be a single-particle basis,i.e., a set of states |φi〉 that are orthonormal and—assuming a discrete basis—such thatany possible quantum state |ψ〉 can be written as |ψ〉 =

∑i αi |φi〉 for a suitable choice

of αi (which is also unique by orthogonality of the basis). Now consider a state |Ψ〉 ofthe system with ni particles in state |φi〉. Without any symmetry requirement, such astate would read

|Ψ〉 = |φ1〉⊗n1 ⊗ · · · ⊗ |φi〉⊗ni ⊗ · · · (3.68)

A more convenient way to represent the state (3.68) is to specify directly the numbersof particles in each of the (previously agreed) states |φi〉, i.e.,

40A positive operator M is such that for all states |φ〉, 〈φ|M |φ〉 ≥ 0.


|Ψ〉 = |n1, . . . , ni, . . .〉 (3.69)

using once more the elegant Dirac’s notation. By the fourth postulate, the wavefunctionfor a collection of particles needs to be symmetrised. The advantage of the occupationnumber formalism then becomes compelling, as the details of the symmetrization canbe hidden in the abstract notation of eqn (3.69). For fermions, all ni equal at most 1,and the exchange of two particles result in a change of the sign of the wavefunction.For bosons, there are no limits in the number of particles, and exchange of two particlesleaves the wavefunction unchanged.

We have introduced with eqn (3.38) the so-called ladder operators a and a† thatannihilate and create, respectively, excitations of the quantum harmonic oscillator. Be-cause the energy levels are equally spaced, cf. eqn (3.53), we can understand such astate, say the nth one with energy (n + 1/2)ω, as consisting of n excitations eachwith energy ω, with a remainder of ω/2 in energy when n = 0, which is thereforeidentified as the energy of the vacuum. The field-theoretic version of the ladder opera-tors are creation and annihilation operators, which bring a state with n particles to onewith n− 1 and n+1 particles, respectively. If there is no particle to annihilate, applica-tion of the annihilation operator yields zero, that is, the process is cancelled or does notappear in the calculation.

Let us give an example to make clear the overall picture. Let us assume three pos-sible quantum states of Bose particles, that we call s, p and i, with associated Boseoperators as,p,i, respectively. The basis is afforded by states |ns, np, ni〉 with ns,p,i ∈ N.Each operator acts on its relevant part of the total state, e.g.,

as |ns, np, ni〉 =√

ns − 1 |ns − 1, np, ni〉 , (3.70)

if ns ≥ 1, 0 otherwise. A process that scatter-off two particles from state p to redistributethem in states s and i would read, in the second quantization formalism:

a†sa

†i a

2p . (3.71)

Such a process would bring an initial state |0, 2, 0〉 into the final state |1, 0, 1〉. Thereverse process that brings back the particle into the state p is the hermitian conjugage ofeqn (3.71). The state |1, 1, 0〉, for instance, vanishes when eqn (3.71) is applied on it, soit decouples from the dynamics and from all the calculations (like transition rates) for anHamiltonian based on eqn (3.71). Physically, such a process corresponds to parametricamplification where a state which is pumped (p) scatters off particles into a signal (s) anda so-called “idler” state (i). Such a dynamics will be amply discussed in the chapters tofollow. Of course, the procedure extends to spaces of infinite dimensions. A two particleinteraction in free space, such as the Coulomb interaction, reads in second quantization:

V =1

2

∑k1,k2k3,k4

〈k3k4|V |k1k2〉 a†k3

a†k4

ak2ak1

. (3.72)

To such a process one can attach a Feynman diagram which displays graphically oneterm of the Born expansion of an interaction, as shown in Fig. 3.1. We have given an


example for the case of bosons, but the same would apply for fermions using σ operatorsinstead, that follow Fermi’s algebra.

|k1 + q〉 Vq = 4πe2/q2

|k1〉 |k2〉

|k2 − q〉

Fig. 3.1: A Feynman interaction for Coulomb interaction whereby two particles of momenta k1 and k2 inthe initial state scatter to final states k3 and k4 by exchanging momentum q.

Second quantization also sheds much light on the dilemma of wave–particle dual-ity and the so-called complementarity41 that is naively perceived as a photon behavingsometimes like a particle, sometimes like a wave. A particle is a quantum state of thefield that reads |0, . . . , 0, 1, 0 . . .〉, the position of the 1 corresponding to which quantumstate the particle is in. Taken in isolation, this always behave as a particle in the classi-cal sense (of one discrete lump of matter or energy). With repeated measurements oversingle-particle states, wavelike behaviour start to emerge. For example if one traces theposition on the screen where a single electron is detected after crossing a double slit,one always detect a single spot corresponding to a single particle. Repeating the mea-surement many time—each time tracking a single electron—results in a wavelike inter-ference patterns for all the collected positions. The behaviour of a collection of particlesgive rise to the notion of many-particle quantum states. We discuss more states in thenext sections. For now, we complete the second quantization of light by promoting theelectromagnetic field modal decomposition of the previous chapter to matrix algebra.

3.2.3 Quantization of the light field

The quantization of the electromagnetic spectrum is known as quantum electrodynamics(QED). For the spectral window of light that corresponds to high frequencies but stillmoderate quanta energies, QED can be framed in a suitable way to make the most ofthe various energy and time scales, which is known as quantum optics and is the topicof this section.

Quantization of the field is made in a vacuum, leaving the complicated problem ofthe interaction with matter to the next chapter that will bring many different techniques

41It is interesting to recall Anderson who comments in Nature, 437, 625, (2005) that “Niels Bohr’s ‘com-plementarity principle’—that there are two incompatible but equally correct ways of looking at things—wasmerely a way of using his prestige to promulgate a dubious philosophical view that would keep physicistsworking with the wonderful apparatus of quantum theory. Albert Einstein comes off a little better because heat least saw that what Bohr had to say was philosophically nonsense. But Einstein’s greatest mistake was thathe assumed that Bohr was right—that there is no alternative to complementarity and therefore that quantummechanics must be wrong.”


to tackle this issue in various approximations. We therefore proceed from the classicalequations obtained in Section 2.1.1. We quantize the field following Dirac (1927) andFermi (1932) by canonical quantization of the variables a related to the field througheqn (2.8a) and that is known from eqn (2.13) to undergo harmonic oscillation. To settlenotations, we regard free space as a large cubic box of size L, with boundary conditionsfor the electromagnetic field of running waves42 ei(ωkt−k · r), as opposed to standingwaves. A wavevector of these solutions is defined by:

k =

(2πnx

L,2πny

L,2πnz

L

), (3.73)

where nx,y,z ∈ N and L is taken as high as necessary for the sought precision. The fieldamplitude in this box is

Ek =

√ωk

2ε0V. (3.74)

Completing the modal expansion with quantized operators, the expression of thefield E (now also an operator) reads:

E =∑k

Ekakei(k · r−ωt) + h.c. (3.75)

The vector nature of a is associated to polarization. We split these two aspects—fieldamplitude and polarization—to deal with them separately, which allows us to applythe results already presented regarding the formal quantum systems of the harmonicoscillator and the two-level system. The vector variable a being transverse, we choosean orthogonal 2D basis in the transverse k plane:

a(k, t) = a↑(k, t)e↑(k) + a↓(k, t)e↓(k) , (3.76)

where (e↑(k), e↓(k),k) form an orthogonal basis of unit vectors. When need ariseswe will use the full vector expression, but without limitation we now focus on oneprojection only, say a↓, which for brevity we shall denote a, thereby coming back to theelementary one-dimensional quantum mechanics, which was our starting point.

3.3 Quantum states

We now discuss some commonly encountered quantum states in the light of the formal-ism of second quantization.

3.3.1 Fock states

An important and intuitive building block is the Fock state, which is the canonical basisstate of the corresponding Fock space. It is the state with a definite number of parti-cles (in the case of the electromagnetic field, a definite number of photons) and is, in

42Boundary conditions of a running wave are such that once the wavefront reaches the end of the freespace within the box and touches the border, it goes through and reappears instantaneously on the otherside of the box. In this sense this boundary condition is not really physically relevant, and should be viewedas a mathematical trick to model the infinite universe in a more intuitive way, with labels L to track ourwavevectors.

QUANTUM STATES 101

Vladimir Fock (1898–1974) gave his name to the Fock space and Fock stateafter building the Hilbert space for Dirac’s theory of Radiation, Zs. f. Phys.,49 (1928) 339.

Many other results and methods due to him such as the Fock proper timemethod, the Hartree–Fock method, the Fock symmetry of the hydrogen atomand still others make him one of the most popular names in quantum fieldtheories.

this respect, a physical state as well as a mathematical pillar of second quantization.The vacuum |0〉 and the single particle |1〉 are two typical, important and “relatively”easy states to prepare in the laboratory.43 The arbitrary case |n〉 with n ∈ N becomesincreasingly difficult with large values of n. Small values have, however, been indeedreported in the literature, and cavities are precious tools to this end, as demonstrated forinstance by Bertet et al. (2002).

From the mathematical point of view, Fock states are useful for many computationalneeds. When quantum electrodynamical calculations are conducted pertubatively, theFock state plays a major role as each successive order of the approximation describesprocesses that increase or decrease the number of photons.

3.3.2 Coherent states

The coherent state was initially introduced by Schrodinger as the quantum state of theharmonic oscillator that minimizes the uncertainty relation (3.16) for the observables Pand X . From the derivation of the inequality of the generalized Heisenberg uncertaintyrelation of Exercice 3.4, we know that eqn (3.16) is the Schwarz inequality in disguiseapplied on vectors

(Ω− 〈Ω〉) |α〉 and (Λ− 〈Λ〉) |α〉 , (3.77)

with now Ω = P and Λ = X , and with uncertainty assumed to be equally distributed44

in both position P and impulsion X . The inequality is optimized to its minimum whenthese vectors are aligned, which yields:

(P − 〈P 〉) |α〉 = i(X − 〈X〉) |α〉 , (3.78)

43To prepare a single photon state it suffices to dispose a thin absorbing media in front of a light withthickness increasing until an avalanche photodiode registers separate detections. Each detection correspondsto a quantum. This is assured to work provided that one is disposed to wait the necessary time. To provide asingle-photon source on demand is an altogether more difficult problem, subject to active research because ofits application in quantum cryptography (see the problem at the end of this chapter).

44When the uncertainty of a coherent state is not equally shared, i.e., ∆X = ∆P , although the prod-uct remains /2, the state is said to be squeezed in the variable whose root mean square goes below /2.In the dynamical picture provided by Exercise 3.8, squeezed states correspond to the ground-state of theharmonic oscillator, i.e., a Gaussian wavefunction, whose width is mismatched with the harmonic potentialcharacteristic length, so that as the state bounces back and forth in the trap, its wavepacket spreads or narrowsperiodically (it is said to “breathe”).


or, written back in terms of a, a†:

a |α〉 =1√2

(〈X〉+ i〈P 〉) |α〉 , (3.79)

so that the coherent state that minimizes the uncertainty relation appears as the eigen-state of the annihilation operator (we come back later to the more general case wherethe uncertainty is not equally distributed).

It was expected that α could be complex since a is not hermitian. From eqn (3.79)it is seen that it actually spans the whole complex space. The phase associated withthis complex number (in the sense of a polar angle in the complex plane) maps to thephysical notion of phase since it arises as a complex relationship between the phase-space variables. The physical meaning of the (unsqueezed44) coherent state is that ofthe most classical state allowed by quantum physics, since it has the lowest uncertaintyallowable in its conjuguate variables, and can be located in the complex plane as theposition in the phase space of the quantum oscillator, with position on the real axisand momentum on the imaginary axis. The Hamiltonian being time independent, theevolution of |α〉 is obtained straightforwardly from the propagator:

|α(t)〉 = e−iω(a†a+1/2)(t−t0) |α(t0)〉= e−iω(t−t0)/2

∣∣∣e−iω(t−t0)α(t0)⟩

,(3.80)

so that the free propagation of the coherent state is rotation in the complex space, orharmonic oscillations in real space. Definitely, it is a state of well-defined phase.

In terms of Fock states, it reads:

|α〉 = exp(−|α|2/2)

∞∑n=0

αn

√n!|n〉 , (3.81)

with α ∈ C.Coherent states are normalized though not orthogonal:45

〈β|α〉 = exp(− 1

2(|α|2 + |β2| − 2αβ∗)

), (3.82)

with indeed 〈α|α〉 = 1 but 〈β|α〉 = δ(α− β), being smaller the farther apart α, β in C.

3.3.3 Glauber–Sudarshan representation

Sudarshan (1963) promoted coherent states as a basis for decomposition of the densitymatrix ρ, replacing it by a scalar function P of the complex argument α of the coherentstate |α〉:

45The non-orthogonality of coherent states close in phase space was one compelling support in favourof Glauber’s (1963) argument that the light emitted by the newly discovered maser called for a quantumexplanation over the classical model advocated by Mandel and Wolf (1961).

QUANTUM STATES 103

Roy Glauber (b. 1925) advocated the importance of quantum theory in op-tics and its implications for the notion of coherence. As such he is widelyregarded as the father of quantum optics and was awarded the 2005 Nobelprize in physics (half of the prize) “for his contribution to the quantum the-ory of optical coherence”.

Before being awarded the Nobel prize, Glauber was an active and long-timesupporter of its parody—the “Ig Nobel” prize—where he traditionally sweptoff the stage the paper planes sent on it by the audience. He made a testimo-nial for the 1998 Ig Nobel Physics prize awarded for “unique interpretationof quantum physics as it applies to life, liberty, and the pursuit of economichappiness”.

ρ(t) ≡∫

P (α, α∗, t) |α〉〈α| d2α , (3.83)

which allows one to carry out quantum computations with tools of functional analysisrather than operator algebra. Sudarshan incorrectly deduced that this proved the equiv-alence of the quantum formulation and classical one. Glauber, who also had in his firstpublication the insight of this decomposition, showed him to be wrong.46

We shall see many applications and usage of the Glauber–Sudarshan (or simplyGlauber) representation later on. Now we proceed to give its expression for some statesof interest.47 For the coherent state, it is straightforward by identification to obtain forrepresentation of |α0〉 that

P (α, α∗) = δ(α− α0) . (3.84)

3.3.4 Thermal states

The incoherent superposition of many uncorrelated sources generates chaotic or a so-called thermal state, as is the case of the light emitted from a lightbulb where each atomemits independently of its neighbour. The convolution rule and the central limit theoremcombine to provide the P function of a such a state: a chaotic state without phaseor amplitude correlations has a Glauber distribution that is Gaussian in the complexplane:

46Upon award of the Nobel prize in 2005 to Glauber, Sudarshan sparked a controversy, questioning thisdecision and writing to the Nobel committee and the Times (who did not publish his input) where he claimedpriority on the representation. An extract of the letter of Sudarshan to the Nobel committee reads “[. . . ]Whilethe distinction of introducing coherent states as basic entities to describe optical fields certainly goes toGlauber, the possibility of using them to describe ‘all’ optical fields (of all intensities) through the diagonalrepresentation is certainly due to Sudarshan. Thus there is no need to ‘extract’ the classical limit [as statedin the Nobel citation]. Sudarshan’s work is not merely a mathematical formalism. It is the basic theory un-derlying all optical fields. All the quantum features are brought out in his diagonal representation[. . . ]”. Heconcludes “Give unto Glauber only what is his.”

47The P representation for Fock states is highly singular, involving generalized functions of a muchhigher complexity than for the coherent state, which somehow restrain its applicability. This can be linked tothe non-classical character of such a field.


P (α, α∗) =1

πnexp(−|α|2/n) , (3.85)

where n = 〈a†a〉 is the average number of particles in the mode. From this expressionthe Fock-state representation can be obtained:

Exercise 3.16 (∗) Show that the density matrix of the thermal state in the Fock statesbasis is:

ρ =∞∑

n=0

nn

(n + 1)n+1|n〉〈n| . (3.86)

Observe that the density matrix eqn (3.86) is diagonal (all the terms |n〉〈m|with n =m are zero). This means that there is no quantum coherence and the superposition isclassical. This is the fully mixed state (cf. eqn (3.67)), where the uncertainty is entirelystatistical. The result can be alternatively obtained directly from the statistical argu-ment: in the canonical ensemble, the density matrix for a system with Hamiltonian Hat temperature T is given by:48

ρ =exp(−H/kBT )

Tr(exp(−H/kBT )), (3.87)

which can be evaluated exactly when H is the Hamiltonian for an harmonic oscillator,given by eqn (3.40):

Exercise 3.17 (∗) Show that eqn (3.87) evaluates to

ρ =

[1− exp

(− ω

kBT

)] ∞∑n=0

exp(− nω

kBT

)|n〉〈n| . (3.88)

for the harmonic oscillator Hamiltonian. Deduce the following relation between theaverage occupancy of the mode and temperature:

n = 〈a†a〉 =1

exp(− ω

kBT

)− 1

(3.89)

The important formula eqn (3.89)—the thermal distribution of bosons at equilibrium(here for a single mode)—is named the Bose–Einstein distribution.49

48This follows from the fundamental postulate for the canonical ensemble, which states that if a system isin equilibrium at temperature T , the probability that it is found with energy En is (1/Q) exp(−En/(kBT ))where Q =

Pn exp(−En/(kBT )) (known as the partition function) and En is the energy of any of

the states in which the system can be found. Therefore, taking |i〉 as the state with energy Ei and Ω anoperator, from the definition of quantum average that we have given, cf. eqn (3.32), it follows that 〈Ω〉 =(1/Q)

P|i〉〈i|Ω |i〉 exp(−Ei/(kBT )).

49The general formula for the Bose–Einstein distribution, with all modes (and possible degeneracy suchas that imparted by polarization) and with a chemical potential is readily obtained along the same lines asExercise 3.17 and will be a central ingredient of the physics of Chapter 8.

QUANTUM STATES 105

3.3.5 Mixture states

An important property of P functions is that the superposition of uncorrelated fieldseach described by its P function amounts to a field whose own P function is obtainedby the convolution of that of the constituting fields. So, for instance, if we superposetwo non-correlated fields described, respectively, by functions P1 and P2, the total fieldis given by P = P1 ∗ P2, or:

P (α, α∗) =

∫P1(β, β∗)P2(α− β, α∗ − β∗)dβdβ∗ . (3.90)

This has application for an important class of states, which interpolate between thecoherent state eqn (3.84) and the thermal state eqn (3.85), it being rare in practice thata state is realistically completely coherent.50 When there is a large number of particles,the physically relevant case is that of an essentially coherent state, with, say, nc particles,to which is superimposed a fraction of a thermal state, with nt particles. Such a stateis obtained in the optical field by interfering ideal coherent radiation with that emittedby a blackbody. In the resulting field, we will conveniently refer to such particles ascoherent and incoherent respectively, though of course once the two fields are merged,a particle no longer belongs to a part of this decomposition but is indistinguishable fromany other. This is just a vivid picture to describe a collective state that has some phaseand amplitude spreading. Only nc + nt = 〈n〉 is well defined.

The P state that results from the convolution of a Gaussian centred about αcoh ∈ C

and a delta function centred about αth is a Gaussian centred about αcoh − αth, so thatthe P state of the mixed state is a Gaussian centred about α0, as depicted in Fig. 3.2. Itsanalytical expression reads:

Pm(α, α∗) =1

πnte−|α−nceiϕ|2/nt , (3.91)

where ϕ is the mean phase of the state, inherited from the phase of the coherent state.We conclude by illustrating the power of the P function as a mathematical tool by

providing a few techniques that we shall use later. With proper notation, eqn (3.79) reads

a |α〉 = α |α〉 , (3.92)

which states that the coherent state is an eigenstate for the annihilation operator (recip-rocal equation is 〈α| a† = 〈α|α∗). Such properties make the evaluation of many statesstraightforward, translating the operator as its eigenvalue. It also results in many sim-plifications of the mathematical analysis of equations, as operator algebra gets mappedto complex calculus. However, during this translation from operators to c-numbers, theneed for the following can (and does) arise:

a† |α〉 , (3.93)

(or the reciprocal 〈α| a). This is made technically easy with a few tricks that can bedevised from the Fock representation of the coherent state, eqn (3.81), as in this case:

50From photodetection theory one can show that the inefficiency of the detector results in a broadeningof the counting statistics of a coherent state of the kind of the mixed states for the density matrix of a Bosesingle mode.


Fig. 3.2: Schematic representation of P representations for a thermal state (Gaussian), a coherent state (δfunction) and a superposition, or mixture, of the two, in which case the Gaussian gets displaced onto the δlocation.

Exercise 3.18 (∗) By going back and forth to the Fock and coherent basis, derive thefollowing rule of thumb:

a† |α〉 =

(∂

∂α+

α∗

2

)|α〉 , (3.94)

and following the same principle, compute the exhaustive set of possible combinationsthat arise in the conversion of the master equation with operators into a so-calledFokker–Planck equation for c-numbers functions:

a |α〉〈α| a† , a†a |α〉〈α| , |α〉〈α| a†a , (3.95a)

|α〉〈α| aa† , a† |α〉〈α| a . (3.95b)

The master equation for a matrix can in this sense be directly rewritten as a Fokker–Planck equation, which is an equation of diffusion and drift for a probability distribu-tion. Many insights into the quantum picture can be gained through this approach as wewill see when investigating the quantum interaction of light with matter in Chapter 5.

3.3.6 Quantum correlations of quantum fields

We emphasized in Section 2.3 how a realistic description of the optical field needs totake into account its statistical character, at the classical level as has been seen in theprevious chapter but also at the quantum level with additional features of a specificquantum nature. The importance of statistics in optics has been realized by Mandeland Wolf (1995) but they missed the importance of quantum mechanics and it was forGlauber (1963) to formalize into the theory of quantum coherence. In all cases it is thefluctuations of light that underly the notion of optical coherence. Since there are alwaysat least the quantum fluctuations of the field, a fundamental definition of coherence isrequired at the quantum level. Glauber provided such a definition by emphasizing therole of correlations. One of his main contributions in this respect was to separate the no-tion of optical coherence from that of monochromaticity, and the notion of interferencefrom that of phase.

QUANTUM STATES 107

A generic property of the light field can depend on arbitrary high-order correlationfunctions. On the other hand, a statistical classical model—that for all relevant detailswould be modelled after a stationary Gaussian stochastic process—has all its informa-tion contained already in its frequency spectrum (first order in the correlators). The firstsuch physical property to require a higher-order correlation than g(1) was the bunchingof photons counted in a Hanbury Brown–Twiss experiment that motivated the work ofGlauber. At a time when the Hanbury Brown–Twiss effect was highly controversial,Purcell (1956) stood in its favour and pointed out the quantum character of the effectfrom the Bose statistics, even predicting antibunching for fermions. In the abstract ofhis paper published in Nature, he comments:

“Brannen and Ferguson (preceding abstract) have suggested that the correlationbetween photons in coherent beams observed by Brown and T. (cf. above), if true,would require a revision of quantum mechanics. It is shown that this correlationis to be expected from quantum mech. considerations and is due to a clumpingof photons. If a similar experiment were performed with electrons a neg. cross-correlation would be expected.”

His semiclassical insight is based on quantum statistics of bosons along the linesor arguments used in Section 3.1.4 on the symmetric states of Bose particles. Let usconsider two particles, a and b that can be detected by any one of two detectors, 1 and 2.In the case where particles are classical (distinguishable), the probability Pc of detectingone particle on each detector is the sum of the probabilities of all possibilities, namelydetecting a in 1 and b in 2 or vice versa, with respective probabilities |〈a|1〉〈b|2〉|2and |〈a|2〉〈b|1〉|2 so that

Pc = 2|ab|2 . (3.96)

If particles are bosons, however, there is quantum interference of their trajectoriesat detection, and the amplitudes sum, rather than the probabilities (which is one tenetof how quantum mechanics extends classical mechanics). The probability remains themodulus square of the total amplitude, so that:

PQ = |〈a|1〉〈b|2〉+ 〈a|2〉〈b|1〉|2 , (3.97a)

= |2ab|2 = 2Pc , (3.97b)

which shows how the probability of joint detection increases for thermal bosons as com-pared to classical particles. As the latter display no correlation, this increase translates asa bunching of particles, that is, a tendency to cluster and arrive together at the detector.

Exercise 3.19 (∗∗) Spell out the procedure outlined above for plane wave modes forthe photons (with |a〉, |b〉 dotting with momentum 〈k| and position 〈r|) and by usingannihilation operators to model the detection. Follow the routes of the particles and howthey interfere destructively. Recover the result of Problem 2.9 obtained in a classicalpicture.

We now systematize this idea to more general quantum states, a necessity followingfrom an insight by Glauber (1963), who inaugurated a series of publications to accountfor the Hanbury Brown–Twiss correlations in a fully quantum picture. His approach


competed with the classical one by Mandel et al. (1964) that put a strong emphasis onstatistics but essentially excluding quantum effects, or at any rate excluding the conceptof non-classical fields that we have touched upon in the previous section. The semi-classical picture, Glauber observes, based on a statistical approach requires solely theknowledge of the power spectrum (2.56), whereas the full quantum picture allows statesthat need to track up to arbitrary high number of correlators.

First, Glauber derives the quantum analogue of eqn (2.60) with quantum fields andoperators and obtains eqn (3.98).

Exercise 3.20 (∗∗) The photodetection can be modelled by the ionization of atoms ofthe active medium of the detector, these being at positions r1 and r2, respectively. Call-ing wi the constant transition probability for an atom excited by the beam show (sum-ming over final electron states) that the probability of coincidence detection w(t1, t2)at t1 and t2 is given by:

w(t1, t2) = w1w2

Tr(ρE(−)(r1, t1)E

(−)(r2, t2)E(+)(r1, t1)E

(+)(r2, t2))

Tr(ρE(−)(r1, t1)E(+)(r1, t1)

)Tr(ρE(−)(r2, t2)E(+)(r2, t2)

) .

(3.98)

A more direct route is to agree on some canonical quantization of the classical no-tions of coherence developed in Section 2.3, indeed still largely valid in the quantumregime. This requires only the field E—which was a c-number quantity in the previouschapter—to now be promoted to its operator form, eqn (3.75). If we consider a singlemode (noting a the associated annihilation operator), we find the expressions of g(n) forquantum fields with n = 1, 2 given by:51

g(1)(τ, t) =

⟨a†(t)a(t + τ)

⟩〈a†(t)a(t)〉 , (3.99a)

g(2)(τ, t) =〈a†(t)a†(t + τ)a(t + τ)a(t)〉

〈a†(t)a(t)〉2 , (3.99b)

which matches with eqn (3.98).Once they are evaluated on the quantum state of a given system, the correlators

eqn (3.99) become c-number functions that can be processed in the same way as before,e.g., the g(1) fed into eqn (2.56) provides the emitted spectra of the system, as in theclassical case, but this time taking into account the quantum dynamics of the systemand quantum fluctuations of the state.

For a single mode, the second-order correlator g(2) is often more interesting, espe-cially in the quantum regime. At zero time delay, it becomes

g(2)(0) =〈(a†)2a2〉〈a†a〉2 . (3.100)

51Higher-order correlations, with n ≥ 3, are defined in the same way but they find little practical valueeven in fields where these quantities have been mastered for a long time. Theoretically, however, Glaubercrowns the definition of coherence by stating that a field is all order coherent if g(n)(t, t + τ) = 1 forall n ∈ N and all τ ∈ R. That is, the correlations all factorize into products of single-operator averages. Thisis the case for a monochromatic wave, for instance, or in terms of states, for a coherent state |α〉.

QUANTUM STATES 109

Fig. 3.3: Bunching (front) and antibunching (back) oflight as observed by Hennrich et al. (2005) by varyingthe average number of atoms in a cavity. At zero delay,g(2) has a peak when light arrives in lumps and a dropwhen photons avoids each other. The curves are sym-metric in time and tend asymptotically (though slowly)towards 1 for long delays.

Observe indeed how this quantity is sensitive to the quantum state considered:

Exercise 3.21 (∗) Show that the following values of g(2)(0) are obtained for the asso-ciated quantum states:

g(2)(0) State

1 − 1

nFock state |n〉

1 Coherent state |α〉 (cf. eqn (3.81))

2 Thermal state (cf. eqn (3.88)) .

The most noteworthy result of exercise (3.21) is g(2)(0) for the Fock state that iszero for n = 1 and is smaller than 1 for all n ∈ N. This contradicts eqn (2.63), if oneremembers that the two-time coincidence probability P2 is proportional to g(2)(τ) andthat g(2)(∞) = 1, which translates as the constraint:

g(2)(0) ≥ 1 (from classical model ). (3.101)

This is because there is no classical state that can describe correctly a Fock state,which thus poses itself as a pure quantum field without any classical analogue. On theother hand, coherent states and thermal states and all the mixed states interpolating be-tween them do have such classical counterparts and the effects they display (like bunch-ing or no correlations) can be explained classically or semiclassically. But antibunching,typically—that is the decrease of probability of a second detection once one has beenregistered—cannot be reproduced by any classical model. How quantization of the fieldbears on this problem can be illustrated as follows: If instead of the light of a star orthe Sun, a single photon is fed to the HBT setup, maximum anticorrelation results fromannihilation of a quantum, which in this case is an abrupt annihilation of the whole sig-nal. Indeed the branch that detects the photon destroys it and the probability to detect it


on the other branch becomes zero (whereas if collecting from a star, detecting a photonmeans an increase of the chance that another one will arrive on the other branch).

How to say whether the “quantum character” of a field is relevant can be decidedsimply from the Glauber P representation: all states whose P representation has a sin-gularity stronger than a Dirac delta function, is a quantum state without any classi-cal counterpart. Indeed, one can show that the P distribution for the Fock state |n〉is a 2n partial derivative of the delta function.52 For other non-classical states, it canbe that P becomes negative. Physically, whenever P cannot be interpreted as a well-behaved probability distribution, the field is quantum. Otherwise, a classical descriptionwith statistical weight of plane waves weighted by the P function will provide an ade-quate substitute. The first report of such a field without a good classical counterpart—byKimble et al. (1977) who observed photon antibunching in resonance fluorescence—still ranks among the few unambiguous pieces of direct evidence of quantization of theoptical field.

Typical experimental results for g(2)(τ) are shown in Fig. 3.3 where light emit-ted from an atomic ensemble coupled to a single cavity mode offers the nice featureof changing from bunching to antibunching with the number of atoms, as reported byHennrich et al. (2005). There is a peak at zero delay in the bunching case (in front ofthe plot) and a dip in the antibunching case (in back). Observe, however, that despitethe dip, g(2)(0) is still higher than 1 and eqn (3.101) is not violated in this case, butg(2)(0) ≥ g(2)(τ) is and this also is a signature of a quantum field. This is one reasonwhy it is interesting to know g(2)(τ) also at nonzero values of τ , although quantumcharacteristics are usually more marked at zero delay and a decay towards uncorrelatedvalues of the field are of course eventually obtained at long times (the timescale of thisdecay is another good reason to measure or compute at nonzero delay).

3.3.7 Statistics of the field

The zero-delay second-order coherence degree, g(2)(0), presents, however, strong ex-perimental and theoretical assets, mainly because it is a single time parameter (althoughit is still g(2)(t, 0)). It embeds a lot of information in a single quantity, in particular itis able to characterize the non-classical character of the field, it has the strongest sig-natures compared to other delays that suffer decoherence (and therefore decorrelation)and theoretically it can be computed easily in the simple quantum-mechanical pictureswe have presented earlier in this chapter.

The quantity that g(2)(0) captures in the most general case is the so-called “countingstatistics” of the field, already discussed, and of which we give a graphical representa-tion in terms of a time series of detection events for three limiting cases in Fig. 3.4,namely a Fock state (antibunching), coherent state (no correlation) and thermal state(bunching). Observe how the upper sequence—associated to Fock states and therefore

52Explicitly, the expression reads

PFock(α, α∗) =exp(|α|2)

n!∂2n

αn,α∗nδ(α) . (3.102)

QUANTUM STATES 111

Fig. 3.4: Computer-generated time series for detection of photons for:i) antibunched light such as emitted by resonance fluorescence of an atom (with g(2) < 1),ii) coherent light such as emitted by a laser or other coherent source (g(2) = 1),iii) bunched light such as emitted by a candle, the Sun or any such incoherent source (g(2) > 1) .

to a field of a highly quantum nature—the chain of detections looks like a stream ofwell-spaced events reaching the target one by one, as if emitted by a “photon gun”.Such emitters are highly prized for quantum information processing.53 If an additionaldegree of freedom like polarization can be controlled for each emission, it is possiblewith such a device to set up a completely secure (unbreakable) cryptographic system,as is studied in the problem at the end of this chapter.

The counting statistics is itself strongly linked to the statistics of the state, whichis the probability of a given state to be found with a given occupancy number n (to befound in the state |n〉). These probabilities are obtained from the diagonal elements ofthe density matrix in the basis of Fock states. Therefore, this statistics is defined as

p(n) = 〈n| ρ |n〉 . (3.103)

From the definition (eqn (3.99b)) and the boson algebra, it is straightforward toobtain the formula that links g(2) to p(n) in the Fock-state basis:

g(2)(0) =

∞∑n=0

n(n− 1)p(n)

( ∞∑n=0

np(n))2

, (3.104)

with ρ the density matrix. It is easy to compute the statistics of most of the states con-sidered so far. The results are compiled in Table 3.1.

Other states present more obscure distributions. This is the case of the mixture state,for instance, which despite being merely a Gaussian with an offset, (cf. eqn (3.91)), ismore involved mathematically.

Exercise 3.22 (∗∗) Consider the state that is the mixture of a thermal state with inten-sity nt and a coherent field with intensity nc, the P representation of which is given byeqn (3.91). Show that the distribution of this state is:

53Antibunched states are more easily obtained with fermions, for instance they are easily formed by elec-trons passing through a large resistance due to Coulomb interaction.


Table 3.1 Probability distribution and moments of basic quantum states.i) First row is the probability to have n excitationsii) Second row is the mean number of excitationsiii) Third row is the varianceiv) Fourth row is the profile for averages 〈a†a〉 = 1, 5 and 10.

Fock state Coherent state |α〉 Thermal state|n0〉 cf. eqn (3.81) cf. eqn (3.86)

Distribution p(n) δn,n0e−|α|2 |α|2n

n!

1

n + 1

(n

n + 1

)n

〈a†a〉 =∑

n np(n) n0 |α|2 n

〈(a†a)2〉 − 〈a†a〉2 0 |α|2 n2 + n

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

Pro

babi

lity

n

<n>=5

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20

n

<n>=15

10

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20

n

<n>=15

10

pm(n) = exp

(− nχ

1 + n(1− χ)

)(n(1− χ))n

(1 + n(1− χ))n+1Ln

(− χ

(1− χ)(1 + n(1− χ))

),

(3.105)where Ln is the nth Laguerre polynomial54 and χ is the coherent ratio (percentage of“coherent particles”):

χ =nc

nc + nt. (3.106)

The distribution (eqn (3.105)) of mixture states is plotted in Fig. 3.5 for values of χranging from 0 to 100% in steps of 10%, also with the two limiting cases of the purecoherent state (χ = 1) and the pure thermal state (χ = 0). The mathematical expressionis heavy but as a function it is well behaved and brings no problem for a numericaltreatment. For analytical efforts, as this distribution flattens very quickly with smallincoherent fractions, it can profitably be replaced by a Gaussian that is a much betterapproximation than to consider pure coherent states, as seen in Fig. 3.5. The first two

54The Laguerre polynomials are a canonical basis of solutions for the differential equation xy′′ +

(1 − x)y′ + ny = 0. They can be written as Ln(x) = ex

n!dn

dxn

`e−xxn

´and computed by car-

rying out the derivation. The first Laguerre polynomials are for n ≥ 0, Ln(x) = 1 (constant),−x + 1, 1

2(x2 − 4x + 2), . . . See E. W. Weisstein, “Laguerre Polynomial.” From MathWorld at

http://mathworld.wolfram.com/LaguerrePolynomial.html.

http://mathworld.wolfram.com/LaguerrePolynomial.html

QUANTUM STATES 113

moments of Pm can be retained exactly, namely the mean n and the variance computedas

σ2 = n + n2t + 2ncnt , (3.107)

which, moreover, links χ and g(2) through:

g(2)(0) = 2− χ2 . (3.108)

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250

p(n)

n

Coherent fraction=0%10%20%30%40%50%60%70%80%90%

100%

Fig. 3.5: Statistics of mixtures of thermal and coherent states in proportions ranging from 0 (thermal state)to 100 per cent (coherent state) coherence, for an average number of particle n = 100. At 90% coherence,the deviation is already very significant. The pure coherent state that has been truncated extends four timeshigher than is displayed, see Fig. 3.6 where this state is fully displayed. On the other hand, the curves with10% coherence and the thermal state are practically superimposed.

3.3.8 Polarization

Elliptical polarization

The polarization state of a photon and of a beam of photons is described by directtransposition of the Jones vector into a quantum superposition of basis states. This canbe described using a second-quantized formalism as well, as is done now in the mostgeneral case of elliptical polarization.

A photon with circular polarization degree given by P ≡ cos2 θ − sin2 θ is thecoherent superposition of a spin-up photon with probability cos2 θ and of a spin-downphoton with probability sin2 θ, therefore, its quantum state can be created from thevacuum |0, 0〉 (zero spin-up and zero spin-down photon) by application of the followingoperator:

|1, θ, φ〉 ≡ (cos θa†↑ + eiφ sin θa†

↓) |0, 0〉 . (3.109)

where the angle φ is the inplane orientation of the axis of the polarization ellipse. Thisdefines a†

θ,φ the creation operator for an elliptically polarized photon as


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 20 40 60 80 100 120 140 160

p(n)

n

Coherent fraction=90%95%99%

99.9%100%

0

0.005

0.01

0 100 200

Fraction=90%Gaussian fit

Fig. 3.6: Statistics of mixtures with high fractions of coherence, in respective proportions of 90% (also dis-played on Fig. 3.5), 95%, 99%, 99.9% and 100% (coherent state) coherence. For high values of coherence,the exact expression (3.105) can be approximated by a simple Gaussian, as illustrated in the inset. The ap-proximation becomes much better at higher ratios of coherence.

a†θ,φ ≡ cos θa†

↑ + eiφ sin θa†↓ . (3.110)

The superposition of n such correlated photons is obtained by recursive application ofthe creation operator:

|n, θ, φ〉 = a†nθ,φ |0〉 =

1√n!

(cos θa†↑ + eiφ sin θa†

↓)n |0, 0〉 , (3.111)

which we have normalized (here |0〉 is the vacuum in the space of elliptically polarizedstates).

Exercise 3.23 (∗∗∗) The BB84 protocol of single photon quantum cryptography.One desirable goal of cryptography is the secure communication of a key, which

is a binary digit list of arbitrary length that two parties A and B should be able tocommunicate at will. The exact information carried by the key is not important as longas it is known by the two parties and them only.

Assume that the two parties use a quantum channel over which they can send singlephotons that reach B in the polarization state they have been encoded by A. They alsouse a classical channel over which they can communicate any information they need,given, however, that this information is thus also made available to potential code-breakers.

By using the concept of conjugate bases, cf. Exercise 3.3, design a process bywhich A is able to generate and communicate to B a key that they both know for surehas not been observed by any third party.

OUTLOOK ON QUANTUM MECHANICS FOR MICROCAVITIES 115

3.4 Outlook on quantum mechanics for microcavities

Quantum physics at large is a significantly active and important area of research today.It is at the outset of numerous series of topics that diverge from each other as they getmore specialized. One of these routes leads to microcavities.

What makes this topic especially attractive is the depth and extent that it affords, aprecious and rather uncommon quality in today’s research, where specialization reducesthe physics to its most intricate details. This point can be illustrated from the photographbelow taken on the occasion of the fifth Solvay conference. The first edition in 1911was also the first international conference (the series is still continued to this date, itis held every three years) on the topic of Radiation and the Quanta. Einstein attendedas the youngest participant (at 25 years old). We have put in bold the names of thescientists whose work is central to the physics that makes the topic of this book, andwithout whose knowledge, one cannot pursue useful research. As one can clearly see,microcavities physics essentially brings again to the fore the fundamental physics of thefathers of modern science.

A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrodinger, E. Verschaffelt, W. Pauli,W. Heisenberg, R.H. Fowler, L. Brillouin, (upper row); P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers,P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr, (middle); I. Langmuir, M. Planck,Mme. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson and O.W. Richardson(lower row) posing for the 1927 Solvay conference.


3.5 Further reading

This chapter has reviewed some of the basic aspects of quantum mechanics and theirrelevance to the optical field to cover our needs for the more specific treatment in whatfollows and in a form and context such as they would typically be encountered in thephysics of microcavities. But dealing with such general issues, this chapter is also themost remote from the object of study of this book and therefore requires much supple-mentary reading to appreciate the subject in some depth. General quantum mechanicscan be obtained from countless sources, e.g., Merzbacher (1998). For quantum fieldtheory, attention should be directed towards texts written with statistical physics orcondensed-matter in mind, as there is little to be gained from the more popular rel-ativistic formulation that has different concerns. Renowned classics are the textbooksby Abrikosov et al. (1963), Negele and Orland (1998) and Fetter and Walecka (2003).For a study with special attention paid to the light field, while still maintaining a strongfield-theoretic approach, one can refer to the textbook by Cohen-Tannoudji et al. (2001).For quantum optics proper, Mandel and Wolf (1995) provide the best reference, ac-companied by a very large literature with such textbooks as those by Loudon (2000)or Scully and Zubairy (2002) among the most significant.

4

SEMICLASSICAL DESCRIPTION OF LIGHT–MATTER COUPLING

In this chapter we consider light coupling to elementary semiconductorcrystal excitations—excitons—and discuss the optical properties ofmixed light–matter quasiparticles named exciton-polaritons, which playa decisive role in optical spectra of microcavities. Our considerationsare based on the classical Maxwell equations coupled to the materialrelation accounting for the quantum properties of excitons.

118 SEMICLASSICAL DESCRIPTION OF LIGHT–MATTER COUPLING

4.1 Light–matter interaction

4.1.1 Classical limit

Although this chapter refers to a “semiclassical” treatment of light–matter interaction,there naturally exists a “pure classical treatment” where matter is put on an equal “clas-sical” footing with light (cf. Chapter 2). This description is, moreover, a very powerfulone and one that will lay down an important concept: the oscillator strength. It is, how-ever, simple enough to serve as an introduction to the so-called semiclassical treatment,where parts of the quantum concepts are involved in a rather vague and intuitive wayinto the material excitation. This helpful and short description in full classical terms iswhat we address now.

Lorentz proposed a fully classical picture of light–matter interactions where lightis modelled by Maxwell’s equations and the atom by a mechanical system of twomasses—the nucleus and an electron—bound together by a spring. ω0 is therefore thenatural frequency of oscillation. This purely mechanical oscillator will carry along manyconcepts into the quantum picture. The spring is set into motion when light irradiatesthe atom.

Hendrik A. Lorentz (1853-1928), who received the 1902 Nobel Prizein physics (with Zeeman) for his work on electromagnetic radiation,provided an insightful classical description of light–matter interactions.

His doctoral thesis in 1875 developed Maxwell’s theory of 1865 toexplain reflection and refraction of light. He is also noted for the Lorentztransformation of space and time dilations and contractions that would be atthe heart of Einstein’s special theory of relativity.

The gist of the physical implications of this assumption is retained in the simplestcase where the atom is fixed and the electron a distance x(t) away, moving under theinfluence of an applied electric field E(t), with an equation of motion:

m0x + m02γx + m0ω20x = −eE(t) , (4.1)

where m0 is the mass of the electron, −e its charge and m0ω20 the harmonic potential

binding the electron (the spring). Excluding γ, this is merely Newton’s equation F =m0a of a dipole whose acceleration a is driven by a force F . The loss term arises in thismodel from the fact that an oscillating dipole radiates energy.55 With this understanding,

55That a moving charge radiates energy is one of the problems with the classical picture of atoms wherethe electron is thought of as orbiting (hence, moving around) the nucleus, therefore doomed to spiral into it asit loses energy. It cannot stay still either because of the gravitational attraction of the nucleus. Bohr postulatedthat the electron cannot move smootly towards its nucleus because of interferences along certain orbits, givingbirth to the original quantum theory.

LIGHT–MATTER INTERACTION 119

its expression can be related to fundamental constants by computing the rate of energyloss of a dipole:

Exercise 4.1 (∗∗) Show that γ = e2ω20/(3m0c

3).

In our case where the oscillator models an atom, it is a good approximation to modelthe exciting field as an harmonic function of time—the vacuum solution of Maxwell’sequations—with E(t) = E0 cos(ωt). One can solve eqn (4.1) and obtain the steady-state of the system, where the electron also oscillates harmonically with the frequencyof the external force but with a different amplitude and a different phase:

x(t →∞) = A cos(ωt− φ) . (4.2)

One can see the intrinsic simplicity of the Lorentz oscillator. We are going to seenow its considerable richness. When we say “different amplitude and phase” of oscil-lation, we mean essentially the change in the response as a function of the excitingfrequency ω (see Fig. 4.1):

A(ω) =−eE0

m0

1√(ω2 − ω2

0)2 + (2γω)2, (4.3a)

φ(ω) = arctan( 2γω

ω20 − ω2

). (4.3b)

The external frequency that maximizes the amplitude A for a given set of param-eters ω0 and γ is the resonant frequency. We find it by taking the derivative of theamplitude function and setting it to zero:

ωres =√

ω20 − 2γ2 . (4.4)

At this frequency, the field is transferring energy most efficiently to the atom’s elec-tron. In the inset of Fig. 4.1 one can see how the resonant frequency ωres(γ) varieswith the damping. There is only a genuine resonance for cases with γ < ω0/

√2, other-

wise the system is overdamped and does not oscillate. On the other hand, if γ = 0 theresonant frequency is simply the natural frequency of the system. In the limiting casewhere γ ω0/

√2 (for frequencies close to ω0), the energy–transfer distribution is a

Lorentzian:γ

(ω − ω0)2 + γ2. (4.5)

A more general method to solve eqn (4.1), is to find the “output” or “answer” x(t)of the system to the “input” field E(t) = exp(iωt) through its transfer functionH:

x(t) = H(ω)E(t) . (4.6)

The linearity of eqn (4.1) allows one to compute H directly (by direct substitution)as


0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

Osc

illat

ion

ampl

itude

(a.

u.)

ω

γ = 0.1, 0.3, 0.7, 10

ω0 =1

0

0.5

1

0 0.2 0.4 0.6

γ

ωRes

0

0.5

1

0 0.2 0.4 0.6

γ

ωRes

Fig. 4.1: The amplitude of the oscillations of the electron, driven by a sinusoidal external field of frequency ω,shows a resonace when ω = ωres. In the inset, we can see how the resonant frequency ωres varies with thedamping coefficient γ, becoming ω0 for the undamped case. The natural frequency of the system is set toone.

H(ω) =e/m0

ω2 − ω20 − 2iωγ

(4.7)

Then, the time evolution of the oscillator for an arbitrary excitation E(t)—other thanexp(iωt) for which eqn (4.6) applies—can be obtained directly as:

x(t) =

∫ ∞

−∞h(t− τ)E(τ) dτ , (4.8)

where h is the Fourier transform ofH. The imaginary part ofH is called the absorptionsignal as it provides similar information as the amplitude A we have analysed in ourparticular case.

4.1.2 Einstein coefficients

A fundamental problem of light–matter interaction that falls out of the scope of clas-sical physics, as noted by the beginning of the twentieth century, is that of blackbodyradiation.56 So fundamental was this problem that any mathematical trick to derive thesolution was providing a physical insight into a worldview going much beyond that af-forded by the then existing models, which would flourish into quantum mechanics. Theattempt by Planck culminated in his law for blackbody radiation:

56A black body is an object that radiates energy originating from intrinsic emission by the object and isnot the result of reflection or transmission from external radiation. Therefore, all these radiations from outsidethe object and that impinge on it are absorbed by it (and later reradiated but only as a result of how the blackbody stores energy to maintain its thermal equilibrium). This is an ideal limiting case (since all objects reflector transmit light to some extent) to investigate the thermodynamic of the electromagnetic field.

LIGHT–MATTER INTERACTION 121

I(ω) =2ω3

πc3

1

eω/(kBT ) − 1, (4.9)

where I is the spectral energy density (with dimensions of joule per cubic meter persecond−1), ω the frequency,57 kB Boltzmann’s constant58 and T the temperature. Itsmain merit is the perfect accord with experimental data, previously afforded only whenseparating short and long wavelengths. Expression (4.9) gave impetus to the concept ofthe photon.59 The attempt by Bose led to the new statistics of bosons and in 1916, theattempt of Einstein himself—already a key actor in the two previous approaches—ledto the fundamental processes of semiclassical light–matter interaction. These are thetopics of this section.

Einstein proposed the existence of three fundamental processes in the interaction oflight with matter, that is—in the view of the time—in the interaction of a photon withan atom. These are:

1. Absorption2. Spontaneous emission3. Stimulated emission.

Fig. 4.2: The three fundamental processes of light–matter interaction in the semiclassical paradigm: Absorp-tion, spontaneous emission and stimulated emission.

The first two processes present no difficulty once the postulate of quantized energylevels is accepted: a photon of energy Ef−Ei can be absorbed by an atom, thus excitingit from energy level Ei to energy level Ef (energy is conserved and—this is the nov-elty arising from the semiclassical/old quantum theory—exchanged in discrete amountdefined by the atom structure).

57Strictly speaking, ω/(2π) is the frequency, and ω is the pulsation. We shall follow the standard usagethat is to call ω the frequency as well.

58Boltzmann’s constant is 1.380 6504 × 10−23 JK−1 and 8.617 343 × 105 eVK−1. It connects manymacroscopic quantities from the microscopic realm, its most fundamental connection being between the en-tropy and the logarithm of the number of microscopic states available to the system. Although this relationis engraved on Boltzmann’s tombstone in the Zentralfriedhof in Vienna, the constant, its name k and itsnumerical value were first provided by Planck some 23 years after the link envisioned by Boltzmann.

59Einstein interpreted Planck’s hypothesis in terms of “quanta” of light—one reason for which he is cred-ited as a founding father of quantum mechanics—which he used for his explanation of the photoelectric effect.The name “photon” itself was coined in by the chemist Gilbert Lewis as a support of a theory that was soonrefuted and abandoned.


Spontaneous emission is the reverse process where energy is released by the atomand a photon created that carries it away. It is called spontaneous because an excitedatom will decay by itself into a lower energy level (until it reaches the ground-state,where it stays until it gets excited again). The average time for this transition is, likethe possible energies of the atom, a property of the atom.60 It will later be for quantummechanics to calculate it; at this stage this is a given constant τsp.

To account for the Planck distribution, however, Einstein requires the introductionof the third process, stimulated emission.61 Because of this process, an excited atom—in addition to the decay by spontaneous emission (first process)—can also decay ifit is interacting with a photon, by emitting a clone of this photon. This is not veryintuitive but we now show that if it is accepted, the blackbody radiation spectrum isreadily derived by mere rate equations and detailed balance arguments. Each of theseprocesses are associated to a probability of occurrence: the probability per unit time ofspontaneous emission is called A, that is, if the density (number per unit volume) ofatoms in the excited (resp. ground) state is n2 (resp. n1), there is a transfer from theexcited to the ground-state populations following

dn1

dt= An2 . (4.10a)

Equation (4.10a) implies that A = τ−1sp .

Absorption is ruled by the Einstein coefficient B12 that gives the probability per unittime and per unit energy density of the radiation field that the atom initially in state 1absorb a photon and jump to state 2 causing a change in the number density of atoms inthe ground state of:

dn2

dt= B12n1I(ω) , (4.10b)

with I(ω) the spectral intensity of the radiation field at the frequency of radiation ω =(E2 − E1)/. This equation merely spells out the definition of B given above, withI(ω) quantifying the number of photons available to excite the atom.

Stimulated emission, as noted above, is induced by a photon bringing an atom inthe excited state to its ground-state. It is therefore the same process as eqn (4.10b) onlyreversed, with an excited atom as the starting state and finishing with one more ground-state atom. Calling the corresponding probability of this event B21

dn1

dt= B21n2I(ω) . (4.10c)

At equilibrium, all these three processes concur to establish the steady-state condi-tions dni/dt = 0. In this regime the average change in the populations of ground andexcited states is zero, being balanced by the losses and gains. The principle of “detailed

60Spontaneous emission is not an intrinsic property of the atom as it can be modified by changing theoptical environment of the atom. This important point is discussed in detail in Chapter 6.

61Historically, one was speaking of “induced” rather than “stimulated” emission, but the new term that isnow prevailing is more common and vivid for other similar manifestations, such as “stimulated scattering”.

OPTICAL TRANSITIONS IN SEMICONDUCTORS 123

balance” postulates that such an equilibrium in population exchanged is reached bypairwise compensations of the type dn1/dt = −dn2/dt, which leads to:

n2A− n1B12I(ω) + n2B21I(ω) = 0 , (4.11)

which provides the energy density as a function of other parameters as

I(ω) =A

(n1/n2)B12 −B21. (4.12)

We are investigating the energy distribution of the radiation field. That of the atoms(which form a classical system like a gas or a solid and nothing like a Bose conden-sate where their statistics could play a role) was well known from the earlier work ofBoltzmann and Maxwell. The kinetic theory of gases that they developed gives the pop-ulations of atomic states after the ratio of their energy to the temperature:

ni = gi exp(− Ei

kBT

)(4.13)

(with gi the degeneracy of the state), so that finally:

I(ω) =A

(g1/g2) exp(ω/kBT )B12 −B21. (4.14)

Identifying this result with eqn (4.9) provides the following expressions for Einsteincoefficients:

A =ω3

π2c3B21 , (4.15a)

g1B12 = g2B21 . (4.15b)

4.2 Optical transitions in semiconductors

An arena of choice for microcavity physics is that of semiconductor physics. A semi-conductor is a solid whose electrical conductivity has behaviour and magnitudes in be-tween metals and insulators. This comes from the energy levels of such systems thatform bands separated by gaps of forbidden energies (or states). Consequently, semicon-ductors afford a great control of electronic excitations. We note here the most essentialfeatures of the structure of optical transitions in semiconductors.62

The discrete electronic levels of individual atoms form bands in crystals where thou-sands of atoms are assembled in a periodic structure. There are also gaps between theallowed bands where no electronic states exist in an ideal infinite crystal. Those crystalsthat have a Fermi level63 inside one of the allowed bands are metals, while the crystals

62Much more information on this subject can be found in Charles Kittel, Introduction to Solid StatePhysics (Wiley: New York, 1996) and Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt:Orlando, 1976).

63The Fermi energy is the energy below which, at zero temperature, all the electronic states are occupiedand above which all the states are empty. The Fermi level is the set of states with Fermi energy.


Table 4.1 Classification of solids.

Fermi level Energy gap width Conductivity (Ω−1m−1)

metals inside the band any Up to 6.3× 107 (silver)semiconductors inside the gap < 4 eV Varies in large limitsdielectric inside the gap 4 eV Can be as low as 10−10

having a Fermi level inside the gap are semiconductors or dielectrics. The difference be-tween semiconductors and dielectrics is quantitative: the materials where the bandgapcontaining the Fermi level is narrower than about 4 eV are usually called semiconduc-tors, the materials with wider bandgaps are dielectrics. In this chapter we consider onlysemiconductor crystals.

The eigenfunctions of electrons inside the bands have a form of so-called Blochwaves. The concept of the Bloch waves was developed by the Swiss physicist FelixBloch in 1928 (see on page 174), to describe the conduction of electrons in crystallinesolids. The Bloch theorem states that a wavefunction of an electronic eigenstate in aninfinite periodic crystal potential V (r) can be written in the form (see Section 2.6.1):

Ψk,n = Uk,n(r)eik · r, (4.16)

where Uk,n (called the Bloch amplitude) has the same periodicity as the crystal poten-tial, k is a so-called pseudowavevector of an electron (hereafter we shall omit “pseudo”while speaking about this quantity), and n is the index of the band.

Substitution of the wavefunction (4.16) into the Schrodinger equation for an electronpropagating in a crystal

− 2

2m0∇2Ψk,n + V (r)Ψk,n = Ek,nΨk,n , (4.17)

with m0 being the free-electron mass, one obtains an equation for the Bloch amplitude:

− 2

2m0∆Uk,n + V (r)Uk,n +

(2k2

2m0+

m0(k · r)) = Ek,nUk,n , (4.18)

where p =

i∇. Consideration of the operators in the parentheses as a perturbationconstitutes the method of the k ·p perturbation theory, which readily enables solvingthe shape of the electronic dispersion in the vicinity of k = 0 points of all bands andthat appears to be strongly different from the free-electron dispersion in vacuum. Theapproximation:

Ek,n ≈ E0,n +2k2

2mn∗ (4.19)

is called the effective mass approximation with mn∗ the electron effective mass in thenth band:64

1

mn∗ =1

m0+

2

m20

∑l =n

|〈U0,l|p|U0,n〉|2E0,l − E0,n

. (4.20)

64In general, the effective mass is a tensor. It reduces to a scalar in crystals having a cubic symmetry.

OPTICAL TRANSITIONS IN SEMICONDUCTORS 125

The frequencies and polarization of the optical transitions in direct gap semiconduc-tors are governed by the energies and dispersion of the two bands closest to the Fermilevel65, referred to as the conduction band (first above the Fermi level) and the valenceband (first below the Fermi level; often several close bands are important).

Fig. 4.3: Zincblende (left) and wurtzite (right) crystal lattices.

Semiconductors can be divided into those with direct bandgaps and those with in-direct bandgaps. In indirect gap semiconductors (like Si and Ge) the electron and holeoccupying the lowest energy states in conduction and valence bands cannot directly re-combine emitting a photon due to the wavevector–conservation requirement. While aweak emission of light by these semiconductors due to phonon-assisted transitions ispossible, they can hardly be used for fabrication of light-emitting devices and studies oflight–matter coupling effects in microcavities. In the following, we shall only considerthe direct gap semiconductor materials like GaAs, CdTe, GaN, ZnO, etc. (see Fig. 4.4).Most of them have either a zincblende or a wurtzite crystal lattice66 (see Fig. 4.3). Inzincblende semiconductors, the valence band splits into three sub-bands referred to asthe heavy-hole, light-hole and spin-off bands (see Figure 4.5). At k = 0 the heavy andlight hole bands are degenerate in bulk crystals, while this degeneracy can be lifted bystrain or external fields. In the wurzite semiconductors the valence band is split intothree non-degenerate subbands referred to as A, B, and C bands.

Dispersion of the light and heavy holes in zincblende semiconductors can be conve-niently described by the Luttinger Hamiltonian67

H =2

2m0[(γ1 +

5

2γ2

)∆− 2γ3(∇J)2 + 2(γ3 − γ2)

(J2

x

∂2

∂x2+ J2

y

∂2

∂y2+ J2

z

∂2

∂z2

)] ,

(4.21)

where Jα = ± 12 ,± 3

2 , α = x, y, z and γ1, γ2, γ3 are Luttinger band parameters depen-dent on the material.

65In semiconductors, the Fermi level is situated in the gap. The width of this gap, Eg, governs the opticalabsorption edge.

66A cubic phase is somewhat more exotic. It is found for GaN, for example.67Proposed in the famous paper by J. M. Luttinger and W. Kohn, Physical Review 97, 869 (1955).


Fig. 4.4: Energy gaps and lattice constants in some direct-bandgap semiconductors and their alloys, fromVurgaftman et al. (2001).

Fig. 4.5: Schematic band structure of a zincblende semiconductor (a) with a conduction band (on the top),degenerated heavy and light hole bands (in the middle) and the spin-off band (at the bottom) and a wurtzitesemiconductor (b) with A, B and C valence subbands.

In the bulk zincblende samples, at k = 0 the degenerate light– and heavy–holestates mean that the probability of an allowed optical transition from a heavy hole to anelectron state is three times higher than the probability of a transition from a light–holestate (see Fig. 4.6). This is why illumination of a semiconductor crystal by circularly po-larized light leads to preferential creation of electrons with a given spin projection. Thiseffect referred to as optical orientation will be discussed in more detail in Chapter 9.

Optical absorption spectra in semiconductors are governed by the density of elec-tronic states in the valence and conduction bands, g(E) = ∂n

∂E , where n is the numberof quantum states per unit area. In bulk crystals, inside the bands the density of statesbehaves as

√E, which results in the corresponding shape of the interband absorption

EXCITONS IN SEMICONDUCTORS 127

Fig. 4.6: Polarization of the interband optical transitions in zincblende semiconductor crystals. Solid, dashedand dotted lines show σ+, σ− and linearly polarized transitions, respectively.

spectra. Besides this, at low temperatures the absorption spectra of semiconductors ex-hibit sharp peaks below the edge of interband absorption (i.e. at frequencies ω < Eg/,where Eg is the band-gap energy). These peaks manifest the resonant light–matter cou-pling in semiconductors. They are caused by the excitonic transitions that will remainthe focus of our attention throughout this book.

4.3 Excitons in semiconductors

4.3.1 Frenkel and Wannier–Mott excitons

In the late 1920s narrow photoemission lines were observed in the spectra of organicmolecular crystals by Kronenberger and Pringsheim (1926) and I. Obreimov and W.de Haas. These data were interpreted by the Russian theorist Frenkel (1931) who in-troduced the concept of excitation waves in crystals and later coined the term exciton(Frenkel 1936).

By definition, the exciton is a Coulomb-correlated electron–hole pair. Frenkel treatedthe crystal potential as a perturbation to the Coulomb interaction between an electronand a hole belonging to the same crystal cell. This scenario is most appropriate in or-ganic molecular crystals. The binding energy of Frenkel excitons (i.e. the energy ofits ionization to a non-correlated electron–hole pair) is typically of the order of 100–300 meV. Frenkel excitons have been searched for and observed in alkali halides by Ap-ker and Taft (1950). At present they are widely studied in organic materials where theydominate the optical absorption and emission spectra.

At the end of the 1930s, Wannier (1937) and Mott (1938) developed the concept ofexcitons in semiconductor crystals where the rate of electron and hole hopping betweendifferent crystal cells much exceeds the strength of their Coulomb coupling with eachother. Unlike Frenkel excitons, Wannier–Mott excitons have a typical size of the orderof tens of lattices constants and a relatively small binding energy (typically, a few meV).

Besides Frenkel and Wannier–Mott excitons, there are a few other types of exci-tons. The charge–transfer excitons are spatially separated Coulomb-bound electron–


Yakov Il’ich Frenkel (1894–1952), Sir Nevill Francis Mott (1905–1996) and Gregory Wannier (1911–1983) gave their names to the two main categories of excitons.

Frenkel was a versatile physicist who made his main contributions in solid-state physics. He wrotethe first paper devoted to the quantum theory of metals. He is now remembered for the exciton bearinghis name and the Frenkel defect. A very prolific writer, Kapitza reportedly once told him “You would be agenius if you published ten times less than you do” (his most noted work is the textbook “Kinetic Theory ofLiquids”). A good overview of his life and work is given by S. L. Lopatnikov and A. H.-D. Cheng. in J. Eng.Mech., 131, 875, (2005).

Mott, the Nobel prize-winner for Physics in 1977 for “fundamental theoretical investigations of theelectronic structure of magnetic and disordered systems”, is most renowned for his mechanism explainingwhy material predicted to be conductors by band theory are in reality (so-called Mott) insulators and fordescribing the transition of substances from metallic to non-metallic states (Mott transition).

Wannier authored a series of important papers on the properties of crystals. His main achievement isa complete set of orthogonal functions, known as “Wannier functions”, which provide an alternativerepresentation of localized orbitals to the usual Bloch orbitals. According to his graduate student D. R.Hofstadter who pays him tribute in Phys. Rep., 110, 237 (1984), “it is not what [he] would want to be knownfor primarily. He was so involved in so many areas of physics, and his breadth was so refreshing, comparedto the narrow range of most physicists today, that I think he would wish to be remembered for that breadthand for his style, a style that stressed beauty and purity and fundamentality”.

hole pairs having a spatial extension of the order of the crystal lattice constant. Thelowest energy charge–transfer exciton usually extends over two nearest–neighbour mo-lecules in a molecular crystal and creates a so-called donor–acceptor complex. See, forinstance Silinish (1980) for a more thorough treatment.

Another example of a few-particle exciton complex in a quantum-confined semi-conductor system is the so-called anyon exciton appearing in the regime of the quantumHall effect as described by Rashba and Portnoi (1993). The energy of anyon excitonictransitions lies in the far-infrared range, thus these quasiparticles cannot be coupled tolight in optical cavities.

Recently, Agranovich et al. (1997) have proposed a concept of hybrid Frenkel–Wannier–Mott excitons that can be formed in mixed organic–inorganic structures. Suchexcitons would combine a huge binding energy and relatively large size. Extensive in-formation on the Frenkel or hybrid excitons and their coupling with light in organicmicrocavities can be found in the recent volume edited by Agranovich and Bassani


Fig. 4.7: A Wannier–Mott exciton is the solid-state analogy of a hydrogen atom, while they have very differentsizes and binding energies. Unlike atoms, the excitons have a finite lifetime.

(2003).In the present section we only discuss the Wannier–Mott excitons in semiconduc-

tor structures. Such excitons can be conveniently described within the effective massapproximation that allows the periodic crystal potential to be neglected and describeselectrons and holes as free particles having a parabolic dispersion and characterized byeffective masses dependent on the crystal material. Usually, the effective masses of car-riers are smaller than the free-electron mass in vacuum m0. For example, in GaAs theelectron effective mass is me = 0.067m0, the heavy-hole mass is mhh = 0.45m0.

Consider an electron–hole pair bound by the Coulomb interaction in a crystal havinga dielectric constant ε. The wavefunction of relative electron–hole motion f(r) can befound from the Schrodinger equation analogous to one describing the electron state in ahydrogen atom:

− 2

2µ∇2f(r)− e2

4πεε0rf(r) = Ef(r) , (4.22)

with µ = memh/(me + mh) the reduced mass, r =√

x2 + y2 + z2 the distance be-tween electron and hole. The solutions of eqn (4.22) are well known as they correspondto the states of the hydrogen atom with the following renormalizations:68

m0 → µ, e2 → e2/ε , (4.23)

68In the hydrogen-atom problem the reduced mass is equal, in good approximation, to the electronmass m0 because of the very large mass of the nucleus.


For example, the wavefunction of the 1s state of exciton reads:

f1s =1√πa3

B

e−r/aB , (4.24)

with the Bohr radius aB given as:

aB =4π2εε0

µe2. (4.25)

The binding energy of the ground exciton state is

EB =µe4

(4π)222εε2=

2

2µa2B

. (4.26)

Given the difference between the reduced mass µ and the free-electron mass, andtaking into account the dielectric constant in the denominator, one can estimate that theexciton binding energy is about three orders of magnitude less than the Rydberg con-stant. Table 4.2 shows the binding energies and Bohr radii for Wannier–Mott excitonsin different semiconductor materials.

Table 4.2 Strongly anisotropic conduction and valence bands, direct transitions farfrom the centre of the Brillouin zone.∗ Strongly anisotropic conduction and valence bands, direct transitions far from the cen-tre of the Brillouin zone.∗∗ In the presence of a magnetic field of 5 T.∗∗∗ An exciton in hexagonal GaN.∗∗∗∗ The ground-state corresponds to an optically forbidden transition, data givenfor n = 2 state.

Semiconductor crystal Eg (eV) me/m0 EB (eV) aB (A)

PbTe∗ 0.17 0.024/0.26 0.01 17 000InSb 0.237 0.014 0.5 860Cd0.3Hg0.7Te 0.257 0.022 0.7 640∗∗

Ge 0.89 0.038 1.4 360GaAs 1.519 0.066 4.1 150InP 1.423 0.078 5.0 140CdTe 1.606 0.089 10.6 80ZnSe 2.82 0.13 20.4 60GaN∗∗∗ 3.51 0.13 22.7 40Cu2O 2.172 0.96 97.2 38∗∗∗∗

SnO2 3.596 0.33 32.3 86∗∗∗∗

The exciton excited states form a number of hydrogen-like series. Observation ofsuch a series of excitonic transitions in the photoluminescence spectra of Cu2O in


Evgenii Fedorovich Gross (1897–1972) andthe experimental discovery of the exciton: thehydrogen-like “yellow” series in emission ofCu2O observed by Gross et al. (1956), withits numerical fit. Besides discovering experi-mentally the exciton, he is also noted for pi-oneering in 1930 the experimental observationof Rayleigh scattering fine structure due toBrillouin–Mandelstam light scattering on acous-tic waves. A short account of his scientificachievements can be found in “Evgeni Fe-dorovich Gross”, B. P. Zakharchenya and A. A.Kaplyanski, Sov. Phys. Uspekhi, 11, 141 (1968).

1951 was the first experimental evidence for Wannier–Mott excitons (see the illustra-tion on the current page). This discovery was made by the Russian spectroscopist Ev-geniy Gross who worked in the same institution—the Ioffe Physico-Technical institutein Leningrad—as Ya.I. Frenkel at that time.

4.3.2 Excitons in confined systems

Since the beginning of the 1980s, progress in the growth technology of semiconduc-tor heterostructures encouraged study of Wannier–Mott excitons in confined systemsincluding quantum wells, quantum wires and quantum dots. The main idea behind thedevelopment of heterostructures was to artificially create potential wells and barriers forelectrons and holes combining different semiconductor materials. The shape of the po-tential in conduction and valence bands is determined in these structures by the positionsof the corresponding band edges in the materials used as well as by the geometry of thestructure. The band engineering in semiconductor structures by means of high-precisiongrowth methods has allowed the creation of a number of electronic and opto-electronicdevices including transistors, diodes and lasers. It has also permitted discovery of im-portant fundamental effects including the integer and fractional quantum Hall effects,Coulomb blockade, light-induced ferromagnetism, etc.

The large size of Wannier–Mott excitons makes them strongly sensitive to nanometre-scale variations of the band-edge positions that can be easily obtained in modern semi-conductor nanostructures. The energy spectrum and wavefunctions of quantum-confinedexcitons can be strongly different from those of bulk excitons. Here, we consider bymeans of an approximate but efficient variational method the excitons in quantum wells,wires and dots (see Fig. 4.8). We will use the effective-mass approximation. When werefer to wavefunctions we always mean the envelope functions, neglecting the Blochamplitudes of electrons and holes. Note that in these examples we neglect the complex-ity of the valence-band structure and consequent anisotropy of the hole effective massthat sometimes strongly affects the excitonic spectrum in real semiconductor systems.More information on excitons in confined systems can be found in the books by Bastard(1988), Ivchenko and Pikus (1997) and Ivchenko (2005).


Fig. 4.8: Reduction of the dimensionality of a semiconductor system from 3D to 0D from a bulk semicon-ductor to a quantum dot. The electronic density of states g(E) = dN/dE—with dN the number of electronquantum states within the energy interval dE—changes drastically between systems of different dimension-alities as is shown schematically in the figure. This variation of the density of states is very important forlight-emitting semiconductor devices.

4.3.3 Quantum wells

The Schrodinger equation for an exciton in a quantum well (QW) reads:(− 2

2me∇2

e −2

2mh∇2

h + Ve(ze) + Vh(zh)− e2

4πεε0|re − rh|)

Ψ = EΨ , (4.27)

with Ve,h(ze,h) the confining potential for electron, hole on the z-axis, which is thegrowth axis of the structure. Solving exactly eqn (4.27) is not an easy task. We approachthe problem variationally over a class of trial functions having the form:

Ψ(re, rh) = F (R)f(ρρρ)Ue(ze)Uh(zh) , (4.28)

whereR =

mere + mhrh

me + mh(4.29)

is the exciton centre of mass coordinate and

ρρρ = ρρρe − ρρρh (4.30)

is the inplane radius-vector of electron and hole relative motion, r = (ρρρ, z). Four com-ponents of the trial function (4.28) describe the exciton centre of mass motion, the rela-tive electron–hole motion in the plane of the QW, and electron and hole motion normalto the plane direction. The factorization of the exciton wavefunction makes sense whenthe QW width is less than or comparable to the exciton Bohr diameter in the bulk semi-conductor. In this case, the electron and hole are quantized independently of each other.


On the other hand, in larger QWs, one can assume that the exciton is confined as awhole particle and keeps the internal structure of a 3D hydrogen atom. Here and laterwe shall consider narrow QWs where eqn (4.28) represents a good approximation. Thefour terms that compose the exciton wavefunction are normalized to unity:∫

|Ue(ze)|2dze = 1,

∫ ∞

0

|f(ρ)|22πρdρ = 1 , (4.31a)∫|Uh(zh)|2dzh = 1,

∫ ∞

0

|F (R)|22πRdR = 1 . (4.31b)

After substitution of the trial function (4.28) and integration over R, eqn (4.27)becomes:

− 2

2me

∂2

∂z2e

− 2

2mh

∂2

∂z2h

− 1

ρ

∂

∂ρ

(2

2µρ

∂

∂ρ

)+ Ve(ze) + Vh(zh)

− e2

4πεε0

√ρ2 + (ze − zh)2

− P 2exc

2(me + mh)− E

f(ρ)Ue(ze)Uh(zh) = 0 , (4.32)

where Pexc is the excitonic momentum, P = 0 for the ground state. Equation (4.32)can be transformed into a system of three coupled differential equations, each definingone of the components of our trial function. The equation for f(ρ) is obtained by mul-tiplication of both parts of eqn (4.32) by U∗

e (ze)U∗h(zh) and integrating over ze and zh.

This yields:−1

ρ

∂

∂ρ

(2

2µρ

∂

∂ρ

)− e2

4πεε0

∫∫ |Ue(ze)|2|Uh(zh)|2√ρ2 + (ze − zh)2

dzedzh

f(ρ) = −EQW

B f(ρ) ,

(4.33)where EQW

B is the exciton binding energy. The electron and hole confinement energiesEe and Eh, and wavefunctions Ue,h(ze,h), can be obtained by multiplying eqn (4.32)by f∗(ρ)U∗

h,e(zh,e) and integrating over ze,h and ρ:

− 2

2me,h∇2

e,h + Ve,h − e2

4πεε0

∫∫ |f(ρ)|2|Uh,e(zh,e)|2√ρ2 + (ze − zh)2

2πρdρdzh,e

Ue,h(ze,h)

= Ee,hUe,h(ze,h) . (4.34)

In the ideal 2D case, |Ue,h(ze,h)|2 = δ(ze,h) and eqn (4.33) transforms into− 2

2µ

1

ρ

∂

∂ρ

(ρ

∂

∂ρ

)− e2

ερ

f(ρ) = E2D

B f(ρ) , (4.35)

which is an exactly solvable 2D hydrogen atom problem. For the ground state:

f1S(ρ) =

√2

π

1

a2DB

exp(−ρ/a2DB ) , (4.36)


witha2DB =

aB

2, (4.37)

and aB the Bohr radius of the three-dimensional exciton given by eqn (4.25). The bind-ing energy of the two-dimensional exciton exceeds by a factor of 4 the bulk excitonbinding energy:

E2DB = 4EB , (4.38)

For realistic QWs, eqns. (4.32) and (4.33) still can be decoupled if the Coulomb term ineqn (4.33) is neglected. This allows the functions |Ue,h(ze,h)| to be found independentlyfrom each other as well as f(ρ). Solving eqn (4.32) with a trial function

f(ρ) =

√2

π

1

aexp(−ρ/a) , (4.39)

where a is a variational parameter, one can express the binding energy as:

EQWB (a) = − 2

2µa2+

e2

4πεε0

∫∫∫ |f(ρ)|2|Ue(ze)|2|Uh(zh)|2√ρ2 + (ze − zh)2

2πρdρdzedzh . (4.40)

Maximization of EQWB (a) finally yields the exciton binding energy in a QW, which

ranges from EB to E2DB and depends on the QW width and barrier heights for electrons

and holes. The binding energy increases if the exciton confinement strengthens. This iswhy the dependence of the binding energy on the QW width is non-monotonic: for widewells the confinement increases with the decrease of the QW width, while for ultranar-row wells the tendency is inverted due to tunnelling of electron and hole wavefunctionsinto the barriers (Fig. 4.9).

Fig. 4.9: Exciton binding energy as a function of the QW width (schema). The insets show the QW potentialand wavefunctions of electron (convex shape) and hole (concave shape) for different QW widths.


4.3.4 Quantum wires and dots

Variational calculation of the ground exciton state energy and wavefunction in quantumwires or dots can be done using the same method of separation of variables and decou-pling of equations as for QWs. There exist a number of important peculiarities of wiresand dots with respect to wells, however.

For a wire, the Schrodinger equation for the wavefunction of electron–hole relativemotion f(z), with z the axis of the wire, reads:

− 2

2µ∇2

z −e2

4πεε0

∫∫ |Ue(ρe)|2|Uh(ρh)|2√z2 + (ρe − ρh)2

dρedρh

f(z) = −EQWW

B f(z) ,

(4.41)

with the kinetic energy for relative motion being along the axis of the wire. Ue,h(ρe,h)

is the electron, hole wavefunction in the plane normal to the wire axis and EQWWB

is the exciton binding energy in the wire. Despite visible similarity to eqn (4.33) forelectron–hole relative motion wavefunction in a QW, eqn (4.41) has a different spec-trum and different eigenfunctions. As a quantum particle in a 1D Coulomb potentialhas no ground-state with a finite energy, the exciton binding energy in a quantum wireis drastically dependent on spreading of the functions Ue,h(ρe,h) and can, theoretically,have any value between EB and infinity. The trial function cannot be exponential (asit would have a discontinuous first derivative at z = 0 in this case). The Gaussianfunction is a better choice in this case. Usually, realistic quantum wires do not have acylindrical symmetry (most popular are “T-shape” and “V-shape” wires, see Fig. 4.10),which makes computation of Ue,h(ρe,h) a separate, difficult task. Moreover, the realis-tic wires have a finite extension in the z-direction that is comparable with the excitonBohr diameter in many cases. Even if the wire is designed to be much longer than theexciton dimension, inevitable potential fluctuations in the z-direction lead to the excitonlocalization. This makes realistic wires similar to elongated quantum dots (QDs).

An exciton is fully confined in a QD, and if this confinement is strong enough itswavefunction can be represented as a product of electron and hole wavefunctions:

Ψ = Ue(re)Uh(rh) , (4.42)

where the single-particle wavefunctions Ue,h(re,h) are given by coupled Schrodingerequations:

− 2

2me,h∇2

e,h + Ve,h − e2

4πεε0

∫ |Uh,e(rh,e)|2|re − rh|

Ue,h(re,h) = Ee,hUe,h(re,h) ,

(4.43)

with Ve,h is the QD potential for an electron, hole. In this case, the exciton bindingenergy is defined as

EQDB = E0

e + E0h − Ee − Eh , (4.44)

where E0e and E0

h are energies of the non-interacting electron and hole, respectively,i.e., the eigenenergies of the Hamiltonian (4.43) without the Coulomb term.


Fig. 4.10: Cross-sections of V-shape (a) and T-shape (b) quantum wires, from Di Carlo et al. (1998).

Fig. 4.11: Transmission electron microscopy image from Widmann et al. (1997) of the self-assembled QDsof GaN grown on AlN

In small QDs Coulomb interaction can be considered as a perturbation to the quantum-confinement potential for electrons and holes. The exciton binding energy can be esti-mated using perturbation theory as

EB ≈ e2

4πεε0

∫∫ |Ue(re)Uh(rh)|2|re − rh| dredrh . (4.45)

As in the wire, the exciton binding energy in the dot is strongly dependent on thespatial extension of the electron and hole wavefunctions and can range from the bulkexciton binding energy to infinity, theoretically. In realistic wires and dots, the binding

EXCITON–PHOTON COUPLING 137

Solomon I. Pekar (1917–1985), John J. Hopfield (b. 1933) and Vladimir M. Agranovich (b. 1929).

Pekar was the leading theorist of the Physical Institute in Kiev. He created the theory of adiabatic polarons in1946 (the term polaron comes from him), and with his work on the so-called additional light wave—statingthat under excitation by monochromatic light, the dielectric constant’s spatial dispersion near an excitonic res-onance gives rise to an additional propagating polariton wave—he originated the theory of exciton-polaritons.

Hopfield, after important contributions to physics, turned his interest to biology, where he made his mostsignificant contribution to science with his associative neural network now know as the Hopfield network. Atthe time of writing, he is professor in the department of Molecular Biology at Princeton University and thePresident of the American Physical Society.

Agranovich is the head of the Theoretical Department of the Institute of Spectroscopy of the Russian Academyof Sciences. He made seminal contributions to the theory of excitons especially in organic crystals and is oneof the founders of the theory of polaritons. A prolific author, he fathered among other important work, “Crys-tal Optics with Spatial Dispersion, and Excitons” with Nobel laureate Ginsburg, a “Theory of Excitons” andrecently the monograph “Electronic Excitations in Organic Based Nanostructures”. He currently holds thespecial position of “Pioneer of Nano-Science” at the University of Texas at Dallas.

energy rarely exceeds 4EB, however. At present, small QDs are mostly fabricated by theso-called Stransky–Krastanov method of molecular beam epitaxy and have either pyra-midal or ellipsoidal shape (see Fig. 4.11). In large quantum dots (“large” meaning “ofa size exceeding the exciton Bohr diameter”) excitons are confined as whole particlesand their binding energy is equal to the bulk exciton binding energy. Good examples oflarge quantum dots are spherical microcrystals that may serve also as photonic dots.

Exercise 4.2 (∗) Find the binding energies of the first excited states of 2D and 1D exci-tons.

4.4 Exciton–photon coupling

It has been clear since the very beginning of experimental studies of excitons that theeasiest way to create these quasiparticles is by optical excitation. In the mid-1950s,theorists understood that coupling to light strongly influences the physical propertiesof excitons and their energy spectrum. The Ukrainian physicist Pekar (1957) was thefirst to describe these changes of the exciton energy spectrum due to coupling to lightin terms of additional waves appearing in the crystal. Almost simultaneously, the termpolariton appeared in the works of Agranovich (1957) (Russia) and Hopfield (1958)


(USA) devoted to the description of photon-exciton coupling.69

We begin the description of exciton-polaritons from equations proposed by Hopfield(1958), adapting his notations to our exposition.

The first equation reads:

εB

c2

∂2

∂t2E(r, t) +∇×∇×E(r, t) = − 1

c2

∂2

∂t2P(r, t) , (4.46)

which directly follows from the wave equation (2.22) and material relation eqn (2.16).εB is the normalized background dielectric constant70 that does not contain the excitoniccontribution. The link between the polarization and the electric field is given by thesecond Hopfield equation:[

∂2

∂t2+ 2γ

∂

∂t+ ω2

0 −ω0

Mx∇2

]P(r, t) = εBω2

pE(r, t) , (4.47)

where ωp is the so-called polariton Rabi frequency, ω0 is the exciton transition energythat is dependent on the difference of energies of an electron and a hole composingthe exciton and on the exciton binding energy, and Mx = me + mh is the excitontranslation mass. The second Hopfield equation is a direct consequence of the Lorentzdipole oscillator model. Formally, it can be derived from eqn 4.1.

Equation (4.47) is derived assuming a linear optical response of the system and con-sidering each exciton as an harmonic oscillator having its eigenfrequency correspond-ing to the energy of the excitonic transition, with damping caused by exciton interactionwith acoustic phonons. The polarization created by excitons is taken to be proportionalto the amplitude of the harmonic oscillator, which constitutes the so-called dipole ap-proximation. A derivation in full details appears in the textbook of Haug and Koch(1990). A double Fourier transform of eqn (4.47) yields

P(ω,k) =εBω2

pE(ω,k)

ω20 − ω2 − 2iωγ + ω0k2/Mx

. (4.48)

In the vicinity of the resonant frequency, one can express the normalized dielectric func-tion from eqn (4.48) as

ε(ω, k) = εB +εBωLT

ω0 − ω + k2/(2Mx)− iγ, (4.49)

where ωLT = ω2p/(2ω0) is the so-called longitudinal-transverse splitting. For γ = 0

and Mx →∞, it is equal to the splitting between the frequencies at which the dielectricconstant goes to infinity (ωT) and to zero (ωL = ωT + ωLT). This splitting is a direct

69The polaritons in their most generally accepted terminology refer to mixed light–matter states in crys-tals. They do not necessarily imply excitons, but can also be formed, typically, by optical phonons. In this textwe use the term “polariton” to mean exciton-polaritons only. The reader can find a recent starting point onphonon-polaritons in, e.g., the text of Stoyanov et al. (2002).

70εB is normalized so that it is equal to unity in vacuum.


measure of the coupling strength between the exciton and light, and is proportional tothe exciton oscillator strength f :

f =4πε0

√εB

π

m0c

e2ωLT . (4.50)

For the ground exciton state in GaAs, ωLT = 0.08 meV, while in wide-band-gapmaterials (GaN, ZnO) it is an order of magnitude larger.

The dependence of the tensor of the dielectric susceptibility of a crystal on thewavevector of light is referred to in crystal optics as spatial dispersion. Equation (4.49)describes the spatial dispersion induced by excitons in semiconductors.

The relation between frequency and wavevector of the transverse polariton modesis given by

k2 = ε(ω, k)ω2

c2, (4.51)

which is a biquadratic equation with solutions:

k21,2 = −Mx

(ω0 − ω − iγ − εB

2Mx

ω2

c2

)

± Mx

√(ω0 − ω − iγ − εB

2Mx

ω2

c2

)2

+2

MxεBωLT

ω2

c2. (4.52)

For the longitudinal modes, the condition ε(ω, k) = 0 yields

k2L =

2Mx

(ω − ω0 − ωLT + iγ) . (4.53)

One can see that at each frequency two transverse and one longitudinal polaritonmodes with different wavevectors can propagate in the same direction. The appearanceof additional light modes in crystals at the exciton resonance frequency as a result ofspatial dispersion was theoretically predicted by Pekar in 1957 and confirmed experi-mentally by Kiselev et al. (1973). Description of additional light modes in finite-sizecrystal slabs requires additional boundary conditions (ABC) that have to be imposed onthe dielectric polarization in spatially dispersive media. Pekar proposed the followingform for the ABC:

P = 0 (4.54)

at the surface of the crystal. The condition comes from the physical argument that theexcitons that are responsible for the appearance of the polarization P exist only in-side the crystal. Physically, the Pekar conditions assume that the exciton wavefunctionis confined within the crystal slab, which acts on the exciton as a potential well withinfinitely high barriers. The choice (4.54) of ABC is not the only possible one. In anumber of studies the concept of a so-called “dead-layer” is used, assuming that the ex-citon centre of mass cannot approach the interface closer than some critical length of theorder of the exciton Bohr radius. In general, Neumann-type conditions on the polariza-tion and its derivative may be formulated. Though Pekar ABC have proven to provide a


good agreement between theoretical and experimental spectra of exciton-polaritons, thedebate on the exact form of ABC still continues.

Here we use the Pekar ABC to obtain the eigenfrequencies of exciton-polaritonmodes in a crystal slab of thickness L. Assuming zero inplane wavevector of all lightmodes (normal incidence geometry), one can readily obtain from eqn (4.54) the condi-tions on wavevectors kj , (j = 1, 2, L):

kj =Nπ

L, N ∈ N . (4.55)

Fig. 4.12 shows the dispersion of transverse (solid lines) and longitudinal (dashedline) polariton modes in GaAs calculated with Eqs. (4.52) and (4.53), respectively. Thevertical dotted lines show those values of the wavevectors which satisfy the condi-tion (4.55) for a given value of L. Crossing points of the dotted lines and the dispersioncurves yield the discrete spectrum of eigenfrequencies of exciton-polaritons in the thinfilm. These frequencies correspond to resonances in reflection or transmission spectra.The splitting between neighboring eigenfrequencies increases with decrease of the ex-citon mass and the thickness L. The fit of optical spectra of exciton-polaritons in thinfilms allows one to obtain the exciton mass with a good accuracy.

Fig. 4.12: Dispersion of the transverse (solid) and longitudinal (dashed) exciton-polariton modes in GaAs ascalculated by Vladimirova et al. (1996). Vertical dotted lines show positions of quantum confined polaritonmodes in a 1148 A thick film of GaAs.

4.4.1 Surface polaritons

Surface polaritons result from exciton coupling with light modes having a componentof wavevector in the plane of the surface, kx > ω/c, i.e., outside of the light cone (weassume that TM-polarized light propagates along the x-axis, in the plane, so that themagnetic field vector of the light-wave is parallel to the y-axis also in the plane). In this


case, the light wave propagating along the surface of the crystal decays in vacuum. If weconsider the right surface, the electric field vector of such a mode in vacuum behaves as:

E+ = E+0 exp

(ikxx−

√k2

x −ω2

c2z

), (4.56)

(z-axis is normal to the surface). The electric field vector in the crystal reads:

E− = E−0 exp

(ikxx +

√k2

x −ω2

c2ε(ω)z

), (4.57)

where ε(ω) is the same as in eqn (4.49). Dependent on the sign and value of ε(ω), theelectric field may decay or not inside the crystal. From Eqs. (4.56) and (4.57), one easilyobtains the ratio of the x- and z-components of the field:

E+z

E+x

=kx

i√

k2x − ω2

c2

,E−

z

E−x

= − kx

i√

k2x − εω2

c2

. (4.58)

The dispersion of surface polariton modes can be obtained from the Maxwell boundaryconditions, which require:

E−x = E+

x , ε(ω)E−z = E+

z . (4.59)

The second condition comes from the continuity of the magnetic field at the surface.From (4.58) and (4.59) we obtain:

ε(ω)

√k2

x −ω2

c2= −

√k2

x − ε(ω)ω2

c2, (4.60)

thus

ω = ckx

√1 + ε(ω)

ε(ω), ε < 0 . (4.61)

To analyse the dispersion equation (4.61), let us consider the limit γ → 0 and Mx →∞. ε(ω) is a real function schematically displayed in Fig. 4.13. Equation (4.61) yieldsthe real exciton-polariton eigenfrequencies if ε(ω) < −1. One can see that this condi-tion can only be satisfied within the frequency range

ω0 ≤ ω < ω0 + ωLT − δ , (4.62)

where δ can be found from the condition

ε(ω0 + ωLT − δ) = −1 . (4.63)

The dispersion of surface polaritons can now be easily understood: it starts at ω =ω0, kx = ω/c and goes to ω → ω0 + ωLT − δ, kx →∞ as shown on Fig. 4.14.


Fig. 4.13: Dielectric constant in the vicinity of the exciton resonance (scheme). The broadening and specialdispersion are neglected.

Fig. 4.14: Schematic dispersions of the surface polariton (solid), the transverse bulk polaritons (dashed)and the longitudinal bulk polariton (dotted). Spatial dispersion and broadening of the exciton resonance areneglected.

Exercise 4.3 (∗∗) A short pulse of light centred at the exciton resonance frequency isreflected from a semi-infinite semiconductor crystal. Calculate the time dependence ofthe intensity of reflected light neglecting the spatial dispersion of exciton-polaritons.

Exercise 4.4 (∗∗∗) A short pulse of light centred at the exciton resonance frequency istransmitted through a semiconductor crystal slab of thickness d. Calculate the time de-pendence of the intensity of transmitted light neglecting the spatial dispersion of excitonpolaritons.

4.4.2 Exciton–photon coupling in quantum wells

In quantum confined structures the second Hopfield equation (4.47) cannot be directlyused as the exciton is no more a free moving particle and its wavevector in the con-finement direction is not defined. On the other hand, the theory of spatial dispersion in


Fig. 4.15: Transmission intensity through a slab of Cu2O of thickness d = 0.91mm as calculated by Panzariniand Andreani (1997).

optical media can still be applied to describe the dielectric response of quantum struc-tures containing excitons, where the exciton wavefunction plays the role of a correlationfunction. Indeed, once an exciton is created, the dielectric polarization changes in allpoints where its wavefunction spreads.

The first theoretical description of exciton-polaritons in 2D structures was givenby Agranovich and Dubovskii (1966). In this Section we will follow the so-called non-local dielectric response theory, developed in the beginning of the 1990s by Andreaniet al. (1991) and Ivchenko (1992) to describe the optical response of excitons in QWs.We consider the simplest case of reflection or transmission of light through a QW inthe vicinity of the exciton resonance at normal incidence. We neglect the differencebetween background dielectric constants of the well and barrier materials (which isusually small) and only take into account the exciton-induced resonant reflection.

The non-local dielectric response theory is based on the assumption that the exciton-induced dielectric polarization can be written in the form:

Pexc(z) =

∫ ∞

−∞χ(z, z′)E(z′)dz′ , (4.64)

whereχ(z, z′) = χ(ω)Φ(z)Φ(z′) , (4.65)

with

χ(ω) =Q

ω0 − ω − iγ, Q = εBωLTπa3

B . (4.66)

Here, Φ(z) = Ψexc(R = 0, ρ = 0, ze = zh = z) is the exciton wavefunction (4.28)taken with equal electron and hole coordinates, ω is the frequency of the incident light,γ is the homogeneous broadening of the exciton resonance, same as in the Hopfieldequations, ωLT and aB are the longitudinal-transverse splitting and Bohr radius of exci-ton in the bulk material. Once the polarization (4.64) is introduced, eqn (4.46) becomes


an integro-differential equation and can be solved exactly using the Green’s functionmethod. In this method the solution of eqn (4.46) is represented in the form

E(z) = E0 exp(ikz) + k20

∫Pexc(z

′)G(z − z′)dz′ , (4.67)

where E0 is the amplitude of the incident light, k0 = ω/c and the Green’s function Gsatisfies the equation:(

∂2

∂z2+ k2

)G(z) = −δ(z), k =

√εBk0 . (4.68)

Bearing in mind that∫∞−∞ f(z′)δ(z − z′)dz = f(z′), one easily checks that G is

given by:

G(z) =i exp(ik|z|)

2k. (4.69)

Equation (4.67) can be solved with respect to E(z). In order to do it, multiply theleft and right parts of eqn (4.67) by Φ(z) and integrate over z. This yields:∫

EΦ(z)dz = E0

∫Φ(z) exp(ikz)dz+

+ k20χ

∫∫Φ(z)Φ(z′)G(z − z′)dzdz′

∫EΦ(z′′)dz′′ (4.70)

which means that∫EΦ(z)dz =

E0

∫Φ(z) exp(ikz)dz

1− k20χ


. (4.71)

We now return to eqn (4.67) and substitute eqn (4.71) into its right-hand side:

E =E0 exp(ikz) + k20χ

∫Φ(z′)G(z − z′)dz′

∫E(z′′)Φ(z′′)dz′′

=E0

[eikz +

k20χ

∫Φ(z′)G(z − z′)dz′

∫eikzΦ(z′′)dz′′

1− k20χ


]. (4.72)

Using eqn (4.69) we finally obtain

E(z) = E0eikz +

ik0

2√

εBQE0

∫Φ(z′′)eikz′′

dz′′∫

Φ(z′)eik|z−z′|dz′

ω0 − ω − iγ −Q ik0

2√

εB

∫∫eik|z′−z′′|Φ(z′)Φ(z′′)dz′dz′′

(4.73)

The amplitude reflection (r) and transmission (t) coefficients of the QW can then beobtained as

r =E(z)− E0(z)eikz

E0(z)e−ikz

∣∣∣∣z→∞

and t =E(z)

E0eikz

∣∣∣∣z→∞

. (4.74)

If we consider a ground exciton state in a QW, Φ(z) is an even function and the integralson the right-hand side of eqn (4.73) can be easily simplified.


Fig. 4.16: Schema of multiple reflections of light within a crystal slab.

In the case z →∞,∫Φ(z′)G(z − z′)dz′ =

i

2k

∫Φ(z′)eik(z−z′)dz′ =

ieikz

2k

∫cos(kz′)Φ(z′)dz′

(4.75)and in the case z → −∞,∫

Φ(z′)G(z − z′)dz′ =i

2k

∫Φ(z′)e−ik(z−z′)dz′ =

ie−ikz

2k

∫cos(kz′)Φ(z′)dz′

(4.76)

∫∫G(z − z′)Φ(z)Φ(z′)dzdz′ =

i

2k

[∫Φ(z) cos(kz)dz

]2

− 1

2k

∫∫Φ(z)Φ(z′) sin(k|z − z′|)dzdz′ . (4.77)

This allows us to obtain the reflection and transmission coefficients of the QW in asimple and elegant form:

r(ω) =iΓ0

ω0 − ω − i(Γ0 + γ), (4.78a)

t(ω) = 1 + r(ω) , (4.78b)

where

Γ0 =Qk0

2√

εB

[∫Φ(z) cos(kz)dz

]2

(4.79)

is an important characteristic further referred to as the exciton radiative broadening, and

ω0 = ω0 +Qk0

2√

εB

∫∫Φ(z)Φ(z′) sin(k|z − z′|)dzdz′ (4.80)

is the renormalization of the exciton resonance frequency due to the polariton effect.


Fig. 4.17: Typical frequency- (a) and time-resolved (b) reflection spectra of a structure containing quantumwells (schema). In frequency-resolved reflectivity, the exciton resonance usually induces a characteristic mod-ulation on the top of strong background reflectivity dependent on the refractive index of the barrier materials.

The radiative broadening Γ0 is connected to the exciton radiative lifetime τ by therelation:

τ =1

2Γ0(4.81)

which follows from the time-dependence of the intensity of light reflected by a QWexcited by an infinitely short pulse of light:

R(t) =

∣∣∣∣ 1

2π

∫ ∞

−∞r(ω)e−iωtdω

∣∣∣∣2

= Γ20e

−2Γ0t . (4.82)

A finite exciton radiative lifetime is a peculiarity of confined semiconductor sys-tems. In an infinite bulk crystal, an exciton-polariton can freely propagate in any di-rection and its lifetime is limited only by non-radiative processes such as scatteringwith acoustic phonons. On the contrary, in a QW the exciton-polariton can disappear bygiving its energy to a photon which escapes the QW plane. The polariton effect (some-times referred to as the retardation effect) consists, in this case, in the possibility for theemitted photon to be reabsorbed once again by the same exciton. The chain of virtualemission-absorption processes leads to a finite value of τ and is also responsible forrenormalization of the exciton frequency (4.80). This renormalization does not exceeda few µeV in realistic QWs, although it becomes more important in quantum dots. Itdoes not play an essential role in microcavities, and we shall neglect it hereafter. Theradiative lifetime τ is about 10ps in typical GaAs-based QWs. Although it is extremelyhard to observe free-excitons in the photoluminescence as often PL is governed mainlyby excitons localized at imperfections of a QW, a lifetime of 10±4ps for a free-excitonhas been measured experimentally by Deveaud et al. (1991) in a record-quality (for thattime) 100 A-thick GaAs/AlGaAs QW (see Fig. 4.18).


Fig. 4.18: Time-resolved photoluminescence spectra of a GaAs/AlGaAs QW measured by Deveaud et al.(1991) at different detunings between the excitation energy and the exciton resonance. These data allow anexciton radiative lifetime of about 10ps to be extracted.

4.4.3 Exciton–photon coupling in quantum wires and dots

Quantum wires and dots scatter light. In the vicinity of the exciton resonance, this scat-tering has a resonant character and polariton effects take place. If a wave of light isincident on an array of quantum wires or dots, the interference of waves scattered bydifferent individual wires or dots results in enhanced signals in reflection and trans-mission directions. A regular array of identical wires or dots would diffract light incertain directions similarly to a crystal lattice diffracting X-rays. In realistic semicon-ductor structures, the inevitable potential disorder leads to random fluctuations of theexciton resonance frequency from wire to wire and from dot to dot. This leads to theinhomogeneous broadening of exciton resonances in reflection or transmission spectra.The stronger the inhomogeneous broadening, the larger the ratio of scattered to reflectedlight intensity.

Polariton effects in systems of quantum wires and dots manifest themselves in fi-nite exciton radiative lifetime, appearance of collective exciton-polariton states due tooptical coupling of different wires or dots and appearance of a specific polarization-dependent fine-structure of exciton resonances. All these phenomena can be conve-niently described within the nonlocal dielectric response theory in a similar way to thatdone for quantum wells.

The dielectric polarization induced by an exciton in a plane array of N quantumwires or dots reads

Pexc =N∑

n=1

TnΦn(r−Rn)

∫E(r′)Φn(r′ −Rn)dr′ (4.83)

with

Tn =εBωLTπa3

B

ωn − ω − iγ, (4.84)


where k0 is the wavevector of the incident light in a vacuum, ωLT is the longitudinal-transverse splitting in bulk, εB is the background dielectric constant, aB is the excitonBohr radius in bulk, γ is the homogeneous broadening and Φn(r) is the exciton wave-function in the nth wire or dot taken with equal electron and hole coordinate r. Solv-ing the wave equation using the Green’s function method, one represents the electricfield as:

E(ω, r) = E0 exp(ik · r)

+ k20

N∑n=1

Tn

∫Φn(r′ −Rn)G0(r− r′)dr′

∫E(r′′)Φn(r′′ −Rn)dr′′ , (4.85)

where k = k0√

εB is the wavevector of the incident light in the medium. In the case ofinfinite quantum wires oriented along the y-axis

G0(x, y, z) =i

4H

(1)0 (kxx + kzz) , (4.86)

where H(1)0 is the Hankel function. G0 is a Green’s function that satisfies the equation

−(

∂2

∂x2+

∂2

∂y2+ k2

)G0 = δ(x)δ(z) . (4.87)

In the case of QDs, eqn (4.85) is the Green’s function for a zero-dimensional system.For realistic systems, the “quantum dot” model is more relevant, as the excitons inquantum wires are inevitably localized in the y-direction due to potential fluctuations.This makes the wires similar to elongated dots. In the following, we therefore consideran array of QDs.

The integral equation (4.85) can be treated analytically, see for instance the dis-cussion by Parascandolo and Savona (2005), but the calculation becomes heavy if thenumber of dots is large. A compact analytical expression for the electric field can beobtained if we assume that the wavefunctions of excitons in all QDs are identical andthat E(ω, r + Rn) ≈ eir ·RnE(ω, r). This would be true for a regular grating of iden-tical QDs characterized by the exciton wavefunction Φ(r). In all other cases, this is amore or less accurate approximation depending on the degree of disorder in the system.Multiplying eqn (4.85) by Φ(r) and integrating over r we obtain in this case:

E(ω, r) = E0 exp(ik · r)

+ k20

N∑n=1

Tneik ·Rn

∫Φ(r′ −Rn)G0(r− r′)dr′

∫E0eik · rΦ(r)dr

1−∑Nn=1 Θn

, (4.88)

where Θn = k20Tneik ·Rn

∫∫G0(r− r′)Φ(r′ −Rn)Φ(r)dr′dr.

The Fourier transform of the electric field (4.88) yields its directional dependence,which can be represented in the form:


Ed(ω,ks) = E0δk,ks+E0

N∑m=1

iΓQD0

ωm − ω − iγ

1

1−∑Nn=1 Θn

exp(iks ·Rm) , (4.89)

where

ΓQD0 =

1

6ωLTk3

0a3B

[∫cos(k · r)Φ(r)dr

]2

, (4.90)

and δk,ks= 1 if k = ks and is zero otherwise.

Fig. 4.19: Exciton radiative decay rate as a function of the radius of a spherical quantum dot R (scheme).

The quantity ΓQD0 is the exciton radiative decay rate in a single QD. This is an im-

portant characteristic of light–matter coupling in QDs. It changes non-monotonicallywith the dot size, as Fig. 4.19 shows. If the size of the dot R is much smaller than thewavelength of light at the exciton resonance frequency λ0, one can neglect the cosinein the right part of eqn (4.90). In this case the radiative damping rate is directly pro-portional to the volume occupied by the exciton wavefunction. This dependence hasbeen obtained theoretically for the first time by Rashba and Gurgenishvili (1962) forimpurity-bound excitons in semiconductors. The minimum of ΓQD

0 corresponds to thestrongest exciton confinement at R ≈ aB. At very small R the exciton is less confineddue to penetration of the electron and hole wavefunctions into the barriers. At larger Rthe exciton is confined as a whole particle inside the dot, and the volume occupied byits wavefunction increases proportionally to the size of the dot. When the size of thedot approaches the wavelength of light, the cosine in the right part of eqn (4.90) can nolonger be neglected, as was shown by Gil and Kavokin (2002). ΓQD

0 has its maximumabout R ≈ λ0/(π

√εB) and than decreases.

Equation (4.89) describes all kinds of coherent optical experiments (reflection, trans-mission and Rayleigh scattering) within the same semiclassical formalism and takesproperly into account the polariton effect. In the specular reflection direction, one canneglect the small portion of scattered light. Substituting summation by integration ineqn (4.89) we obtain the reflection and transmission coefficients of an array of QDs,rQD and tQD, as:


rQD =β

1− βand tQD =

1

1− β, (4.91)

where

β = iΓr

∫f(ν)

ν − ω − iγdν , (4.92a)

Γr =k

2d2ωLTπa3

B

∫[Φ(r) cos(k · r)dr]2 , (4.92b)

where d is the average distance between the dots, f(ν) is the distribution function ofexciton resonance frequencies in the dots (

∫∞−∞ f(ν)dν = 1). The ratio of reflected and

average scattered intensities can be estimated from eqns. (4.89)–(4.90), assuming thatthe light waves scattered by QDs lying within a Γr vicinity interfere positively in thereflection direction:

Ir

Is≈ N

(Γr

∆

)2

, (4.93)

where N is given by the number of QDs within the spot of light that illuminates thesample. In real systems, Γr ΓQD

0 , which means that optically coupled QDs emitlight much faster than single QDs. The radiative lifetime of an exciton in a single QDvaries in large limits as a function of the QD size, typically in the range 10−10–10−8s.

Exercise 4.5 (∗∗) Find the reflection coefficient of a system of two identical quantumwells parallel to each other and separated by a distance d.

4.4.4 Dispersion of polaritons in planar microcavities

Fig. 4.20: An electron microscopy image of a GaAs microcavity with GaAlAs/AlAs Bragg mirrors and thecalculated profile of electric field of the cavity mode.

4.4.4.1 Bulk microcavities: In Chapter 2, we considered the eigenmodes of planarmicrocavities in the absence of exciton–photon coupling. Imagine that the cavity is


now made from a material having an excitonic transition at the frequency close to theeigenfrequency of the photonic mode of the cavity. Neglecting the spatial dispersion ofexciton-polaritons in the cavity layer, one can solve the eigenvalue problem using thetransfer matrix technique as we did in Chapter 2. Imposing the boundary condition ofno light incident from left and right sides on the cavity, one can readily obtain the matrixequation for the eigenfrequencies:(

cos(kLc)i√ε(ω)

sin(kLc)

i√

ε(ω) sin(kLc) cos(kLc)

)(1 + rB√

εB(rB − 1)

)= A

(1 + rB√

εB(rB − 1)

)(4.94)

where Lc is the cavity width and

ε(ω) ≈ εB +εBωLT

ω0 − ω − iγ, (4.95)

as follows from eqn (4.49) if the wavevector-dependent term in the denominator is ne-glected. Eliminating A we obtain

rBeikLc = ±1 . (4.96)

Near the frequency ω, which is the centre of the stop-band of the mirrors, rB can beapproximated by

rB =√

R exp

[incLDBR

c(ω − ω)

], (4.97)

where LDBR is the effective length of the mirror (see Section 2.5). Assuming ω0 = ω,ωLT ω0, Γ ω0 and 1 − R 1 one can reduce eqn (4.96) to the two-coupledoscillator problem

(ω0 − ω − iγ)(ωc − ω − iγc) = V 2 , (4.98)

where ωc − iγc is the complex eigenfrequency of the cavity mode in the absence ofexciton–photon coupling, and

V =

√2ω0ωLTd

LDBR + d. (4.99)

Equation (4.98) has two complex solutions:

ω1,2 =ω0 + ωc

2− i

2(γ + γc) (4.100a)

±√(

ω0 − ωc

2

)2

+ V 2 −(

γ − γc

2

)2

+i

2(ω0 − ωc)(γc − γ) . (4.100b)

The parameter V has the sense of the coupling strength between the cavity photonmode and the exciton.

If ω0 = ωc, the splitting between the two values is given by√

4V 2 − (γ − γc)2.


If

V >

∣∣∣∣γ − γc

2

∣∣∣∣ , (4.101)

the anticrossing takes place between the exciton and photon modes, which is charac-teristic of the strong-coupling regime. In this regime, two distinct exciton-polaritonbranches manifest themselves as two optical resonances in the reflection or transmissionspectra. The splitting between these two resonances is referred to as the vacuum-fieldRabi splitting. It reaches 4–15 meV in current GaAs-based microcavities, up to 30 meVin CdTe-based microcavities, and is found to be as large as 50 meV in GaN cavities.

If

V <

∣∣∣∣γ − γc

2

∣∣∣∣ , (4.102)

the weak-coupling regime holds, which is characterized by crossing of the exciton andphoton modes and an increase of the exciton decay rate at the resonance point. Thisregime is typically used in vertical-cavity surface-emitting lasers (VCSELs).

Fig. 4.21: Real parts of the eigenfrequencies of the exciton-polariton modes in the weak-coupling regime(left) and strong coupling regime (right).

The spatial dispersion of exciton-polaritons leads to the appearance of additionalresonances in reflection and transmission and additional eigenmodes of the microcavity.In order to take into account the spatial dispersion one should use the expression (4.49)for the dielectric function, which links it not only with the frequency but also with thewavevector. The wavevector is no longer a unitary function of frequency, so that thesimple transfer matrix method described in Chapter 2 and used to obtain eqn (4.98)fails. In order to calculate the optical spectra of semiconductor films containing excitonresonances we apply the generalized transfer matrix method. As was discussed in Sec-tion 4.4, the usual Maxwell boundary conditions are not sufficient in this case. We usePekar’s additional boundary conditions:

P |z=±Lc/2 = 0 , (4.103)


where P is the exciton-induced dielectric polarization, and corresponds to the bound-aries of the cavity. Equation (4.103) yields a series of exciton-polariton eigenmodes(see Fig. 4.12). Their energies depend on Lc and the exciton mass. The modes can becharacterized by a quantum number N equal to the number of nodes in F (R) for thecorresponding state plus one. These are polariton modes uncoupled to the cavity photonmode. To introduce the coupling one should substitute the amplitudes of all existingmodes of electromagnetic field into the Maxwell boundary conditions. For simplicity,we consider only the normal incidence geometry (inplane wavevector equal to 0) andonly the transverse light modes. In order to write down the transfer matrix taking into ac-count spatial dispersion of exciton-polaritons, one should describe propagation of fourwaves with amplitudes E±

1 , E±2 at the beginning and the end of the layer, respectively,

where “+” denotes a wave propagating in the positive direction and “−” denotes a wavepropagating in the negative direction.

The matrix Q connecting E±1 and E±

2 so that(E+

2

E−2

)= Q

(E+

1

E−1

), (4.104)

can be written as

Q = −ε1

ε2

(λ+

2 λ2

λ2 λ+2

)−1 (λ+

1 λ1

λ1 λ+1

), (4.105)

with εj = c2k2j /ω2 − εB and

λj = exp

(ikj

d

2

), λ+

j = exp

(−ikj

d

2

), (4.106)

kj , (j = 1, 2) are wavevectors of the transverse polariton modes given by eqn (4.52).The transfer matrix across the cavity layer, i.e., the matrix that connects the inplanecomponents of the electric and magnetic fields at the boundaries, can be written as:

T =

[(λ+

1 λ1

n1λ+1 −n1λ1

)+

(λ+

2 λ2

n2λ+2 −n2λ2

)Q

]

×[(

λ1 λ+1

n1λ1 −n1λ+1

)+

(λ2 λ+

2

n2λ2 −n2λ+2

)Q

]−1

, (4.107)

where nj = ckj/ω. Once its elements are known, the eigenmodes of the cavity can befound using the standard procedure described in Section 2.5 from the equation:

T

(1 + rB√

εB(rB − 1)

)= A

(1 + rB√

εB(1− rB)

). (4.108)

Generalization of this equation into the oblique incidence case requires substantialmodifications of the transfer matrix (4.107). In the case of TM-polarized light, the longi-tudinal polariton modes can be excited, and their amplitudes should also be substitutedinto the Maxwell boundary conditions. We address the interested reader to the paper by


Vladimirova et al. (1996) and show here the dispersion curves of exciton-polaritons ina model bulk GaAs microcavity calculated in that study (Fig. 4.22). Remarkably, onlythe modes with even N are coupled to light: this is because the exciton wavefunctionshould be of the same parity as the cavity mode to be coupled.

Fig. 4.22: Calculated exciton-polariton eigenenergies as a function of the incidence angle of light in TE-(triangles) and TM- (solid line) polarizations, from Vladimirova et al. (1996).

One can see that the light mode of the cavity goes through different polariton res-onances in the layer of GaAs forming a series of anticrossings. The splitting (Rabisplitting) at the lowest of them (N = 2) is given with a good accuracy by 2V , where Vis defined by eqn (4.99).

The Rabi splitting for higher exciton states (ΩN ) is related to the splitting for thelowest state approximately as the ratio of squared overlap integrals of the excitonicpolarization and electric field of the light mode:

ΩN

Ω2=

(PN

P2

)2

, N = 2, 4, 6 . . . , (4.109)

where

PN ≈∫ π/2

−π/2

sin(Nx) sin xdx =2

N2 − 1. (4.110)

This result easily follows from the quantum model as the matrix element of light-excitoncoupling is proportional to PN . In real bulk microcavities, the absorption of free e–hpairs above Eg produces a large imaginary component to εB and washes out the higheranticrossings.

4.4.4.2 Microcavities containing quantum wells: Microcavities containing quantumwells (QWs) are most commonly used in practice. They allow for observation of thestrong coupling of a single exciton resonance and the cavity mode. The use of multiple


QWs also helps to enhance the Rabi splitting and make the cavity polaritons more sta-ble. In order to obtain the dispersion equation of the polaritons in structures containingQWs it is convenient to use the transfer matrices written in the basis of amplitudes ofelectromagnetic waves propagating in positive and negative directions along the axis ofthe cavity.

Let us represent the electric field at the point z of the structure as:

E(z) = E+(z) + E−(z) , (4.111)

where E±(z) is the complex amplitude of a light wave propagating in the positive/nega-tive direction. One can define the transfer matrix Ma by its property:

Ma

(E+(0)E−(0)

)=

(E+(a)E−(a)

). (4.112)

Consider a few particular cases.If the refractive index n is constant across the layer a, the transfer matrix has the

simple form:

Ma =

(eika 00 e−ika

). (4.113)

The transfer matrix across the interface between a medium with refractive index n1 anda medium with refractive index n2 is

Ma =1

2n2

(n1 + n2 n2 − n1

n2 − n1 n1 + n2

). (4.114)

It can be obtained using the condition (4.112) applied to the light waves incident fromthe left side and right side of the interface, keeping in mind the well-known expressionsfor the reflection and transmission coefficients of interfaces:

r =n1 − n2

n1 + n2, t =

2n1

n1 + n2. (4.115)

A transfer matrix across a structure containing m layers has the form

M =

2m+1∏j=1

M2m+2−j , (4.116)

where j = 1, . . . , 2m + 1, labels all the layers and interfaces of the structure from itsleft to right side. The amplitude reflection and transmission coefficients (rs and ts) ofa structure containing m layers and sandwiched between two semi-infinite media withrefractive indices nleft, nright before and after the structure, respectively, can be foundfrom the relation

M

(1rs

)=

(ts0

), (4.117)

as

rs =m21

m11and ts =

1

m11. (4.118)

Note that here the refractive indices nleft, nright do not appear explicitly as they arecontained in the transfer matrices across both surfaces of the structure.


If the reflection and transmission coefficients for light incident from the right-handside and left-hand side of the layer are the same and nleft = nright = n, Maxwell’sboundary conditions for light incident from the left and right sides of the structure yield,in addition to eqn (4.117), the following equation:

M

(0ts

)=

(rs

1

). (4.119)

In this case, the transfer matrix across a symmetric object (a QW embedded in thecavity, for example) can be written as:

M =1

ts

(t2s − r2

s rs

−rs 1

). (4.120)

Consider a symmetric microcavity with a single QW embedded in the centre. In thebasis of amplitudes of light waves propagating in positive and negative directions alongthe z-axis, the transfer matrix across the QW has the form

TQW =1

t

(t2 − r2 r−r 1

), (4.121)

where r and t are the angle- and polarization-dependent amplitude reflection and trans-mission coefficients of the QW derived previously. The transfer matrix across the cavityfrom one Bragg mirror to the other one is the product:

Tc =

(eikLc/2 0

0 e−ikLc/2

)1

t

(t2 − r2 r−r 1

)(eikLc/2 0

0 e−ikLc/2

), (4.122)

where Lc is the cavity width. The matrix elements read:

T c11 =

t2 − r2

teikLc , T c

12 =r

t, (4.123)

T c21 = −r

t, T c

22 =1

te−ikLc . (4.124)

To find the eigenfrequencies of the exciton-polariton modes of the microcavity, oneshould search for non-trivial solutions of Maxwell’s equations under the requirement ofno light incident on the cavity from outside. This yields

Tc

(rB

1

)= A

(1rB

), (4.125)

where rB is the angle-dependent reflection coefficient of the Bragg mirrors for lightincident from inside the cavity, introduced in Section 2.6.

Eliminating the coefficient A from eqn (4.125), one obtains the following equationfor polariton eigenmodes:

T c21rB + T c

22rB

T c12rB + T c

11rB= rB . (4.126)

This is already a dispersion equation because the coefficients of the transfer matrixand rB are dependent on the inplane wavevector of light. Substituting the coefficients


(4.123) into eqn (4.126), one can represent the dispersion equation in the followingform: (

rB(2r + 1)eikLc − 1) (

rBeikLc + 1)

= 0 . (4.127)

Here, we have used the relation t − r = 1, cf. eqn (4.78b). Solutions of eqn (4.127),coming from zeros of the second bracket on the left-hand side, coincide with pure oddoptical modes of the cavity. These modes have a node at the centre of the cavity wherethe QW is situated. Therefore, they are not coupled with the ground exciton state havingan even wavefunction. The first bracket on the left-hand side of eqn (4.127) contains thereflection coefficient of the QW, which is dependent on excitonic parameters. The zerosof this bracket describe the eigenstates of exciton-polaritons resulting from coupling ofeven optical modes with the exciton ground state. From now on we shall consider onlythese states, neglecting excited exciton states that may be coupled to odd cavity modes.

For the even modes and normal incidence, if we take

rB = r exp(iα(ω − ωc)) ≈ r(1 + iLDBRnc

n(ω − ωc)) , (4.128)

where r is close to 1 (see Section 2.5) and assume eikLc ≈ 1 + i(ω − ωc)/cncLc, weobtain, using the explicit form for the reflection coefficient r:

r(1+ i(LDBR +Lc)nc

c(ω−ωc))(ω0−ω− i(γ−Γ0)) = ω0−ω− i(γ +Γ0) , (4.129)

which finally yields, after trivial transformations:

(ω0 − ω − iγ)(ωc − ω − iγc) = V 2 , (4.130)

where

γc =1− r

r nc

c (LDBR + Lc), (4.131)

V 2 =1 + r

r

Γ0c

nc(LDBR + Lc). (4.132)

Here, quadratic terms in (ωc − ω) have been omitted. In all further calculationswe assume for simplicity ω0 = ω0. Equation (4.130) is an equation for eigenstates ofa system of two coupled harmonic oscillators, namely, the exciton resonance and thecavity mode. In this form eqn (4.130) was published for the first time by Savona et al.(1995) while its general form (4.126) was later obtained by Kavokin and Kaliteevski(1995). Its solutions have the form (4.100). The weak-to-strong coupling threshold isdefined in the same way as for the bulk cavities (cf. eqns. (4.101) and (4.102)). Notethat all the above theory neglects the effect of disorder on the exciton resonance. Takinginto account the inevitable inhomogeneous broadening of the exciton resonance andRayleigh scattering of exciton-polaritons, one should also modify the criterion for theweak-to-strong coupling threshold. The detuning of bare photon and exciton modesin a microcavity is an important parameter that strongly affects the shape of polaritondispersion curves in the strong coupling regime as Fig. 4.23 shows.


2 1 0 1 2

3.50

3.55

3.60

3.65

3.70

Ene

rgye

V

2 1 0 1 2

Wavevector 105 cm1

2 1 0 1 2

Fig. 4.23: Energies of exciton-polaritons at (a) negative, (b) zero and (c) positive detuning between thebare photon and exciton modes. Solid lines show the inplane dispersion of exciton-polariton modes. Dashedlines show the dispersion of the uncoupled exciton and photon modes. The calculations have been made withparameters typical of a GaN microcavity, from Kavokin and Gil (1998), with a photon mass of 0.5×10−4m0

and a Rabi splitting of 50 meV.

4.4.4.3 Oblique incidence case: Equation (4.130) can be easily generalized for theoblique incidence case by introduction of the dependence of the cavity and excitoneigenmode frequencies ωc and ω0 on the inplane wavevector kxy:

ωc =k2

xy

2mph, ω0 =

k2xy

2Mexc, (4.133)

where Mexc is the sum of the electron and hole effective masses in the QW planeand mph is the photon effective mass. In an ideal λ-microcavity, the normal-to-planecomponent of the wavevector of the eigenmode is given by kz = 2π/Lc. The energy ofthe mode is

ωc =c

nc

√k2

xy + k2z ≈

c

nckz

(1 +

k2xy

2k2z

)=

2πc

ncLc+

k2xy

2mph, (4.134)

and thus mph = hnc/(cLc). This mass is extremely light in comparison to the exci-ton mass as it usually amounts to 10−5–10−4m0, where m0 is the free-exciton mass.Note also that the inplane wavevector kxy is related to the angle of incidence of lightilluminating the structure, ϕ, by the relation:


kxy =ω

csin ϕ . (4.135)

By measuring the angle dependence of the resonances in the reflection or trans-mission spectra of microcavities, one can restore the true dispersion curves of exciton-polaritons.

Fig. 4.24: Theoretical (lines) and experimental (squares) values of longitudinal-transverse splitting of the up-per (UP) and lower (LP) polariton branches in a GaAs-based microcavity with embedded QWs, by Panzariniet al. (1999b).

The coupling constant V is renormalized in the case of oblique incidence and it be-comes polarization dependent. In TE-polarization, Γs

0 = Γ0/ cos ϕc, where ϕc is thepropagation angle within the cavity. In TM-polarization, Γp

0 = Γ0 cos ϕc. The effec-tive length of the Bragg mirrors, LDBR, is also angle and polarization dependent: it de-creases with angle in TE-polarization but increases in TM-polarization. Also, the coeffi-cient r slightly depends on the angle (see eqns. (2.150)–(2.152)). All these factors makethe coupling constant increase with angle in TE-polarization, while in TM-polarizationthe opposite tendency occurs. This is why in TM-polarization at some critical anglethe strong coupling regime can be lost. Note also that in TM- and TE-polarizations theeigenfrequencies of pure cavity modes ωTE,TM

c are split (see eqn (2.153)). Fig. 4.24by Panzarini et al. (1999a) shows the longitudinal-transverse splitting of cavity polari-ton modes calculated and experimentally measured in a GaAs microcavity containingInGaAs QWs.

The longitudinal-transverse splitting becomes essential at large inplane wavevec-tors, and it has an impact on the spin-relaxation of exciton-polaritons, as we shall dis-cuss in detail in Chapter 9. The splitting is dependent on two main factors: the TE–TMsplitting of the bare cavity mode and the detuning between cavity and exciton modes.


Therefore, it can be efficiently controlled and tuned within a wide range by changingthe cavity width.

4.4.5 Motional narrowing of cavity polaritons

Motional narrowing is the narrowing of a distribution function of a quantum particlepropagating in a disordered medium due to averaging of the disorder potential overthe size of the wavefunction of a particle. In other words, a quantum particle, which isnever localized at a given point in space but always occupies some nonzero volume, hasa potential energy that is the average of the potential within this volume. This is why, ina random fluctuation potential, the energy distribution function of a particle is alwaysnarrower than the potential distribution function (see Fig. 4.25).

Fig. 4.25: Due to their finite De Broglie wavelength the quantum particles see the averaged potential. Theaveraging is done on the scale of the particle’s wavefunction. This is why the distribution function of quan-tum particles localized within some fluctuation potential is usually narrower than the potential distributionfunction.

Motional narrowing of exciton-polaritons in microcavities was the subject of sci-entific polemics at the end of the 1990s. The debate was initiated by an experimentalfinding of the Sheffield University group reported by Whittaker et al. (1996). Measuringthe sum of full-widths at half-minimum (FWHM) for two exciton-polariton resonancesin reflection spectra of microcavities as a function of incidence angle, the experimental-ists found a minimum of this function at the anticrossing of exciton and cavity modes(see Fig. 4.26). This result contradicts what one could expect from a simple model oftwo coupled oscillators. Actually, if the dispersion of microcavity polaritons is given byeqn (4.130) then the sum of the imaginary parts of the two solutions of this equationis always −(γ + γc), independently of the detuning δ = ω0 − ωc. This follows from amore general property of any system of coupled harmonic oscillators to keep constantthe sum of eigenfrequencies, independently of the coupling strength.

Clearly, eqn (4.130) is no longer valid if, instead of a single free-exciton transition,one has an infinite number of resonances distributed in energy. This is what happens inrealistic QWs, where inplane potential fluctuations caused by the QW width and alloyfluctuations induce the so-called inhomogeneous broadening of an exciton resonance.An idea has been proposed that exciton-polaritons, having a smaller effective mass thanbare excitons, are less sensitive to the disorder potential, which is a manifestation of


Fig. 4.26: Left: linewidth of lower (solid) and upper (open) polaritons in an InGaAs microcavity as a functionof exciton fraction (or detuning), demonstrating linewidth narrowing on resonance, reported by Whittakeret al. (1996). Right: full-width at half-minimum of two polariton resonances in reflectivity spectra of a GaAs-based microcavity with an embedded QW, from Whittaker et al. (1996), as a function of the detuning δ =ω0 −ωc (circles correspond to the upper branch, squares correspond to the lower branch) in comparison witha theoretical calculation accounting for (solid lines) or neglecting (dashed lines) asymmetry in the excitonfrequency distribution from Kavokin (1997).

their motional narrowing. Thus, the inhomogeneous broadening of exciton-polaritonmodes is less than that of a pure exciton state, which is a consequence of the polaritonmotional narrowing effect. At the anticrossing point this effect is especially strong, sinceat this point both upper and lower-polaritons are half-excitons half-photons.

Further analyses by Whittaker et al. (1996) and Ell et al. (1998) have shown, how-ever, that experimentally observed narrowing of polariton lines at the anticrossing pointis indeed caused by exciton inhomogeneous broadening, but not by the motional nar-rowing effect. On the other hand, the motional narrowing may manifest itself in resonantRayleigh scattering or even photoluminescence.

In order to understand the inhomogeneous broadening effect on the widths of polari-ton resonances in microcavities, let us first consider its influence on the optical spectraof QWs.

We suppose that the in plane wavevector of any exciton interacting with the incidentlight is the same as the inplane wavevector of light q, while the frequency of excitonresonance ω0 is distributed with some function. This is a particular case of a microscopicmodel considering all exciton states as quantum-dot like and assuming no wavevectorconservation. It is well adapted for the description of reflection or transmission, i.e.,experiments that conserve the inplane component of the wavevector of light. Note thatmost scattered light does not contribute to reflection and transmission spectra, while asmall part of the scattered light can re-obtain the initial value of q after a second, third,etc. scattering act. The main reason why motional narrowing has almost no influenceon reflection spectra is that it is an effect that originates from the finite inplane size ofthe exciton-polariton wavefunction, or, in other words, implies scattering of exciton-


polaritons in the plane of the structure. The impact of scattering is, however, negligiblysmall in reflection and transmission experimental geometries.

As in Section 4.4.2, we shall operate with the dielectric susceptibility in eqn (4.65),while also taking into account that the exciton resonance frequency is distributed withsome function f(ω − ω0). We assume that in this case the dielectric susceptibility of aQW can be written in the form

χ(ω) =

∫χ(ω − ν)f(ν − ω0) dν . (4.136)

If f is a Gaussian function, this integral can be computed analytically:

χ(ω) =1√π∆

∫χ(ω − ν) exp

[−(

ν − ω0

∆

)2]

dν =i√

πΘ

∆exp(−z2)erfc(−iz) ,

(4.137)with

χ(ω − ω0) =ε∞ωLTπa3

Bω20/c2

ω0 − ω − iγ, Θ = ε∞ωLTπω2

0a3B/c2 (4.138a)

and z =ω − ω0 + iγ

∆(4.138b)

where erfc is the complementary error function, and ∆ is a width parameter of theGaussian distribution that describes exciton inhomogeneous broadening. We assume ∆,γ > 0 and consider a normal incidence case for simplicity. Substituting the susceptibil-ity (4.136) into eqn (4.64) and carrying out the same transformations as in Section 4.4.2,we obtain finally the amplitude reflection and transmission coefficients of a QW in theform:

r =iαχ

1− iαχ, t = 1 + r , (4.139)

where α = Γ0/Θ and Γ0 is the radiative damping rate of the exciton in the case of noinhomogeneous broadening. This yields, using eqn (4.137):

r = −√

πΓ0 exp(−z2)erfc(−iz)

∆ +√

π(Γ0 + i(ω0 − ω0)) exp(−z2)erfc(−iz), t = 1 + r . (4.140)

We shall neglect renormalization of the exciton resonance frequency due to the polaritoneffect, since it is much less than Γ0, and also assume ω0 − ω0 ≈ 0. In the limit of smallinhomogeneous broadening, the complementary error function becomes:

lim|z|→∞

exp(−z2)erfc(−iz) =i√πz

, (4.141)

which allows one to reduce eqn (4.140) to the “homogeneous” formula (4.78) of An-dreani et al. (1998).

In realistic narrow QWs, the exciton resonance frequency may have a more com-plex non-Gaussian distribution. Quite often it is asymmetric due to the so-called exci-tonic motional narrowing effect (to be distinguished from the motional narrowing of


exciton-polaritons). This effect comes from the blueshift of the lower-energy wing ofthe excitonic distribution due to the lateral quantum confinement of localized excitons.

Exciton inhomogeneous broadening of any shape necessarily modifies the equationfor exciton-polariton eigenmodes in a microcavity. Equation (4.130), obtained for themodel of two coupled oscillators, is no longer valid, and one should use the generalformula (4.128) instead. For the coupled exciton and photon modes it reduces to:

rB(2r + 1)eikLc = 1 . (4.142)

Using the same representation for the reflection coefficient of the Bragg mirror as ineqn (4.129) and eqn (4.139) for r with χ given by eqn (4.102), we obtain after transfor-mations analogous to those in section 4.4.2:

ωc − ω − iγc = V 2

∫ ∞

−∞

f(ν − ω0)

ν − ω − iγdν . (4.143)

In the limit of a small inhomogeneous broadening, it reduces to eqn (4.130). As foreqn (4.143), it has between zero and two complex solutions depending on the shape ofthe distribution function f(ν−ω0). In the general case, the sum of the imaginary parts ofits eigenfrequencies varies as a function of detuning, which is not the case for the puretwo coupled oscillator problem. Figure 4.27 shows the dependencies of the FWHM oftwo polariton resonances on detuning δ = ω0 − ωc, in calculated reflection spectra ofa GaAs-based microcavity with an embedded QW, in comparison with experimentaldata. One can see that the broadenings of the two polariton modes coincide near thezero-detuning point. At this point, both in calculation and experiment, the sum of thetwo FWHM has a pronounced minimum.

This minimum is a specific feature of an inhomogeneously broadened exciton statecoupled to the cavity mode. It can be interpreted in the following way. The coupling tolight has a different strength for excitons from the centre and from the tails of an inho-mogeneous distribution. As the density of states of excitons has a maximum at ω = ω0,these excitons have the highest radiative recombination rate and the strongest couplingto light. Now, the strong-coupling regime holds only for excitons situated in the vicin-ity of ω0, while the tails remain in the weak-coupling regime. The central part of theexcitonic distribution is of course less broadened than the entire distribution. Therefore,the two polariton modes that arise due to its coupling with the cavity photon are alsonarrower than one would expect for the case of all excitons equally coupled to light.At zero detuning, both polariton modes are far enough from the bare exciton energy, sothat the tails of the exciton resonance give no contribution to the FWHM of polaritonresonances. On the contrary, for strong negative or positive detunings, one of two polari-ton states almost coincides in energy with a bare exciton state, so that the lineshape ofthe corresponding spectral resonance is necessarily affected by the tails of the excitonicdistribution.

Note that this interpretation does not involve any motional narrowing. On the otherhand, specific effects of motional narrowing play a role in resonant Rayleigh scatter-ing of the cavity polaritons, and may also provide narrowing of the photoluminescence


Fig. 4.27: Schematic energy distributions functions of bare excitons, bare photons and exciton-polaritons inmicrocavities. The sum of the broadenings of two polariton modes at the anticrossing point is less than thesum of exciton and photon mode broadenings because of the preferential photon coupling with the excitonstates close to the centre of the inhomogeneous distribution of excitons.

lines. Only wavevector-conserving optical spectroscopies such as reflection and trans-mission are not sensitive to the motional narrowing.

Another remark concerns the asymmetric behaviour of the broadenings of the twopolariton peaks in Fig. 4.26. This comes from the asymmetry of the excitonic distri-bution. It is a manifestation of the exciton motional narrowing that we have describedabove. Actually, the lower-polariton branch is the result of mixing between the photonmode and the lower part of the excitonic distribution, which is sharper than the upperpart. This is why the lower branch has a narrower linewidth at the anticrossing condi-tion.

4.4.6 Microcavities with quantum wires or dots

Planar microcavities with embedded quantum wires or quantum dots can exhibit theweak or strong exciton–photon coupling regime similarly to bulk cavities or microcav-ities with embedded quantum wells. An essential difference comes from the enhancedresonant scattering of light by excitons in these structures. If the scattering is strong,exciton states in quantum dots or wires are coupled to the whole ensemble of the cavityphoton modes with different inplane wavevectors. In this case, the polariton eigenstatecan hardly be represented as a linear combination of plane waves, and calculation ofthe spectrum of exciton-polaritons becomes a non-trivial task. However, for the cavitieswith embedded arrays of quantum wires or dots, scattering of light can be neglected ifthe spacing between neighbouring wires (dots) is less than the wavelength of light andthe variation of the exciton resonance frequency from wire to wire (from dot to dot)is less than the Rabi splitting. In this case, one can still use eqn (4.127) for polaritoneigenfrequencies, replacing the QW reflection coefficient r by the reflection coefficientof the array of wires or dots in eqn (4.91).


Fig. 4.28: Schematic diagram by Kaliteevski et al. (2001) of a spherical microcavity with an embeddedquantum dot. The central core of radius rQD is surrounded by a spherical Bragg reflector constructed fromalternating layers with refractive indices na and nb.

Microcavities with a three-dimensional photonic confinement can exhibit strongcoupling between their confined photon modes and an exciton resonance in a single QD.Experimental evidence for strong coupling of a single QD exciton to a cavity mode hasbeen reported by Reithmaier et al. (2004), Yoshie et al. (2004) and Peter et al. (2005)(see Section 6.1). Here, we consider the simplest model of a spherical microcavity ofradius rQD with a spherical QD embedded in its centre (Fig. 4.29).

The eigenfrequencies of exciton-polariton modes can be found by the transfer matrixmethod generalized for spherical waves. Using the Green’s function technique to resolvethe Maxwell equations for a spherical wave incident on the QD in a similar way as weused above to describe scattering of a plane wave by a QD, one can obtain the reflectioncoefficient

rQD = 1 +2iΓQD

sp

ω0 − ω − i(γ + ΓQDsp )

, (4.144)

as described in more detail by Kaliteevski et al. (2001), with

ΓQDsp =

2

3πk4V 2

0 ωLTa3B , (4.145)

V0 =∫

Φ(r)j0(kr)dr and k = ncω/c, (j0 is the zero-order spherical Bessel functionand Φ the exciton wavefunction in the QD taken with equal electron and hole coor-dinates and assumed to have a spherical symmetry). The term 1 in the right part ofeqn (4.144) comes from the fact that for a spherical wave incident on a centre, the trans-mission contributes to reflectivity. The generalized transfer matrix method yields an


equation for the eigenfrequencies of the cavity polaritons, being a spherical analogue ofeqn (4.127):

h(2)1 (kR0) = rBrQDh

(1)1 (kR0) , (4.146)

where the functions h are related to the spherical Hankel functions H by h(1,2)l (x) =√

π/2xH(1,2)l+1/2(x) and rB is the reflection coefficient of the Bragg mirror for the spher-

ical wave incident from inside the cavity.

Fig. 4.29: Schematic distribution by Kaliteevski et al. (2001) of the magnitude of the electric-field intensity(grey scale) for the l = 1, m = 1 TM mode in a spherical microcavity with an embedded QD. The arrowsshow directions of the electric-field vector of the mode.

Equation (4.146) yields the dispersion of all existing cavity modes including thosecoupled to the QD exciton. Each mode can be characterized by an orbital quantumnumber l and magnetic number m and also TE- or TM-polarization (corresponding tothe modes where electric or magnetic fields have no radial component, respectively). Adetailed classification of spherical polariton modes has been given by Ajiki et al. (2002).Note that there exist no allowed optical mode having a perfect spherical symmetry (l =0) and the photon mode with l = 1 has the lowest allowed energy (see Fig. 4.29). Inparticular cases, eqn (4.146) can be reduced to the problem of two coupled harmonicoscillators familiar in planar microcavities. Assuming zero broadening of both excitonand photon modes and approximating rBR by

rBR ≈ exp(iβω − ωB

ωB) , (4.147)

with β = πnanb/nc(nb − na), one can rewrite eqn (4.146) as

βω − ωb

ωb− 2ΓQD

sp

ω − ωex+ 2kR0 = 2π(N + 1) , N ∈ N . (4.148)


In particular, if the uncoupled cavity mode has a frequency equal to ωB, the eigen-frequencies of exciton-polariton modes are given by

ω1,2 = ωb ±√

2ΓQDsp ωb

β + 2ncωb

c R0. (4.149)

If the splitting exceeds the line broadening of exciton and photon modes, the strongcoupling regime can be observed.

Exercise 4.6 (∗∗∗) Find the frequencies of exciton-polariton eigenmodes in a planarmicrocavity with an embedded periodical grating of infinite quantum wires (QWWs).Consider diffraction effects to the first order.

5

QUANTUM DESCRIPTION OF LIGHT–MATTER COUPLING INSEMICONDUCTORS

In this chapter we study with the tools developed in Chapter 3 the basicmodels that are the foundations of light–matter interaction. We startwith Rabi dynamics, then consider the optical Bloch equations that addphenomenologically the lifetime of the populations. As decay andpumping are often important, we cover the Lindblad form, a correct,simple and powerful way to describe various dissipation mechanisms.Then we go to a full quantum picture, quantizing also the optical field.We first investigate the simpler coupling of bosons and then culminatewith the Jaynes–Cummings model and its solution to the quantuminteraction of a two-level system with a cavity mode. Finally, weinvestigate a broader family of models where the material excitationcreation operator differs from the ideal limits of a Bose and a Fermifield.

170 QUANTUM DESCRIPTION OF LIGHT-MATTER COUPLING

5.1 Historical background

Many important concepts of light–matter interaction and especially their terminologyis rooted in the physics of nuclear magnetic resonance (NMR), where the effects werefirst observed and the models first developed. During World War II there was a stronginterest in radars and researchers investigated the radio-frequency range of the electro-magnetic field with great scrutiny, especially the means of creation and the efficientdetection of such waves. Purcell, then at the Massachusetts Institute of Technology,noted that magnetic nuclei—such as 1H, 13C or 31P—could absorb radiowaves whenplaced in a magnetic field of specific strength. The perturbation of this state allows ac-curate measurements that are the basis of NMR and derived techniques. As the effectwas understood to be linked to the intrinsic magnetic properties of the nucleus, the dy-namics of its spin under the action of strong electromagnetic fields was studied. Thephysics of spin–radiowave interaction shares many similarities to that of atom–light in-teraction: the two levels of the spin-up/spin-down configurations become the groundand excited states of an atomic resonance, and the electromagnetic field merely changesin frequency, so that many results and concepts obtained in the former context reappearin light–matter interaction. We start our investigation of the physics of light–matter in-teraction with Rabi’s approach to the problem of nuclear induction. He had the simplestmodel, minimally quantized. His model cannot be further simplified without reducing tothe Lorentz oscillator. He neglected lifetime, pumping, decoherence, and he eliminatedall “complications”, even those he could have taken into account easily. For this reason,he unravelled the most important features of the problem and had his name pinned in thephysics of light–matter coupling. All subsequent approaches rely on his simple result,as we shall see in this chapter.

5.2 Rabi dynamics

Rabi investigated the coupling of a quantum two-level system driven by a sinusoidalwave, modelling a classical optical field interacting with a spin. We shall use in thefollowing the terminology of atoms. In the approximations of Rabi, we write the atomicwavefunction as:

|ψ(t)〉 = Cg(t) |g〉+ Ce(t) |e〉 , (5.1)

the dynamics of which is entirely contained in the two complex coefficients Cg and Ce,and the Hamiltonian as:

H = Eg |g〉〈g|+ Ee |e〉〈e| (5.2a)

−(Vge |g〉〈e|+ Veg |g〉〈e|)E(t) . (5.2b)

Throughout, subscripts “g” and “e” refers to “ground” and “excited” (level, energy. . . )Veg is the dipole moment of the transition (discussed more later). Only the atom istreated quantum mechanically so the light energy does not enter into the Hamiltonian,it only acts as a time-dependent interaction through the c-function E. In the dipoleapproximation, the electric field is:

E(t) = E0 cos ω0t , (5.3)

RABI DYNAMICS 171

where the amplitude E0 and the frequency ω0 are constant. To emphasize the temporaldynamics, we introduce the frequency associated to the levels energies:

Eg = ωg , Ee = ωe . (5.4)

Exercise 5.1 (∗) Show that for the system (5.1)–(5.4), Schrodinger’s equation (3.1) be-comes:

Ce = −iωeCe + iΩRe−iφ cos(ω0t)Cg

Cg = −iωgCg + iΩReiφ cos(ω0t)Ce(5.5)

with φ the complex phase of Veg = |Veg|eiφ and in terms of the Rabi frequency

ΩR =Veg|E0|

. (5.6)

The quantity given by formula (5.6) is an important parameter for the quantum dy-namics of a two level system. It is often referred to as an energy, ΩR (the Rabi energy),and more often still as twice this quantity under the name of Rabi splitting. We have al-ready encountered this in the semiclassical treatment of Chapter 4 in eqn (4.47) as aterm quantifying the magnitude of the splitting (hence the name) of two resonances inthe polarization. This was obtained without reference to quantum dynamics, as this is aresult that in most cases can also be derived from a classical perspective, but the originof this popular term71 is in the quantum treatment, eqns (5.5).

In the interaction picture (or in the language more suited to this problem, in “rotatingframes”), the interesting dynamics is in the slow evolution due to the coupling ΩR notin the rapid and trivial one imparted by the optical frequency, so that we redefine

cg = Cgeiωgt and ce = Cee

iωet , (5.7)

which lead to: ⎧⎪⎪⎨⎪⎪⎩

ce = iΩR

2e−iφ[e−i(ω0−ωg)t + ei(ω0+ωg)t]cg

cg = iΩR

2eiφ[e−i(ω0−ωe)t + ei(ω0+ωe)t]ce

, (5.8)

where we have written the cosine as (eiω0t + e−iω0t)/2 to deal with complex expo-nentials only. Despite their simple appearance, these equations are not easy to solveexactly because of the many terms that appear by combinations of the two terms onthe right-hand sides. They become straightforward however if a single term is retained.Now let us remind ourselves that the purpose of eqns (5.7) is precisely to separate theslow dynamics of c coefficients from the rapid oscillation at rates ω = ωe − ωg, so thaton a small time interval over which eqns (5.8) are integrated, the temporal evolution

71Numerous other accommodations of the adjective apply, such as “Rabi flop” or “Rabi oscillation” todenominate the dynamics of Fig. 5.1)


of exp(i(ω0 + ωe/g)t

)ce/g is essentially given by the exponential that is oscillating so

quickly that it averages to zero even over a small interval where c changes by a littleamount. This is called the rotating wave approximation (RWA) because of the way fre-quencies contrive to stabilize or cancel the time evolution depending on whether theyoscillate with or against the frame rotating with the field. In the RWA, the solutionfollows quickly:

cg(t) =(a1e

iΩt/2 + a2e−iΩt/2

)ei∆t/2 , (5.9a)

ce(t) =(b1e

iΩt/2 + b2e−iΩt/2

)e−i∆t/2 , (5.9b)

with constants of integrations computed from initial conditions:

Exercise 5.2 (∗) Show that in solving eqns (5.8), the following is obtained:

a1 =1

2Ω

[(Ω−∆)ce(0) + ΩRe−iφcg(0)

], (5.10a)

a2 =1

2Ω

[(Ω + ∆)ce(0)− ΩRe−iφcg(0)

], (5.10b)

a3 =1

2Ω

[(Ω + ∆)cg(0) + ΩReiφce(0)

], (5.10c)

a4 =1

2Ω

[(Ω−∆)cg(0)− ΩReiφce(0)

], (5.10d)

and more importantly, in terms of newly introduced quantities:

The detuning: ∆ = ω − ω0 , (5.11)

The generalized Rabi frequency: Ω =√

Ω2R + ∆2 . (5.12)

From eqns (5.9)–(5.12), we can finally complete the description of the wavefunc-tion (5.1):

ce(t) =

(ce(0)

[cos(Ωt/2)− i

∆

Ωsin(Ωt/2)

]+ i

ΩR

Ωe−iφcg(0) sin(Ωt/2)

)ei∆t/2 ,

cg(t) =

(cg(0)

[cos(Ωt/2) + i

∆

Ωsin(Ωt/2)

]+ i

ΩR

Ωeiφce(0) sin(Ωt/2)

)e−i∆t/2 .

(5.13a)

From the interpretation of the wavefunction in quantum mechanics, |cg|2 and |ce|2are probabilities to find the atom in its ground or excited state, respectively. If the atomis initially in its ground-state (at t = 0), the probability to find it in its excited state attime t is, from eqns (5.13):

Pe(t) = |ce|2 =ΩR√

Ω2R + ∆2

sin2

√Ω2

R + ∆2

2t . (5.14)

The Rabi frequency, which at resonance is proportional to the amplitude of the lightfield, E0, and to the matrix element Veg, becomes renormalized with detuning. However,

BLOCH EQUATIONS 173

0 5 10 15 20Rt

3

2

1

0

1

2

3

R

2 4 6 8 10Rt

0.2

0.4

0.6

0.8

1Pe t

R1

0

R10

Fig. 5.1: Rabi oscillations in the probability Pe (greyscale coding, light colour corresponds to high probabil-ity) as a function of time and detuning (in units of ∆/ΩR and ΩRt). Departing from resonance increases thetransition rate but spoils its efficiency.

the transition efficiency gets spoiled, as seen in Fig. 5.1. The probability oscillates intime, so that continuously exciting a system is not the best way to excite it. Once theexcitation is created, further continuous excitation now works towards bringing backthe system to its ground state. Note that no mechanism of decay or relaxation has beenincluded in the simplest of the pictures and that if the external excitation is shut off,the system stays where it is forever. In effect, this tendency of an external excitationto induce the atom to de-excite is the stimulated emission foreseen by Einstein andintroduced in Chapter 3, for which we have just provided a microscopic derivation.

5.3 Bloch equations

The model developed by Rabi contains the key elements of the dynamics, associated tothe Rabi frequency ΩR. However, it is not realistic in many respects, if only because itlacks any form of dissipation. Losing excitation from the system or coupling it to someexterior reservoir will result in dephasing of the state,72 which is clearly not the casein the previous model where the quantum state is a wavefunction and therefore a purestate. Dissipation will induce a loss of quantum coherence in the sense of losing thequantum correlations between states that are embedded in the off-diagonal elements ofthe density matrix in this basis. A way to modify this limitation in the above approachis therefore to rewrite the equation with a density matrix instead of with a wavefunctionand add phenomenological decay terms that reduce elements of the density matrix. First,rewriting the equation in terms of

ρ = |ψ〉〈ψ| , (5.15)

gives, in the simple case of eqn (5.1):

72The interaction of the system with reservoirs that dephase it imply that even in the case where theinitial state of the system is well known (pure state), it evolves with time into a mixture of states where onlyprobabilistic information remains.


Edward Purcell (1912–1997) and Felix Bloch (1905–1983), the 1952 Nobel prize-winners in physics for“their development of new methods for nuclear magnetic precision measurements”.

Purcell worked during World War II at the MIT Radiation Laboratory on the development of the mi-crowave radar under the supervision of Rabi. Back in Harvard, he discovered nuclear magnetic resonance(NMR) with Pound and Torrey, published in Phys. Rev, 69, 37 (1946). Other similar and importantcontributions include spin-echo relaxation and negative spin temperature. In another field but still revolvingabout the radio spectrum, he made with Ewen the first detection of the famous “21cm line”, due to thehyperfine splitting of hydrogen. This seeded the field of radioastronomy and allowed a breakthrough in thestudy of galactic structure. Most importantly for the cavity community, he gave his name to the enhancementor inhibition of dipole radiation by boundary conditions (see next chapter).

Bloch’s education—like that of Purcell—was initially that of an engineer, but he soon turned tophysics which gave him the opportunity to study with Schrodinger, Weyl and Debye, among others. WhenSchrodinger left Zurich, Bloch worked with Heisenberg instead and promptly after with Pauli, Bohr, Fermiand Kramers as parts of fellowships he earned. He formulated in 1928 in his doctoral dissertation the theorembearing his name (that we presented on page 124) to describe the conduction of electrons in crystallinesolids. He met with Purcell in 1945 at a meeting of the American Physical Society where they realized theunity of their work on nuclear resonance. They agreed to share its experimental investigations, in crystals forPurcell’s group, in liquids for Bloch.

ρ =

( |Cg(t)|2 Cg(t)Ce(t)∗

Cg(t)∗Ce(t)

∗ |Ce(t)|2)

, (5.16)

and subsequently, its time equation of motion:

Exercise 5.3 (∗) Derive the following dynamics from eqn (3.33) applied to eqns (5.15)and (5.1): ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρgg =i

(VegEρeg − c.c.

),

ρee = − i

(V ∗

egEρge − c.c.),

ρge = −iω0ρeg − i

VegE(ρee − ρgg

),

ρeg = iω0ρge +i

V ∗

egE(ρee − ρgg

).

(5.17)

This is, up to now, strictly equivalent to eqns (5.5) as all elements of the densitymatrix can be reconstructed from the knowledge of the two amplitudes Cg/e. Also ob-serve the symmetries among the matrix elements, namely, ρgg = −ρee and ρge = ρ∗eg.

BLOCH EQUATIONS 175

The latter follows from hermiticity of the density matrix. The former is an expressionof the conservation of probabilities: if the atom gets excited, the ground-state gets de-populated. Because these matrix elements are real numbers, the complex conjugation isnot necessary (in the equations it is corrected by the off-diagonal element).

Lifetime and dephasing are now added phenomenologically to eqn (5.17), by the in-troduction of two characteristic times, known as T1 and T2.73 They account for lifetime(decay in population) and dephasing (decay in phase), respectively. Two typical phys-ical mechanisms responsible for these terms are spontaneous emission and atom–atomcollision, respectively. The spontaneous decay rate also dephases the system so that atypical expression for T2 could be:

1

T2=

1

2T1+

1

T ∗2

, (5.18)

where T ∗2 is the mean time between atomic collisions that reset the phases of the emit-

ters. A microscopic derivation of these quantities is possible but of course more involvedmathematically. In the former case it requires quantization of the optical field (whichhere is still a c-function E) and in the latter introduction of a bath of other carriers and astatistical treatment. On the other hand, the phenomenological treatment is clear enough(we shall see more about their microscopic origin in what follows). Equations (5.17)read: ⎧⎪⎨

⎪⎩ρee = − i

(V ∗

egEρge − c.c.)− ρee

T1,

ρge = −iω0ρeg − i

VegE(ρee − ρgg

)− ρge

T2,

(5.19)

where we have also limited the discussion to independent terms only and performed therotating wave approximation.74

Equations (5.19) are called the optical Bloch equations, after Bloch who derivedthem for a spin in an oscillatory electric field in a form suitable to be mapped onto theBloch sphere (or Poincare sphere). They do not admit analytical solution in the generalcase but limiting cases of practical interest exist and are presented now.

Exercise 5.4 (∗∗) Consider trial solutions of the form ρij(t) = ρij(0) exp(λt). Showthat at zero detuning and for Veg ≥ (1/2T1), the linearization thus afforded admitssolutions:

λ1 = 0, λ2 = − 1

T1, λ3 = −3

2

1

T1+ iλ and λ4 = −3

2

1

T1− iλ , (5.20)

where:

λ =√|Veg|2 − 1/(2T1)2 . (5.21)

Consider now the case Veg 1/T1 where the same method yields:

73In NMR, T1 and T2 are called the longitudinal and transverse relaxation times, respectively, becausethey cause the decay of orthogonal projections of the magnetic spin along its precession axis.

74See the discussion on page 172.


λ1 = 0, λ2 = − 2

T1, λ3 = − 1

T1+ i(ω0 − ω) and λ4 = − 1

T1− i(ω0 − ω) .

(5.22)

5.3.1 Full quantum picture

We now turn to a complete quantum treatment where the optical field gets quantized aswell. Successful descriptions of light–matter interactions in full quantum-mechanicalterms is of course a complicated problem. It culminates in the theory of quantum elec-trodynamics (QED), which is to date the most succesful, accurate and inspiring phys-ical theory, among other points having served as a blueprint for a vast family of sistertheories—collectively known as the “standard model”—to describe the force experi-mentally observed in nature (QED being the quantization of the electromagnetic force).Only gravity is resisting the axiomatization and interpretation of QED.

Back to our concern of light–matter interaction, one starting point75 is to acknowl-edge that matter—being essentially neutral—couples with light through the electric fieldcomponent arising from fluctuations in the electric charges (since the total charge can-cels on the average). In the multipolar expansion of the field, the first nonzero fluctua-tions are the dipolar fluctuations for most cases of interest, including the simplest andgeneric case of an atom made up from a positive nucleus and an orbiting negative elec-tron. This case naturally also depicts the exciton, cf. Fig. 4.7. The potential energy Uof a classical dipole76 d in an electric field E is easily derived from Newton’s law andelectrostatic energy, to yield

U = −d ·E . (5.23)

We have already quantized the field E, cf. eqn (3.75). We now consider the atomdipolar moment d. We will simplify the above interaction by restraining to only twoatomic levels and a single mode of the electromagnetic field. This is a bold approxima-tion that leaves out a lot of the physics described by eqn (5.23) that, even for the simplestatom—the electron–proton pair bound as hydrogen—describes the coupling of two in-finite sets of energy levels. The considerable simplification that we shall make proveshowever to be sound especially for microcavities whose merit is to filter out modes ofthe electromagnetic field by imposing boundary conditions. For the atom, it suffices toselect two levels whose energy difference matches the energy singled out by the opticalmode to make all other transitions negligible in comparison. In the absence of a cavitythis can be approached by using laser light that mimics a single mode thanks to smallenergy fluctuations.

75Another, even more popular quantization scheme, is that of the Hamiltonian H = V (r) + (p −qA(r, t))2/2m of the vector potential A. This Hamiltonian, or that derived from eqn (5.23), are essentiallyequivalent and are exactly so at resonance. The so-called Ap Hamiltonian (of this footnote) is better adaptedto delocalized systems like electrons and holes in a bandstructure. Localized systems, on the contrary, likeatoms or quantum dots, find a better starting point with the dipole Hamiltonian.

76A dipole is most simply visualized as the limiting case of two point-like particles of opposite charges +qand −q, and vanishing separation δ, being d = qδud, where ud is the unit vector pointing from the plusto the minus charge. Physical dipoles are two actual opposite charges separated by some distance small incomparison to the distance at which the dipole is observed.

BLOCH EQUATIONS 177

We recall our notations for the two atomic modes |e〉 and |g〉 with energies Ee

and Eg for “excited” and “ground”, respectively. At this stage and with such nomencla-ture we have virtually already completely abstracted the “real” or “physical” atom away.Let us keep in mind, though, that |g〉 can equally well be the lower of two excited statesof the physical atom. In the spirit of Dirac, |g〉 is a simple notation for a potentiallycomplicated wavefunction. To illustrate the point we shall consider the |Ψ100〉 state ofthe hydrogenoid atom as the ground |g〉 state and |Ψ211〉 as the excited |e〉 state. The ex-pressions for these states follow from the Schrodinger equation with Coulomb potentialas:

Ψ100(r) =1√πa3

B

exp(− r

aB

), (5.24a)

Ψ211(r) =

√1

64πa3B

(r

aB

)exp

(− r

2aB

)sin θeiφ , (5.24b)

with r = (r, θ, φ) in eqn (5.24b).

Exercise 5.5 (∗∗) Show that the dipole element operator77 for states (5.24) is:

d = |e〉〈g|deg + |g〉〈e| (deg)∗ , (5.25)

where:

deg = e

∫Ψ∗

211(r)rΨ100(r) dr , (5.26a)

= −27

35eaB(x− iy) . (5.26b)

For the following it is therefore mathematically advantageous to consider the com-plicated “object” d as a mere two by two matrix on a Hilbert space spanned by:

|e〉 =

(10

), 〈e| = (

1, 0),

|g〉 =

(01

), 〈g| = (

0, 1).

In this space, d can be expressed as a function of σ = (σx + iσy)/2 and σ† (seeeqn (3.12)) to become:

d = degx(σ† + σ) , (5.27)

where, in this case, the orientation of the dipole has been arbitrarily taken along the x-axis so as to cancel the imaginary part (orientation along the latter would result in a

77A note of caution on notations: in eqn (5.25), d is an operator, while deg is a cartesian vector. Someauthors use a hat to denote explicitly an operator, i.e., they would write d = |e〉〈g|deg + h.c. As a matterof fact, d is also a vector, as is seen in eqn (5.26b), whose components are operators. We do not follow thispractice because it is often clear enough from the context what is the mathematical nature of a variable, and itis helpful to retain the scope of an equation in both classical and quantum domains.


minus sign in eqn (5.27) and a complex matrix element). So finally, the light–matterinteraction reads g(σ† + σ)(a− a†) where:

g =

√ω0

2ε0Vdeg . (5.28)

Equation (5.28) is a “typical” result for the cases we have been considering. It specif-ically depends on the system at hand and how it is modelled. In this case we have evenleft the dipole element in its general form. If considering the complete, multimode lightfield, eqn (3.75) should be used instead, as will be the case in the next chapter wherethe continuum of modes is needed to compute the radiative lifetime of a dipole in thevacuum.

With the free energy of the optical mode ωC and the atom ωX added, the “generic”complete Hamiltonian is:

H = ωCa†a +1

2ωXσz + g(σ† + σ)(a− a†) . (5.29)

This system is the basis for what follows. To start with, we investigate quantumcoupling in simpler cases than eqn (5.29), which despite its apparent simplicity is highlynon-trivial. We shall in all cases reduce the problem in this chapter to single modes.This could be the case when large splitting of energies separate higher states from thoseconsidered, typically one cavity mode near resonance with an excitonic transition. Moreelaborate multimode couplings are treated in subsequent chapters. For now, however,neglecting off-resonant terms78 like a†b†, the Hamiltonian becomes:

H = ωCa†a + ωXb†b + g(ab† + a†b) . (5.30)

We denote by ∆ the detuning in the eigenfrequencies:

∆ = ωC − ωX , (5.31)

One mode, a, which describes light, will be in all cases a pure Bose operator with com-mutation relation (3.39). The operator b that describes the material excitation, on theother hand, depends on the model of the matter field, and could range—in the extreme—from another Bose operator following the same commutations relations as a, to a Fermioperator with anticommutation relations. It can also be, more generally, a more com-plicated expression that follows from the rich structure of the particles involved, e.g.,eqn (5.118) for the exciton in the approximations outlined there.

These two limiting cases—coupling a single cavity mode to a bosonic or fermionicfield—are also of tremendous importance and many experimental configurations referto them in some approximations. Their simple mathematical forms allow exact solutionsto be obtained and therefore many insights to be gained.

78Neglecting terms that do not conserve energy like a†b† or ab—since they simultaneously create orannihilate two excitations, respectively—is the second quantized version of the rotating wave approximation.

BLOCH EQUATIONS 179

5.3.2 Dressed bosons

The most elementary problem of the kind of eqn (5.30) is the case where b is also aBose operator, namely

[b, b†] = 1 (5.32)

(cf. eqn (3.39)). This is the elementary problem of coupling of two linear oscillators(hence the problem is referred to as linear and this linearity will transpire in all thatfollows). We investigate the solutions in the various pictures of quantum mechanicsintroduced in Chapter 3.

The analysis of eqn (5.30) can be made directly in the basis |i, j〉 with i excitationsin the matter field and j in the photonic field, i, j ∈ N. These are called bare states incontrast to the dressed states that we consider hereafter. The value of this approach isthat the excitation, loss and dephasing processes generally pertain to the bare particles.For instance, matter excitations are usually created by an external source (pumping) andlight excitations can be lost by transmission through the cavity mirror. This physics isbest expressed in the bare-states basis.

As the system is linear, the integration is straightforward in the bare-states basis. Inthe Heisenberg picture:

Exercise 5.6 (∗) Show that the time evolution of the operators a and b under the dy-namics of Hamiltonian (5.30) is given by:

a(t) = exp(− iωt

)(a(0)

[cos

Gt

2− i

∆

Gsin

Gt

2

],−2ib(0)

g

Gsin

Gt

2

), (5.33a)

b(t) = exp(− iωt

)(− 2ia(0)g

Gsin

Gt

2+ b(0)

[cos

Gt

2+ i

∆

Gsin

Gt

2

]), (5.33b)

where ω = (ωa + ωb)/2 and G =√

4g2 + ∆2.

Note that the commutation relation (3.39) remains well behaved at all times thanksto the intermingling of a and b operators. This can be illustrated in the case ∆ = 0where the expressions (5.33) simplify considerably to:

a(t) = e−iωt[a(0) cos(gt)− ib(0) sin(gt)] , (5.34a)

b(t) = e−iωt[b(0) cos(gt)− ia(0) sin(gt)] , (5.34b)

in which case one gets

[a(t), a†(t)] = [a(0), a†(0)] cos2(gt) + [b(0), b†(0)] sin2(gt) = 1 . (5.35)

We shall see in the Schrodinger representation how this result manifests in complicatedcorrelations between the two states. This result should also make clear that equal-timecommutations hold for Bose operators. Observe indeed how [a(0), a†(t)] oscillates be-tween 0 and 1 with time.

Observables can be obtained directly in the Heisenberg picture from the solutions(5.33). For instance, the population is obtained in the exercise below.


Exercise 5.7 (∗) Show that the population operators for the coupling of two oscillatorswith initial condition |Ψ〉 = α |1, 0〉+ β |0, 1〉 are given by:

(a†a)(t) = |α|2 cos2Gt

2+|∆α + 2βg|2

G2sin2 Gt

2−(αβ∗)

4g

Gcos

Gt

2sin

Gt

2, (5.36)

and use this result to recover the Rabi oscillations of Fig. 5.1.

On the other hand, eqn (5.30) assumes a straightforward expression in the basis ofso-called dressed states that diagonalize the Hamiltonian as we now show explicitly go-ing back to the Schrodinger picture. The most general substitution that can be attemptedis:

p = αa + βb , (5.37a)

q = γa + δb , (5.37b)

with α, β, γ, δ ∈ C. If we require that p and q remain boson operators, i.e.,

[p, p†] = [q, q†] = 1 , (5.38)

implying|α|2 + |β|2 = |γ|2 + |δ|2 = 1 , (5.39)

as well as[p, q] = [p, q†] = 0, . (5.40)

The second one impliesαγ∗ + βδ∗ = 0 , (5.41)

while the first is automatically satisfied.Relations (5.37) reversed read

a =δp− βq

αδ − βγ, (5.42a)

b =−γp + αq

αδ − βγ. (5.42b)

Their substitution in eqn (5.30) yields

(αδ − βγ)2H = p†pω(|δ|2 + |γ|2)− g(δ∗γ) (5.43a)

+ q†qω(|β|2 + |α|2)− g(β∗α) (5.43b)

+ p†qω(−βδ∗ − γ∗α) + g(αδ∗ + βγ∗)+ h.c. (5.43c)

We require that line (5.43c) be zero, which reduces to

αδ∗ + βγ∗ = 0 . (5.44)

Fitting the above conditions, we are led to α = cos θ, β = sin θ, γ = − sin θ and δ =cos θ, ensuring that αδ − βγ = 1 (canonical unitary transformation):

BLOCH EQUATIONS 181

p = cos θa + sin θb , (5.45a)

q = − sin θa + cos θb , (5.45b)

where

cos θ =∆ + G√

2∆2 + 8g2 + 2∆G, (5.46)

θ is known as the mixing angle.Then H reads

H = ωpp†p + ωqq†q , (5.47)

where

ωp/q = ω ± ∆

2G . (5.48)

The eigenfrequencies ωp/q given by eqns (5.48) are plotted in Fig. 5.2.

-10 -5 0 5 10

Fre

quen

cy

∆/g

ωp(∆/g)

ωq(∆/g)

mode b

mode a

Fig. 5.2: Thick lines: Eigenfrequencies of the system eqn (5.30) as a function of the detuning ∆ (in unitsof the interaction strength g and with ωa set to 0). At zero detuning, an anticrossing is observed. At largedetunings, the bare modes (thin lines) are recovered.

We shall later see how this approach is the archetype of so-called Bogoliubov trans-formations, first considered in connection with high-density, weakly interacting Bosecondensates.

For clarity we shall note ||i, j〉〉 the dressed states, i.e., the eigenstates of eqn (5.53)with i dressed particles of energy ωp and j of energy ωq. We call a manifold the setof states with a fixed number of excitations. In the dressed-states basis it reads for thecase of n exitations:

HN = ||n,m〉〉 ; n,m ∈ N with n + m = N . (5.49)


Its energy diagram appears on the left of Fig. 5.3 for manifolds with zero (vacuum),one, two and seven excitations. When an excitation escapes the system while in mani-fold HN , a transition is made to the neighbouring manifold HN−1 and the energy dif-ference is carried away, either by the leaking out of a cavity photon, or through excitonemission into a radiative mode other than that of the cavity, or a non-radiative process.The detailed analysis of such processes requires a dynamical study, but as the cavitymode radiation spectra can be computed with the knowledge of only the energy-leveldiagrams, we shall keep our analysis to this level for the present work. The importantfeature of this dissipation is that, though such processes involve a or b (rather than por q), they nevertheless still result in removing one excitation out of one of the oscil-lators. Hence, only transitions from ||n,m〉〉 to ||n − 1,m〉〉 or ||n,m − 1〉〉 are allowed,bringing away, respectively, ωp and ωq of energy, accounting for the so-called Rabidoublet (provided the initial n and m are nonzero in which case only one transition is al-lowed). From the algebraic point of view, this of course follows straightforwardly fromeqn (5.52) and orthogonality of the basis states. Physically, it comes from the fact that,as in the classical case, the coupled system acts as two independent oscillators vibratingwith frequencies ωp/q.

In the case of vacuum-field Rabi splitting, a single excitation is shared between thetwo fields, and so the manifold H1 is connected to the single line of the vacuum mani-fold. In this case there is obviously no possibility beyond a doublet. It is straightforwardto compute the transition amplitudes between the two manifolds by mean of the bare-states annihilation operators. The rates, which are proportional to the four componentsof the dressed states, are:

M1 = 〈0, 0| a||1, 0〉〉 = α(∆/g) , (5.50a)

M2 = 〈0, 0| a||0, 1〉〉 = γ = −α(−∆/g) , (5.50b)

M3 = 〈0, 0| b||1, 0〉〉 = β = α(−∆/g) , (5.50c)

M4 = 〈0, 0| b||0, 1〉〉 = δ = α(∆/g) . (5.50d)

The amplitudes of transitions Mi given by eqns (5.50) are displayed in Fig. 5.4.Their square is the physical quantity one is interested in:

I1 = |α(∆/g)|2 , (5.51a)

I2 = |α(−∆/g)|2 . (5.51b)

The physical sense of these results is that when the mode annihilates an excitationin one of the dressed states, it only “sees” it through its weight in the total (or dressed)state, so the strength comes out as proportional to these amplitudes. Intuitively, if the“polariton” becomes more “exciton-like” because of detuning, the cavity emission dis-appears. The intensities degenerate into two lines out of four for the amplitudes becauseof the symmetry of the two modes (here we have two harmonic oscillators, in our casethe two modes are different). The transition rates are antisymmetric with ∆, and as onegoes from 0 (from −∞) to 1 (from +∞), the other does the reverse. The emission ofone given mode is therefore two lines, which degenerate at zero detuning (where therates match) and with the detuning, one vanishes while the other increases.

BLOCH EQUATIONS 183

...

...

||1, 0〉〉||0, 1〉〉

−g+g

||0, 2〉〉

||2, 0〉〉||1, 1〉〉

−2g

+2g

−7g−5g−3g

+5g+7g

Bose limit

||0, 0〉〉

||7, 0〉〉||6, 1〉〉||5, 2〉〉||4, 3〉〉||3, 4〉〉||2, 5〉〉||1, 6〉〉||0, 7〉〉

...

−g+g

Fermi limit

−√2g

+√

2g

−√7g

+√

7g

2√ 7

ω

2√ 2

ω

ω

Fig. 5.3: Energy diagrams of the two limiting cases of dressed bosons (left) and fermions (right). In the firstcase the nth manifold has constant energy splitting of 2g between all states and couples to the (n − 1)thmanifold by removal of a quantum of excitation with energy ω ± g, which leads to the Rabi doublet, (a),with splitting 2g. In the second case, each manifold is two-fold with a splitting that increases like a squareroot. All four transitions are allowed, leading to the Mollow triplet, (b), for high values of n when the twomiddle transitions are close in energy. The distance from the central peak goes like 2g

√n and the ratio

of peaks is 1:2. The two lowest manifolds (dashed) are the same in both cases, making vacuum-field Rabisplitting insensitive to the statistics.


-1

-0.5

0

0.5

1

-10 -5 0 5 10

Tra

nsiti

on a

mpl

itude

∆/g

α = δ = cosθ

γ = -sinθ

β = sinθ

0

0.5

1

-10 -5 0 5 10

Inte

nsity

∆/g

I1 = |α|2 = |δ|2

I2 = |γ|2 = |β|2

0

0.5

1

-10 -5 0 5 10

Inte

nsity

∆/g

I1 = |α|2 = |δ|2

I2 = |γ|2 = |β|2

Fig. 5.4: Amplitudes Mi (left) and corresponding intensities |Mi|2 (right) of transitions between the mani-folds of one excitation and the vacuum, as a function of the detuning ∆ in units of the interaction strength g.

For instance, the simplest case where p† and q† creates a coherent superposition ofbare states, respectively in and out of phase:

p = (a + b)/√

2, q = (a− b)/√

2 , (5.52)

which are the eigenstates of zero detuning, with its corresponding diagonalized Hamil-tonian:

H = (ω − g)p†p + (ω + g)q†q . (5.53)

Back to the general case, we note as before ||n,m〉〉 the n, m Fock excitations in p, qoscillators and |i, j〉 the i, j Fock excitations in a, b oscillators:

H||n,m〉〉 = En,m||n,m〉〉 , (5.54)

withEn,m = (nωp + mωq) . (5.55)

From eqn (5.42) it follows that, in our case of eqn (5.45a),

a† = cos θp† − sin θq† , (5.56a)

b† = sin θp† + cos θq† . (5.56b)

So if we have ||n, 0〉〉 particles, this corresponds in bare states to

||n, 0〉〉 =1√n!

p†n||0, 0〉〉 , (5.57a)

=1√n!

(cos θa† + sin θb†)n |0, 0〉 , (5.57b)

=1√n!

n∑k=0

(n

k

)cosk θ sinn−k θa†k

b†n−k |0, 0〉 , (5.57c)

=n∑

k=0

√(n

k

)cosk θ sinn−k θ |k, n− k〉 . (5.57d)

BLOCH EQUATIONS 185

The general case should be computed along the same lines

||n,m〉〉 =1√

n!m!p†

nq†

m||0, 0〉〉 , (5.58)

(5.59)

to finally obtain

||n,m〉〉 =

n+m∑ν=0

ν∑l=0

(−1)l

(n

ν − l

)(m

l

)√(n+m

n

)(n+m

ν

)× cosν−l θ sinl θ sinn−ν+l θ cosm−l θ |ν, n + m− ν〉 . (5.60)

Exercise 5.8 (∗) Derive eqn (5.60) above.

This result gives the probability to measure the bare state |ν, n + m− ν〉 when thesystem is prepared in the dressed state ||n,m〉〉:

p(ν) =

∣∣∣∣∣ν∑

l=0

(−1)l

(n

ν − l

)(m

l

)√(n+m

n

)(n+m

ν

) cosm+ν−2l θ sinn−ν+2l θ

∣∣∣∣∣2

. (5.61)

Let us consider transition amplitudes between ||n,m〉〉 and ||n− 1,m〉〉:〈〈m,n− 1||p||n,m〉〉 =

√n , (5.62a)

〈〈m,n− 1||a||n,m〉〉 =√

n/(2 cos θ) . (5.62b)

The same appears with q and b.We conclude this section with the basis of coherent states, that is, we provide the

time evolution of the P representation of the oscillators (cf. eqn (3.84)). If we call αthe variable relating to oscillator a and β that relating to b, the density matrix ρ of thecoupled system (5.30) decomposes as

ρ =

∫∫P (α, α∗, β, β∗, t) |αβ〉〈βα| dαdβ . (5.63)

Following the procedure detailed in Section 3.3.3, the operator equation (5.30) pro-vides its c-number counterpart:

P =iω(α∂α − α∗∂α∗ + β∂β − β∗∂β∗)P

+ig(α∂β − α∗∂β∗ + β∂α − β∗∂α∗)P ,(5.64)

which can be integrated to yield the exact solution:

P (α, α∗, β, β∗, t) = F( i

2(α∗ + β∗)e−i(ω+g)t,

i

2(−α∗ + β∗)e−i(ω−g)t,

i

2(α− β)e−i(−ω+g)t,

i

2(−α− β)ei(ω+g)t

), (5.65)

where F is a differentiable function.


Exercise 5.9 (∗) Check that the partial differential equation axux + byuy + czuz +dwuw = 0 has solutions of the type

u(x, y, z, w) = F (ya/xb, za/xc, wa/xd) (5.66)

where F is a differentiable function. Therefore, by diagonalization of eqn (5.64), obtaineqn (5.65).

By matching eqn (5.65) with the initial conditions, one can obtain the time evolutionof the complete wavefunction. For instance, the important case of two states that aremixtures of coherent and thermal states (in the sense of Section 3.3.5), with respectiveintensities |α0|2 and n for one oscillator, and |β0|2 and m for the other, leads to thefollowing evolution of their wavefunction:

P (α, α∗, β, β∗, t) =1

πmexp

(− |α|2

[cos2(gt)

m+

sin2(gt)

n

]

−α

[−α∗0e

iωt cos(gt)

m+−iβ∗

0eiωt sin(gt)

n

]

−α∗[−α0e

−iωt cos(gt)

m+

iβ0e−iωt sin(gt)

n

]

−|α0|2[

1

m

])

× 1

πnexp

(− |β|2

[sin2(gt)

m+

cos2(gt)

n

]

−β

[−iα∗0e

iωt sin(gt)

m+−β∗

0eiωt cos(gt)

n

]

−β∗[iα0e

−iωt sin(gt)

m+−β0e

−iωt cos(gt)

n

]

−|β0|2[

1

n

])

× exp(− α∗β

[i cos(gt) sin(gt)

m+−i sin(gt) cos(gt)

n

]

−αβ∗[−i cos(gt) sin(gt)

m+

i sin(gt) cos(gt)

n

]).

(5.67)

This result shows how, due to the coupling, even though it is linear and of the sim-plest kind that can affect two modes, complex correlations build up between the twomodes. The main behaviour of the coupled system remains that of two mixture statesoscillating in their populations, which are represented by the first two exponentials ofthe above product, with variables α and β factoring out, but a third exponential inextri-cably links them. This approach is the basis for justifications of truncated schemes thatwill be the basis for master and rate equations to be developed in the following chapters.

LINDBLAD DISSIPATION 187

5.4 Lindblad dissipation

We have been able to introduce dissipation going from the Rabi picture dealing witha pure state to the Bloch one dealing with a density matrix. In this case, though, thedecay was phenomenological, adding a memoryless decay term to the matrix elementreproducing the sought effect. We now describe a popular formalism to introduce dissi-pation in a quantum system, which also gives many insights into the origin and natureof dissipation in a quantum system. We shall illustrate our point on the simple systemof an harmonic oscillator, cf. eqn (3.40), whose dynamics has already been investigatedfrom many different viewpoints.

The postulates of quantum mechanics do not accommodate well indirect attempts to“reproduce” dissipation. For instance, canonical quantization of the equations of motionof the damped oscillator eqn (4.1) with the methods of Chapter 3 yields:79

[X(t), P (t)] = e−γt[X(0), P (0)] , (5.68)

and the commutation relation and its derived algebra—which define the quantum fieldand such properties as its statistics—are lost in time. The dissipation introduced in thisway washes away the quantum character of the system as well as its dynamics. This isnot a very satisfying picture: an atom that relaxes to its fundamental state should stillremain an atom.

In the previous section where we have investigated the dynamics of two coupledquantum oscillators, we have seen how the excitation of one was transferred to the other.This is a dissipation insofar as the first system is considered over this interval of time.Then the Hamiltonian dynamics brings back the excitation of the second oscillator tothe first one and the process continues back and forth cyclically. Here lies the key ideaof a correct model for dissipation in quantum mechanics. It is not a fundamental char-acteristic of a system that needs to be quantized, but it is a feature of its dynamics: thequantum system gives its energy away and takes it back when it couples to another state.Imagine now that the system is coupled not to one, but to numerous other modes, whichtogether form a reservoir, so that as the system’s energy is exchanged with the reservoir,each mode has little probability to be the recipient but the system has high probability tolose its excitation. Once enough time has elapsed so that energy is with high probabilityin the resevoir, it will continue being exchanged in an oscillatory way, as follows fromHamiltonian dynamics that demands that all configurations of the systems be visitedwith the same weight. However, the system is so insignificant as compared to the reser-voir that the probability of a full return of its initial expenditure is vanishingly smallwith increasing size of the reservoir. In effect, this accounts for dissipation in quan-tum mechanics in the same way that, in classical physics, the Newtonian dynamics thatviolates thermodynamics (being reversible) is supressed by the law of large numbers.

We detail the mathematical form in an explicit, simple case now before turning tothe general expression. Consider a single harmonic oscillator a coupled linearly to a set

79The Hamiltonian equations with dissipation being X = P/m0 and P = −γP − m0ω2X , the corre-spondence principle eqn (3.9) makes d[X, P ]/dt = XP + PX − XP − XP = −γ[X, P ] that integratesto the result eqn (5.68) from eqn (3.17).


of oscillators bi that all merge together to form the bath. All oscillators a and bi followthe algebra of eqn (3.38). The coupling Hamiltonian reads:

H = ωaa†a +∑

i

ωib†i bi +

∑i

(g∗i ab†j + gia†bj) . (5.69)

We have made the same approximations as in the previous section. The dynamics of thissimple system is clear physically: the system a loses its excitation that is transferred toone of the bi modes through the process ab†i weighted by the strength of the coupling gi.The reverse process a†bi where the excitation is destroyed in the mode bi and put backin the system exists. It follows, as we have already emphasized, from the Hamiltoniancharacter of this dynamics. Now we proceed to make the approximations that are inorder when the set of bi is treated as a reservoir. Namely, first of all we neglect thedynamics of the reservoir (being in equilibrium) and its direct correlations to the system.We write the total density matrix for the combined system described by eqn (5.69) as:

(t) = ρ(t)⊗R0 , (5.70)

where R0 is the time-independent density matrix of bosons at equilibrium:

R0 =⊗

i

(1− exp

(− ωi

kBT

))exp

(− ωib†i bi

kBT

). (5.71)

Note that we have assumed the separability of the density matrix into a product ofthe density matrix of the system and of the reservoir. This is called the “Born approxi-mation”. It is physically reasonable for a reservoir, as each mode quickly washes awaythe coherence that it develops with the system through its interaction because it alsointeracts with many other modes of the reservoir. In fact, note that R0 in eqn (5.71) isalso decorrelated for all modes. The important changes retained are the dephasing andloss of energy of the system alone. The latter now, moreover, accounts for the entiretime dependency. To make it more sensible still, we shift to the interaction picture sothat the rapid and trivial oscillation due to the optical frequency is removed. From nowon, therefore:

a(t) = a−iωat and a†(t) = a†eiωat , (5.72)

as detailed in Section 3.1.6 (a(0) and a of the Schrodinger picture being equal). Theinteracting Hamiltonian becomes:

H(t) = [a(t)B†(t) + a(t)†B(t)

], (5.73)

where we have introduced the “reservoir operators”:

B(t) =∑

i

gibie−iωit and B†(t) =

∑i

g∗i b†ieiωit . (5.74)

It remains to eliminate the complicated structure of operators (5.74) and focus onthe dynamics of ρ alone. This can be done by averaging over the degrees of freedom of


the reservoir. Let us first obtain the equation of motion for ρ. If we formally integratethe Liouville–von Neumann equation, cf. eqn (3.33) for , we get:

i(t) = i(−∞) +

∫ t

−∞[H(t), ρ(t)] dt , (5.75)

where, for a while, we write everywhere the time dependency. We have assumed theinitial condition is at t → −∞.

Exercise 5.10 (∗) By inserting the exact result eqn (5.75) back into eqn (3.33) (for ),show that the following integrodifferential is obtained:

(t) = − 1

2

∫ τ

−∞[H(t), [H(τ), (τ)]] dτ . (5.76)

It is enough in this case to limit to order two in the commutator, which yields thedynamics of the populations while neglecting that of the fluctuations. Later in this book,e.g., when we investigate the dynamics of the laser field in weak-coupling microcavities,we shall pursue this kind of expansion of eqn (5.75) further. They are known as Bornexpansions.

Exercise 5.11 (∗∗) Carry out the algebra of [H(t), [H(τ), (τ)]] with definitions givenby eqns (5.73) and (5.74) and show that under the approximation of separability givenby eqn (5.70):

ρ = − 1

2

∫ t

0

[(a2ρ(τ)− aρ(τ)a)e−iωa(t+τ)〈B†(t)B†(τ)〉+ h.c. (5.77a)

+ (a†2ρ(τ)− a†ρ(τ)a†)eiωa(t+τ)〈B(t)B(τ)〉+ h.c. (5.77b)

+ (aa†ρ(τ)− a†ρ(τ)a)e−iωa(t−τ)〈B†(t)B(τ)〉+ h.c. (5.77c)

+ (a†aρ(τ)− aρ(τ)a†)eiωa(t−τ)〈B(t)B†(τ)〉+ h.c.]dτ , (5.77d)

with, for the reservoir correlators, 〈B†(t)B†(τ)〉 =∑

i,j g∗i g∗j ei(ωit+ωjτ)Tr(R0b†i b

†j)

and 〈B(t)B(τ)〉 =∑

i,j gigje−i(ωit+ωjτ)Tr(R0bibj), and more importantly:

〈B†(t)B(τ)〉 =∑

i

|gi|2eiωi(t−τ)n(ωi) , (5.78a)

〈B(t)B†(τ)〉 =∑

i

|gi|2e−iωi(t−τ)(n(ωi) + 1) , (5.78b)

where n is given by the Bose–Einstein distribution, cf. eqn (3.89) and eqn (5.71).

Lines (5.77)a and d cancel exactly because of repeated annihilation on the diagonaldensity operator eqn (5.71). Evaluation of eqns (5.78) is more involved. The physicalidea that we wish to emphasise over direct attempts towards evaluating the mathemat-ical expressions, is that given a time evolution for ρ(τ) that is much smaller than that


of e±iωa(t−τ) and 〈B†(t)B(τ)〉, 〈B(t)B†(τ)〉 also with rapid dependence of the typee±iωi(t−τ), it is a good approximation to write

ρ(τ) ≈ ρ(t) (5.79)

in eqn (5.77), which is called the Markov approximation, since the density matrix ρ(t) attime t on the left-hand side of eqn (5.77) loses the dependency on its value at a previoustime τ < t, i.e., it has no memory of its past. It is especially clear in the form below:

Exercise 5.12 (∗) Combining the arguments given above and using the representation∫ei(ωi−ωa)(t−τ) dτ = πδ(ωi − ωa), reduce eqn (5.77) to the following first-order dif-

ferential, Markovian equation:

ρ = A(2a†ρa− aa†ρ− ρaa†) (5.80a)

+ B(2aρa† − a†aρ− ρa†a) (5.80b)

where:

A = π∑

i

|gi|2n(ωi)δ(ωa − ωi) , (5.81a)

B = π∑

i

|gi|2(n(ωi) + 1)δ(ωa − ωi) . (5.81b)

Coefficients (5.81) are those that would be obtained by Fermi’s golden rule. Theyexhibit a delta-singularity because of the Markov approximation and the long-time av-erage of the interaction between the system and the reservoir that requires conservationof the energy. By broadening the modes of the reservoir, they provide two rates that weassociate in the following way to the master equation:

ρ = nγ

2(2a†ρa− aa†ρ− ρaa†) (5.82a)

+ (n + 1)γ

2(2aρa† − a†aρ− ρa†a) . (5.82b)

where γ = 2πσ(ωa)|g(ωa)|2, σ(ω) being the density of states of the oscillators in thereservoir at frequency ωa (σ is a smooth function of γ that is the continuous limit of ωi).Expression (5.82) is a popular one in quantum optics. Note that now in the interactionpicture the present form eqn (5.82) is completely devoid of Hamiltonian dynamics. Aswe expect and as was the aim of the construction, it precisely describes dissipation, asis investigated further below. Before concentrating on this form, however, let us give theresult of the most general possible form for a master equation under the assumption thatthe evolution is Markovian (as is the case for many models). It is known as the Lindbladmaster equation and reads:

ρ = − i

[H, ρ]− 1

∑n,m

hn,m

(ρLmLn + LmLnρ− 2LnρLm

)+ h.c. , (5.83)

where the part from the Schrodinger equation − i[H, ρ], has been put back, and the

terms on the right, Lm, are operators being, along with the constants hn,m, defined


Goran Lindblad with his work “on the generators of quantum mechanicalsemigroups”, Comm. Math. Phys. 48 (1976), 119, attached his name to termsthat turn the Liouville–von-Neuman equation into a dissipative system.

Lindblad is a mathematical physicist and Emeritus Professor of the KTH(“Kungliga Tekniska hogskolan”, the Royal Institute of Technology) inStokholm, from which he retired in 2005.

by the system. In the case of the harmonic oscillator coupled to the bath, L0 = aand L1 = a†, all others being zero. In the thermal case, coefficients A and B ofeqn (5.80) are interrelated (see eqn (5.82)). They can be made independent in thecontext of an external pumping where B remains the finite lifetime and A becomesthe rate an external incoherent pumping. The equation is then valid for long times inthe case where A < B as the number of particles increases without bounds and thesystem does not reach a steady state (the number of particles in the steady state be-ing 〈n〉ss = A/(B −A)).

Equation (5.83)—which we have derived from the Schrodinger equation in the ther-mal case after a physical model of coupling of a small system to a big reservoir—correctly reproduces features of dissipation, as can be seen by computing the equationof motions of the observables of interest. Consider, for example, the average number ofexcitations in an oscillator part of a reservoir, 〈n〉 = 〈a†a〉 = Tr

(ρa†a

). Its equation of

motion is given by (we assume we are now in the Schrodinger picture to keep a timeindependent but the case of the interaction picture follows straightforwardly):

∂〈n〉∂t

= Tr

[(∂ρ

∂t

)a†a

](5.84a)

= nγ

2Tr

[(2a†ρa− aa†ρ− ρaa†) a†a

](5.84b)

+ (n + 1)γ

2Tr

[(2aρa† − a†aρ− ρa†a

)a†a

]. (5.84c)

The approach of computing equations of motion of operators from the master equa-tion is outlined above. It only remains to simplify the expression (5.84) using algebraicrelations presented in Chapter 3. Let us carry out explicitly the case of line (5.84b):by cyclic permutation of the trace,80 the density matrix can be factored out to give(nγ/2)Tr

(ρ(2aa†aa†− a†aaa†− aa†a†a)

)that give rise to new operators that simpli-

fies further still in terms of quantities already known, for instance by noting that:

80Operators can be permuted cyclically in the trace, i.e., for arbitrary operators A, B and C:

Tr(ABC) = Tr(BCA) = Tr(CAB) , (5.85)

whereas it is not true, in general, for other permutations, e.g., Tr(ABC) = Tr(BAC).


2aa†aa† − a†aaa† − aa†a†a = aa†(aa† − a†a) + (aa† − a†a)aa† , (5.86a)

= aa†[a, a†] + [a, a†]aa† , (5.86b)

= 2aa† , (5.86c)

using expressions like eqn (3.59). In this way, the total expression finally reduces tonγ〈a†a〉 and if the same is carried out for line (5.84c), one gets:

∂〈n〉∂t

= −γ(〈n〉 − n) , (5.87)

i.e., the population relaxes to the Bose distribution n. This integrates immediately to

〈n〉(t) = 〈n〉(0)e−γt + n(1− e−γt) , (5.88)

which is the exact behaviour one would expect for dissipation of an harmonic oscillator:a decay from its initial value (or intensity) 〈n〉(0) towards the mean value of the reser-voir n. If the oscillator is initially in vacuum, it gets populated by thermal contact to thereservoir and comes to equilibrium with it on a timescale γ−1, whereas on the contraryif it has more excitations, it loses them to thermalize with the reservoir.

Note how line (5.82a) is linked to the decay in the sense that it empties the mode,whereas line (5.82b) also has the effect of a pump that can populate the mode. TheLindblad terms can be used to that effect in a large number of systems investigated inthe Schrodinger picture.81

Exercise 5.13 (∗∗) Show that the master equation (5.82) written for the Glauber Pfunction becomes (cf. relations (3.95)):

∂P

∂t=

[γ

2

(∂

∂αα +

∂

∂α∗α∗)

+ γn∂2

∂α∂α∗

]P . (5.89)

Exercise 5.14 (∗) Check that the following expression for P satisfies eqn (5.89) withinitial condition δ(α− α0) in P space (|α0〉 with Dirac notation).

P (α, α∗, t) =1

πn(1− exp(−γt))exp

[−|α− α0 exp(−(γ/2)t)|2

n(1− exp(−γt))

]. (5.90)

The dynamics of this solution is represented in Fig. 5.5, together with the solutioneqn (5.67) where, modelled after an Hamiltonian, the time evolution is cyclic in time.

5.5 Jaynes–Cummings model

All the previous material has been leading us towards the full quantum treatment of thetwo-level system interacting with a light mode, where both the atom (or exciton) and

81In the Heisenberg picture, the average of the reservoir operator serves as a fluctuating force that, bythe action of the “fluctuation–dissipation” theorem, also results in a decay of the averages. There is lessemphasis on the quantum aspect of the decay in this case since there is always the possibility or temptationto understand the dynamical averages as semiclassical and the whole formalism becomes one that favoursclassical interpretations or analogies.

JAYNES–CUMMINGS MODEL 193

Isidor Isaac Rabi (1898–1988) and Benjamin Mollow

Rabi studied in the 1930s the hydrogen atom and the nature of the force binding the proton to the atomic nu-clei, out of which investigations he conceived molecular-beam magnetic-resonance, a new and very accuratedetection method for which he was awarded the exclusive Nobel Prize for Physics in 1944, “for his resonancemethod for recording the magnetic properties of atomic nuclei”. He wanted to be a theorist and had acceptedonly because the invitation was from the prestigious Stern. In his biography he is quoted as saying “Wheneverone of my students came to me with a scientific project, I asked only one question, ‘Will it bring you nearer toGod?’”

Mollow is presently a Physics Professor at the University of Massachusets Boston where he studies quantumoptics.

the photon are quantized, according to eqn (5.29). The cases we have been dealing withso far, where the material excitation is modelled as a harmonic quantum oscillator, donot encompass the fermionic limit, which is the more important in the case of materialexcitations.

Different physics occurs when such excitations are described by fermionic ratherthan bosonic statistics, only in the nonlinear regime where more than one excitationresides in the system, so that higher energy states can be probed, the ground and firstexcited being identical for both Bose and Fermi statistics. With the advent of lasers, suchcases are, however, not difficult to realize experimentally. In the case of cavity QED thematerial is usually a beam of atoms passing through the cavity, and a single excitation isthe independent response of the atoms to the light-field excitation. The simplest situationis that of a dilute atomic beam where a single atom (driven at resonance so that it appearsas a two-level system) is coupled to a Fock state of light with a large number of photons.This case has been described theoretically by Jaynes and Cummings (1963) to yield themodel that now bears their name.

The model is similar to the Bloch equations except that the classical field is now up-graded to a quantum field, and for our single mode, to the Bose annihilation operator a.We rewrite the Hamiltonian (5.29) in the interaction picture and with the approximationof Jaynes and Cummings:

H = g(ei∆tσa† + e−i∆tσ†a) , (5.91)


Fig. 5.5: Evolution of the centre of mass of the P distribu-tion in the complex plane C (horizontal plane) as a functionof time t (vertical axis) of an oscillator a starting in a coher-ent state |α〉 in the cases of the coupling to another quan-tum oscillator, cf. eqn (5.67)—yielding beats—and of thecoupling to a reservoir, cf. eqn (5.82)—converging to thevacuum state. This shows the difference between a Hamil-tonian system and one in the presence of dissipation. Theoscillations are Rabi oscillations this time visualized in thecomplex plane. The closer the centre of mass to the centralaxis, the greater the decoherence.

where we have again denoted ∆ the detuning in energy between the cavity mode (withboson operator a) and the atom (with fermion operator σ). Observe that at resonance,∆ = 0, the Hamiltonian is time independent, which allows a direct solution throughquite painless algebra. Out of resonance, more mathematics are involved but in the limitof very high detunings, conservation of the number of photons and excitation in theatom is recovered and another limiting case is worthy of interest.

In the resonant case the Hamiltonian (5.91) becomes:

H = g(σa† + σ†a) , (5.92)

where σ† is a Pauli matrix that transfers one excitation from the radiation field to theatom, so that in matrix representation:

σ† =

(0 10 0

), (5.93)

with |0〉 = (0, 1)T and |1〉 = (1, 0)T . Because the excitation can only be transferredfrom the atom to the cavity or vice versa, but none is ever lost or created under thedynamics of eqn (5.92), all the dynamics of a fixed number n of total excitations iscontained within the manifold

Hn = |0, n〉 , |1, n− 1〉 , (5.94)

provided that n ≥ 1. The associated energy diagrams appear on the right of Fig. 5.3,with two states in each manifold (in our conventions |0, n〉 refers to the bare states with


Edwin Thompson Jaynes (1922–1998) and Frederick Cummings

Jaynes challenged the mainstream of physics, most notably with his proposition to use Bayesian probabilitiesto formulate a reinterpretation of statistical physics as inferences due to incomplete information. In quantumoptics, he rejected the Copenhagen interpretation that he qualified as “a considerable body of folklore”. Hewrote of Oppenheimer (whom he called Oppy and felt would be an unsuitable Ph.D. advisor to carry outindependent research):

Oppy would never countenance any retreat from the Copenhagen position, of the kind advocated bySchrodinger and Einstein. He derived some great emotional satisfaction from just those elements ofmysticism that Schrodinger and Einstein had deplored, and always wanted to make the world still moremystical, and less rational. This desire was expressed strongly in his 1955 BBC Reith lectures (of which Istill have some cherished tape recordings which recall his style of delivery at its best). Some have seen thisas a fine humanist trait. I saw it increasingly as an anomaly—a basically anti-scientific attitude in a personposing as a scientist—that explains so much of the contradictions in his character.

In quantum optics, he questioned the need of full quantization of the optical field, e.g., to explain effectssuch as blackbody radiation, spontaneous emission or the Lamb shift, the latter two he claimed could beobtained in the realm of his so-called neoclassical theory, whose fields offer the additional advantage to be“conspicuously free from many of the divergence problems of quantum electrodynamics.” Ironically, he istoday most remembered in quantum optics for the Jaynes and Cummings (1963) model the main merit ofwhich is to be an integrable fully-quantized system. This “drosophila” of quantum mechanics was initiallyderived in a form and with approximations allowing for comparison with a classical description in favour ofwhich Jaynes was ready to bet, like for the origin of the Lamb shift (the outcome of his bet with Lamb wasleft undecided).

Frederick Cummings was Jaynes’ student and is now a professor Emeritus at the University of California,Riverside. His interest turned to biophysics in the mid-1980s.

the atom in the ground-state and n photons, while |1, n− 1〉 has the atom in the excitedstate and n − 1 photons) and the total wavefunction |Ψ(t)〉 is a superposition of thesestates. If the number of excitations is not fixed, one can still decouple independent dy-namics of |Ψ〉 by projecting the state ontoHn, which, because they are not coupled, canalways be solved independently following the procedure below, and put back togetheragain at the end.

The formal solution of Schrodinger equation reads


|Ψ(t)〉 = exp

(− i

Ht

)|Ψ(0)〉 (5.95a)

= exp(−ig(σa† + σ†a)t

) |Ψ(0)〉 . (5.95b)

We now compute an expression for exp(−ig(σa† + σ†a)t

). By definition, exp(−ωΩ) =∑

n≥0(−ω)nΩn/n! with ω ∈ C and Ω ∈ ⊗n≥0Hn. The decomposition into real andimaginary parts of the exponential gives:

exp(−ig(σa† + σ†a)t

)= c− is , (5.96)

where

c =

∞∑n=0

(−1)n(gt)2n

(2n)!(σa† + σ†a)2n , (5.97a)

s =∞∑

n=0

(−1)n(gt)2n+1

(2n + 1)!(σa† + σ†a)2n(σa† + σ†a) , (5.97b)

and we are reduced to algebraic computation of the type (σa† + σ†a)2n:

Exercise 5.15 (∗∗) Taking advantage of the algebra of σ, especially of such propertiesas σ2 = σ†2 = 0, show that

(σa† + σ†a)2n =

((aa†)n 0

0 (a†a)n

)(5.98)

in the basis of bare states and consequently that c and s as defined by eqns (5.97) havethe following matrix representations:

c =

(cos(gt

√aa†) 0

0 cos(gt√

a†a)

)and s =

⎛⎜⎜⎝

0gt√

aa†√

aa† a

gt√

a†a√a†a

a† 0

⎞⎟⎟⎠ , (5.99)

where we noted, e.g., (√

a†a)2n = (a†a)n so as to carry out the series summationexactly with the new operator defined unambiguously (and as one can check, correctly)on the Fock-state basis.

Observe that the time propagator has off-diagonal elements. They correspond tovirtual processes where an odd number of excitations is exchanged between the twofields. The knowledge of eqns (5.99) provides the dynamics, including such effects,by direct computation. Let us consider the case of fixed n and the atom in a quantumsuperposition of ground and excited states:

|Ψ(0)〉 = (χg |g〉+ χe |e〉)⊗ |n〉 , (5.100)

which is separable because there is no summation to entangle with other configurations.The label n does not intervene directly and one can write:


|Ψ(0)〉 = χg |g, n〉+ χe |e, n〉 , (5.101)

which, from eqn (5.95b), leads to:

|Ψ(t)〉 = (c− is)(χg |g, n〉+ χe |e, n〉) (5.102a)

= χg[cos(√

ngt) |e, n〉 − i sin(√

ngt) |g, n− 1〉] (5.102b)

+ χe[cos(√

n + 1gt) |e, n〉 − i sin(√

n + 1gt) |g, n + 1〉] . (5.102c)

This dynamics bears much resemblance to the Rabi oscillations of Section 5.2 butthe more pronounced quantum character results in a more complicated dynamics fea-turing virtual transition off-diagonal dynamics of the density matrix.

If the dynamics is constrained from the initial condition to hold in a manifold, e.g.,if the initial state is

|Ψ(0)〉 = |e, n− 1〉 , (5.103)

then the only other state available to the dynamics under eqn (5.92) is |g, n〉 as one cancheck from the closure relations:

σa† |e, n− 1〉 = |g, n〉 , σ†a |e, n− 1〉 = 0 , (5.104a)

σa† |g, n〉 = 0 , σ†a |g, n〉 = |e, n− 1〉 . (5.104b)

The dynamics is therefore closed inHn where the Hamiltonian can be diagonalizedexactly. For the resonant condition the dressed states for this manifold are split by an en-ergy

√ng. In the general case, all four transitions between the states in manifolds Hn

and Hn−1 are possible, and this would result in a quadruplet in the emitted spectrum.It is difficult to resolve this quadruplet, but it has been done in a Fourier transform oftime-resolved experiments by Brune et al. (1996). It is simpler to consider photolumi-nescence directly under continuous excitation at high intensity (where the fluctuationsof particles number have little effect). In this case, with n 1, the two intermedi-ate energies are almost degenerate and a triplet is obtained with its central peak beingabout twice as high as the two satellites. This is the Mollow (1969) triplet of resonancefluorescence. We investigate this problem in more detail in Section 5.7.

Exercise 5.16 (∗) Consider now the out-of-resonance case where the two-level transi-tion does not match the cavity photon energy, i.e., ∆ = 0 in eqn (5.91). The Hamiltonianis now time dependent and the formal integration eqn (5.95a) is no longer possible. Tak-ing advantage of the closure relations (5.104), consider the ansatz

|Ψ(t)〉 =

∞∑n=0

[ψg,n+1(t) |g, n + 1〉+ ψe,n(t) |e, n〉] , (5.105)

where the time dependence is in the coefficients ψg,n+1, ψe,n only. Show therefore thatthe Schrodinger equation with Hamiltonian (5.91) applied on state (5.105) yields:

ψg,n+1 = −ig√

n + 1ei∆tψe,n

ψe,n = −ig√

n + 1e−i∆tψe,n .(5.106)


Evoking as they do the flopping between two states, those equations are reminiscentof Rabi dynamics.

Exercise 5.17 (∗∗) Use a technique of your choice to solve eqns (5.105). Show that theresult reads:

ψg,n+1(t) = ei∆t/2

[(cos

(1

2Ωnt

)− i

∆

Ωnsin

(1

2Ωnt

))ψg,n+1(0)

− i2g√

n + 1

Ωnsin

(1

2Ωnt

)ψe,n(0)

],

(5.107a)

ψe,n(t) = e−i∆/2

[(cos

(1

2Ωnt

)+ i

∆

Ωnsin

(1

2Ωnt

))ψe,n(0)

− i2g√

n + 1

Ωnsin

(1

2Ωnt

)ψg,n+1(0)

],

(5.107b)

in terms of the generalised Rabi frequencies:

Ωn =√

∆2 + 4(n + 1)g2 . (5.108)

Equations (5.107) and (5.108) represent one of the most important results of quantumoptics. They are the crowning achievement towards which all the results converge, eitherfrom simplified models such as Bloch optical equations or Rabi flopping between twostates, or from more refined theory such as non-rotating wave Hamiltonians that elabo-rate around this dynamics. This general solution connects directly to the resonant casetreated previously from operator algebra. In the detuned case, where 2g

√n + 1 |∆|,

a serial expansion of ψe,n(t) shows that the atom is inhibited in its transition and re-mains in the same state up to some phase fluctuations induced by the perturbation of theinteraction.82

5.6 Dicke model

Closely related to the linear coupling of the previous section lies the Dicke (1954) modelthat yields qualitatively similar results at low densities. In this model the matter exci-tation gets upgraded to creation operator J+ for an excitation of the “matter field” that

82Explicitly, it is found that

ψg,n+1(t) ≈ exp

„−i

g2(n + 1)

∆t

«ψb,n+1(0) and ψe,n(t) ≈ exp

„ig2(n + 1)

∆t

«ψe,n(0) .

(5.109)This can be used a posteriori to define a better ansatz for eqn (5.105).


Robert Dicke (1916–1997) was a versatile experimentalist, havingcontributed to the fields of radar physics, atomic physics, quantumoptics, gravity, astrophysics, and cosmology. His contribution to su-perradiance displays his aptitudes in the theoretical field as well. Hewrote his autobiography, dated 1975, but it has not been published (itis now possessed by NAS).

He once wrote: “I have long believed that an experimentalist shouldnot be unduly inhibited by theoretical untidiness. If he insists on hav-ing every last theoretical T crossed before he starts his research thechances are that he will never do a significant experiment. And themore significant and fundamental the experiment the more theoreti-cal uncertainty may be tolerated.”

distributes the excitation throughout an assembly of N identical two-level systems de-scribed by fermion operators σi, so that b† in eqn (5.30) maps to J+ with

J+ =

N∑i=1

σ†i . (5.110)

One checks readily that J+ and J− = J†+ thus defined obey an angular momentum

algebra with magnitude N(N + 1) (and maximum z projection of Jz equal to N ). Inthis case, the Rabi doublet arises in the limit where the total number of excitations µ(shared between the light and the matter field) is much less than the number of atoms,µ N , in which case the usual commutation relation [J−, J+] = −2Jz becomes[J−/

√N, J+/

√N ] ≈ 1, which is the commutation for a bosonic field. This comes

from the expression of a Dicke state with µ excitations shared by N atoms given asthe angular momentum state |−N/2 + µ〉. Therefore, the annihilation/creation opera-tors J−, J+ for one excitation shared by N atoms appear in this limit like renormalizedBose operators

√Na,

√Na†, resulting in a Rabi doublet of splitting 2g

√N . Such a

situation corresponds, e.g., to an array of small QDs inside a microcavity such that ineach dot electron and hole are quantized separately, while our model describes a singleQD that can accommodate several excitons. The corresponding emission spectra areclose to those obtained here below the saturation limit µ N , while the nonlinearregime N 1, µ 1 has peculiar behaviour, featuring non-Lorentzian emission line-shapes and a non-trivial multiplet structure, like the “Dicke fork” obtained by Laussyet al. (2005).

5.7 Excitons in semiconductors

In the following chapters we pursue the investigation of light–matter coupling in boththe semiclassical and quantum regimes, putting more emphasis on specificities of mi-crocavities. To bring forward these results we give now more elements on the materialexcitations of semiconductors that parallel the exposition of the previous chapter, butfrom a quantum-mechanical perspective. At this stage we shall change notations for the


fields to follow popular customs, so that for instance, a that was previously a cavitymode (annihilating a photon) will now typically refer to the polariton. The new nota-tions will be introduced as we proceed.

5.7.1 Quantization of the exciton field

The second-quantized Hamiltonian of a semiconductor at the fermionic level reads, inreal space:

H =

∫Ψ(r)†

(− 2

2m∇2 + V (r)

)Ψ(r) dr (5.111a)

+1

2

∫ ∫Ψ(r)†Ψ(r′)†

e2

|r− r′|Ψ(r′)Ψ(r) dr dr′ , (5.111b)

where Ψ(r) is the electron annihilation field operator and V the Coulomb potential. Weexpand Ψ in terms of ϕi,k(r) = 〈r|i,k〉 the single-particle wavefunction labelled by thequantum number k in the ith semiconductor band, in terms of the electron annihilationoperator ei,k:

Ψ(r) =∑

i∈c,v

∑k

ϕi,k(r)ei,k . (5.112)

Because of interactions and correlations, the determination of ϕi,k(r) is a difficulttask, typically solved numerically. The full many-body problem (5.111) can be approxi-mated to an effective single-body problem through the so-called Hartree–Fock approxi-mation, which introduces an effective potential Veff . The resulting “Schrodinger” equa-tion with Hamiltonian HHF = −2∇2/2m + Veff is nonlinear since the potential de-pends on the wavefunction ϕ, and so the problem remains one of considerable difficulty.Bloch’s theorem, however, allows a statement of general validity:

ϕi,k(r) ∝ eik · rui,k(r) , (5.113)

with u having the same translational symmetry as the crystal.Once all the algebra has been gone through, the semiconductor Hamiltonian (5.111)

becomes, in reciprocal space:

H =∑

i∈c,v

∑k

Ei(k)e†i,kei,k (5.114a)

+1

2

∑i∈c,v

∑k,p,q =0

V (q)e†i,k+qe†i,p−qei,pei,k (5.114b)

+∑

k,p,q =0

V (q)e†c,k+qe†v,p−qev,pec,k , (5.114c)

with Ei the dispersion relation for the ith band and V (q) the Fourier transform of theCoulomb interaction.


The limit of very low density (in fact in the limit where the ground state is devoidof conduction-band electrons) allows some analytical solutions to be obtained after per-forming approximations. One simplification, both conceptual and from the point of viewof the formalism, is the introduction of the hole (fermionic) operator h as

hk = e†v,−k (5.115)

(spin is also reversed if granted). This allows elimination of the negative effective massof valence electrons and enables us to deal with an excitation as an “addition” of aparticle rather than annihilation (in terms of valence electrons, the ground-state is fullof electrons and gets depleted by excitations). Conceptually, it replaces a sea of valenceelectrons by a single particle, making it easier to conceive the exciton as a bound state.In terms of electrons ek and holes hk, eqn (5.114) now reads

H =∑k

[Ee(k)e†kek + Eh(k)h†khk] (5.116a)

+1

2

∑k,p,q =0

V (q)[e†k+qe†p−qepek + h†k+qh†

p−qhphk] (5.116b)

−∑

k,p,q =0

V (q)e†k+qh†p−qhpek , (5.116c)

with explicit expression for electron and hole dispersion (as a function of their effectivemass):

Ee(k) = Egap +(k)2

2m∗e

, (5.117a)

Eh(k) =(k)2

2m∗h

. (5.117b)

In the low-density limit, if one neglects line (5.116b) in the Hamiltonian, it can bediagonalized by introducing the exciton operator

Xν(k) ≡∑p

ϕν(p)hk/2−pek/2+p , (5.118)

with ϕν(p) the Fourier transform of Wannier equation eigenstates (with ν the quantumnumbers, as for the hydrogen atom; the spectrum of energy we call Eν).

The exciton Hamiltonian becomes:

H =∑ν,k

EνX(k)X†

ν(k)Xν(k) , (5.119)

withEν

X(k) = Eν + Ee(k) + Eh(k) . (5.120)


5.7.2 Excitons as bosons

Excitons behave as true bosons when the commutator of the field operators satisfies therelation:

[Xν(k),X†µ(q)] = δν,µδk,q . (5.121)

Direct evaluation of the commutator with explicit expression (5.118) yields

[Xν(k),X†µ(q)] = δν,µδk,q −

∑p

|ϕ1s(p)|2(c†kcq + h†−kh−q) , (5.122)

so that the diagonalization is legitimate at low densities. In particular, 〈[X0, X†0]〉 =

1 − O(Na2B), where N is the density of excitons and aB is the Bohr radius asso-

ciated with ϕ1s. One can therefore treat excitons as bosons with confidence, in thelimit Na2

0 1.

5.7.3 Excitons in quantum dots

We now show one possible route of extending the results of the previous sections. Wehave investigated the limiting cases where the material excitation that couples to the fieldis either a boson, or a fermion. In actual systems, composite particles are neither exactlyone nor the other. The importance of this distinction can become important in a quantumdot (QD), where the excitations are located in a tiny region of real space, so that theirwavefunctions overlap appreciably. If the confining potential of the dot is much strongerthan the Coulomb interaction, electrons and holes, which are elementary excitations ofthe system, will be quantized separately, whereas if Coulomb interactions dominate overthe confinement, one electron–hole pair will bind as an exciton and therefore behaverather like a boson. Here, we investigate a model of interactions of light with excitonsin QDs of varying size, where their boson or fermion character is tuneable.

It is more relevant to carry out the analysis in real space since the QD states arelocalized. We note

ϕene

(re) = 〈re|ϕene〉 and ϕh

nh(rh) = 〈rh|ϕh

nh〉 (5.123)

the set of their basis wavefunctions with re and rh the positions of the electron and hole,respectively. Subscripts ne and nh are multi-indices enumerating all quantum numbersof electrons and holes. The specifics of the three-dimensional confinement manifestsitself in the discrete character of ne and nh components. We restrict our considerationsto direct-bandgap semiconductors with non-degenerate valence bands. Such a situationcan be experimentally achieved in QDs formed in conventional III-V or II-VI semi-conductors, where the light-hole levels lie far below, in energy, the heavy-hole onesdue to the effects of strain and size quantization along the growth axis. Therefore, onlyelectron–heavy-hole excitons need to be considered. Moreover, we will neglect the spindegree of freedom of the electron–hole pair and assume all carriers to be spin polarized.To carry out the same formalism as presented in the previous sections, we need to buildthe second quantized operator for the QD. We define it as:

X† =∑

ne,nh

Cne,nhe†ne

ς†nh, (5.124)


where eneand hnh

are fermion creation operators for an electron and a hole in state∣∣ϕe

ne

⟩and

∣∣ϕhnh

⟩, respectively:

e†ne|0〉 =

∣∣ϕene

⟩, h†

nh|0〉 =

∣∣ϕhnh

⟩, (5.125)

with |0〉 denoting both the electron and hole vacuum fields. The (single) exciton wave-function |ϕ〉 results from the application of X† on the vacuum. In real-space coordi-nates:

〈re, rh|ϕ〉 = ϕ(re, rh) =∑

ne,nh

Cne,nhϕe

ne(re)ϕ

hnh

(rh). (5.126)

At this stage we do not specify the wavefunction (that is, the set of coefficients Cne,nh),

which depends on various factors such as the dot geometry, electron and hole effectivemasses and dielectric constant. Rather, we consider the n-excitons state that results fromsuccessive excitation of the system through X†:

|Ψn〉 = (X†)n |0〉 . (5.127)

The associated normalized wavefunction |n〉 reads

|n〉 =1

Nn|Ψn〉 , (5.128)

where, by definition of the normalization constant

Nn =√〈Ψn|Ψn〉. (5.129)

The creation operator X† can now be obtained explicitly. We define αn the nonzeromatrix element that lies below the diagonal in the exciton representation:

αn = 〈n|X†|n− 1〉 , (5.130)

which, by comparing eqns (5.127) and (5.130) turns out to be

αn =Nn

Nn−1. (5.131)

The coefficients αn can be linked to the coefficients Cne,nh(the latter assuming a

specific value when the system itself is known):

Exercise 5.18 (∗∗∗) Show that the normalization coefficients N necessary to computethe matrix elements αn (through eqn (5.131)), can be computed by the following recur-rent relation:

N 2n =

1

n

n∑m=1

(−1)m+1βmN 2n−m

m−1∏j=0

(n− j)2, (5.132)

with N0 = 1 and βm the irreducible m-excitons overlap integrals, 1 ≤ m ≤ n:

βm =

∫ (m−1∏i=1

ϕ∗(rei, rhi

)ϕ(rei, rhi+1

)

)ϕ∗(rem

, rhm)ϕ(rem

, rh1)

dre1. . . drem

drh1. . . drhm

. (5.133)


The procedure to calculate the matrix elements of the creation operator is as fol-lows:83 One starts from the envelope function ϕ(re, rh) for a single exciton. Thenone calculates all overlap integrals βm as given by eqn (5.133), for 1 ≤ m ≤ nwhere n is the highest manifold to be accessed. Then the norms can be computed witheqn (5.132). Finally the matrix elements αn are obtained as the successive norms ratio,cf. eqn (5.131).

The limiting cases of Bose–Einstein and Fermi–Dirac statistics are recovered in thelimits of large and shallow dots, respectively. This is made most clear through consid-eration of the explicit case of two excitons (n = 2). Then the wavefunction reads

Ψ2(re1, re2

, rh1, rh2

) = ϕ(re1, rh1

)ϕ(re2, rh2

)− ϕ(re1, rh2

)ϕ(re2, rh1

) , (5.134)

with its normalization constant (5.129) readily obtained as

N 22 =

∫|Ψ2(re1

, re2, rh1

, rh2)|2dre1

. . . drh2= 2− 2β2 , (5.135)

where β2, the two-exciton overlap integral, reads explicitly

β2 =

∫ϕ(re1

, rh1)ϕ(re2

, rh2)ϕ(re1

, rh2)ϕ(re2

, rh1)dre1

. . . drh2. (5.136)

This integral is the signature of the composite nature of the exciton. The minussign in eqn (5.135) results from the Pauli principle: two fermions (electrons and holes)cannot occupy the same state. Assuming ϕ(re, rh) is normalized,N1 = 1, so accordingto eqn (5.131),

α2 =√

2− 2β2. (5.137)

Since 0 ≤ β2 ≤ 1 this is smaller than or equal to√

2, the corresponding matrix elementof a true boson creation operator. This result has a transparent physical meaning: sincetwo identical fermions from two excitons cannot be in the same quantum state, it is“harder” to create two real excitons, where the underlying structure is probed, thantwo ideal bosons. We note that if L is the QD lateral dimension, β2 ∼ (aB/L)2 1when L aB. Thus, in large QDs the overlap of excitonic wavefunctions is small, soα2 ≈

√2 and the bosonic limit is recovered. On the other hand, in a small QD, where

Coulomb interaction is unimportant compared to the dot potential confining the carriers,the electron and hole can be regarded as quantized separately:

ϕ(re, rh) = ϕe(re)ϕh(rh) . (5.138)

In this case all βm = 1 and subsequently all αm = 0 with the exception of α1 = 1.This is the fermionic limit where X† maps to the Pauli matrix σ+.

83The numerical computation of the βm and αn values needs to be carried out with great care. Thecancellation of the large numbers of terms involved in eqn (5.132) requires a very high precision evaluationof βm.


5.7.3.1 Gaussian toy model We now turn to the general case of arbitrary-sized QDs,interpolating between the (small) fermionic and (large) bosonic limits. We do not at-tempt for this conceptual presentation to go through the lengthy and complicated taskof the numerical calculation of the exciton creation operator matrix elements for a real-istic QD. Rather, we consider a model wavefunction that can be integrated analyticallyand illustrates some expected typical behaviours.

We consider a QD strongly confined in one direction (along the z-axis) and hav-ing a symmetrical shape in the xy-plane with, possibly, larger dimensions. This cor-responds to realistic self-assembled semiconductor QDs (Widmann et al. 1997). Weassume a Gaussian form for the wavefunction that allows evaluation analytically of allthe required quantities. This follows from a harmonic confining potential, as has beenconsidered for instance by Que (1992). As numerical accuracy is not the chief goal ofthis work we further assume inplane coordinates x and y to be uncorrelated to ease thecomputations. The wavefunction reads:

ϕ(re, rh) = C exp(−γer2e − γhr

2h − γehre · rh) , (5.139)

properly normalized with

C =

√4γeγh − γ2

eh

π, (5.140)

provided that γeh ∈ [−2√

γeγh, 0] with γe, γh ≥ 0. The γ parameters allow inter-polation between the large and small dot limits within the same wavefunction (cf. Sec-tion 4.3.3). To connect these parameters γe, γh and γeh to physical quantities, eqn (5.139)is regarded as a trial wavefunction that is to minimize the Hamiltonian HQD confiningthe electron and hole in a quadratic potential where they interact through Coulomb in-teraction:

HQD =∑

i=e,h

(p2

i

2mi+

1

2miω

2r2i

)− e2

4πεε0|re − rh| . (5.141)

Here, pi is the momentum operator for the electron and hole, i = e, h, respectively, me,mh the electron and hole masses, ω the frequency that characterizes the strength of theconfining potential, e the charge of the electron and ε the background dielectric constantscreening the Coulomb interaction. This Hamiltonian defines the two length scales ofour problem, the 2D Bohr radius aB and the dot size L:

aB =4πεε02

2µe2, (5.142a)

L =

√

µω, (5.142b)

where µ = memh/(me+mh) is the reduced mass of the electron–hole pair. To simplifythe following discussion we assume that me = mh, resulting in γe = γh = γ. The trial


wavefunction (5.139) separates as ϕ(re, rh) = CΦ(R)φ(r), where r = re − rh is theradius-vector of relative motion and R = (re + rh)/2 is the centre-of-mass position:

Φ(R) =

√2(2γ + γeh)√

πexp

(−R2[2γ + γeh]), (5.143a)

φ(r) =

√2γ − γeh√

2πexp

(−r2

[2γ − γeh

4

]), (5.143b)

Equation (5.143a) is an eigenstate of the centre-of-mass energy operator and equat-ing its parameters with those of the exact solution yields the relationship 2γ + γeh =2/L2. This constraint allows us to minimize eqn (5.143b) with respect to a single pa-rameter, a = −γeh/2 + 1/(2L2), which eventually amounts to minimizing 4aB/a2 +aBa2/L4 − 2

√π/a. On doing so we obtain the ratio −γeh/γ as a function of L/aB,

displayed in Fig. 5.6. The transition from the bosonic to the fermionic regime is seen tooccur sharply when the dot size becomes commensurate with the Bohr radius. For large

Fig. 5.6: Ratio of parameters −γeh and γ (with γ = γe = γh) as a function of L/aB. For large dots,where L aB, −γeh/γ ≈ 2, which corresponds to the bosonic limit where the electron and hole arestrongly correlated. For shallow dots, where L aB, −γeh/γ ≈ 0, with electron and hole quantizedseparately. The transition is shown as the result of a variational procedure, with an abrupt transition when thedot size becomes comparable to the Bohr radius.

dots, i.e., for large values of L/aB, the ratio is well approximated by the expression

−γeh/γ = 2 − (aB/L)2 , (5.144)

so that in the limit of big dots where aB/L → 0, eqn (5.139) becomes ϕ(re, rh) ∝exp(−(

√γere − √

γhrh)2) with vanishing normalization constant. This mimics a free-exciton in an infinite quantum well. It corresponds to the bosonic case. On the otherhand, if L is small compared to the Bohr radius, with γeh → 0, the limit (5.138) isrecovered with ϕ ∝ exp(−γer

2e) exp(−γhr

2h). This corresponds to the fermionic case.


The trial wavefunction is of course not exact84 but the exciton operator that it yieldsis exact, as are all the intermediates.85

Together with eqns (5.140), (5.148) and (5.149), expression (5.145) in footnote 85,provides the βm in the Gaussian approximation. One can see the considerable com-plexity of the expressions despite the simplicity of the model wavefunction. Even anumerical treatment meets with difficulties owing to manipulations of a series of large

84One can check that eqn (5.139) gives, in the case γeh → −2√

γeγh, an exciton binding energy thatis smaller by only 20% than that calculated with a hydrogenic wavefunction, which shows that the Gaussianapproximation should be tolerable for qualitative and semiquantitative results.

85 It can be seen, for instance, that the overlap integrals (5.133) take a simple form in terms of multivariateGaussians as a function of a matrix A defined below:

βm = C2m

Zexp(−xT Ax) dx

Zexp(−yT Ay) dy , (5.145)

where

xT = (xe1 , xe2 , . . . , xem , xh1, xh2

, . . . , xhm ) , (5.146a)

yT = (ye1 , ye2 , . . . , yem , yh1, yh2

, . . . , yhm ) , (5.146b)

are the 2m-dimensional vectors that encapsulate all the degrees of freedom of the m excitons-complex, and Ais a positive-definite symmetric matrix that equates eqn (5.133) and (5.145), i.e., which satisfies

xT Ax = 2γe

mXi=1

x2i + 2γh

2mXi=m+1

x2i + γehxmxm+1

+ γeh

mXi=1

xixm+i + γeh

m−1Xi=1

xixm+i+1 , (5.147)

and likewise for y (to simplify notation we have not written an index m on x, y and A, but these naturallyscale with βm). The identity for 2m-fold Gaussian integrals isZ

exp(−xT Ax) dx =πm

√det A

. (5.148)

The problem is now reduced to the determinant of A, which, being a sparse matrix, also admits an analyticalsolution, though this time a rather cumbersome one. The determinant of the matrix A can be computed as:

det A = γeh2m

mXk=0

m−kXl=0

(−1)m/2+kAm(k, l)

„γeγh

γeh2

«k

. (5.149)

Here, we introduced a quantity

Am(k, l) = A′m(k, l) +

mXi=1

`A′m−i(k, l − i) −A′

m−i−1(k, l − i)´

, (5.150)

and

A′m(k, l) =

p(l)Xη=1

(P

i νlη(i))!Q

i νlη(i)!

×“ m − lP

i νlη(i)

”“m − l − Pi νl

η(i)

k − Pi νl

η(i)

”, (5.151)

with k ∈]0, m], l ∈ [0, m] and p(l) and νη(i) already introduced as the partition function of l and thenumber of occurences of i in its ηth partition. For the case k = 0 the finite size of the matrix implies a specialrule that reads Am(0, l) = 4δm,lδm≡2,0.


quantities that sum to small values. αn are obtained by exact algebraic computations tofree coefficients from numerical artifacts.

Fig. 5.7: (a) Matrix elements αn of the exciton creation operator X† calculated for n ≤ 15 for variousGaussian trial wavefunctions corresponding to various sizes of the dot. The top curve shows the limit of truebosons, where αn =

√n, and the bottom curve the limit of true fermions, where αn = δn,1. Intermediate

cases are obtained for values of γeh from −1.95√

γeγh down to −0.2√

γeγh, interpolating between theboson and fermion limits. (b) Magnified region close to the fermion limit. Values displayed are everywheregiven in units of

√γeγh.

Figure 5.7 shows the behaviour of αn for different values of γeh interpolating fromthe bosonic case (γeh = −2

√γeγh) to the fermionic case (γeh = 0). The crossover

from the bosonic to the fermionic limit can be clearly seen: for γeh close to −2√

γeγh,the curve behaves like

√n, the deviations from this exact bosonic result becoming more

pronounced with increasing n. For γeh close to −2√

γeγh, the curve initially behaveslike

√n, the deviations from this exact bosonic result becoming more pronounced with

increasing n. The curve is ultimately decreasing beyond a number of excitations thatis smaller the greater the departure of γeh from −2

√γeγh. After the initial rise, as

the overlap between electron and hole wavefunctions is small and bosonic behaviouris found, the decrease follows as the density becomes so large that Pauli exclusion be-comes significant. Then, excitons cannot be considered as structureless particles, andfermionic characteristics emerge. With γeh going to 0, this behaviour is replaced by amonotonically decreasing αn, which means that it is “harder and harder” to add exci-tons in the same state in the QD; the fermionic nature of excitons becomes more andmore important.

5.8 Exciton–photon coupling

The polaritons discussed in Section 4.4.4.2 can be seen in a simplified but very accuratemodel as the new eigenstates that arise from the coupling of two oscillators, i.e., thephoton and the exciton. Vividly, the polariton is then seen as the chain process wherethe exciton annihilates, emitting a photon with the same energy E and momentum k,which is later reabsorbed by the medium, creating a new exciton with the same (E,k),


and so on until the excitation finds its way out of the cavity (resulting in the annihilationof the polariton), or the electron or hole is scattered.

In this part, we neglect for brevity spins and states other than 1s for the exciton. Thedipole moment −er of the electron–hole pair couples to the light field E and adds thefollowing coupling Hamiltonian to (5.111):

HCX =

∫Ψ†(r)[−er ·E(r)]Ψ(r) dr . (5.152)

The same procedure to obtain eqn (5.116) from eqn (5.111) including eqn (5.152) leadsto the following exciton–photon coupling Hamiltonian:

H =∑k

EC(k)B†kBk +

∑k

EX(k)X†kXk +

∑k

g(k)(B†kXk + XkB†

k) , (5.153)

with g(k) = µcvϕ1s(0)√

EC(k)/(2ε0ε), µcv being the dipole matrix element dottedwith electron and hole, B the photon annihilation operator and X its exciton coun-terpart. Hamiltonian (5.153) can be diagonalized provided that X operators obey thebosonic algebra of eqn (5.121), following exactly the same procedure as those employedpreviously for two linearly coupled harmonic oscillators, starting with eqns (5.37). Insuch an approximation, the Hamiltonian (5.153), bilinear in bosonic operators, is diag-onalized with the Hopfield transformation:

aLk ≡ XkXk − CkBk, (5.154a)

aUk ≡ CkXk + XkBk, (5.154b)

where the so-called Hopfield coefficients Ck and Xk satisfy C2k + X 2

k = 1, so that thetransformation is canonical and a operators follow the bosonic algebra as well. As pre-viously with excitons, Hamiltonian (5.153) reduces to free-propagation terms only

H =∑k

EU(k)aU†k aU

k +∑k

EL(k)aL†k aL

k , (5.155)

for upper and lower-polariton branches, with second quantized annihilation opera-tors aU and aL, respectively. The dispersion relations for these branches are:

EUL(k) =

1

2(EX(k) + EC(k))± 1

2

√∆2

k + 42g(k)2 , (5.156)

(the U subscript is associated with the plus sign, L with minus), where EC is givenby expression (4.134) and EX by eqn (5.120), and ∆k is the energy mismatch, i.e.,detuning, between the cavity and exciton modes:

∆k ≡ EC(k)− EX(k) . (5.157)

The Hopfield coefficients used to diagonalize this Hamiltonian are related to the mixingangle, cf. eqn (5.46) and Fig. 5.4. Their square correspond to the photon (resp. exciton)


fraction of the polariton, that is, to the probability for the mixed-exciton–photon particleto be found in either one of these states. These probabilities are conveniently given as afunction of the dispersion relations:

|Ck|2 =EU(k)EX(k)− EL(k)EC(k)(

EC(k) + EX(k))√

∆2k + 42g(k)2

, (5.158a)

|Xk|2 =EU(k)EC(k)− EL(k)EX(k)(

EC(k) + EX(k))√

∆2k + 42g(k)2

. (5.158b)

Again, because of their connection with the mixing angle, a plot of eqns (5.158) appearson Fig. 5.4b.

5.8.1 Dispersion of polaritons

Equation (5.156) is one of the major results of microcavity polaritons physics for thevarious consequences this relation bears on many key issues that we are going to addressin the next chapter. It is plotted in solid lines in Fig. 4.23 where are also plotted in dashedlines the dispersions for the exciton, eqn (5.120), and the photon, eqn (4.134). The firstand third of these figures display negative (where bare dispersions cross each other) andpositive (where they do not) detunings, respectively, while the central figure displayszero detuning (resonance at k = 0).

As the result of the exciton–photon interaction, there is an avoided crossing (anti-crossing) of energies. The polariton arises as a coherent mixture of the photon and ex-citon states whose fractions are given by Hopfield coefficients (5.158). As already said,the polariton is the true eigenstate of the system, whereas photon and exciton modes aretransient states, exchanging the energy at the Rabi frequency ΩR. In this simple picturethe anticrossing appears however weak the interaction. This can be made more real-istic, taking into account the broadening of exciton and photon resonances, by addingto eqns (5.120) and (4.134) imaginary components −iγx and −iγc, respectively. γx isthe broadening caused by exciton interactions (interparticle or with phonons), while γc

reflects the finite reflectivity that is inversely proportional to the quality factor of thecavity. At zero detuning eqn (5.156) then becomes

EUL(k) =

1

2(EX(k) + Eγ(k)− iγx − iγc)± 1

2

√2Ω2

R − (γx − γc)2. (5.159)

This expression depends crucially on the sign of the expression below the square root,demonstrating that the physical behaviour of the system depends on the interrelationbetween the strength of the exciton–photon coupling and dissipation. If ΩR > (γx −γc), EU

Lexhibits the energy splitting already encountered, the so-called Rabi splitting

that corresponds to the strong coupling regime, where the correlations between excitonand photon are important and their interaction cannot be dealt with in a perturbativeway. An altogether new behaviour of the system is expected and should be describedin terms of polaritons. The term vacuum Rabi splitting was introduced by Sanchez-Mondragon et al. (1983) to differentiate this from Rabi splitting when the oscillationis between two populated modes (rather than with the vacuum of the other excitation).


This terminology is now generally accepted, though some would refer to “normal modesplitting”. To put an emphasis on the specificity of microcavities, the denominations“dressed exciton splitting” or “polariton splitting” have also been used, but are nowencountered only for stylistic purposes.

Note that the splitting presented so far relates to the spectrum of energy and is rathera theoretical notion. Experimentally, this splitting relates in a more subtle way to split-tings in reflectivity (R), transmission (T), absorption (A) and photoluminescence (PL).Indeed this splitting is in general different in all these cases and is also different fromthe exciton–photon coupling constant g. Yet it can be shown that the general followingrelation holds:

∆ER ≥ ∆ET ≥ ∆EA , (5.160)

and also Ω ≥ ∆EA. Only the splitting in absorption unambiguously proves strong cou-pling, while the splitting in transmission or reflectivity is a necessary but not sufficientcondition. It might therefore be the most important experimental expression, which wegive here:

∆EA =

√Ω2 − (γ2

c + γ2x)

2, (5.161)

along with the PL splitting

∆EPL =

√2ΩR

√Ω2

R + 4(γx + γc)2 − Ω2R − 4(γx + γc)2 , (5.162)

so that it is possible that although in the strong-coupling regime (where the device wouldexhibit effects expected from polaritons), the splitting cannot be resolved from PL ob-servations. Savona et al. (1998) give an excellent and more detailed discussion on thesepoints.

On the other hand, if ΩR < |γx − γc|, the square root becomes imaginary and thusthe (real) energy anticrossing disappears. This is the weak-coupling regime where thesystem can be described in terms of weakly interacting photon and exciton. Now theenergies are degenerate at some particular k and the broadenings (imaginary part ofeqn (5.159)) do differ. More detailed analyses show that with the reflectivity going tozero, the broadening of the photon mode diverges (the photon does not remain in thecavity) and the broadening of the exciton mode approaches the bare QW spontaneousemission rate, describing in effect weakly-interacting photons and excitons.

5.8.2 The polariton Hamiltonian

Now that the free-polariton Hamiltonian has been obtained, one can proceed with deriv-ing next-order or additional processes that will account for the dynamics of polaritons.Such additions include polariton–polariton interactions (from the underlying exciton–exciton interaction coming from Coulomb interaction), polariton–phonon interactionand possibly such terms as polariton-electron interaction if there is residual doping, po-lariton coupling to the external electromagnetic field giving them a chance to escape thecavity, a term that is ultimately responsible for decay, and its counterpart that injects


particles, behaving as a pump. Below we provide the expression for these terms withoutdetailing their derivations. A thorough account is given by Savona et al. (1998).

In terms of the annihilation and creation operators ak, a†k for polaritons and bk, b†k

for phonons (where k is a two-dimensional wavevector in the plane of the microcavity),obeying the usual bosonic algebra, our model Hamiltonian in the interaction represen-tation reads

H = Hpump + Hlifetime + Hpol−phon + Hpol−el + Hpol−pol . (5.163)

The usual models considered for these various terms are as follow:

Hpump =∑k

g(k)(Kpumpa†k + K∗

pumpak) (5.164)

This term describes the pumping of the system by a classical light field of ampli-tude Kpump (a scalar as opposed to ak, which is an operator). g(k) is the wavevector-dependent coupling strength between the two fields. This Hamiltonian is adapted todescribe the resonant pumping or the non-resonant pumping case depending on the ap-proximations performed.

Hlifetime =∑k

γ(k)(αka†k + α†

kak) . (5.165)

Hlifetime describes the linear coupling between the polariton field and an empty ex-ternal light field responsible for the polariton lifetime. What happens in reality is thatphotons escape the discrete cavity mode into a continuum from where their probabilityof return is zero. This translates as a decay. αk, α†

k are the annihilation-creation op-erators of the external light field. γ(k) is the wavevector-dependent coupling strengthbetween the two fields.

Hpol−phon =∑

k =0,q =0

U(k,q)ei(E(k)+ωq−E(k+q))ta†

kbqak+q + h.c. (5.166)

describes the coupling between the polariton a and the phonon b fields. U is the Fouriertransform of the interaction potential for polariton–phonon scattering. ωq is the phonondispersion, E(k) is the dispersion of the lower-polariton branch.

Hpol−el =1

2

∑k=0,p=0

q=0

U el(k,p,q)ei

(E(k+q)−E(k)+

2

2me(|p−q|2−|p|2)

)ta†

k+qake†p−qep

(5.167)

describes the coupling between the polariton a and the electron e fields. ek, e†k are theannihilation-creation operators of the electron field, me is the electron mass, U el is theFourier transform of the interaction potential for polariton-electron scattering.


Hpol−pol =1

2

∑k=0,p=0

q=0

V (k,p,q)ei

(E(k+q)+E(p−q)−E(k)−E(p)

)ta†

k+qa†p−qakap

(5.168)

describes the polariton–polariton interaction. V is the Fourier transform of the interac-tion potential for this scattering.

This Hamiltonian will be a starting point for Chapters 7 and 8 where various ap-proximations are made to single out the resonant dynamics where polariton–polaritoninteractions dominate, or out-of-resonance where relaxations bring the system in quasi-equilibrium with reservoirs as a function of pump and decay. In Chapter 9, an extensionof eqn (5.168) will be carried out to include the spin degree of freedom.

5.8.3 Coupling in quantum dots

Although very close in principle, cavity QED (cQED) in atomic cavities and in a solid-state system present many differences.86 A good overview of the cQED with QDs isgiven by Imamoglu (2002), which provides in the abstract the main appealing featureof the solid state realization, namely that “since quantum dot location inside the cav-ity is fixed by growth, this system is free of the stringent trapping requirements thatlimit its atomic counterpart” and further comment that “fabricating photonic nanos-tructures with ultrasmall cavity-mode volumes enhances the prospects for applicationsin quantum information processing.” Experimental findings by Reithmaier et al. (2004)and Yoshie et al. (2004), published as two consecutive letters to Nature magazine, andthat of Peter et al. (2005), have reached the final step of strong coupling (see Chap-ter 7). Although the exciton-polariton in microcavities as first observed by Weisbuchet al. (1992)—a long time ago and nowadays well understood theoretically as well asfinely controlled experimentally—had been initially thought to provide the solid-staterealization, we have discussed at length how the polaritons represent in fact a more com-plicated system. One might think that the delay to strip down this complexity to finallyachieve the exact counterpart of one atomic-like system coupled to a mode of radiationwithout further degrees of freedom was merely a technical problem, and that the theorywas already laid down in twenty years of literature.

However there is much evidence that the semiconductor case brings forward manyspecificities of its own. The most important one playing in favour of the solid-stateimplementation being indeed that in this case one has a much better control of theatomic-like excitation. A QD stuck in the cavity can be kept immobile, while in theatomic cavities case, the excitations are beams of atoms with much difficulty to singleout one atom or to deal with it for prolonged periods of time. In the case of semicon-ductors, for instance, one can expect a much better investigation of coherent exchangesbetween many QDs through the radiation field, as is modelled by the Dicke model of ntwo-level atoms interacting through a single mode of radiation. All variations of these,

86In the words of Weisbuch et al. (1992): “Besides its relying on a much simpler implementation—thesolid-state system is monolithic—the effect should lead to useful applications.”


from phase-mismatched leading to subradiant configurations where the excitation oftwo QDs is exchanged back and forth and does not escape, to the cooperative one—inphase—where the excitation is collectively shared leading to superradiant emission, canbe obtained by proper identification of the mapping of QDs in the structure and tuningthe excitation accordingly. The model developed by Dicke becomes all-important in thiscase. These are directions of research that are just emerging and show how the field ofsolid-state microcavities has a significant future ahead.

To illustrate this richness we present in Fig. 5.8 a density plot of the emission spectraof a QD as modelled in Section 5.7.3, as a function the size of the dot (we refer to thissection for the notations). Two limiting cases of the fundamental physics of light–matterinteraction—namely the Rabi doublet (coupling of the cavity mode to another Bosefield) and the Mollow triplet (coupling to a Fermi field)—are obtained in the spectrumof emission of the semiconductor QD. In the intermediate region, complex multipletstructures arise. More detailed are given by Laussy et al. (2006a).

Fig. 5.8: Density plot on logarithmic scale (to discriminate the peaks, their positions and splitting) of spectrumof emission by a QD in strong coupling with a single-cavity mode for γeh = −2

√γeγh (cf. eqn (5.139)),

recovering, respectively, the Rabi doublet (on the left) and Mollow triplet (on the right). In the intermediateregion, intricate and rich patterns of peaks appear, split, merge, or disappear.

6

WEAK-COUPLING MICROCAVITIES

In this chapter we address the optical properties of microcavities in theweak-coupling regime and review the emission of light frommicrocavities in the linear regime. We present a derivation of the Purcelleffect and stimulated emission of radiation by microcavities, andconsider how this develops towards lasing. Finally, we briefly considernonlinear properties of weakly coupled semiconductor microcavities.The functionality of vertical-cavity surface-emitting lasers (VCSELs) isalso described.

216 WEAK-COUPLING MICROCAVITIES

6.1 Purcell effect

6.1.1 The physics of weak coupling

We are interested in this chapter in the so-called weak coupling of light with matter inthe sense that the effect of the radiation field can be dealt with as a perturbation on thedynamics of the system. The dynamics we have in mind is typically the spontaneousemission of the system initially in its excited state. As emphasized by Kleppner (1981)in the atomic case, the atom releases its energy because of its interaction with the vac-uum of the optical field, so that if the interaction could be “switched off”, the atomwould remain forever in its excited state. This idea is an extension of one reported muchearlier by Purcell (1946) who was seeking the opposite effect, namely to increase theinteraction so as to speed up the release of the excitation.87 Intuitively, if the dipole isresonant with the cavity mode, the photon density of states seen by the dipole is in-creased with respect to the vacuum density of states. The spontaneous emission rate istherefore enhanced: the dipole decays radiatively faster than in vacuum and the pho-tons are emitted in the cavity mode. On the other hand, if the dipole is placed out ofresonance, namely in a photonic gap, the photon density of state seen by the dipole issmaller than in vacuum and the spontaneous emission rate is reduced. The Purcell ef-fect therefore perfectly illustrates the role played by an optical cavity that is to locallymodify the photon density of states. The control of spontaneous emission through thePurcell effect is a way to reduce the threshold of lasers and the effect has been activelylooked for with atoms placed in cavities, for instance by Goy et al. (1983), and morerecently with quantum dots placed in micropillars, microdisks or photonic crystals, forinstance in the work of Gerard et al. (1998). We review in more detail the experimentalrealizations later.

The above description of the emission neglects reabsorption. For dipoles in freespace, one can easily believe intuitively that this effect is weak. In fact, it is respon-sible for the energy shift known as the Lamb shift that is indeed orders of magnitudesmaller than the radiative broadening. In quantum electrodynamics, this shift is inter-preted as the influence of virtual photons emitted and reabsorbed by the dipole. Thesituation changes dramatically when the dipole is placed in a cavity. Photons emittedare then reflected by the mirrors and remain inside the cavity. This increases the prob-ability of reabsorption of the photons by the dipole. If the confinement is so good thatthe probability of re-absorption of a photon by the dipole is larger than its probabilityof escaping the cavity, the perturbative weak-coupling regime breaks and instead theso-called strong coupling takes place. This means that the eigenmodes of the coupleddipole-cavity system are no longer bare modes but mixed light–dipole modes. Their en-ergies are strongly modified with respect to the bare modes. The strong-coupling regimeis addressed in detail in Chapters 7 and 8, but in discussing weak coupling, one should

87Purcell was motivated by a practical goal in nuclear magnetic resonance: to bring spins into thermalequilibrium at radio frequencies, with the relaxation time for the nuclear spin in vacuum of the order of 1021s.He calculated that the presence of small metallic particles would, thanks to the Purcell enhancement, bereduced to order of minutes. The much-quoted reference that records this landmark of QED—E. M. Purcell,Phys. Rev., 69, 681, (1946)—is actually a short proceedings abstract.

PURCELL EFFECT 217

bear in mind that strong coupling is regarded as implying richer physics from the fun-damental point of view88 and that it is typically harder to obtain than the weak couplingthat “historically” precedes it in a given system until enough control of the system andits interactions are reached to give preponderance to the quantum hamiltonian dynamicsover the dissipations.

6.1.2 Spontaneous emission

We now give a mathematical derivation of the above ideas, relating spontaneous emis-sion with weak (or perturbative) coupling to the optical field. Our starting point iseqn (3.75) for the electric-field operator that we write here in the dipolar approxima-tion and for a given state of polarization:

E =∑k

uk

√ωk

2ε0L3ek(a†

k + ak) . (6.1)

We recall that photon modes are resulting from the quantization of the electromag-netic field in a box of size L. From there we go to the Jaynes–Cummings model (seeSection 5.5) for the multimode field operator eqn (6.1), which turns the coupling termeqn (5.91) into, written directly in the interaction picture:

V =∑k

√ωk

2ε0L3(ek ·d)e−i(ω0−ωk)t(a†

kσ + σ†ak) . (6.2)

We consider as the initial state the atom in an excited state and all photon modesempty, |e, 0k1

, 0k2, · · · , 0kn

, · · · 〉. The final states are states with the atom in the ground-state and one photon in one of the final states, |g, 0k1

, 0k2, · · · , 1km

, · · · 〉. We also takeinto account only the term a†

kσ of the Hamitonian, which destroys the atomic excitationand creates a photon. The reverse process akσ† is neglected. This means that we assumethat the photon is escaping quickly far away from the atom and cannot be reabsorbed.The matrix element between the initial state and one of the final state therefore reads:

Mkn= 〈e, 0k1

, 0k2, · · · , 0kn

, · · · |H |g, 0k1, 0k2

, · · · , 1kn, · · · 〉 =

√ωk

2ε0L3(ek ·d) .

(6.3)One can then apply the Fermi golden rule that stands as (we denote “at” for atom):

Γat0 =

2π

∑k

|Mk|2δ(E0 − Ek) , (6.4)

where the sum is on the set of final states, which becomes a continuum as L →∞. Wenow pass to the thermodynamic limit, making the size of the system go to infinity. Thesum is replaced by an integral using the rule:

88It is not so obvious from the experimental point of view that strong coupling is intrinsically better thanweak coupling; all the physics of lasers that is addressed later in this chapter pertains to the weak coupling.


∑k

→ 2(2πL

)3∫ 2π

0

dφ

∫ π

0

sin θdθ

∫ ∞

0

k2dk . (6.5)

The factor 2 in the numerator stands for the two different polarizations, ek ·d =d cos θ with θ the angle between the dipole axis and the electric field. This gives:

Γat0 =

c

4π2ε0d2

∫ 2π

0

dφ

∫ π

0

sin θ cos2 θdθ

∫ ∞

0

δ(E0 − Ek)k3dk (6.6a)

=c

3πε0d2

∫ ∞

0

δ(E0 − Ek)k3dk . (6.6b)

We then use k = E/(c), which yields:

Γat0 =

1

3πc23ε0d2

∫ ∞

0

δ(E0 − Ek)E3dE , (6.7)

which finally gives the usual formula for the emission of an atom in the vacuum:

Γat0 =

ω30

3πc3ε0d2 . (6.8)

An approach involving the dynamics—also based on the Jaynes–Cummings Hamil-tonian and known as the Wigner–Weisskopf theory—also assumes that at time t = 0 theatom is in the excited state and the field modes are in the vacuum state. The state vectortherefore reads:

|ψ(t)〉 = ce(t) |e, 0k〉+∑k

cg,k(t) |g, 1k〉 , (6.9)

with the initial time amplitude of probabilities given by ce(0) = 1 and cg,k(0) = 0.We now determine the state of the atom and the state of the light field at some

later time, when the atom starts to emit. We therefore write the Shrodinger equation(∂/∂t) |ψ(t)〉 = −(i/)V |ψ(t)〉 with V given by eqn (6.2). Projecting this equation onthe different basis vectors, one gets the equations of motion for the probability ampli-tudes:

ce(t) = −i∑k

Mkei(ω0−ωk)tcb,k(t) , (6.10a)

cb,k(t) = −iMkei(ω0−ωk)tce(t) . (6.10b)

We formally integrate eqns (6.10b) and substitute the result in eqn (6.10a) yielding:

ce(t) = −∑k

|Mk|2∫ t

0

ei(ω0−ωk)(t−t′)ce(t′)dt′ . (6.11)

This expression is still exact. We now perform the so-called “Wigner–Weisskopf”approximation, which amounts to the following substitution:

PURCELL EFFECT 219

∫ t

0

ei(ω0−ωk)(t−t′)ce(t′)dt′ ≈ ce(t)

∫ ∞

0

ei(ω0−ωk)t′dt′ . (6.12)

This approximation is also known more generally as a Markov approximation, whichwe shall use repeatedly in subsequent chapters to derive kinetic equations. This approx-imation consists physically in neglecting memory effects, that is, to make the time evo-lution of a quantity depend on its value at the present time and not on its history (itsvalues in the past). Mathematically, it consists in replacing ce(t

′) by ce(t) in the integralof eqn (6.12). The second approximation that is performed is to replace the upper boundof the integral by infinity. This approximation is justified if ω0t 1. We now use theCauchy formula, which is:

1

2π

∫ ∞

0

eiωtdt =1

2

(δ(ω)− 1

iπP( 1

ω

)), (6.13)

where P stands for the principal value of the integral.89 Equation (6.12) becomes

ce(t) =

(−1

2Γ0 + i∆ω

)ce(t) , (6.14)

where Γ0 = (2π/)∑

k |Mk|2δ(E0 − Ek) has the same expression as in eqn 6.7, and

∆ω = −1

∑k

|Mk|2P(

1

E0 − Ek

). (6.15)

This last quantity is the Lamb shift. It is, as already discussed, the renormalizationof the frequency of emission of the atom induced by the reabsorption of the light by theatom after its initial emission. This frequency shift can be calculated using the Wigner–Weisskopf approach, whereas it is naturally absent from derivations based on the Fermigolden rule. This shift is, however, usually small.

6.1.3 The case of QDs, 2D excitons and 2D electron–hole pairs

This aspect has been treated at length in Chapter 4 and we merely discuss here how themain results compare with respect to the case of atoms. The QD Hamiltonian and theprocedure that can be used to calculate the decay of a QD excitation is exactly similar tothe one we have just detailed. The only difference comes from the shape of the matrixelement of coupling. The situation is slightly different for QWs excitons for which thedecay has been found in Chapter 4 as a solution of Maxwell’s equations. The operatorsdescribing excitons are bosonic and not fermionic as for atoms. This has, however, noimpact on the result since we are dealing with occupation numbers smaller than one.Another important difference is that a QW exciton with a given inplane wavevector iscoupled to a continuum of states in a single direction of the reciprocal space, instead of

89See footnote 10 on page 32.


three for atoms or QDs. In the framework of the Fermi golden rule, the decay rates of aQW exciton having a null wavevector in the plane reads:

ΓQW0 =

2π

∑kz

|MQWkz

|2δ(E0 − Ekz) , (6.16)

with the matrix element of interaction between photons and the QW exciton now being

MQWkz

=µcv

2πa2DB

√Ekz

εε0L. (6.17)

where L is the QW width. Going to the thermodynamic limit as before, this gives:

ΓQW0 =

2nL

h2c|MQW

E0|2 =

nµ2cvE0

4π2h2cεε0(a2DB )2

. (6.18)

For electron–hole pairs in a quantum well the matrix element of interaction simplyreads:

M ehkz

= µcv

√Ekz

2εε0L3, (6.19)

which, with the same procedure as above, yields:

Γeh0 =

nµ2cvE0

h2cεε0L2. (6.20)

6.1.4 Fermi’s golden rule

Both inhibition and enhancement of the spontaneous emission are essentially related toFermi’s golden rule.90 For the case of an electric dipole d interacting at point r andtime t with the light field E(r, t), the spontaneous emission rate for an emitter withenergy ωe, reads

1

τ=

2π

2|d ·E(r, t)|2ρ(ωe) , (6.22)

with ρ(ωe) the photon density of states at the energy ωe of the emitter. In the vacuum,it is given by

ρv(ω) =ω2V n3

π2c3. (6.23)

90Fermi’s golden rule states that if |i〉, |f〉 are eigenstates of a Hamiltonian H0 subject to a perturba-tion H(t), the probability of transition from an initial state |i〉 to a continuum of final states |f〉 is given bythe formula:

1

τ=

2π

2| 〈f |H′ |i〉 |2ρf , (6.21)

with ρf the density of final states. If H′ is time independent, ρf becomes an energy-conserving delta function.

PURCELL EFFECT 221

Fig. 6.1: Density of states of the vacuum,ρv (thick dashed line), and of a cavity sin-gle mode (thick solid), ρc, as a function ofthe frequency ω. The lines of two emitters,e.g., quantum dots, are sketched in a con-figuration where the emitter is in resonancewith the cavity (A) with an enhanced prob-ability of emission into the cavity mode, orstrongly detuned with no final state to de-cay, resulting in an increase of its lifetime.

In a single-mode cavity, however, with energy ωc and quality factor Q, the densityof states becomes a Lorentzian:

ρc(ω) =2

π

∆ωc

4(ω − ωc)2 + ∆ω2c

. (6.24)

Both densities are sketched in Fig. 6.1. The localization of modes available for thefinal state (decay) of the emitter around the cavity mode allows us to enhance (resp.inhibit) spontaneous emission by tuning (resp. detuning) the emitter with the cavitymode. As a result of its reduced (resp. increased) lifetime, the line gets correspondinglybroadened (resp. sharpened). When the characteristic emission time given by eqn (6.22)for the vacuum and cavity case are compared, one gets:

Γc

Γ0=

3Q(λc/n)3

4π2Veff

∆ω2c

4(ωe − ωc)2 + ∆ω2c

|E(r)|2|Emax|2

(d ·E(r)

dE

)2

. (6.25)

Equation (6.25) is the central equation behind the Purcell effect. It puts neatly to-gether all the modifications that the cavity forces on the lifetime of an enclosed emitter:

• The term 3Q(λc/n)3/(4π2Veff) depends only on parameters of the cavity, itsquality factor Q, wavelength λc, refractive index n and effective volume Veff . Assuch, it is a figure of merit of the cavity, which quantifies the efficiency of Purcellenhancement for an ideal emitter coupled in an ideal way to the cavity. It is thisquantity that appears in Purcell et al.’s (1946) seminal paper, in honour of whomit is now called the Purcell factor:

FP =3Q(λc/n)3

4π2Veff. (6.26)

• The term ∆ω2c/[4(ωe − ωc)

2 + ∆ω2c

]comes from the density of states of a single

mode in Fermi’s golden rule formula. It shows the effect of the detuning on theefficiency of the coupling, on which it acts through a phase-space-filling effect.As this quantity is smaller than one, it contributes always towards inhibition ofemission. In this approach, fully focused on the Purcell effect, it means that thelifetime of the dot can be made arbitrarily long by detuning it farther from the


Fig. 6.2: Time-resolved photoluminescence of a quantum dot when placed, (a) in the bulk of the material(GaAs matrix), and (b&c) in a pillar microcavity of the same material, as observed by Gerard et al. (1998).In case (b) the dot is in resonance with the single mode of the cavity and decays about five times quicker thanwhen it is coupled to a continuum of modes, case (a). In case (c) the dot still inside the cavity is detuned withthe mode and as a result displays only a small enhancement of its lifetime as compared to the bulk case.

cavity. In a real situation, there are other modes to which the emitter couplesas well, especially leaky modes that act as a dissipation. For instance, emissionthrough the sides of a pillar put a limit to emission inhibition, as observed byBayer et al. (2001) who could partially enhance the situation by coating the sides.Finally, there is always a channel of nonradiative decay, which adds a constantterm to eqn (6.25) (as is the case in eqn (1.5) where it is appears as +f ).

• The term |E(r)|2/|Emax|2 underlines the importance of fluctuations in the sys-tems. In the solid-state case, a given sample is more stable than its atomic coun-terpart as the dot stays fixed at the same position. However, the possible con-figurations change from sample to sample and it can require many trials until agood sample is found where the location, oscillator strength and other propertiesof the dot are suitable to display nontrivial physics (this is especially true whentrying to achieve strong coupling, as we shall see in the subsequent chapters). Thetheoretical fits in Fig. 6.2, for instance, were made by averaging over eqn (6.25).

• The last term (d ·E)2/(dE)2, which in some cases or for some authors would fallin the same category as above, as adding an element of randomness weakening theoptimal Purcell enhancement as quantified by the Purcell factor. However, recentresults such as those obtained by Unitt et al. (2005) indicate that a determinis-tic pining of the dipole along crystallographic axis allows polarization-selectivePurcell enhancement.

The Purcell enhancement of spontaneous emission is neatly demonstrated in Fig. 6.2where emission is released according to curve (b) in resonance or (c) out of resonance,displaying a much quicker decay in the former case as opposed to the latter that isessentially the same as without mode coupling (curve (a)).

An important issue is dimensionality. An atom and its semiconductor counterpart,

PURCELL EFFECT 223

the quantum dot, is naturally emitting spherical light waves. In a plane-wave representa-tion of light, this means that an atom is intrinsically coupled to all possible plane-wavedirections. The decay of such a system will be well described by the application ofFermi’s golden rule. On the other hand, the achievement of the Purcell effect or ofstrong coupling will require light to be confined in the three directions of space, namelyto use a three-dimensional optical cavity. The case of bulk excitons considered at lengthin Chapter 4 is radically different. In the bulk, an exciton is a plane wave character-ized by a well-defined wavevector. This exciton is coupled to a single photon state. Thestrong coupling is achieved and eigenstates of the system are exciton-polaritons thatonly decay because the photon ultimately escapes through the edges of the sample. InQWs the situation becomes similar again to the atom/dot case. Indeed, the transitionalinvariance is broken in one direction and kept in the other two. In these latter two di-rections, the one-to-one coupling holds, whereas in the third direction, the QW excitonis coupled to a continuum of photonic modes. QW excitons therefore radiatively decayin this direction. The achievement of the Purcell effect or of strong coupling in this lat-ter case requires the use of a planar microcavity that confines the light in the directionperpendicular to the QW.

6.1.5 Dynamics of the Purcell effect

We now investigate a toy model illustrating the Purcell effect from a quantum dynam-ical point of view. We consider the case of an atom placed within a three-dimensionaloptical cavity of volume Vc = L3

c . The fundamental mode of this cavity has resonanceenergy ωc and is characterized by a Q factor Q = ωc/γc, where γc is the width ofthe cavity mode. We recall that a picturesque definition of the quality factor Q is that itnumbers how many round-trips light makes in the cavity before leaking out. Formally,the atom is now coupled to a single decaying mode. The Fermi golden rule can no longerbe used. We describe the coherent coupling of the atom with a single cavity mode thatis dissipatively coupled to a bath of external photon modes. The system we consider issketched in Fig. 6.3.

Fig. 6.3: Sketch of the physics involved in the coupling of one atom in a leaky cavity mode: there is coherentcoupling between states |e, 0〉 and |g, 1〉 through Jaynes–Cummings dynamics, and dissipative coupling to areservoir, bringing the system to the vacuum |g, 0〉. These three states form a basis of states called |1〉, |2〉and |3〉, respectively.

We write the atom–light states involved in coherent/dissipative couplings of the firstmanifold as follow:

|1〉 = |e, 0〉 , |2〉 = |g, 1〉 , and |3〉 = |g, 0〉 . (6.27)


With these notations, the density matrix of the system reads:

ρ =

3∑i=1

3∑j=1

ρi,j |i〉〈j| . (6.28)

The equation of motion for the density matrix taking into account Lindblad dissipa-tion of Section 5.4 (now for the atom case) gives, in this special case where the Hilbertspace is spanned by the basis |1〉 , |2〉 , |3〉:

∂

∂tρ = − i

[H, ρ] + Lρ , (6.29)

with,

H =

⎛⎝ω0 M 0

M ωc 00 0 0

⎞⎠ and L =

⎛⎝ 0 − ωc

2Q 0

− ωc

2Q −ωc

Q 0

0 0 ωc

Q

⎞⎠ . (6.30)

It follows for the matrix elements:

ρ11 = iM

(ρ12 − ρ21) , (6.31a)

ρ22 = −iM

(ρ12 − ρ21)− ωc

Qρ22 , (6.31b)

ρ12 − ρ21 = iM

(ρ11 − ρ22) +

[i(ω0 − ωc)− ωc

2Q

](ρ12 − ρ21) , (6.31c)

ρ33 =ωc

Qρ22 . (6.31d)

The Fourier transform of this linear system gives the equations determining theeigenfrequencies, reminiscent of the renormalizations already encountered several timesthroughout this book. They read:

ω± =1

2

(ω0 + ωc + i

ωc

Q±√(

ω0 − ωc − iωc

Q

)2

+ 4(M

)2)

. (6.32)

One can see that in the general case, the real and imaginary parts of the modes areboth renormalized with respect to the case of the bare states. We now analyse someparticular cases.

First, we consider the resonance between the dipole and the cavity mode: ω0 = ωc.In this case two different regimes take place, depending on the sign of the quantity thatis below the radical. If ωC/Q > 2M/, the square root in eqn (6.32) is imaginary. Thetwo eigenmodes have the same real part but different imaginary parts, one larger andone smaller than ωc/(2Q). This regime is the weak coupling. If ωc/Q 2M/, theexpansion of the square root gives

ω+ ≈ ωc + iQ

ωc

(M

)2

= ωc + iγc , (6.33a)

ω− ≈ ωc(1 + i/Q) . (6.33b)

PURCELL EFFECT 225

One mode has the energy and decay of the bare cavity mode. The other mode hasthe same energy as the bare mode but the decay constant is enhanced by the presence ofthe cavity, which is the manifestation of the Purcell effect. Using the explicit expressionfor M , the decay of an atom is usually given as:

γc =Q

ωc

(M

)2

= FPΓ0 , (6.34)

where Γ0 is the free atom decay and FP is the Purcell factor given by eqn (6.26).One can see that FP—the Purcell enhancement factor—is proportional to the Q

factor of the cavity and is maximal for small cavity volumes that maximize the overlapbetween the confined photon mode and the atom. Physically, this means that the atomwill emit light only in the cavity mode and will decay much faster than in the vacuum.

On the other hand, if ωc/2 < 2M/, the radical of eqn (6.32) is real. In this case,the two eigenmodes have the same imaginary part but two different real parts. Thisregime is the strong coupling. A photon emitted by the atom does not tunnel out butis virtually reabsorbed and re-emitted several times by the atom before it leaks out.The eigenmodes of the system are therefore mixed light–matter modes. In the limitingcase where ωc/Q 2M/, the splitting between the two eigenfrequencies is givenby 2M/. The strong coupling regime is presented in detail in the subsequent chapters.

In the strong off-resonance case, when |ω0 − ωc| M/ and also |ω0 − ωc| ωc/Q, the square root of eqn (6.32) can be developed keeping only terms contributingto the imaginary part:

ωc

Q±√(

ω0 − ωc − iωc

Q

)2

+ 4(M

)2

(6.35)

≈ (ω0 − ωc)

(1− i

(ωc

Q(ω0 − ωc)2+

(4(M/)2 − (ωc/Q)2

)(ωc/Q)

2(ω0 − ωc)3

)).

In contrast with the other case, the photonic mode therefore remains unperturbedwhereas the emission from the atom is strongly inhibited.

6.1.6 Case of QDs and QWs

The formalism presented above to calculate the Purcell enhancement factor is valid forany single mode emitter coupled to a single optical mode. The value achieved by theenhancement factor only depends on the dimensionality. Hence, the enhancement factorof a QD in a 3D cavity is the same as that calculated above for an atom. The case ofQWs excitons and electron–hole pairs is, however, different. We use the formula andwe replace the optical quantization length L of the equation by the cavity thickness Lc,and we find:

F 2DP =

1

4π

λc

nLcQ . (6.36)

This formula has the same form as its zero-dimensional counterpart. In normal mi-crocavities, the cavity length is of the order of the wavelength, which makes an enhance-ment factor of the order of Q/(4π).


Our previous analyses showed that the effect depends on the value achieved by thematrix element of interaction that itself is composed of an intrinsic part (used in theHamiltonian) and of a part induced by the non-ideal overlapping between the cavitymode and the dipole mode. It is easy and instructive to consider as an example the im-pact of the position of a QW in a planar microcavity. For a cavity surrounded by perfectmirrors, the amplitude of the electric field along z goes like E(z) = E0 cos(2πz/L).This is valid if the well is placed at the maximum of the electric field. Otherwise, oneshould go back to the formula replacing M by M cos(2πz/L). The Purcell enhance-ment factor therefore becomes:

F 2DP (z) =

1

4π

λc

nLcQ cos2

(2πz

L

). (6.37)

One can see that an inappropriate placement of the well inside the cavity can lead toa strong decrease of the Purcell factor and even to a complete suppression of emission.In practice, this constraint is not that strong in planar microcavities where a very goodcontrol of the QWs position can be achieved. This constraint is, however, much strongerfor QDs in 3D cavities. Indeed, the position of QDs is only very weakly controlled bycrystal-growth techniques and the quality of overlapping between an optical mode anda QD is often governed by chance, so far.

6.1.7 Experimental realizations

Fig. 6.4: (a) Atomic cavity experiment from Heinzen et al. (1987) for Purcell modification of (b) spontaneousemission with cavity open (i) or blocked (ii).

Experimental realizations of the Purcell effect have been held back because typicalstrong dipoles emit at high rates (lifetimes of order 1 ns) which implies that high Q-factors and small cavities (cavity lifetimes Qc/L) are needed to perturb the photondensity of states (DoS) sufficiently.

For atoms in an optical microcavity, this required the development of cold atomicbeams and suitable continuous-wave tuneable lasers. The first realization of the Purcellexperiment in a larger macrocavity (L = 5 cm) by Heinzen et al. (1987) showed afew per cent change in the emission rates, primarily due to the low solid angle in theconfocal cavity mode (Fig. 6.4). In the same year, experiments by Jhe et al. (1987) in

PURCELL EFFECT 227

the near-infrared on plane-plane Au-coated microcavities separated by 1.1 µm showedthat the µs decay times of Cs atoms could be increased by 60%. Also in the same year,de Martini et al. (1987) demonstrated the same effect for dye molecules flowing in asolution between two plane-plane dielectric mirrors whose tuneable separation (downto below 100 nm) produced photoluminescence lifetime enhancements of up to 300%.

Fig. 6.5: Optically pumped dielectric DBR planar microcavity laser with flowing dye

More recently, similar effects for semiconductor quantum dots embedded in micro-cavities have been demonstrated by Bayer et al. (2001). In 5 µm GaAs microdisks theemdedded InAs quantum dot lifetimes could be 300% longer, and corresponding effectshave been seen with quantum dots in 1–20-µm diameter micropillars. The difficulty ofgreatly increasing the spontaneous lifetimes in semiconductors is the extra contributionfrom other non-radiative recombination processes, which in general also increase whenpatterning devices into optical microcavities. As a result, such experiments have beencarried out at low lattice temperatures, T ≈ 4 K reducing the excited phonon modeoccupations of the solid.

Subsequent improvements in this type of experiment have allowed emission ex-periments on single ions or single semiconductor quantum dots inside microcavities.However, the Purcell factors have not been greatly improved, and it remains difficultto control precisely the spatial position of the emitter within the microcavity—in orderto have the slowest emission, the emitter should be in a field minimum. More recentexperiments have shown that photonic crystal defect cavities with semiconductor quan-tum dots have great potential for realizing the Purcell effect. This is mainly because themode volume is so small that the cavity linewidth is much smaller than the linewidthof the electronic transition at room temperature, implying that the number of photonround-trips before phase scattering inside the solid is what controls the emission char-acteristics.

In all the above examples, the cavity–emitter system is in the weak-coupling regime.However, it is possible to increase the coupling strength between light and matter suffi-ciently that the normal modes of the system become polaritons (Sections 4.4 and 5.8).The first observations of this so-called “vacuum-Rabi splitting” for atomic systems in1989 (Fig. 6.6) by Raizen et al. (1989) showed that the atomic decay rate was halvedon resonance, because the atoms spent half their time in the de-excited state with theenergy in the cavity photon field. Because of the small optical cross-section for atoms,much improved cavity finesse was needed before. McKeever et al. (2003) demonstrated


atoms

empty

Fig. 6.6: Transmission spectra (around the 3P–3S transition at 589 nm) through a L=1.7 mm cavity withfinesse of 20 000 that is (a) empty, and (b) contains sodium atoms at zero detuning ∆=0, from Raizen et al.(1989).

emission and lasing from a single atom in the strong-coupling regime. Recently, in2004, three realizations were achieved for strong coupling using single semiconduc-tor quantum dots: in photonic crystal defect microcavities by Yoshie et al. (2004), inwhispering-gallery microdisks by Peter et al. (2005) and planar micropillars by Reith-maier et al. (2004) (see Fig. 6.7).

Further progression on such experiments demands the ability to both spectrally tunethe microcavity resonant mode and dipole transition frequency independently, and alsoto control their spatial overlap. A current advantage of solid implementations comparedto atom systems over and above their utility and portability is the fixed embedding ofthe emitters—atoms move around or need trapping, and also have to be injected in abeam, hence restricting how much the cavity can be shrunk.

6.2 Lasers

The laser is a direct application of Einstein’s theory of light–matter interaction based onthe A and B coefficients that have been presented in Section 4.1.2. Prokhorov (1964)discusses in his Nobel lecture why this “obvious” application—the principle of whichhad been realized by many—took so much time for its realization, and through the skillsof the radio wave community rather than from optics. The main principle of generat-ing gain in an inverted population by overcoming absorption with emission thanks tostimulation was, however, clear. Moreover, the wide range of possible types of lasers,including semiconductors, was realized early on. In fact, in the Nobel lectures accom-panying that of Prokhorov the emphasis is made simultaneously and equally both foratoms by Townes (1964) and for semiconductors by Basov (1964). Those main princi-ples having to do with the interplay of gains and losses are presented now. Later, weturn more specifically to the semiconductor laser, which was rapidly demonstrated asBasov envisioned it showing that the main ideas were sound, but quickly required in-genious manufacturing to operate efficiently. The description of these elaborations andspecificities are discussed next.

LASERS 229

Fig. 6.7: (a) Electron micrograph of microcavity pillar containing quantum dots from Reithmaier et al. (2004),with (b) emission spectrum from the single quantum dot tuned across the cavity resonance by changing thetemperature. (c) Photonic crystal single quantum-dot microcavity from Yoshie et al. (2004) and (d) microdiskmicrocavity from Peter et al. (2005).

6.2.1 The physics of lasers

Einstein’s A and B coefficients are fundamental parameters of the system. The equi-librium considerations in Section 4.1.2 served as a useful particular case to investigatethem, but they still apply out of equilibrium when the system is excited or driven in someway. In the following for simplicity of notations we consider non-degenerate cases sothat B12 = B21, which we will denote simply B.

If an excited atom can be “induced” to emit a photon by another photon, there is thepossibility to start a chain reaction in a population of inverted atoms with each additionalphoton stimulating another one, in turn stimulating other photons. Quantitatively, in apopulation of n = n1 +n2 atoms (per unit volume), n1 of which are in the ground-stateand n2 in the excited state, one has n1BI photons absorbed by the atoms (with I thephoton energy density) and n2BI emitted by stimulation, hence δnBI photons numberof photons gained per second and per unit volume, with δn = n2 − n1. The numberof photons per unit area exchanged (emitted or absorbed) with those already presentand accounting for a flux φ in the small distance ∆z is δnBI∆z. Going to the limit ofinfinitesimal distance (and flux) dφ = δnBIdz, which integrates as a function of z as


Charles Townes (b. 1915), Nikolai Basov (1922–2001) and Aleksandr Prokhorov (1916–2002), the 1954Nobel Prize in Physics for “fundamental work in the field of quantum electronics, which has led to the con-struction of oscillators and amplifiers based on the maser-laser principle”.

Townes gained expertise with microwaves techniques from his design of radar bombing systems during WorldWar II. From spectroscopy he gradually went on to develop the concept but also the physical realization ofthe maser (he also gave the device its name). In the late 1950s, he discussed with Gordon Gould, then a Ph.D.(that was never completed), about the optical maser, nowadays known under the term that was used for thefirst time by Gould: the laser. Gould referred to his knowledge of optical pumping to bring the maser intothe optical window, while Townes and his brother-in-law, Schawlow, were undertaking important foundingwork on the topic. This sparked one of the most famous patent fights in history that lasted over thirty year andfinally saw Gould victorious over Townes’ design, deemed by the court as not eventually working. Gould alsowon his court battles against laser manufacturers and became a multimillionaire from royalties of his patents.Controversy, especially sparked by Townes and Schawlow, still surround Gould as the inventor of the laser.

Prokhorov was Basov’s Ph.D. advisor. They both served in the Red Army during the war (Prokhorov beingwounded twice) and earned many distinctions from the Soviet union. Prokhorov was chief editor of the GreatSoviet Encyclopedia from 1969. They made the breakthrough in the maser effect simultaneously and inde-pendently from Townes, using a cavity reflecting light at both ends to amplify a microwave beam. Their workdisplays an harmonious mastery of both experimental and theoretical treatments.

φ(z) = φ(0) exp(δnIBz) . (6.38)

According to the sign of δn, eqn (6.38) will display either exponential attenuationor amplification. The condition δn > 0 is called population inversion and is realizedwhen there are more atoms in the excited than in the ground-state.

The amplification by stimulation results from the interplay of emission and absorp-tion, the former having spontaneous emission in addition to the stimulated channel thatin other respects is similar to absorption. To quantify this, we define Is the radiationenergy density that equates spontaneous emission and stimulated emission:

A = IsB . (6.39)

As we discussed in Section 4.1.2, the detailed balance of the rate equation dn1/dt =−dn2/dt = n2A + (n2 − n1)IB becomes, at equilibrium and in terms of Is:

n2Is + (n2 − n1)I = 0 , (6.40)

solving for atomic populations:

n1 =Is + I

Is + 2In and n2 =

I

Is + 2In , (6.41)

LASERS 231

with n = n1 + n2. These equations show that n2 < (1/2) < n1 for all values of I: itis impossible to have more atoms in the excited state than in the ground-state by directpumping as the result of the intrinsic nature of light–matter interactions that have achannel of stimulated decay in addition to the intrinsic or spontaneous one. The moreintense the radiating field to excite atoms on the one hand, the more it stimulates decayof the atoms already excited on the other.

This shows that a two-level system, that is, one with only two populations of atomicstates, cannot lase: arbitrary high excitation will only approach the equal populationconfiguration (and displays optical transparency as a beam will cross the material with-out being absorbed or emitted). It is, however, possible to obtain inversion of populationif there are other transient states. For instance, if the final state 1 of laser radiation haszero lifetime and is always empty, decaying toward the ground-state 0, however small isthe population of an excited state 2 that decays into 1, it will realize an inversion of pop-ulation. Amplification is therefore obtained by turning to at least three-level systems, aswe show now.

Consider a population of n atoms with possible energy states 0, 1 and 2 with ener-gies E0, E1 and E2, respectively, such that E0 < E1 < E2. The population of state i isdenoted ni.

τ sp

τ nr

τ21

τ 20

τ 1

τ2

1

2

R2

R1

W

Fig. 6.8: Sketch of transition rates of a three-level system suitable for amplification by stimulated emission.The lasing transition is 2–1. Level 2 has decay channels into level 1 labelled sp for spontaneous emissionand nr for all other nonradiative transitions, and a decay channel into the ground-state 0 labelled 20. Theyall contribute to a total lifetime τ2 for this level. Other such transitions are labelled in the same way. Alsomentioned are the pumping rates R2 populating the excited state and R1 depopulating it. Finally, when theradiation field starts to become important, stimulated emission and absorption W also enter the picture.

The transition of interest remains the lasing transition, shown in Fig. 6.8 with en-ergy E2 − E1. Level 2 decays into mode 1 with associated lifetime τ21 and into theground-state with lifetime τ20. If there are many channels of decay for a given transi-tion, typically one decay by spontaneous emission with characteristic time τsp and anonradiative decay with time τnr, then the total lifetime builds up as follows:

τ−121 = τ−1

sp + τ−1nr . (6.42)

Without pumping, the system quickly decays into its equilibrium state with all atoms


in the ground-state. Efficient and typical pumping schemes involve pumping the ex-cited state of the laser transition at a rate R1 and depopulating it at a rate R2. The rateequations taking into account these transitions only, are, by definition of the quantitiesinvolved:

dn2

dt= R2 − n2

τ2, (6.43a)

dn1

dt= −R1 − n1

τ1+

n2

τ21. (6.43b)

The steady-state solution is readily found in this case and the population differ-ence δn = n2 − n1 reads

δnss = R2τ2

(1− τ1

τ21

)+ R1τ1 , (6.44)

where subscript “ss” means “steady-state”. Large values of δnss (and therefore highvalues of optical gain) are obtained when the pumping rates Ri are high and when τ2

is large so as to build up a high population of excited states relative to level 1, and witha small value of τ1 if R1 < (τ2/τ21)R2. Otherwise, the population of level 1 becomesdetrimental and it is better if the level is quickly depopulated thanks to the short lifetimeto compensate an inefficient R1.

When the radiation field builds up it triggers transitions between levels 1 and 2by absorption (depopulating 1 for 2) and stimulated emission (inducing the oppositetransition from 2 to 1). The equations now become:

dn2

dt= R2 − n2

τ2− n2W + n1W , (6.45a)

dn1

dt= −R1 − n1

τ1+

n2

τ21+ n2W − n1W. (6.45b)

Observe that the new term δnW cancels in the sum of eqns (6.45). The populationdifference for this case is also obtained readily in the steady-state:

∆nss =δnss

1 + τsW, (6.46)

where δnss is the population difference in the absence of the radiation field, eqn (6.44);and τs is the so-called saturation time constant:

τs = τ2 + τ1

(1− τ2

τ21

). (6.47)

Of course δnss and ∆nss coincide when W → 0.

Exercise 6.1 (∗) Show that τs is a well-behaved time that is always positive. As a resultshow that

∆nss ≤ δnss . (6.48)

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Equation (6.47) shows a very important aspect of light–matter interactions as theresult of Einstein processes that is displayed in Fig. 6.9. Stimulated emission is a de-sirable effect for amplification (and its coherent properties as the result of cloning thestimulated emitted photon with the stimulating one), but it is detrimental to the popula-tion inversion. When the radiation field becomes important enough, stimulated emissionand absorption dominate, with equal weights as they have equal probabilities. In thiscase the dilemma of the two-level system springs up again and strong radiation tendstowards equalization of populations (this time from above, though).

0

0.5

1

1e-08 1e-06 1e-04 0.01 1 100τsW

∆nss/δnss

Fig. 6.9: Normalized population difference in the steady state ∆nss/δss as a function of the radiation fieldintensity W . When W = τ−1

s , ∆nss has been halved from its optimum value δnss, the population inversionin the absence of radiation.

6.2.2 Semiconductors in laser physics

Semiconductors are important materials in light–matter physics thanks to the radiativerecombination of electrons and holes, although not at equilibrium where their densi-ties are too small to produce a detectable quantity of light even with high doping.91

It is, however, easy to operate a semiconductor out of equilibrium by applying anelectric voltage to it and generating huge populations of carriers. Indeed, a forward-biased p-n gallium arsenide junction generates strong light in the infrared as reportedin the early 1960s by Hall et al. (1962), Nathan et al. (1962) and Quist et al. (1962).92

Holonyak and Bevacqua (1962) could obtain emission in the visible window93 by using

91The first connection of light to semiconductors was made in the beginning of twentieth century withcat’s whisker detectors by a collaborator of Marconi, H. J., Round, who reported his finding of a green glowfrom SiC in “A note on carborundum” in Elect. World 19, 309 (1907).

92Biard and Pittman could prevail on the observation of radiation by a junction while working on GaAsdiodes, dating it to 1961. Not expecting light emission, they noticed it using an infrared microscope. Theircontribution has been acknowledged for the record by patent issues (for which they received $1 each) al-though their oldest publication on this topic under the title “GaAs Infrared Source” By Biard, Bonin, Carr andPittman, is also dated 1962 (in the PGED Electron Device Conference.)

93Bringing radiation emission from semiconductors “in the visible spectrum where the human eye sees”,to quote Holonyak, made his paper the most quoted of Applied Physics Letters.


Zhores Alferov (b. 1930) and Herbert Kroemer (b. 1928)

The 2003 Nobel prize-winners (with Kilby), enabled semiconductors the means to revolutionize laser physicsand shape the era of telecommunications.

Kroemer, now a Professor at University of California, Santa Barbara, proposed the concept of the doubleheterostructure in 1963. A major publication on this topic was rejected for Applied Physics Letters and pub-lished in Proc. IEEE instead. His favourite saying—as claimed in his Nobel lecture—is “If in discussing asemiconductor problem, you cannot draw an energy band diagram, then you don’t know what you are talkingabout.”

Alferov’s name is an icon for the Ioffe institute in Saint Petersburg, where he worked from 1953, and as itsdirector in the period between 1987 and 2004. A gifted administrator, he managed to save the institute in-frastructures from disaster in the 1990s. In 1971 he received the USA Franklin’s institute gold medal for hispioneering works on semiconductor heterolasers. In its history, the Franklin medal has been awarded to fourRussian physicists: Kapitsa, Bogoliubov, Sakharov and Alferov, all but Bogoliubov having later received theNobel prize.

the GaAsP compound. By increasing the pumping of the structure to the point whereelectrons and holes undergo an inversion of population, the diode reaches the stagewhere gain by stimulated emission overcomes losses, and an input signal on the activeregion is amplified. It remains to engineer the device so that this input is levied fromits output to trigger the laser oscillations. The cavity in this case is provided by thesemiconductor crystal itself whose facets have been cleaved, i.e., terminated along thecrystal axis to create a perfectly flat endface, perpendicular to the axis of the junction94.When the light generated by recombination of electrons and holes gets to this surface,it is partially reflected back by internal reflection. The reflectivity is consequently quitelow for such lasers, about 30% (the facets can be coated for better reflection).

These preliminary diode sources were not efficient lasers as the active region whereelectron and hole recombine is spread out across the junction with great losses andrequiring significant threshold currents to compensate. Only the short-pulse regime ispossible before melting the device. A solution was envisioned on how to constrain car-riers effectively, theoretically by Kroemer (1963) and later realized in the Ioffe instituteby Alferov: the double heterostructure (DH). It consists of a thin region of semiconduc-

94The idea to use the crystal itself as the cavity comes from Hall who polished the surfaces. He shared itwith Holonyak who was planning to use an actual external cavity. Holonyak had the further idea to cleave thecrystal. It proved too difficult, however, and Hall came first with the semiconductor laser.

LASERS 235

tor with a small energy gap sandwiched between two oppositely doped semiconductorswith a wider bandgap. When forward biased, carriers flow into the active region andrecombine more efficiently because the potential barriers of the heterostructure confinethe carriers to the active region. Practical and soon efficient operation was achieved andthe device became one key element in the computer and information era, with maybeits most significant impact in the data storage with optical reading of CD and DVDtypes of optical disks (still widely used today for these applications.)95 These structuresthat are now called classical heterostructures rely on the profile of the energy bands forproviding potential traps for the carriers. Their size varies in the range between a fewhundred µm and a few mm. The idea was pushed forward by reducing further still thearea of localization to the point where size-quantization plays a role, opening the wayto quantum heterostructures, quantum wells, quantum wires and quantum dots.

Fig. 6.10: Sketch of semiconductor lasers: a) the p-n junction where electron–hole recombinations at theinterface serve as the active population; this scheme is more viable for LED operation, b) the edge-emittinglaser where the active region is confined by a heterostructure and c) the VCSEL where localization is pushedto the quantum limit and emission made from the surface.

These various schemes of lasing with semiconductors are sketched in Fig. 6.10.Lasing with a simple junction is a brute-force approach that requires high thresholdcurrents.96 Consequently, the device can only be operated in pulsed mode with muchloss in the conversion. By confining the carrier in the active region with the heterostruc-ture potential trap (simulating the action of a quasi-electric field—a term coined byKroemer—a feast that no genuine external field can achieve), the lasing could be oper-ated for threshold currents Jth reduced by two orders of magnitude, a trend that has con-tinued by following the road towards evermore quantization, as is illustrated in Fig. 6.11where a further two orders of magnitude gain on the current threshold is attained withQDs as the most recent realization.

95Other applications of semiconductor lasers for public use include fibre-optics communication, laserprinters, laser surgery, barcode readers, laser pointers. . . They are also useful for research and the military.

96Holonyak and Bevacqua (1962) report 11 × 103 A/cm2 as a threshold to superlinear emission andlinewidth narrowing.


Fig. 6.11: Evolution of the threshold current Jth of semiconductor lasers, from Alferov (2001).

The main advantage of edge-emitting lasers is that the size of the active regionallows one to store significant amounts of energy as compared to the microscopic orsmaller volumes involved in diodes where the active region lies in a quantum het-erostructure. For this reason, the light–matter interaction must be compensated by con-straining the photons inside the active region. This is achieved by shifting the opticalaxis from the plane of the structure (where photons travel a few times over millimeters)to the growth axis (where photons travel back and forth thousands of times over a mi-crometre). The mirror that, in the case of the edge-emitting laser is provided by indexcontrast, must now have a reflectivity of the order of 99.9%. This is possible only withdielectric mirrors, namely Bragg mirrors, that form a microcavity that confine the ac-tive region. Such structures are known as VCSELs (for vertical-cavity surface-emittinglaser).

6.2.3 Vertical-cavity surface-emitting lasers

Until the late 1970s, semiconductor lasers exclusively used the stripe geometry withcavity lengths longer than 100 µm. However, with the production of high-quality inte-grated DBR mirrors, it became possible to rotate the orientation of the emission so thatit emerged normal to the growth-layer planes. The principal advantage of this surface-emission geometry of vertical-cavity surface-emitting Lasers (VCSELs) is the abilityto make and test large numbers of lasers on a single wafer without having to cleavethe wafer into individual lasers with facets, with a large reduction in the cost of qual-ity control, manufacture and ease of packaging devices. Another advantage is the highspatial quality and non-astigmatic laser emission, which makes it easy to also match tooptical fibres. Subsequently, it was realized that very fast modulation could be obtainedin VCSELs with low power consumption.

However, in order to get these devices to work, the hundred-fold reduction in cavity

LASERS 237

length (and hence round-trip gain) has to be recovered in decreased round-trip loss (andhence the need for very high mirror reflectivities). This became possible with the intro-duction of lattice-matched DBR designs producing 99% to 99.9% intensity reflectioncoefficients. The cavity lifetime typically can reach τc = 1 ps (i.e. Q values up to 1000),which is similar to that in conventional stripe lasers in which the photons reflect off thecavity mirrors hundreds of times less often.

In order to efficiently pump the device, and retain single-mode operation, the cur-rent transport and optical emission within the planar microcavities have to be laterallyconfined. Typically, this is achieved in two ways: either by etching a circular mesa withan annular top contact injecting holes into the upper p-type material, or by using an Al-rich layer within the active region that is oxidized from an outside trench to produce aninsulating annulus that forces current through the centre of the device. However, otherfabrication procedures can give the same effect including growing in buried holes andproton bombardment. The oxide technology has some useful advantages in that as wellas confining the injection current, it also confines photons (since the central core has ahigher refractive index than aluminium oxide annulus), which results in better overlapof the light and electron–hole pairs.

Typically, to get maximum gain, quantum wells or quantum dots are used as theactive material, and placed within the structure at the peak of the antinodes of the in-tracavity electric field. To overcome the round-trip loss, the gain needs to be as high aspossible and thus the carrier density in the active region is universally above the ioniza-tion threshold for excitons. Hence, these devices operate without excitonic contributionsto their gain spectrum, distinguishing them from the polariton-lasers discussed later.

One of the advantages of VCSELs is that their active volume can be made verysmall. Typical small mode areas can be 1–10 µm2, and active cavity material volumescan be as low as V =0.05 µm3, which is about 100 times smaller than conventional stripelasers. Hence, the threshold in a VCSEL is correspondingly small since the thresholdcurrent is approximately Ith = eV Nt/τc when the cavity loss rate is small, controlledby the need for carrier densities above the transparency value Nt 1018cm−3. Thresh-olds below 10 µA have recently been reported for quantum dot VCSELs by Zou et al.(2000).

The growth of such multilayers is now at a highly sophisticated level, with spe-cial designs being able to simultaneously control the electrical transport through theDBR layers (using specific doping profiles that provide useful band bending), to controlthe photon confinement (through the oxide apertures), to control the thermal manage-ment (by matching the thermal impedence mismatches through the DBR stacks) andto control the electrical carrier spatial distributions to maximize the gain. The resultingoverlaps with the optical fields can be of the order of 5% in the vertical direction (over-lap with QWs at the centre of cavity) and up to 80% in the lateral direction, leading toconfinement factors of 4% or so.

The power conversion (or wallplug) efficiency of VCSELs is also very high whenpumped above threshold, because little voltage drop exists across the p-n junction ex-cept at the active region, leading to η > 50%. However, as the current increases, thetransverse mode of the VCSEL has a tendency to switch, due to spatial hole burning


Fig. 6.12: Light output curve for a quantum dot VCSEL, as reported by Zou et al. (2000).

and current spreading. To prevent this, structures can be designed that are antiguidingand operate well at higher power. In addition, another switching can appear at higherpower due to the birefringence of the devices, originating either from shape anisotropiesof intrinsic strain in the layers. The polarization output of these devices is always linear,but can switch axes between two near degenerate cavity modes as the temperature ofthe device rises. Full polarization dynamics of VCSELs have been discussed by Gahlet al. (1999).

We now provide a more thorough discussion of the Boltzmann equations (i.e. semi-classical rate equations) following an analysis by Bjork and Yamamoto (1991) andBjork et al. (1993), which is modified to provide a clear comparison with the inversion-less polariton-lasers discussed in Chapter 8. We define generally a population densityNx pumped at a rate P , producing laser emission from state N0 through the steady-staterate equations:

VdNx

dt=

P

ω− ΓnrV Nx −N0R(Nx)− S(Nx) = 0 , (6.49a)

dN0

dt= −Γ0N0 + N0R(Nx) + S(Nx) = 0 , (6.49b)

where N0 is the number of photons in a conventional laser, and the number of polaritonsin a matter-wave laser. For a conventional semiconductor laser of active volume V , thespontaneous, S, and stimulated, R, scattering rates are

LASERS 239

S = βΓsV Nx , (6.50a)

R = βΓsV (Nx −Nt) , (6.50b)

with non-radiative decay rate Γnr = 1/τnr + (1− β)Γs, where Γ0 is the photon cavityescape rate, Γs is the spontaneous emission rate, and τnr is the non-radiative lifetime.The fraction of photons that are spontaneously emitted into the lasing mode is β, whichis a key parameter for enhancing the emission from microcavities.

These equations can be simply solved to yield the output power as the pump rate isincreased, giving a lasing threshold,

Pth =ωΓ0

β(1 + Γnr/Γs)

(1 +

βΓsV Nt

Γ0

). (6.51)

Bjork and Yamamoto (1991) have shown how this threshold behaves in a verticalmicrocavity laser in the weak-coupling regime, and emphasized that the values of bothβ and Γnr are critical to low threshold action. Typically in a semiconductor Fabry–Perot laser, thresholds are around 10 mW in a 300 µm long device. This contrasts withVCSELs with optimized β∼10−2, allowing thresholds below 1 mW (Fig. 6.13).

0.2

0.1

0.0

-0.1

-0.2

Rat

e (p

s-1)

1x1018

0.50

Carrier density (cm-3

)

R

S(a)

Nt10

-6

10-4

10-2

100

102

Out

put p

ower

(m

W)

10-4

10-2

100

102

104

Pump Power (mW)

(b)

Fig. 6.13: VCSEL characteristics: (a) spontanteous (S) and stimulated (R) scattering rates from excitonsto photons as the carrier density increases. τnr = 50 ps, Γ−1

0 = 30 ps, Γ−1s = 3 ps, β = 3.6×10−3,

V =0.15 µm3. (b) Output vs. input power for the conditions in (a) (solid), for τnr =50 ns (dashed), and fora Fabry–Perot laser with τnr >10 ns, Γ−1

0 =3 ps, Γ−1s =3 ns, β=3×10−5, V =60 µm3 (dotted).

The form of these equations is the same for most lasers, and is underpinned by thedeep relationship between spontaneous and stimulated emission through the Einsteincoefficients of a transition. The result is that population inversion is a necessary con-dition for lasing, and in a semiconductor laser this requires a sufficient carrier densityNt to bleach the absorption. At these densities above the Mott density, screening by theCoulomb interaction and phase space filling is sufficiently strong that the electron–holeplasma screens out the exciton binding, and thus exciton lasing is impossible. The situ-ation is completely altered when the polariton pair scattering discussed in Chapter 7 isused to feed energy into a lasing transition that is in the strong-coupling regime.


One further important characteristic of the VCSEL is the possibility of turning iton and off at high frequencies. Typically, this rate is controlled by the relaxation oscil-lation frequency, fr, the natural oscillation rate between cavity photons and electronicexcitations produced by the coupled equations (6.49b):

fr =Γs

2π

√1

Γsτc

(I

Ith− 1

). (6.52)

For typical parameters this frequency exceeds 10 GHz in VCSELs compared to factorsof 10 smaller for conventional lasers.

We also note here that there has been some discussion about the extent to whichcoherence in the electronic system can play a role in semiconductor lasers, and thecoherence in the system remains difficult to calculate due to the many-body nature ofthe problem.

6.2.4 Resonant-cavity LEDs

Recently, a popular alternative to VCSELs for strong light emission that is incoher-ent has been the resonant-cavity light-emitting diode (RCLED). Because the gain ofa wavelength-thick semiconductor layer is typically small, high-quality mirrors are re-quired to produce effective lasers, and these require sophisticated fabrication of manyprecise semiconductor layers. On the other hand, for many applications, the coherenceof the light emission from the device is not so important, and it is the efficiency and di-rectionality that is key. Typical applications are in display technologies, in xerography,in biophotonics, and in general lighting applications. In an RCLED, the semiconduc-tor emitting layer is clad with low-finesse mirrors (unbalanced so they have smallerreflectivity on the top side), which modify the angular emission pattern, and suppressemission into inplane waveguide modes that do not escape efficiently from the sample.With these modifications, the LED efficiencies can exceed 50% and make such emit-ters competitive even with incandescent lighting. The rise of such devices has trackedthat of the GaN-based technologies so that UV/blue emitters can now be tuned acrossthe visible spectrum using either phosphor-based light conversion or In doping of theemitting GaN to produce ranges of colours.

Typically the design of RCLEDs is similar to VCSELs. To enhance the emission,the active p-n junction is placed at the antinodes of a microcavity whose length is closeto a few optical wavelengths. This ensures that the linewidth of the cavity mode setby the Q-factor is as large as possible, while the angular emission is beamed as muchas possible in the surface normal direction. In thick cavities of refractive index n, thefraction of resonances that exist within the light cone (from which light can escape) isgiven by 1/2n2, and this emission is shared between the number of possible microcavityresonances (see Fig. 6.14). This also shows intuitively that the enhancement will bebetter for active LEDs with a narrower spectral width.

The simplest design for a RCLED uses a buried conducting DBR and a metal mirroras the top contact, with the light extracted through the substrate (which therefore hasto be transparent). In such a structure the extraction efficiency can exceed 20%. The

LASERS 241

Fig. 6.14: (a) Cavity dispersion (curves) and LED active emission spectrum (shaded box) for (a) wavelength-scale and (b) thick microcavity, showing different emission patterns inside the light cone (dashed).

remainder is emitted into guided modes (which are absent as they are below the cutoffin half-wavelength thick cavities) or remains trapped in the substrate.

6.2.5 Quantum theory of the laser

In the above sections we have developed theories of the laser dealing with averagevalues both of populations of atomic (or carrier) populations and of the radiation fieldintensity (number of photons).

A full quantum treatment of the laser requires some approximations that one canconsider in some depth in the classical or semiclassical theories, such as multimodeoperation, spatial inhomogeneities, temporal drifts, inhomogeneous broadenings and soon. The mathematical complications brought by dealing with operators in the quantumcounterpart makes it awkward to draw a clear parallel. All that pertain to approximationsthat concern the average populations can be stripped from a quantum perspective as theyappear as classical averages anyway. The simplest Hamiltonian of interaction is soughtand a-posteriori investigations show that for most purposes it is enough to consider:

H = g

aσ†u(r)ei(ω0−ω)(t−t0) + aσu(r)e−i(ω0+ω)(t−t0) + h.c.

(6.53)

in the interaction picture, where a is the photon annihilation operator, σ the atom annihi-lation operator, u(r) the normalized cavity mode function and g the interaction strengthderived in Chapter 4. We have already considered a system close to this one to derivethe Bloch equations, but an important approximation has been retained here that hints atthe difference in the laser case, namely the rotating wave approximation of the secondterm.

The density matrix of the system is the combined atom (carrier)–photon field sys-tem. At initial time in the absence of correlations between them, it reads:

ρ(t0) = ρA(t0)⊗ ρF(t0) , (6.54)


with ρA (resp. ρF) the density matrix for the atom (resp. photon field). The state of ρF isobtained by tracing over the atomic variables. If we write the Liouville–von Neumannequation of motion as its Born expansion to infinite order before doing so, we get:

ρF(t) = ρF(t0) + TrA

( ∞∑n=1

1

(i)r

∫ t

t0

∫ t1

t0

· · ·∫ tr−1

t0

[H(t1), [H(t2), [. . . [H(tr), ρ(t0)] . . .]]]dt1dt2 · · · dtr

). (6.55)

To successfully describe laser action—even in the simplest setting—with eqns (6.53)and (6.55), one must carry out the algebra to high order in the commutators. One gets

[H, ρ] = g[σa†ρFu∗(r)− σ†ρFau(r)] , (6.56)

and successively, up to the fourth-order commutator that yields

[H, [H, [H, [H, ρ]]]] = (g)4|u(r)|4[σ†σ(aa†aa†ρF + 3aa†ρFaa† + h.c.)

− 4σσ†(a†aaa†ρFa + h.c.)] . (6.57)

To proceed one considers the evolution from the initial atomic position in bothground and excited states. For instance, in the excited state, eqn (6.57) becomes, tracingover atomic variables:

TrA([H, [H, [H, [H, ρ]]]]) = (g)4|u(r)|4(aa†aa†ρF

+ 3aa†ρFaa† − 4a†aaa†ρFa + h.c.). (6.58)

Up to now we have dealt with a single atom interacting with a single mode of thecavity mode. The latter approximation is reasonable but the single atom does not de-scribe a conventional laser,97 which properly involves an assembly of atoms as its ac-tive media. Rigorous but heavy methods have been developed, for instance by LambJr. (1964) or Scully and Lamb Jr. (1967). Even textbooks specializing in this topic findit difficult to attain such feats of meticulousness. Mandel and Wolf (1995) propose asa shortcut—which a posteriori proves to be essentially equivalent—to consider the ef-fects of an assembly of atoms by considering a coarse-grained lifetime average withprobability distribution P (∆t) = e−∆t/T2/T2 where T2 is the lifetime of the excitedstate (level 2), and multiplied by R2 the pumping rate of this level. Also, the equationis averaged spatially over the active medium. An equation of motion is obtained for thegain mechanism (since the atom was in its excited state). The same procedure can bestarted again for the atom initially in its ground state, yielding another master equation.

97The necessity to consider an assembly of atoms in the model of a laser is made more stringent by thefact that there exist single-atom lasers that would dispense with these requirements, demanding in exchangea more thorough consideration of its quantum-mechanical features.

LASERS 243

The sum of which (since there is no coherence between the atoms) provides the finalmaster equation for the photon field, which reads:

∂ρF

dt= −1

2A[aa†ρF − a†ρFa + h.c.]− 1

2C[a†aρF − aρFa† + h.c.]

+1

8B[aa†aa†ρF + 3aa†ρFaa† − 4a†aa†ρFa + h.c.] , (6.59)

where the above-mentioned derivation (introducing quantities such as η(r) the densityof active atoms and η1 that of loss atoms) provides coefficients:

A = 2(R2/N)(gT2)2

∫η(r)|u(r)|2 dr , (6.60a)

B = 8(R2/N)(gT2)4

∫η(r)|u(r)|4 dr , (6.60b)

C = 2(R1/N)(gT2)2

∫η1(r)|u(r)|2 dr . (6.60c)

Coefficients (6.60) characterize gain, nonlinearity and losses of the laser, respec-tively. Observe that B is of the order of the square of A and C.

Equation (6.59) is a typical single-mode laser master equation. Coefficients wouldvary for other systems derived under other approximations (or more rigorously derived)but the main principles remain with nonlinear terms displaying such asymmetric repar-tition about the density operator. This results in coupling the diagonal elements to off-diagonal elements and plays a role in the coherence of the field. As for the diagonalelements, their equation of motion is readily obtained by dotting the master equation toget the equation of motion of p(n, t) = 〈n| ρF |n〉 as:

∂p(n, t)

∂t= −A(n + 1)

(1− B

A(n + 1)

)p(n, t) + An

(1− B

An)p(n− 1, t)

+ C(n + 1)p(n + 1, t)− Cnp(n, t) ,(6.61a)

≈ − A(n + 1)

1 + (B/A)(n + 1)p(n, t) +

An

1 + (B/A)np(n− 1, t)

+ C(n + 1)p(n + 1, t)− Cnp(n, t) .

(6.61b)

Equation (6.61) is a rate equation similar to those already encountered in the classicaltheory of lasers but this time describing the flow in probability space rather than foraverages (the latter can of course be obtained by summing the weighted probabilities).In the same way, detailed balance can be applied to obtain the steady-state solution ofeqn (6.61) from the compensation of configurations differing by one photon. This is seenclearly in eqns (6.61) where a term involving A cancels with the term involving C of theopposing sign. The other two terms also cancel in this way; they are in fact equivalentsubstituting n for n + 1:


A(n + 1)

1 + (B/A)(n + 1)p(n, t) = −C(n + 1)p(n + 1, t) , (6.62a)

An

1 + (B/A)np(n− 1, t) = −Cnp(n, t) . (6.62b)

The corresponding transitions are sketched in Fig. 6.15

An1+(B/A)n

p(n− 1)

|n〉

|n− 1〉

|n + 1〉

A(n+1)1+(B/A)(n+1)

p(n)

Cnp(n)

C(n + 1)p(n + 1)

Fig. 6.15: Flow of probabilities between configurations with n − 1, n and n + 1 photons. At equilibrium,steady-state is established by detailed balancing of the neighbouring terms that equate to each other throughsubstitutions n → n + 1.

Equation (6.62) can be solved by recurrence, yielding p(n) knowing p(n − 1)through:

p(n) =A/C

1 + (B/A)np(n− 1) , (6.63)

which repeated application yields

p(n) = p(0)

n∏i=1

A/C

1 + i(B/A), (6.64)

with p(0), the starting point, being determined by normalization condition

∞∑n=0

p(n) = 1 . (6.65)

Exercise 6.2 (∗) Show that the polynomial expansion for the equation of motion of 〈n〉 =∑n np(n) derived from eqn (6.61b) is of the type:

d〈n〉dt

= α〈n〉 − β〈n2〉+ γ . (6.66)

Analyse this equation providing physical meanings of the parameters α, β and γ andlink them to microscopic parameters.

LASERS 245

0

0.02

0.04

0.06

0 20 40 60 80

Pro

babi

lity

n

Coherent

Laser

<n>=40

Fig. 6.16: Statistics p(n) of a laser given by eqn (6.64) as compared to the Poisson distribution of a coherentstate. Coefficients A/C = 1.2 and B/C = 0.05 result in an average number of photons 〈n〉 = 40. Evenabove threshold a laser still has large deviation from the ideal coherent case.

In the semiclassical theory, the linewidth is obtained from the Fourier transformof 〈E(t)〉, where E is the photon field operator so that, in the Schrodinger picture:

〈E〉(t) =√

ω/2eV sin(kz)Tr(ρ(t)a†)eiνt (6.67a)

∝ sin(kz)

∞∑n=0

√n + 1ρn,n+1(t)e

iνt , (6.67b)

where ρn,n+1 = 〈n| ρ |n + 1〉 is the upper diagonal element of the density matrix.Conversely to diagonal elements, the off-diagonal elements do not form a closed setof equations and couple to all other elements of the density matrix (including diagonalelements), showing that the dynamics of coherence that is of a quantum character ismore complicated than the dynamics of population that is of a classical character.

Dotting eqn (6.59) with |n〉 and |n + 1〉, one gets for the equation of motion:

ρn,n+1 = −[(

A−B(n +3

2))(n +

3

2) +

1

8B + C(n +

1

2)

]ρn,n+1

+(A−B(n +

1

2))√

n(n + 1)ρn−1,n (6.68)

+ C√

(n + 1)(n + 2)ρn+1,n+2 .

High above threshold where n assumes high values over which ρ varies smoothly,ρn,n+1 can be approximated by its neighbour value ρn,n, for which recursive closedrelations are known, cf. eqn (6.64). This turns eqn (6.2.5) into:


ρn−1,n ≈ C√

(A−Bn)(A−B(n + 1))ρn,n+1 (6.69a)

ρn+1,n+2 = C√

(A−B(n + 1))(A−B(n + 2))ρn,n+1 . (6.69b)

Injecting this expression back into eqn (6.2.5) gives

ρn,n+1 = −1

2Dρn,n+1 , (6.70)

with

D ≈ 1

2

A

〈n〉 . (6.71)

From the Fourier transform of eqn (6.67), the lineshape of the laser turns out to be:

S(ω) =|〈E(0)〉|2

(ω − ν)2 + (D/2)2, (6.72)

that is, it is a Lorentzian centred on the laser transition with width D. The notablefeature is that D varies inversely with the photon field intensity: the laser has a verynarrow line as a result of the photon compression in phase space, an effect first realizedby Schawlow and Townes. This is, however, more of a theoretical limit as other factorsbroaden the line much beyond the value given by eqn (6.71).

6.3 Nonlinear optical properties of weak-coupling microcavities

By placing a material inside a microcavity, its nonlinear optical properties are enhanced.The first enhancement arises simply from the enhancement in internal optical intensitydue to the finesse, which thus reduces the external threshold light intensity to get a cer-tain nonlinear optical response. The advantage of using a microcavity is that the builduptime for the optical field is short, as well as the transit time, and hence the device opera-tion remains nearly as fast as that of the intrinsic nonlinear material. Another advantageis that the refractive part of the nonlinear response is converted into a transmission non-linearity due to the optically induced spectral shifting of the cavity modes.

Fig. 6.17: (a) Microcavity with enhanced optical intensity within a nonlinear medium, (b) Optically inducedspectral shift of cavity modes from refractive-index change produces a change in transmission at a near-resonant wavelength.

NONLINEAR OPTICAL PROPERTIES OF WEAK-COUPLING MICROCAVITIES 247

The nonlinear optical process may arise directly from occupation of the upper stateof two-level systems (in atoms, or in semiconductors) that depends on the fermionicstatistics of electrons. Or it may arise from Coulomb interactions between optically ex-cited states (such as the exchange interaction). It also is typically divided into “real”and “virtual” processes, which correspond to whether switching optical energy ulti-mately ends up absorbed inside the medium (the former), or the energy remains withinthe optical field (the latter). In effect, the “virtual” process is a transient state inside themedium that reemits the optical energy (for instance if an atom is strongly excited off-resonantly), but transiently produces nonlinear responses as above. Such a process willbe intrinisically faster than a “real” process, in which the absorbed energy must be lost(e.g through recombination or spontaneous emission) before the next switching eventcan take place.

The use of microcavities in these applications has been studied since the 1980s,typically in semiconductor Fabry–Perot interferometers, for possible ultrafast opticalswitching elements. A typical example is a gold-coated semiconductor slab containinga quantum well that is optically excited to the long-wavelength side of the exciton reso-nance. The creation of a virtual population of excitons blueshifts the exciton resonance,thus changing the refractive index at the cavity resonance and producing an enhancedultrafast response. In general, the problem with such devices is that the nonlinear re-sponse from a small-volume microcavity is limited.

6.3.1 Bistability

New effects occur when the cavity mode can be spectrally shifted by more than the cav-ity linewidth. In this case, optical bistability can occur in which there are conditions forwhich two stable states of the cavity transmission exist, “high” and “low”. The idea is toset up the cavity response and the pump-laser tuning in such a way that an increase in in-cident optical power spectrally shifts the cavity closer into resonance with the excitationlaser. This further increases the power fed into the cavity, and thus provides a positivefeedback that clamps the transmission to maximum. The reverse situation occurs as theincident power is reduced, in that the internal optical field is sufficiently strong so thatthe cavity resonance remains closer to the incident laser wavelength than expected fromthe incident optical power alone, until a critical minimum power at which the wholeeffect switches off. Two regimes are possible for bistability in microcavities, with thenonlinear response primarily either absorptive or dispersive, as shown by Gibbs (1985).Bistability is also observed in the atom-filled microcavity system in the strong-couplingregime, in which case measurements of the transmission at higher power become dis-torted and shifted, as seen in Fig. 6.18(c) by Gripp et al. (1997).

A bistable optical response can also be seen in polarization switching in VCSELs. Insuch lasing microcavities, only one mode lases at any time, however, perturbations (forinstance, in current injection, or incident light) can switch the lasing between two or-thogonally polarized nearly degenerate lasing wavelengths. The balance between thesestates is controlled by the spin relaxation of the excitons inside the active quantum wellregions, as well as strain within the fabricated pillar microcavities.

For any microcavity system that is bistable, there remain the transverse degrees


Fig. 6.18: (a) Optically induced spectral shift of cavity modes locks cavity to input laser wavelength at highpower, (b) transmitted intensity response vs. input intensity showing the region of bistability (shaded). (c)Transmission through strongly coupled atom–cavity showing hysterisis as the incident light is tuned, fromGripp et al. (1997).

of freedom that allow optical pattern formation within the microcavity. At one extreme,this can lead to the formation of spatial solitons, in which the light within a region of themicrocavity that is switched “high” suffers nonlinear diffraction in such a way that thelateral shape of the resonant optical mode within the region is preserved. In other cases,2D grating patterns can emerge either statically, or in a constantly changing dynamicpattern evolution. The exact response depends critically on the illumination conditions,the cavity length and mode spectrum, and the boundary conditions, as discussed byHachair et al. (2004).

Fig. 6.19: Pattern formation in electrically contacted microcavities: (a) spatial soliton in liquid-crystal micro-cavity as observed by Hoogland et al. (2002) and (b) pattern formation and (c) seven stable spatial solitons insemiconductor VCSELs reported by Hachair et al. (2004).

A number of realizations of pattern formation within microcavities have been demon-strated including (a) atoms on resonance (though this is not in a microcavity but ex-tended over cm lengths), (b) liquid crystals within planar microcavities, and (c) semi-conductor quantum wells in large area VCSELs (Fig. 6.19). In general, pattern forma-tion is a sensitive phenomena and is thus perturbed strongly by imperfections in themicrocavity properties. Use of this phenomena, for instance for switching of pixels, isthus problematic.

CONCLUSION 249

6.3.2 Phase matching

One further use for nonlinear microcavities has been to act as optimized optical fre-quency-doubling devices. By carefully controlling the Bragg reflector mirror stack, it ispossible to produce a microcavity that is resonant at both ω and 2ω, with a selectablephase difference between the two per round-trip due to different DBR penetrations. Thiscan thus act as a phase-matching device, when non-critical phase matching is difficult.

An equivalent use of microcavities has been as a pulsed photodiode to measure ul-trashort optical pulses, using two-photon absorption to generate a dc electrical currenteven at small input intensities. By surrounding the active region of a two-photon photo-diode with a microcavity tuned to the input wavelength, the electrical current measured,which depends on the peak field of the pulses, is amplified by 104, while the short cavitylength ensures minimal broadening of the temporal response.

6.4 Conclusion

In this chapter, a basic overview of emission from microcavities in the weak-couplingregime shows a number of their benefits including lower thresholds, fast response, andcontrollable emission characteristics. In the next chapter we show how these are modi-fied in the strong-coupling regime.

7

STRONG COUPLING: RESONANT EFFECTS

This chapter presents experimental studies performed on planarsemiconductor microcavities in the strong-coupling regime. The firstsection reviews linear experiments performed in the 1990s that haveevidenced linear optical properties of cavity exciton-polaritons. Thechapter is then focused on experimental and theoretical studies ofmicrocavity emission resonantly excited. We mainly describeexperimental configuration at which the stimulated scattering wasobserved due to formation of a dynamical condensate of polaritons.Pump-probe and cw experiments are described as well. Dressing of thepolariton dispersion and bistability of the polariton system because ofinter-condensate interaction are discussed. The semiclassical and thequantum theories of these effects are presented and their resultsanalysed. The potential for realization of devices is also discussed.

252 STRONG COUPLING: RESONANT EFFECTS

7.1 Optical properties background7.1.1 Quantum well microcavities

In 1992, the strong-coupling regime was first identified by Weisbuch et al. (1992) insemiconductor microcavities. In fact, their goal had been to optimise the superradiantemission of quantum wells inside microcavities, and the splitting in reflection that theyobserved was not really expected, because it was previously thought that the light–matter coupling was too small for strong coupling. The correct identification of this asstrong coupling led to a number of investigations of the emission characteristics of thesedevices.

Fig. 7.1: (a) Strong-coupling reflection spectra in a planar semiconductor microcavity, and (b) normal incidentpolariton energies vs. cavity detuning (scanning across sample), as observed by Weisbuch et al. (1992).

Besides the complete formulation of such a multilayer structure presented in Section2.7, there are several simple models for the strong coupling that are appropriate for theintuition they provide. A sharp exciton (or atomic) transition produces a characteristicresonant absorption and dispersion lineshape (Fig. 7.2(a)). If this is inserted into a mi-crocavity, then the total round-trip phase as a function of wavelength acquires an extracontribution (Fig. 7.2(b)), which means that there are now three resonant conditions.

The upper and lower resonant conditions occur where the absorption is small and sohave narrow linewidths, while in this picture the central constructive condition remain-ing at the resonance energy occurs with strong absorption and is not observed. Thissimple picture of a net refractive index corresponds to the non-local dielectric suscep-tibility model presented in Section 4.4.2 so that excitons feel the optical field from thecavity together with the polarization from all other excitons around them.

The key characteristic of semiconductor microcavities in the strong-coupling regime

OPTICAL PROPERTIES BACKGROUND 253

Fig. 7.2: (a) Dispersion and absorption of resonance, producing (b) net microcavity round-trip phase, φ,without (dashed) and with (solid) resonant medium. The lower (•) and upper () polaritons are at energieswhere φ = 2π.

is the dispersion relation. This maps how the resonant polariton modes shift with theinplane wavevector (or angle of incidence). The derivation for the multilayer (eqns 1.9,2.147 and 4.134) can be simply realized from the resonant condition on the wavevectorperpendicular to the planar cavity mirrors (see Sections 1.3, 2.7 and 4.4.4.2). Whilethe angular dispersion near k = 0 can be expanded quadratically giving a very lightpolariton mass, the full dispersion is often crucial for the effects reported and is thesolution of eqns (4.130), producing the characteristic shape in Fig. 7.3. We have termedthe centre of this dispersion a k-space polariton trap as the energy of lower-polaritonshere is below that of all other electronic excitations, and scattering out of the trap isdifficult for polaritons if kBT < Ω/2.

Fig. 7.3: Angular dispersion of a typical semiconductor microcavity at zero detuning, (a) on a log scaleshowing the constrast between light polaritons and heavier excitons and (b) showing the critical regionsaround the polariton trap.


7.1.2 Variations on a theme

Quantum well microcavities that incorporate InGaAs microcavities exhibit the clearestpolaritonic features because the strain within the InGaAs energetically splits the heavy-and light-hole excitons so that only the simple j=3/2 heavy-hole polaritons are resolved.Even in this case the spin-degeneracy and residual lattice strain along [110] producescomplicated polariton interactions. Microcavities in which GaAs quantum wells are in-corporated have narrower linewidths (typically below 0.1 meV) because of the elim-inated alloy disorder, however, the light-hole excitons with a third of the oscillatorstrength are only a few meV to higher energy and also strongly couple to the cavitymode, producing a more complicated polariton dispersion. Microcavities that incorpo-rate GaN are even more complicated, since there are A,B and C excitons that all coupleto the cavity mode with different polarization dependences.

Besides microcavities that use quantum wells for the excitonic coupling to the cavitymode, it is also possible to use wavelength-thickness layers of bulk semiconductors.Because binary semiconductors do not have alloy disorder, their excitonic linewidthscan be narrow, although larger thicknesses need to be used to overcome the weakeroscillator strength, see for instance the discussion by Tredicucci et al. (1995). Bothreflectivity and luminescence show similar strong coupling to QW microcavities, withclear Rabi splittings of typically 4 meV, as well as extra polaritonic modes from thequantization of the exciton centre of mass within the finite thickness layer (Fig. 7.4).

Fig. 7.4: Angular dispersion of a λ/2 GaAs bulk semiconductor microcavity showing the strong couplingand additional centre-of-mass polariton modes, from Tredicucci et al. (1995).


It is not even strictly necessary to use a microcavity to produce such polaritonicdispersions, since unwrapped the microcavity looks like a periodic array of quantumwells.

The first theoretical study of the “Bragg arranged quantum wells” was undertakenby Ivchenko et al. (1994), who showed that the Bragg arrangement leads to an ampli-fication of the exciton–light coupling strength proportional to the number of quantumwells. Nowadays, this effect is also exploited in 2D and 3D resonant photonic crystals.

Experimentally, “Bragg-arranged quantum wells” have been explored by severalgroups, see for instance the publications by Hubner et al. (1996) and Prineas et al.(2002). They show many analogous features to semiconductor microcavities. However,it is in practice harder to produce many quantum wells (up to 100 are needed in GaAs)all of exactly the same thickness, spacing and composition, and this is even harder inother material systems. However, even in a small number of closely spaced quantumwells, polaritonic effects can be observed, as reported by Baumberg et al. (1998).

Typically, it is useful to study different detuning conditions of the microcavity,where the detuning is the energy difference between the normal incidence uncoupledcavity mode and the exciton energy ∆ = ωC − ωX at θ = 0. One way in which thisis achieved is by increasing the growth variation between different areas of the wafer(typically by eliminating the wafer rotation in the growth reactor), which produces anincreasing thickness of the cavity length across the wafer. Hence it is possible to findareas in which zero detuning is present, and either side of this detunings greater or lessthan zero. The weaker dependence of the QW energy on the well width means that thismethod is quite effective. Another possibility is to use temperature to control the de-tuning, since the expansion of the lattice shifts both exciton energy and cavity mode tolower energy, though the exciton shifts about three times faster. This is seen on Fig. 7.5from the work of Fisher et al. (1995). At temperatures above 100 K the thermal ioniza-tion of excitons becomes sufficient to broaden the excitons in III-V semiconductors andwash out strong coupling, limiting the effective range of tuning. However, this techniquehas been frequently used, often to tune localized excitons in quantum dot microcavities.

However, there are situations in which one would like to remain in a specific positionon the sample and tune the cavity mode or exciton energy. Tuning of the exciton energyis possible using either electric or magnetic applied fields. By growing the microcavityin a p-i-n device, for instance with the DBR mirror stacks doped as in a VCSEL, avertical electric field can be applied that produces a quantum-confined Stark shift of theexciton energy. Up to 35 kV/m, the oscillator strength of the exciton transition decreasesby only 30%, while the excitons redshift by 20 meV, allowing them to be scanned acrossthe cavity mode to demonstrate the strong-coupling regime, as in Fig. 7.6 from Fisheret al. (1995). On the other hand, magnetic fields split the heavy-hole exciton into spin-up and spin-down components, and the magneto-exciton can have a larger oscillatorstrength due to its more compact binding. This allows a weak to strong coupling tran-sition to be observed with applied magnetic field, as reported by Tignon et al. (1995).In addition, although the magneto-splitting is only 1 meV for B = 10 T and hence lessthan the Rabi splitting, the individual spin-down and spin-up polaritons can be resolvedusing circularly polarized light, as reported by Fisher et al. (1996).


Fig. 7.5: Temperature tuning of an InGaAs microcavity by Fisher et al. (1995).

7.1.3 Motional narrowing

Another effect of strong coupling is to change the effect of disorder in the exciton andphoton modes. This arises because the length scale over which polaritons average overdisorder can be different from the lengthscales of disorder in the components. Typi-cally, excitons, even in high-quality quantum wells, are localized on the 10–100 nmlengthscales, on the order of 10 Bohr radii. The unavoidable variation in the width ofthe quantum wells (so-called monolayer fluctuations, which are different on the twosides of the quantum well) means that there is a population of excitons with differentenergies in different spatial locations. When these excitons are all coupled to the samecavity mode, the resulting polariton averages over all their energies producing an inho-mogeously broadened polariton much narrower than the exciton distribution. This effecthas been termed “motional narrowing”, as seen in Section 4.4.5. Another way to see thiseffect is that the lower-energy polariton has a linewidth given by the imaginary part ofthe dielectric constant that is much reduced further away from the centre of the exci-ton distribution. Measurements of the polariton linewidth in the strong-coupling regimeindeed show this effect, with a reduced inhomogeneous distribution for polaritons com-pared to excitons, see Fig. 4.26.

7.1.4 Polariton emission

One of the first observations concerning the polariton radiative emission from InGaAsquantum wells in GaAs semiconductor microcavities was that although the photolu-minescence mapped onto the predicted dispersion relation, the intensity of this lumi-nescence was rather different from the typical thermalized emission seen from barequantum wells. Similar results were seen in CdTe-based microcavities, for instance by


Fig. 7.6: Electric field tuning of an InGaAs microcavity by Fisher et al. (1995).

Muller et al. (1999) (see Fig. 7.7). The reason for this is that the excitons generatedimmediately after non-resonant excitation with a pump laser relax quickly to the high-k part of the lower-polariton dispersion (often termed the “exciton reservoir”). Theircooling to lower energies and lower k, and particularly into the polariton trap, is thenrestricted by the need to lose large amounts of energy with very little simultaneous re-duction in k. Very few quasiparticles exist within the semiconductor that can removethis combination of energy and momentum, and hence the exciton-polaritons collect atthe “bottleneck” region in the vicinity of the trap (Fig. 7.3(b)). Hence, instead of thegreatest luminescence intensity emerging at the lowest energies, much more lumines-cence emerges from this bottleneck spectral region. Note also that unlike quantum wells,the luminescence spectrum is also angularly dependent due to the polariton dispersion.

From angular measurements of the luminescence as a function of detuning (whichchanges the depth of the trap) and temperature, one can estimate that more than fiveacoustic phonon scattering events are needed to cool a carrier into these 3-meV polaritontraps (for GaAs-based microcavities), which is significantly slower than the radiative


Fig. 7.7: Emission intensity and emission rate as a function of the emission angle from a non-resonantlypumped CdTe-based microcavity, from Muller et al. (1999).

lifetime in the bottleneck region. Hence, in the linear regime, emission from strong-coupled semiconductor microcavities is reduced rather than enhanced, besides beingstrongly angle dependent.

At the end of the 1990s, experiments such as those of Le Si Dang et al. (1998)and Senellart and Bloch (1999) began to show that the bottlenecked luminescence fromthe trap states was highly nonlinear with the injected laser power. Full understandingof this behaviour required an overview of the scattering processes available to exciton-polaritons, which we treat in the next chapter.

7.2 Near-resonant-pumped optical nonlinearities

7.2.1 Pulsed stimulated scattering

In quantum wells, many experiments have shown how the injection of excitons or freeelectron–hole pairs leads to changes in the exciton absorption spectrum. These resultfrom scattering processes between excitons (generally repulsive) and between excitonsand free carriers (which can ionize the exciton). However, the experimental difficultyin studying excitons within a quantum well is that the optically accessible states arenot distinguishable by changing the angle of incidence (due to their almost flat disper-sion), so that the inhomogeneous broadening dominates. Equally problematic is that de-spite the best efforts to grow smooth-walled atomically flat interfaces, the disorder fromroughness and alloying of quantum wells produces excitonic states that are at most de-localized over a few hundred nanometres, many times the exciton Bohr radius (≈ 15 nmin GaAs) but much less than the optical wavelength. Hence, these exciton states emitand absorb in all directions. Thus, it is not possible to directly observe exciton collisionprocesses using conventional spectoscopy in quantum well (or bulk) samples, and onehas to resort to indirect methods.

NEAR-RESONANT-PUMPED OPTICAL NONLINEARITIES 259

On the other hand, the clear dispersion of exciton-polaritons in semiconductor mi-crocavities allows polaritons at different angles and energies to be distinguished. Hence,it became possible to perform resonant nonlinear experiments on microcavities, bypumping excitons at one angle and measuring how fast they scatter into other states.

The first pump-probe experiments on semiconductor microcavities were performedwith two beams under normal or quasinormal incidence with as an underlying objectiveto modify the system using an intense pump pulse and to record the resulting polar-ization by measuring the reflection, transmission, absorption or scattering of a weakprobe pulse. The main goal of such preliminary investigations, such as those by Jahnkeet al. (1996) (an extensive list of references is given by Khitrova et al. (1999)) was toelucidate the mechanisms responsible for the loss of the strong-coupling regime. Afterthis early stage, the understanding of nonlinear optical properties of microcavities hasprogressed considerably. This progress has been mainly due to use of advanced spec-troscopy techniques, allowing one to tune the angle, energy and time delay betweenpulses independently. The breakthrough came from an experiment performed by Sav-vidis et al. (2000) and discussed at length below. This experiment has evidenced thebosonic behaviour of cavity polaritons. It has also shed much light on the main mech-anisms governing optical nonlinearity in microcavities. An avalanche of experimentaland theoretical studies followed that of Savvidis revealing rich and deep physical phe-nomena. Most of these results are now being discussed.

One first indication of the peculiarities of the polariton interactions was how themeasured optical nonlinearities depended not just on the energies of the polaritons,but on their full dispersion, see the discussion by Baumberg et al. (1998). This con-firmed that angular tuning and position tuning (in which the cavity mode energy variesacross the sample due to a low-angle wedged thickness variation) were not equivalent.In 2000, the group at Southampton first reported experiments by Savvidis et al. (2000)that definitively showed that the scattering of polaritons was influenced by pre-existingpopulations of polaritons—in other words that scattering could be a stimulated process.While it is well known that photons can stimulate photon emission, the process of stim-ulated scattering is much less studied.

By injecting a pump pulse at a particular angle of incidence (k) and energy (ω),the time-resolved evolution of the scattering of polaritons can be tracked using a weakbroadband probe pulse to measure the reflection spectrum at different times (Fig. 7.8),as done by Savvidis et al. (2000). For particular conditions, reflectivities much largerthan 100%—corresponding to extremely large amplifications—were observed, reach-ing 10 000%. These gains persisted only while the pump-injected polaritons remainedinside the microcavity. Moreover, the gain of the seeded probe pulse is extremely sen-sitive to the incident pump angle—termed the magic angle (see Fig. 7.9)—and pumppower. These features are the signature of the polariton pair-scattering process shownin Fig. 7.9(c) in which two polaritons injected by the pump have exactly the right (k, ω)to mutually scatter sending one down to the bottom of the trap (at k=0, often called the“signal”) and the other to 2k (the “idler”).


Fig. 7.8: (a) Microcavity dispersion, showing pump pulse injecting polaritons at (k,ω) and scattering to otherstates on the dispersion, (b,c) Reflection spectrum before and as pump pulse arrives showing strong gain onthe lower-polariton at k=0, and (d) Time response of the gain, as published by Savvidis et al. (2000).

Fig. 7.9: (a) Resonant gain (at k=0) as the pump angle is varied, (b) exponential observed pump-powerdependence of the gain and (c) schematic polariton pair scattering, from Savvidis et al. (2000).

Three clear new features are shown in this experiment:

• polaritons can scatter strongly from each other providing that both energy conser-vation and momentum conservation can be simultaneously satisfied in the two-quasiparticle collision,

• polariton scattering can be enhanced by occupation of the final state. In otherwords, that polariton scattering can be stimulated, as expected for bosons,

• polaritons are stable at the bottom of the polariton trap, in comparison to theirlifetime (governed by their escape from the cavity).

While all bosons should possess the particle statistics that produce this effect, it hasnot been observed for excitons in semiconductors. This is because the exciton disper-sion is normally so flat that many other processes can scatter excitons (e.g. disorder,


phonons), and hence k is not a good quantum number for excitons; a macroscopic pop-ulation of a single quantum state is unlikely for excitons. On the other hand, for cavitypolaritons, because of the tendency for bosons to occupy the same state, the gains mea-sured can exceed 106 cm−1, larger than in any other material system.

In the language of nonlinear optics, such processes are said to be parametric scat-tering processes, and are commonly observed for parametric downconversion (wherea photon at 2ω transforms into two photons at ω + ε and ω − ε). Polariton scatter-ing in this experiment is equivalent to a four-wave mixing process (or near-degenerateparametric conversion) where two pump polaritons create a signal and idler polariton,which emerge from the sample at different angles. Because this description via four-wave mixing only deals with the incident and emitted photons, it describes nothing ofthe solid-state coherence within the semiconductor microcavity and is thus a limitedtool for understanding polariton scattering and coherence.

In a similar way, using the exciton and cavity photon basis for understanding po-lariton scattering is also limiting. For zero cavity detuning, both the lower and upper-polaritons are composed of half a photon and half an exciton, however the scatteringproperties of these polaritons are completely different, due to their different energiesand the density of states into which they can scatter. In an exciton/photon basis the onlydifference is the sign with which their wavefunctions are combined.

Further evidence for the coherent nature of the lower-polariton signal state at k=0 isprovided by coherent control experiments that show how the signal polaritons amplifiedby a first seed pulse may be destroyed by a subseqently oppositely phased reset pulse.Such experiments have been done by Kundermann et al. (2003).

Fig. 7.10: Stimulated polariton scattering peak gain as a function of temperature for different microcavitysamples, from Saba et al. (2003).

Stimulated scattering at the magic angle has been observed in many different semi-conductor microcavities, with single or multiple quantum wells, of different materials,and in patterned mesa microcavities (which have different dispersions with the extratransverse mode). The main effect of using different materials is to change the tem-perature at which the stimulated scattering process switches off (see Fig. 7.10 and itsdiscussion by Saba et al. (2003)). The current model of temperature-dependence is that


because of the parametric process, both signal and idler polaritons together (in a jointcoherent state) generate the stimulation (see Section 7.2.2). Scattering of the idler po-laritons, which occurs at elevated temperatures thus destroys the polariton stimulation.If the idler is too close energetically to the electron–hole continuum, then the fast scat-tering (very similar to that of excitons) resumes, which destabilizes the idler polaritons.This motivates the current experimental push to building strong-coupling microcavi-ties that are based on ZnSe, GaN and ZnO semiconductors, since these are predictedto provide strong stimulated scattering at room temperature. This would open the wayto building more complex optoelectronic devices (such as coherent interferometers andswitches) from semiconductor microcavities.

Fig. 7.11: Stimulated scattering processes in (a,b) photonic wires, from Saba et al. (2003), and (c) from twopump beams launched at equal angles either side of k=0.

Stimulated scattering also occurs in a variety of geometries. For instance, when aplanar microcavity is patterned into photonic wires, quantization perpendicular to thewire produces a nested series of lower-polariton dispersion curves for k along the wire(see Fig. 7.11 and discussion by Dasbach et al. (2003)). These produce a new range ofpossibilities for stimulated scattering, involving more than one branch of the dispersion,with final idler states existing at lower energies, thus reducing the scattering that con-strains the temperature of operation. Stimulated scattering of polaritons is not limitedto the magic-angle condition in which the two initial polaritons are at the same (k,ω).It can also be observed for polaritons that are in initially different states, for instanceon either side near the bottom of the polariton trap (Fig. 7.11b), see the discussion byRomanelli et al. (2005). All these schemes have suggested novel ways in which to effi-ciently produce entangled photons as part of signal and idler beams.

One clear signature of the stimulated scattering process (to be discussed in detailin the theory of Section 7.3) is the rigid blueshift of the whole lower-polariton disper-sion as it becomes macroscopically occupied. This results from a self-scattering term tothe energy, but it has the effect of modifying dynamically the tuning of incident lasersand dispersion as scattering occurs. This rigid energy shift of the polariton dispersion


Fig. 7.12: (a) Blueshifted lower-polariton disperion (dashed to solid thin lines) subsequently produce stimu-lated scattering. The macroscopic signal and pump polariton occupation (•) generates new off-branch polari-tons (), observed at the indicated output angle (dash-dot vertical) as (b) new peaks in emission (arrows).

is only the first-order effect, and is proportional to the total population of polaritons.A second-order term means that occupation of the dispersion changes the shape of thedispersion—a highly nonlinear process. One effect of this is that when pump and sig-nal polaritons become macroscopically occupied, new scattering processes appear forwhich one of the final states is off-branch (see Fig. 7.12 and the discussion by Savvidiset al. (2001)). The polariton dispersion is distorted, and produces a flat region aroundk=0, whose onset also signals the destabilization into spatial solitons (mentioned in Sec-tion 6.3.1). Thus there remain many confusing and novel features about the stimulatedscattering process in both space and time that need to be further explored.

7.2.2 Quasimode theory of parametric amplification

In this section, we address the theories of CW parametric scattering in the dynamicregime. This identifies, at each moment in time, the transient eigenstates of the pairpolaritons that independently experience the gain or loss. We assume a slowly varyingpolariton amplitude (a reasonable approximation for narrow spectral linewidth cavities),and also work in the limit of negligible pump depletion (i.e. at low probe powers). Inthis case the equations governing the slowly varying envelope of signal (S) and idler (I)can be written

∂S

∂t= −γSS − ΛI∗ , (7.1a)

∂I∗

∂t= −γII

∗ − Λ∗S , (7.1b)

where Λ(t) = iV P 2(t)eiνt accounts for the coupling. Here, V is the polariton–polaritoninteraction constant, P is the dynamic pump polariton occupation, and ν = 2ωP−ωS−ωI is the frequency mismatch from the magic-angle condition. We look for solutions


Fig. 7.13: Quasimode calculations as a function of real time (ps) for (a) eigenvaules of M, N , (b) mixingparameter ψ, (c) fractional amount of signal and idler components in M , (d) dynamics of eigenmodes M , Nand (e) of signal and idler, when the probe pulse is at t = −1 ps, pump at t = 0 ps.

corresponding to gain: S, I∗ ∝ eqt. Solving the determinant of eqn(7.1b) produces thetwo solutions for the damping:

γ± = −γS + γI

2±√

α2 + |Λ|2 , (7.2)

with α = (γS − γI)/2. These solutions are time dependent, with q± < 0 away fromthe pump pulse corresponding to the individual damping of signal and idler. They re-pel strongly when the pump arrives, to produce transient gain (q+ > 0, Fig. 7.13(a)).The eigenvectors of these solutions correspond to the two mixed modes (M,N ) thatexperience these gains.

M = C−eiφS + eψI∗

, (7.3a)

N = CeψS + eiφI∗

, (7.3b)

where we have defined sinhψ = α/ |Λ|, φ = arg(Λ), and the normalization C =1/√

1 + e2ψ . This mixed complex transformation of the signal and idler is controlledby the phase mismatch, φ = νt, and a mixing parameter, ψ(t). The mode M is ampli-fied when the pump pulse arrives, while the mode N is deamplified. The gain of thesemodes is given by q± = λ ± Λcosh ψ, with the average damping, λ = (γS + γI)/2.


The incident probe couples into both modes, giving new instantaneously decoupled dy-namical equations:

∂M

∂t= q+M − CeiφSprobe(t) , (7.4a)

∂N

∂t= q−N − CeψSprobe(t) . (7.4b)

In the vicinity of the pump pulse, the modes M,N contain roughly equal admixturesof the signal and idler (Fig. 7.13(c)): in other words, when the pump is present, thetrue modes of the system are not S, I but M,N . The dynamics of the quasi-uncoupledmodes and the signal and idler are shown in Figs. 7.13(d) and (e) for a probe pulse thatis 1 ps before the pump, and with damping of signal and idler, γS,I = 0.2, 0.4 meVcorresponding to the experiments.

From these equations it can be seen that the amplification of the population of po-laritons in the M -mode is roughly given by:∣∣∣∣Mout

Min

∣∣∣∣2

= exp 2q+T = exp 2 |Λ|T = exp 2V IpumpT , (7.5)

where T is the pulselength and Ipump is the pump power. This recovers the experimentalresult. It is also not what might be intuitively expected from a pair scattering processthat in an uncoupled system would have a gain proportional to the square of the pumpintensity. The completely mixed nature of signal and idler polaritons is what makes theparametric amplification so sensitive to dephasing of the idler component.

7.2.3 Microcavity parametric oscillators

While the multiple effects of stimulated scattering are clearest for pulsed excitation, theyare also observed in continuous wave (CW) excitation. A pump beam incident at themagic angle first generates spontaneous parametric pairs to signal and idler states, whichthen act as the seed for further stimulated scattering, see Fig. 7.14 and its discussionby Baumberg et al. (2000). After this threshold (where the signal polariton populationexceeds unity), scattering then proceeds exponentially with pump power until saturationoccurs. A set of spectra obtained in this configuration by Stevenson et al. (2000) atvarious angles of detection and at different powers is displayed in Fig. 7.15, where thefeatures at k = 0 completely dominate above threshold.

The system behaves as a microparametric oscillator (µOPO), an integrated equiv-alent to the cm- to m-scale bulk parametric oscillators (normally based on paramet-ric downconversion, which is a three-photon and not a four-photon process). Typicalthresholds for this micro-OPO device are in the 10 mW range, although the physicalgain length is some 10 000 times smaller than conventional OPOs.

The output of the µOPO is coherent, narrow spectral linewidth (≈ 1 GHz), andemitted into a narrow angular beam (width ∼ 5). While the signal output phase is in-dependent of the pump laser phase, the sum of signal and idler phases is locked to thatof the pump (as in a normal parametric oscillator). However there are some peculiarnovel features of the µOPO system. One of these is that because of the energy shifts


Fig. 7.14: (a) Geometry for microparametric oscillator with (b) spectrum at k=0 and (c) power dependence,from Baumberg et al. (2000).

possible in the lower-polariton dispersion as it becomes occupied, the device can orga-nize itself to optimize the stimulated scattering. Thus, while pulsed stimulated scatteringonly occurs close to a magic angle, in the CW case the device adjusts to produce µOPObehaviour over a wide range of pump angles (Fig. 7.16).

One result of the twin photon production of signal and idler is that they are quantum-mechanically correlated, or entangled. This can be most simply understood from theirsimultaneous origin with correlated phases from colliding pump polaritons, even thoughthey emerge with different energies in different directions. Such correlations can beextracted from experiments in which the two beams are mixed with a local oscillator ontwo balanced photodiodes, as has been done by Messin et al. (2001). Theoretically, suchexperiments can only slightly (by a few %) circumvent the quantum noise limit, due tothe degradation of the perfect polariton correlation when they convert into photons onexiting the sample.

More recently, there have been proposals by Ciuti (2004) and Savasta et al. (2005)for generating more useful entangled photon pairs from semiconductor microcavity po-lariton pair scattering, using geometries in which both signal and idler are lower inenergy than the exciton reservoir. However, it remains a challenge to generate brighthigh-efficiency correlated photon beams from these devices.

The effect of disorder is getting increasing attention in recent research both in theresonant experiments studied in this chapter but also in the off-resonant case exposed inChapter 8. In the case of the µOPO, a pseudoperiodic potential due to strain—which ispresent in every sample from CdTe to GaAs—results in local differences in the refrac-


Fig. 7.15: Spectra observed by Stevensonet al. (2000) in the resonant CW pumping ofa microcavity, as a function of power. Thesystem is below threshold in (a), approach-ing threshold in (b) and increasingly abovethreshold in (c), (d) and (e). Each figure dis-plays spectra collected at different angles.The strong feature around 1.456 eV is in-duced by the laser. Below threshold, exci-tations relax to states close to the pump.Above threshold, strong emission is ob-served at the signal and idler states cor-responding to 0 and 32, respectively.The k = 0 emission quickly becomes byfar the dominant one (the importance of theidler is seen in the insets).

tive index of the microcavity. Due to this disorder—or so-called “photonic potential”—the formation of the signal of the µOPO is strongly influenced by the minima of the po-tential wells. Sanvitto et al. (2006) recently reported that when the power is increased,the signal occupies different regions in real space.98 This might prove to be a key pointin the formation of OPO and related physics.99 For instance, Sanvitto et al. (2005)reported that locally, the Q factor could increase up to 30 000 from a nominal valueof 10 000 just by restricting to a region of 5 µm2.

In a final review of microcavity parametric scattering, we note that recently a µ-OPO regime has been observed in the weak-coupling regime by Diederichs and Tignon(2005). In these devices, three microcavities are stacked such that their cavity photonscan mix between them, while quantum wells still provide the nonlinear scattering pro-cess. Instead of relying on the distorted dispersion of the lower-branch polaritons, the

98Disorder is an open door to richer physics in relatively well understood systems. In the case of the OPO,for instance, “bistability” (discussed in further detail in Section 7.4.3) manifests strikingly by “switching”the OPO on and off and results in depopulating some regions as others get populated under the influence ofpotential traps in the disorder.)

99Effect and potential importance of disorder in the case of spontaneous condensation will be discussedin Chapter 8 on page 294. The intimate links between these two limits as regards disorder are not yet fullyunderstood.


Fig. 7.16: (a) Power dependence of signal and idler near the magic angle, and (b) threshold for signal atdifferent pump angles, from Butte et al. (2003).

three-photon cavity mode branches provide a phase-matched photon-stimulated emis-sion for all pump, signal and idler at k=0. Hence, in a similar way to the strong-coupledmicrocavities, these cavities lower the emission energies for the participating modes be-low that of the dissipative excitons, however, at the expense here of returning to photonand not polariton quasiparticle scattering processes.

7.3 Resonant excitation case and parametric amplification

This section presents the theoretical description of microcavity emission for the case ofresonant excitation. We focus on the experimental configuration of this chapter wherestimulated scattering is observed due to formation of a dynamical condensate of po-laritons. Pump-probe and cw experiments are both described. Dressing of the polaritondispersion because of inter-condensate interaction is discussed as well as its main con-sequence, which is the bistable behaviour of this system. The semiclassical and thequantum theories of these effects are presented and their results analysed.

7.3.1 Semiclassical description

We describe parametric amplification experiments using rate equations.100 The advan-tages of such a description with respect to the parametric amplifier model (classical orquantum), which will be presented next, is that it allows us to account for stimulatedscattering and to include easily all types of interactions affecting exciton-polariton re-laxation. Its disadvantage is that dispersion dressing of polaritonic energies—an impor-tant feature of parametric amplification—cannot be easily accounted for in this model.In the resonant configuration, one can single out states where energy-momentum trans-fer is very efficient and dominates the dynamics. We assume the simplest case of athree-level model:101 the ground or signal state, the pump state and the idler state (cf.

100Rate equations of populations are closely linked to so-called “Boltzmann equations”, which will bestudied in Chapter 8 where relaxation of polaritons will be the central theme of study.

101In the complete picture including all states to which we shall return in Chapter 8, the generic Boltzmannequation (see footnote 100) for a state of wavevector k is given by eqn (8.35). The equation to be solveddescribing the polariton dynamics is formed by the ensemble of Boltzmann equations written for all allowed

RESONANT EXCITATION CASE AND PARAMETRIC AMPLIFICATION 269

Section 7.3.4). The names arise from similar physics in nonlinear optics. The main lossprocesses for these states are radiative losses and elastic scattering processes driven bydisorder, which can both be included in the same loss constant even if their nature isvery different. The radiative loss means the disappearance of the particles, while thedisorder scattering implies transfer of a particle towards other states that are neglectedin this model. The interaction with phonons in this framework is similar to the disor-der interaction, and it can also be included as a loss with its appropriate constant. Thephonon contribution is often negligible at low temperatures in typically used cavities. Itmay cause a significant broadening of the polariton states in experiments performed athigh temperatures. The broadening, or loss parameter, can be written as:

1

Γk=

|Xk|2∆ + Γphonons

+|Ck|2γc

, (7.6)

where Xk and Ck are the exciton and photon Hopfield coefficients, respectively, ∆ is theexciton inhomogeneous broadening, Γphonons is the phonon-induced broadening and γc

is the cavity-photon broadening. At low temperature, Γphonons ∆. Moreover, in mostof the cavity samples studied experimentally, ∆ ≈ γc, which yields Γ0 ≈ Γp ≈ Γi = Γ.In this framework, the system can be described by a set of three coupled equations:

n0 = P0 − Γn0 − αn0ni(np + 1)2 + α(n0 + 1)(ni + 1)n2p , (7.7a)

np = Pp − Γnp + 2αn0ni(np + 1)2 − 2α(n0 + 1)(ni + 1)n2p , (7.7b)

ni = Pi − Γni − αn0ni(np + 1)2 + α(n0 + 1)(ni + 1)n2p , (7.7c)

where

α =2π

2

|M |2πΓ/2

,

where M is the polariton–polariton matrix element of interaction, which is here approx-imately equal to one fourth of the exciton–exciton matrix element of interaction. Thissystem of equations can be easily solved numerically. Moreover, if one considers the cwexcitation case, P0 = P2p = 0, this gives n0 = n2p. The system (7.7) thus becomes:

n0 = −Γn0 − αn0(np + 1)2 + α(n0 + 1)n2p + αn2

p , (7.8a)

np = P − Γnp + 2αn20(np + 1)2 − 2α(n0 + 1)2n2

p . (7.8b)

7.3.2 Stationary solution and threshold

In the stationary regime, n0 = np = 0. Before proceeding further with the formalism,we have to discuss how to define correctly the threshold condition for amplification in

values of the inplane wavevector. It can be solved numerically, choosing suitable initial conditions. Theseconditions are, for a pump-probe experiment, n0(0) = nprobe, nkp (0) = npump and Pk = 0. For cwexperiments, these initials conditions are nk(0) = 0 and Pkp (t) = P0. In Chapter 8, cylindrical symmetryof the distribution function will be assumed. In the parametric amplification experiments of interest in thischapter, the resonant excitation conditions break this symmetry and a two-dimensional polariton distributionfunction should be assumed. However, just below and above the amplification threshold a good description ofthe ground-state population can be performed assuming only the three states of the text.


the stationary case. Very often, it is believed that a good empirical criterion is that thepopulation of a given state reaches one. Indeed, the evolution equation for the ground-state population formally reads:

n0 = Win(n0 + 1)−Woutn0 . (7.9)

Win is supposed to include all channels used by incoming polaritons and Wout allchannels for their departure from the ground-state. The +1 in brackets corresponds tothe spontaneous scattering process, n0 in brackets describes the stimulated scatteringand−Woutn0 the loss term. Therefore, the condition n0 = 1 means that the stimulationterm is as large as the spontaneous scattering term and that the amplification threshold isreached. This point of view is, however, quite misleading. The equation for the ground-state population can indeed be rewritten as

n0 = n0(Win −Wout) + Win . (7.10)

Thus, the threshold is given by the condition Win −Wout = 0. Equation (7.10) yieldsin this case:

−Γ− αn0(np + 1)2 + α(n0 + 1)n2p = 0 , (7.11)

which implies:

n0 =αn2

p − Γ

α(2np + 1). (7.12)

where n0 is a population, so it should be positive or zero. In the latter case

np =

√Γ

α. (7.13)

Below threshold, np ≈ P/Γ, which gives:

Pthres = Γ

√Γ

α= Γ

Γ

2|M | . (7.14)

Using the conventional threshold condition leads to a similar formula for the thresh-old power (see Exercice 7.1). It is noteworthy that using two apparently independentthreshold conditions, one recovers exactly the same value of the amplification thresh-old. Assuming an exciting laser spot size of 50 micrometres, Γ = 1 meV and for thetypical GaAs parameters, Pthres ≈ 106Γ ≈ 50 µW. This is in good agreement withexperimental data.

Exercise 7.1 Assuming np n0, find the solution of the system (7.8). Find the thresh-old assuming as a threshold condition n0 = 1.

7.3.3 Theoretical approach: quantum model

Our starting point is the Hamiltonian (5.163). We neglect interactions with phonons orfree carriers. The framework used historically to describe this configuration, e.g., by


Louisell et al. (1961) for the general problem and by Ciuti et al. (2000) or Ciuti et al.(2001), for microcavities, is the one of the Heisenberg formalism rather than the densitymatrix approach. We will therefore follow this trend.

To obtain the equation of motion for polariton operators ak and a†k, we write the

Heisenberg equation:

idak

dt= [ak,H] = ELP(k)ak +

∑k,k′′

Eintk,k′,k′′a

†k′+k′′−kak′ak′′ + P (k) , (7.15a)

ida†

k

dt= [a†

k,H∗]= E∗LP(k)a†

k −∑k,k′′

Eintk,k′,k′′ak′′ak′a†

k′+k′′−k + P (k) , (7.15b)

where ELP is the lower-polariton branch dispersion relation

Eintk,k′,k′′ =

1

2

(Vk′,k′′,k−k′ + Vk′,k′′,k′′−k

), (7.16)

and P (k) the polarization amplitude induced by an external pumping field.

7.3.4 Three-level model

A three-level model has been proposed by Ciuti et al. (2000) to explain the Savvidis–Baumberg experiment described in Section 7.2.1. Its starting point is eqn (7.15), con-sidering only the three most important states, namely the pumped state kp, the groundor signal state k0 and the idler state 2kp. The authors assumed these three states to becoherently and macroscopically populated. In other words, they assumed these statesto behave as classical coherent states and they replaced the operators a0, akp

, a2kp

and their adjoint by c-numbers. This ansatz was proposed in the 1950s by Bogoliubov(1947) to describe superfluids (see also Bogoliubov’s (1970) textbook). He diagonal-ized a Hamiltonian equivalent to eqn (5.163), considering the existence of a macro-scopically occupied ground-state (the superfluid). He assumed that only interactionsinvolving the ground-state were important and also proposed to neglect fluctuationsof the ground-state because of its macroscopic occupation. His argument is that forthe ground-state [a, a†] N , where N is the ground-state population. Therefore, thenonzero value of the commutator can be neglected and the ground-state operators can bereplaced by complex numbers. Ciuti et al. proposed a similar approximation but threecondensates instead of one were assumed.

In this section, we consider that only the pumped-state operators reduce to com-plex numbers, keeping the operator nature of signal and idler. In this framework, thesystem (7.15) can be reduced to just three equations:

−ia0 = ELP(0)a0 + Einta†2kp

P 2kp

+ Pprobe(t) , (7.17a)

−iPkp= ELP(kp)Pkp

+ EintP∗kp

a0a2kp+ Ppump(t) , (7.17b)

ia†2kp

= ELP(2kp)a†2kp

+ E∗inta0P

∗2kp

, (7.17c)

where


ELP(0) = ELP(0) + 2V0,kp,0|Pkp|2 , (7.18a)

ELP(kp) = ELP(kp) + 2Vkp,kp,kp|Pkp

|2 , (7.18b)

ELP(2kp) = ELP(2kp) + 2V2kp,kp,0|Pkp|2 , (7.18c)

and

Eint =1

2

(Vkp,kp,kp

+ Vkp,kp,−kp) . (7.19)

The advantage of this formalism with respect to rate equations of populations inSection 7.3.1 is that it allows one to account for the energy renormalization processesdriven by interparticle interactions. Here, a blueshift of the three states considered is in-duced by the pump intensity. Replacing all operators by complex numbers, this equationsystem can be solved numerically for any pump and probe configuration.

We now consider the steady-state excitation case where a stationary pump of fre-quency ωp excites the system, without a probe. This pump drives the pump polarizationgiven by:

Pkp(t) = Pkp

eiωpt , (7.20)

with Pkp∈ C. The system of eqns (7.18) reduces to two coupled equations:

−ia0 = ELP(0)a0 + Einta†2kp

P 2kp

e2iωpt , (7.21a)

−ia†2kp

= −E∗LP(2kp)a†

2kp− Einta0P

∗2kp

e−2iωpt . (7.21b)

We define:

ω0 =1

(ELP(0)) , ωi =

1

(ELP(2kp)) ,

Γ0 =2

(ELP(0)) , Γi =

2

(ELP(2kp)) ,

and introduce the two rescaled quantities:

a0 = a0e−iω0t , a†

2kp= a†

2kpeiωit ,

andβ = |β|e2iϕp = EintP

2kp

.

The two previous equations become:

−i ˙a0 = −Γ0

2a0 + βa†

2kpei(2ωp−ω0)t , (7.23a)

−i ˙a†2kp

= −Γi

2a†2kp− β∗a0e

i(2ωi−ωp)t . (7.23b)

This equation is simply a quantum-mechanical equation for parametric processes firstwritten and solved by Louisell et al. (1961). Replacing all quantum operators in thisequation system by complex numbers is equivalent to treating the classical paramet-ric oscillator studied in the last century by Faraday and Lord Rayleigh, as has been


pointed out by Whittaker (2001). This equation system has been widely studied in re-cent decades. It can be solved in the Heisenberg representation in the time domain, asdetailed in the expositions of Mandel and Wolf (1961) and Louisell et al. (1961) or inthe frequency domain, as discussed by Loudon (2000) and in the case of microcavitiesby Ciuti et al. (2000).

For simplicity, we assume that the resonance conditions are satisfied and that theloss coefficients are the same for signal and idler:

ω0 + ωi − 2ωp = 0 , Γ0 = Γp = Γi = Γ .

Equations (7.23) become

˙a0 = −Γ

2a0 + iβa†

2kp, (7.24a)

˙a†2kp

= −Γ

2a†2kp− iβ∗a0 . (7.24b)

Equations (7.24) imply

˙a†2kp

=1

iβ(¨a0 +

Γ

2˙a0) , (7.25a)

¨a0 + Γ˙a0 +(Γ2

4− |β|2

)a0 = 0 . (7.25b)

The solutions of the characteristic equation associated with eqn (7.25b) are

r± = −Γ

2± |β| . (7.26)

The solutions of the system (7.25) are therefore

a0(t) = e−Γ2 t(a0 cosh(|β|t)− ia†

2kpsinh(|β|t)ei2ϕp

), (7.27a)

a2kp(t) = e−

Γ2 t(a†2kp

cosh(|β|t) + ia0 sinh(|β|t)e−i2ϕp). (7.27b)

Note that the right-hand side of eqn (7.27a) is back in terms of a rather than a (sinceat t = 0, operators coincide).

For t > 1/|β|, cosh and sinh can be approximated by exponentials with positiveargument. Therefore:

a0(t |β|−1) =1

2e(|β|−Γ/2)t

(a0 − ia†

2kpei2ϕp

), (7.28a)

a2kp(t |β|−1) =

1

2e(|β|−Γ/2)t

(a†2kp

+ ia0e−i2ϕp

). (7.28b)

The “particle number” operator for the signal is:

a†0(t)a0(t) =

1

4e(2|β|−Γ)t

(a†0a0+a†

2kpa2kp

−i(a†0a

†2kp

ei2ϕp−a2kpa0e

−2iφp)). (7.29)


If the signal and idler states are initially in the vacuum state, the average number ofparticles is therefore:

〈a†0(t)a0(t)〉 = 〈0, 0| a†

0(t)a0(t) |0, 0〉 =1

4e(2|β|−Γ)t . (7.30)

However,

〈a0(t)〉 = 〈0, 0| a0(t) |0, 0〉 = 0 . (7.31)

Equations (7.30) and (7.31) show that a ground-state, initially symmetric in thephase space, will have its population growing exponentially, while its amplitude remainszero. This shows that the symmetry of the ground-state is not broken by the pumpinglaser. To illustrate our purpose we consider that the system is initially in a state otherthan the vacuum.

7.3.5 Threshold

The threshold condition to stimulated scattering is given by Γ = 2|β|, that is,

|Pkp|2 =

Γ

2Eint. (7.32)

With such a pump polarization, the energy shift of the signal at threshold is equal to thepolariton linewidth. This theoretical result is in good agreement with the available ex-perimental data. It is instructive to compare the criterion (7.32) with the threshold con-dition obtained in Section 7.3.2 from the population rate equations (Boltzmann equa-tions).

The relation between the pumping power and the coherent polarization is Γ|Pkp|2 ≈

P and the threshold condition for the pump power is thus:

P =Γ2

2Eint. (7.33)

Assuming, as in Section 7.3.2, that the broadening Γ is independent of the wavevector,and that Eint ≈ |M |, the polariton–polariton matrix element of interaction, eqn (7.33),becomes:

Pthres = ΓΓ

2|M | . (7.34)

This value is exactly the same as the one obtained in Section 7.3.2, illustrating theequivalence of the semiclassical and quantum models in this aspect.

7.4 Two-beam experiment

7.4.1 One-beam experiment and spontaneous symmetry breaking

We assume that a cw pump laser excites the sample, together with an ultrashort probepulse that seeds the probe state. Therefore, at t = 0, the probe state is a coherent

TWO-BEAM EXPERIMENT 275

state |α0〉 with α0 = |α0|eiϕ0 . The idler state is initially unpopulated (vacuum state).The initial state of the signal⊗idler system is denoted |α0, 0〉. With such initial states:

〈a†0(t)a0(t)〉 =

1

4(1 + |α0|2)e(2|β|−Γ)t , (7.35)

〈a0(t)〉 =1

2α0e

(|β|−Γ/2)t . (7.36)

The phase ϕ0 of the order parameter does not depend on the value of the pumpphase ϕp and is determined by the probe phase. We define the coherence of the systemas

η =|〈a0(t)〉|2〈a†

0(t)a0(t)〉. (7.37)

We then find:

η =|α0|2

1 + |α0|2 . (7.38)

This coherence is constant for any phase relationship between pump and probe. It isclose to one if the probe introduces a coherent seed population much larger than one. Inthis case, the symmetry of the system is broken by the probe.

We have seen in the previous paragraph that the wavefunction of the initially sym-metrical system will remain symmetrical during its temporal evolution. Now we aregoing to artificially break this symmetry, assuming that the initial state is a coherentstate characterized by a small but finite amplitude. Since the signal and idler are nowcompletely identical, we consider that they are both initially in a coherent state withthe same amplitude α0 = |α|eiϕ0 , α2kp

= |α|eiϕ2kp but different phases. The averagesignal polarization and population are:

〈a0(t)〉 =1

2e(|β|−Γ/2)t

(α0 − iα∗

2kp

)=

1

2|α|e(|β|−Γ/2)teiϕ0

(1− ei(2ϕp−ϕ0−ϕ2kp )

). (7.39)

The signal polarization strongly depends on the phase relation between pump, probeand idler. Namely, it vanishes if

2ϕp − ϕ0 − ϕ2kp= 0 mod 2π , (7.40)

and achieves its maximum if

2ϕp − ϕ0 − ϕ2kp= π mod 2π . (7.41)

This last equation is the phase-matching condition for parametric oscillation to takeplace. A similar equation can be written for the signal population:

〈a†0(t)a0(t)〉 =

1

4e(2|β|−Γ)t

(1 + |α0|2 + |α2kp

|2 − i(α∗0α

∗2kp

ei2ϕp − α2kpα0e

−i2ϕp))

=1

4e(2|β|−Γ)t

(1 + |α|2(1− cos(2ϕp − ϕ0 − ϕ2kp

)))

. (7.42)

One can see that the population has a minimum (but does not vanish) if condi-tion (7.40) is fulfilled, while it has a maximum if condition (7.41) is fulfilled.


The coherence then reads:

η =2|α|2(1− cos(2ϕp − ϕ0 − ϕ2kp

))

1 + 2|α|2(1− cos(2ϕp − ϕ0 − ϕ2kp))

. (7.43)

An initial coherent state is expected to appear because of the system fluctuations.It is difficult to describe such fluctuations theoretically and to quantify |α|. It is, how-ever, clear that the system will choose to grow on the most “favourable” fluctuation,respecting the constructive phase-matching condition (7.41). It is essential to note thatthe phase of the signal and idler is not fixed by the phase of the pumping laser togetherwith the phase matching condition, as was proposed by Snoke (2002). One can see thatonly the quantity ϕ0 + ϕ2kp

is actually fixed. Therefore, there is a well-defined phaserelation between signal and idler but all the values of the signal phase are equivalentfor the system. This signal phase is not a priori determined by the pump phase, but itis randomly chosen by the system from experiment to experiment. Choosing its phase,the system “breaks its symmetry”. This symmetry-breaking effect is common to thelaser phase transition, superconducting phase transition, and Bose–Einstein condensa-tion (BEC). To summarize, it is a common feature of phase transitions induced by thebosonic character of the particles involved. This is not a BEC, however, because, as isthe case with lasers, it is an out-of-equilibrium phase transition, so that, for example, achemical potential cannot be defined.

7.4.2 Dressing of the dispersion induced by polariton condensates

As already mentioned, stimulated scattering experiments have shown new emissionpeaks surprisingly far from the polariton dispersion. This “off-branch emission” is in-duced by strong interactions taking place between macroscopically populated states,which are the pump, signal and idler states. Interaction between these states is not only aperturbation in the sense that it leads to a dressing of the polariton dispersion. Ciuti et al.(2000) have provided the theoretical interpretation. In this section we briefly summarizethis theory. We shall consider all scattering processes that involve two macroscopicallypopulated states as initial states, and, as final states, one state on the polariton branchand one off-branch state. We shall require conservation of energy and wavevector. Asan example of such a transition, one can consider scattering events having two pumppolaritons as initial states. The wavevector- and energy-conservation laws give in thiscase:

kp, kp → k, 2kp − k , (7.44a)

2ELP(kp) = ELP(k) + Eoffpp (2kp − k) . (7.44b)

Equation (7.44b) defines a new dispersion branch Eoffpp . The appearance of a polari-

ton on this branch is possible because the corresponding scattering event is fast enough(see Section 7.2.1). Four other branches corresponding to signal–pump, pump–idler,signal–signal and idler–idler scattering can be defined. The observation of off-branchemission, which can be associated with the existence of macroscopically populatedpolariton states, is a characteristic feature of phase transitions of weakly interacting

TWO-BEAM EXPERIMENT 277

bosons. Its experimental observation confirms once again that microcavity polaritonsare quasiparticles suitable for the observation of collective bosonic effects.

7.4.3 Bistable behaviour

As discussed in the previous section, an important feature of the resonant excitationscheme is the renormalization of the polariton energies. This renormalization is ob-served in the polariton emission, but it also plays a key role in the absorption of thepump light. Two situations can be distinguished. If the laser is below the bare polaritonenergy, the absorption is simply reduced by the pump-induced blueshift. If the laser isabove the bare polariton energy, the pump gets closer to the absorption energy becauseof the blueshift that in turn increases the shift that enhances the absorption and so on.There are two different regimes. At low pumping, the pump energy remains above therenormalized polariton energy. At higher pumping, the polariton energy jumps above thepump energy that results in a dramatic increase of the population of the pumped state.The threshold between the two regimes is called a bistable threshold since it comes fromthe existence of two possible polariton populations and energies for the same pump en-ergy and intensity. Bistability in strongly coupled microcavities was predicted in 1996by Tredicucci et al. (1996) and observed eight years later by Baas et al. (2004). Thisthreshold yields a very abrupt jump of the population of the pumped state and can initi-ate the parametric scattering process. The coexistence of two different nonlinear phys-ical effects (bistability and stimulated parametric scattering) makes this configurationextremely rich to analyse, as shown by Gippius et al. (2004) and Whittaker (2005). Inthe following we present the formalism describing the pumping of a single state thatleads to bistability only.

We can use eqns (7.17b) and (7.20) describing the dynamics of the pump state, butwithout the coupling term with idler and signal, in other words the nonlinear term of theHamiltonian reduced to a†

kpa†

kpakp

akp:

˙Pkp= i(ωkp

− ωp + iΓkp)Pkp

+ i2

Vkp,kp,kp

|Pkp|2Pkp

+ Ap . (7.45)

This last quantity should be zero in the stationary regime. Then, multiplying eqn (7.45)by its complex conjugated and replacing Pkp

by the population of the pump state,and |Pkp

|2 by the pump intensity, one gets:[((ωkp

− ωp) +2

Vkp,kp,kp

Np

)2

+ Γ2kp

]Np = Ip . (7.46)

The plot Np versus Ip is shown in Fig. 7.17 from Baas et al. (2004). The dashedpart of the curve is unstable. The plot clearly exhibits the hysteresis cycle taking placewhen the pump intensity is successively increased and decreased. The position of thetwo turning points can be found from the condition dIp/dNp = 0, which yields:

3(2

Vkp,kp,kp

)2N2p + 4(ωkp

− ωp)2 + Γ2kp

= 0 . (7.47)

Bistability takes place if there are two positive different solutions for the quadratic equa-tion (7.47), which gives the following condition:


Fig. 7.17: Bistability of the polariton amplifier.

ωp > ωkp+√

3Γkp. (7.48)

Equation (7.48) says that the only condition to get bistability is to pump about onelinewidth above the bare polariton energy. Of course in a real situation, the pumpingcannot be too high in energy since it would require an enormous pump intensity toreach the bistable threshold. If eqn (7.48) is fulfilled the solutions for the turning pointsread:

Np =2(ωp − ωkp

)±√

(ωp − ωkp)2 − 3Γ2

kp

6Vkp,kp,kp

. (7.49)

The solution with the minus sign corresponds to the turning point with the higherpumping, namely the one that can be found on increasing the pumping power. Thesolution with the plus sign corresponds, on the other hand to the turning point that canbe found on decreasing the pumping power. Therefore, the − solution of eqn (7.49) canbe injected into eqn (7.46) in order to find the pumping threshold intensity.

8

STRONG COUPLING: POLARITON BOSE CONDENSATION

In this chapter we address the rich physics revolving about the notion ofa Bose–Einstein condensation of exciton-polaritons, for instancepolariton lasing. From these discussions it is clear that Bosecondensation at room temperature is a practical goal, and could be usedfor a new generation of opto-electronic devices. The way toward thisbreathtaking perspective and the most serious obstacles on this way,which are not yet overcome, are addressed in this chapter.

280 STRONG COUPLING: POLARITON BOSE CONDENSATION

8.1 Introduction

As discussed in Chapter 5, cavity polaritons, although they are a mixture of excitons andphotons, behave as bosons in the low-density limit. One would therefore expect themto exhibit bosonic phase transitions such as Bose–Einstein condensation (BEC). Thisfascinating possibility would represent the first clear example of Bose condensation ina solid-state system.102 The effective realization of this effect also opens the way tothe realization of a polariton laser, which we shall introduce as a new type of coherentlight emitter. Imamoglu and Ram (1996) were the first to point out how the bosoniccharacter of cavity polaritons could be used to create an exciton-polariton condensatethat would emit coherent laser light. The buildup of a ground-state coherent popula-tion from an incoherent exciton reservoir can be seen as a phase transition towards aBose condensed state, or as a polariton-lasing effect resulting from bosonic stimulatedscattering. This conceptual proposal was followed in 2000 by the observation of polari-ton stimulated scattering in resonantly pumped microcavities as shown in Chapter 7. Apolariton laser is, however, different from a polariton parametric amplifier. In the for-mer case, the system is excited non-resonantly—optically or electronically—resultingin a cloud of electrons and holes that form excitons, which subsequently thermalize attheir own temperature mainly through exciton–exciton interactions. They reduce theirkinetic energy by interacting with phonons and relax along the lower-polariton branch.They finally scatter to their lower-energy state, where they accumulate because of stimu-lated scattering. The coherence of the condensate therefore builds up from an incoherentequilibrium reservoir and the associated phase transition can be interpreted as a BEC.Once condensed, polaritons emit coherent monochromatic light. As the light emissionby a polariton quasicondensate is spontaneous, there is no population inversion condi-tion in polariton-lasers, absorption of light does not play any role and ideally there is nothreshold for lasing. Concerning this latter point, the argument is that it is sufficient tohave two polaritons in the ground-state to create a condensate, which will subsequentlydisappear with the emission of two coherent photons. Moreover, because of the smallpolariton mass, critical temperatures larger than 300 K can be achieved. All these char-acteristics as a whole make polariton-lasers ideal candidates for the next generation oflaser-light emitting devices.

8.2 Basic ideas about Bose–Einstein condensation

8.2.1 Einstein proposal

A fascinating property of bosons is their tendency to accumulate in unlimited quantity ina degenerate state. Einstein (1925) made an insightful proposition based on this propertyin the case of an ideal Bose gas that led him to the prediction of a new kind of phasetransition. Let us consider N non-interacting bosons at a temperature T in a volume Rd,where R is the system size and d its dimensionality. The bosons are distributed in energyfollowing the Bose–Einstein distribution function:

102Superconductivity works in the BCS limit and as such does not qualify as Bose condensation.

BASIC IDEAS ABOUT BOSE–EINSTEIN CONDENSATION 281

Einstein (1879–1955) and Bose (1894–1974) shortly after the publication of “Plancks Gesetz und Lichtquan-tenhypothese” in Zeitschrift fur Physik, 26, 178 (1924), written by Bose (1924) and translated by Einstein,who would latter complement it with the concept of what is now known as “Bose–Einstein” condensation,although Bose did not take part in this aspect of the theory.

fB(k, T, µ) =1

exp

(E(k)− µ

kBT

)− 1

, (8.1)

where k is the particle d-dimensional wavevector, E(k) is the dispersion function of thebosons, kB is Boltzmann constant and µ is the chemical potential, which is a negativenumber if the lowest value of E is zero. −µ is the energy needed to add a particle to thesystem. Its value is given by the normalization condition for the fixed total number ofparticles N ,

N(T, µ) =∑k

fB(k, T, µ) . (8.2)

Before going to the thermodynamic limit,103 it is convenient to separate the groundstate from the others:

N(T, µ) =1

exp

(− µ

kBT

)− 1

+∑

k,k =0

fB(k, T, µ) . (8.3)

In the thermodynamic limit, the total polariton density is given by bringing the sumto converge into an integral over the reciprocal space:

n(T, µ) = limR→∞

N(T, µ)

Rd= n0 +

1

(2π)d

∫ ∞

0

fB(k, T, µ)dk , (8.4)

where

103The thermodynamic limit is the limiting process by which the system size and the number of particlesincrease indefinitely but conjointly so that the density remains constant.


n0(T, µ) = limR→∞

1

Rd

1

exp

(− µ

kBT

)− 1

. (8.5)

If µ is nonzero, the ground-state density vanishes. On the other hand, the integral onthe right-hand side of eqn (8.4) is an increasing function of µ. So, if one increases theparticle density n in the system, the chemical potential also increases. The maximumparticle density that can be accommodated following the Bose distribution function istherefore:

nc(T ) = limµ→0

1

(2π)d

∫ ∞

0

fB(k, T ) dk . (8.6)

This function can be calculated analytically in the case of a parabolic dispersion re-lation. It converges for d > 2 but diverges for d ≤ 2, i.e., in two or less dimension(s), aninfinite number of non-interacting bosons can always be accomodated in the system fol-lowing the Bose distribution, the chemical potential is never zero and there is no phasetransition. In higher dimensions, however, nc is a critical density above which it wouldseem no more particles can be added. Einstein proposed that at such higher densities theextra particles in fact collapse into the ground-state, whose density is therefore given by:

n0(T ) = n(T )− nc(T ) . (8.7)

This is a phase transition characterized by the accumulation of a macroscopic num-ber of particles—or equivalently by a finite density—in a single quantum state. Theorder parameter is the chemical potential, which becomes zero at the transition.

8.2.2 Experimental realization

This proposal was not immediately accepted and understood by the scientific commu-nity, principally because of Uhlenbeck’s thesis, wherein it was argued that BEC was notrealistic because it would not occur in a finite system. The interest in BEC saw a revivalin 1938 with the first unambiguous report of Helium-4 superfluidity. A few months afterthis observation, London first proposed the interpretation of this phenomenon as a man-ifestation of BEC. This link between BEC and He-4 superfluidity marked the beginningof an impressive amount of scientific activity throughout the twentieth century, whichis still being pursued. Another early field where Einstein’s intuition found potential andpractical applications is superconductivity. However, the link was only properly under-stood in the 1950s with the advent of the Bardeen, Cooper, and Schrieffer (BCS) theory.In both cases (He and superconductivity) the total particle density is fixed. It is thuspossible to define a critical temperature Tc given by the solution of

nc(Tc) = n . (8.8)

A remarkable indication of the validity of Einstein’s hypothesis (as was immediatelypointed out by London) is that its direct application yields a BEC critical temperatureof 3.14 K for He, very close to the experimental value of 2.17 K. However, a majordifficulty is that these two systems are strongly interacting, in fact already in their liquid

BASIC IDEAS ABOUT BOSE–EINSTEIN CONDENSATION 283

Pyotr Leonidovich Kapitza (1894–1984) and Fritz London (1900–1954)The pioneers of quantum hydrodynamics.

Kapitza worked in the Cavendish Laboratory in Cambridge with Ernest Rutherford, whom he called “thecrocodile”. He had one carved on the outer wall of the Mond Laboratory, now the emblem of the prestigiousresearch centre. The following year he visited the Soviet Union where he was held forcibly to continue hisresearch. He died the only member of the presidium of the Soviet Academy of Sciences who was not a mem-ber of the communist party. He was awarded the Nobel prize in 1978 for his work in low-temperature physics.

London’s (1938) proposition to root superfluidity in Bose condensation was highly controversial inhis time, if only because he based his argument on an ideal gas, whereas a fluid is a strongly interactingsystem. He, more than any other, took the Bose condensation seriously, even at the time when Einsteinhimself seemed to accept its rebuttal by the scientific community.

phase, and therefore particles interactions are expected to play a fundamental role, thusmaking them poor realizations of Einstein’s ideal gas. Consequently, the objective ofmost theoretical efforts of the 1940s–1960s was to describe condensation of stronglyinteracting bosons. Stimulated emission of light and laser action is also induced bythe bosonic nature of the particles involved—the photons—which do not interact. This,however, means that they cannot self-thermalize and a photon assembly represents fun-damentally a non-equilibrium system. Consequently, laser action is a non-equilibriumphase transition that cannot be directly interpreted as a BEC. In fact, the first clear man-ifestation of condensation in a weakly interacting Bose gas was performed recently byAnderson et al. (1995) with trapped alkali atoms. This discovery, crowned by the 2001Nobel prize for physics, has given a strong revival to this field.

8.2.3 Modern definition of Bose–Einstein condensation

Research on BEC was extremely intense in the period 1938–1965, especially on the the-oretical side, where it allowed for many deep advances in understanding. In particular,these efforts led to a new definition of the BEC criterion. BEC is now associated withthe appearance of a macroscopic condensate wavefunction ψ(r), which has a nonzeromean value 〈ψ(r)〉:

〈ψ(r)〉 =√

ncond(r)eiθ(r) , (8.9)


〈ψ(r)〉 is the order parameter for this phase transition. It is a complex number withan amplitude—the square root of the condensate density—and a phase. The systemHamiltonian is invariant under an arbitrary phase change of ψ(r) (a property referred toas global gauge invariance). However, at the phase transition this symmetric solutionbecomes unstable and the system breaks this symmetry by choosing a specific phasethat is assumed throughout the whole condensate, which is therefore completely phase-coherent. Penrose and Onsager (1954) proposed the following criteria for BEC:

〈ψ†(r)ψ(r′)〉 −−−−−−−→|r−r′|→∞

〈ψ(r′)〉∗〈ψ(r)〉 , (8.10)

which is now generally accepted. Its significance has emerged gradually through theefforts of many theorists. Goldstone (1961) and Goldstone et al. (1962) advanced theidea of spontaneous symmetry breaking, Yang (1962) termed the phenomenon “off-diagonal long-range order” (ODLRO) and Anderson (1966) emphasized the notion ofphase coherence. The superfluid velocity can be defined from eqn (8.9) as

mvs(r, T ) = ∇θ(r, T ) . (8.11)

A system is therefore “superfluid” if two arbitrary spatial points are connected by aphase-coherent path, allowing for frictionless transport, i.e., no scattering.

8.3 Specificities of excitons and polaritons

Depending on their density and on temperature, excitons behave as either a weakly in-teracting Bose gas, a metallic liquid, or an electron–hole plasma. It has been understoodby Moskalenko (1962) and Blatt et al. (1962) that excitons remain in the gas phase atlow densities and low temperatures, and are therefore good candidates for observation ofBEC in the way envisioned by Einstein. At that time there were no experimental exam-ples of BEC of a weakly interacting gas and a great deal of research effort was dedicatedto the problem of exciton BEC. A number of theoretical works on excitonic condensa-tion and superfluidity have appeared, with major publications such as those by Keldyshand Kozlov (1968), Lozovik and Yudson (1975, 1976a, 1976b), Haug and Hanamura(1975) and Comte and Nozieres (1982). In most of these, the fermionic nature of exci-ton composite quasiparticles is also addressed. The starting point of these models is asystem of degenerate electrons and holes of arbitrary densities that is treated in the spiritof the BCS theory. A key point of all the formalisms that have been developed is thatthey assume an infinite lifetime for the semiconductor excitations. In other words, thesetheories are looking for steady-state solutions of the Schrodinger equation of interactingexcitons. It is indeed clear that to have enough time to Bose-condense, excitons musthave a radiative lifetime much longer than their relaxation time. Thus, the use of “dark”(uncoupled to light) excitons seems preferable. This is the case for bulk Cu2O paraex-citons, whose ground-state spin is 2, or of excitons in coupled quantum wells, wherethe electron and hole are spatially separated. These two systems have been subject tointense experimental studies that have sometimes claimed achievement of exciton BECor superfluidity, see for instance the publications by Butov et al. (2001), Snoke et al.

SPECIFICITIES OF EXCITONS AND POLARITONS 285

(2002) and Butov et al. (2002). However, careful analysis by Tikhodeev (2000) andLozovik and Ovchinnikov (2002) showed that clear evidence of excitonic BEC has notyet been achieved in such systems. The difficulties of Bose-condensing excitons aretwo-fold. The first reason is linked to the intrinsic imperfections of semiconductors.Because of an unavoidable structural disorder, dark excitons non-resonantly excited areoften trapped in local minima of the disorder potential and can hardly be consideredas free bosons able to condense. The second source of difficulty is connected with theproblem of detection of the condensed phase. The clearest signature of exciton Bosecondensation should be the emission of coherent light by spontaneous recombination ofcondensed excitons. Such emission is a priori forbidden for a system of dark excitons.

On the other hand, “bright” excitons, that are directly coupled to light, might also begood candidates for condensation, despite their short lifetimes. In bulk semiconductors,this coherent coupling gives rise to a polarization wave that can be considered from aquantum-mechanical point of view as a coherent superposition of pure exciton and pho-ton states (polaritons). Bulk polaritons are stationary states that transform into photonsonly at surfaces. Polaritons also being bosons, they can, in principle, form condensatesthat would emit spontaneously coherent light. Typical dispersion curves of bulk polari-tons are shown in Fig. 4.14. In the vicinity of the exciton–photon intersection point,the density of states of polaritons is strongly reduced and the excitonic contribution tothe polariton is decreased. One should note that strictly speaking, a k = 0 photon doesnot exist, and that consequently the k = 0 polaritonic state of the LPB does not existeither. The polariton dispersion has no minimum so that a true condensation processis strictly forbidden. Polaritons accumulate in a large number of states in the so-calledbottleneck region. The situation is drastically different in microcavities. The cavity pre-vents the escape of photons and allows the formation of long-lifetime cavity polaritons.Conversely to the bulk case, the inplane cavity polariton dispersion exhibits a well-defined minimum located at k = 0, but since they are two-dimensional quasiparticlesthey cannot exhibit a strict BEC phase transition, but rather a local condensation orso-called Kosterlitz–Thouless phase transition towards superfluidity. They have, more-over, an extremely small effective mass around k = 0, allowing for polariton lasingat temperatures that could be higher than 300 K. Experimental discovery of stimulatedscattering of polaritons in microcavities (see Chapter 7) has proved that a microcavityis probably a very suitable system to observe effects linked to the bosonic nature ofpolaritons, and probably BEC. The recent paper by Kasprzak et al. (2006) shows thatBEC of polaritons is indeed possible in CdTe-based microcavities at temperatures upto about 40 K. Much experimental and theoretical effort followed Imamoglu’s proposalof a polariton-laser. We shall describe in detail these efforts throughout the rest of thischapter. A fundamental peculiarity and difficulty of a microcavity is the finite polaritonlifetime that may be responsible for a strongly non-equilibrium polariton distributionfunction. The relaxation kinetics of polaritons plays a major role in this case.

8.3.1 Thermodynamic properties of cavity polaritons

In this section we discuss the thermodynamic properties of microcavity polaritons con-sidered as equilibrium particles, i.e. particles having an infinite lifetime. Even though


this approximation is very far from reality, mainly governed by relaxation kinetics, itis instructive to examine the BEC conditions in this limiting case. We shall, moreover,assume that polaritons behave as either an ideal or a weakly interacting boson gas, sothat the following analysis is valid only in the low-density limit.

As mentioned in Section 8.2.1, the critical condensation density is finite for nonzerotemperature if d > 2. However, this density diverges in the two-dimensional case. Thus,a non-interacting Bose gas cannot condense in an infinite two-dimensional system, andthe same statement turns out to be true when interactions are taken into account. A rig-orous proof of the absence of BEC in two dimensions has been given by Hohenberg(1967). An equivalent statement known as the Mermin–Wagner theorem after the workof Mermin and Wagner (1966), asserts that long-range order cannot exist in a system ofdimensionality lower than two. Finally, it has been shown that spontaneous symmetrybreaking does not occur in two dimensions, see for instance the discussion by Cole-man (1973). However, a phase transition between a normal state and a superfluid statecan still take place in two dimensions as predicted by Kosterlitz and Thouless (1973)in the framework of the XY spin model. Such a second-order phase transition is for-bidden for ideal bosons, but according to Fisher and Hohenberg (1988) can take placein systems of weakly interacting bosons such as low-density excitons or polaritons,as shown by Fisher and Hohenberg (1988). The case of excitons has been especiallyinvestigated by Lozovik et al. (1998) and Koinov (2000). In the next section, we intro-duce the Kosterlitz–Thouless phase transition and its application to the cavity polaritonsystem. We describe the effect of interactions on bosons through a presentation of theBogoliubov formalism as a guide. The problem of local condensation is addressed inSection 8.3.3.

8.3.2 Interacting bosons and Bogoliubov model

In order to explain the properties of superfluid He, a phenomenological model was de-veloped by Landau (1941) (extended in 1947) who introduced an original energy spec-trum, displayed in Fig. 8.1. This spectrum is composed of two kinds of quasiparticles:phonons and rotons. These quasiparticles are collective modes in the “gas of quasipar-ticles” and are associated with the first and second sound. Such a spectrum introduced“by hand” allowed Landau to describe most of the peculiar properties of superfluidHe. Bogoliubov’s work of 1947 was a real breakthrough. He presented a microscopicdescription of the condensed weakly interacting Bose gas. As we shall see below, heshowed how BEC is not much altered in a weakly interacting Bose gas, something thatwas not obvious at the time. He also showed how interactions completely alter the long-wavelength response of a Bose gas. He recovered qualitatively the spectrum assumedby Landau for the quasiparticle dispersion relation. Most of the further theoretical de-velopments in the field are based on the Bogoliubov approach.

Bogoliubov considered the Hamiltonian describing an interacting Bose gas:

H =∑k

E(k)a†kak +

1

2

∑k,k′,q

Vqa†k+qa†

k′−qakak′ , (8.12)


Lev Davidovich Landau (1908–1968) and Nikolai Nikolaevich Bogoliubov (1909–1992) provided the mostlasting phenomenological and microscopical interpretations of superfluidity.

Landau made numerous contributions to theoretical physics, especially in Russia where the “Landau School”is still referred to. Alexey Abrikosov, Lev Pitaevskii, Isaak Khalatnikov are among his most famous students.He put special emphasis on broad qualifications for a physicist as opposed to narrow expertise. A childprodigy, he said he couldn’t remember a time when he was not familiar with calculus. He kept a list ofphysicists whom he graded, with his contemporaries Bohr, Heisenberg and Schrodinger falling into the firstcategory along with Newton, but he made an exception for Einstein who he ranked in a superior category ofhis own. He put himself in the second category. He was the victim of a severe car accident in January 1962that would ultimately claim his life. He abandoned his research activities during this last period. He receivedthe 1962 Nobel Prize in Physics for his work on superfluidity.

Bogoliubov published his first paper at 15. He was one of the first to have studied nonlinearities inphysical systems at a time where absence of computers made them forbidding. In the late 1940s and 1950she studied superfluidity, successfully taking into account the nonlinear terms that others had deemed toocomplicated. He introduced the Bogoliubov transformation in quantum field theory. In the 1960s he turnedhis interest to quarks in nuclear physics.

with a†k, ak the creation and annihilation bosonic operators and Vq the Fourier trans-

form of the interaction potential for the boson–boson scattering. His objective was todiagonalize this Hamiltonian making reasonable approximations.

At zero Kelvin an ideal Bose gas should be completely condensed. Bogoliubov as-sumed that interactions are only responsible for weak condensate depletion. In otherwords, most system particles are assumed to be still inside the condensate. This im-plies 〈a0〉, 〈a†

0〉 ≈√

N0 and therefore

[a0, a†0] 〈a†

0a0〉 , (8.13)

where N0 is the condensate population. Bogoliubov proposed to neglect the conden-sate fluctuations and to replace the operators a0, a†

0 by complex numbers A0, A∗0. The

condensate is thus treated classically as a particle reservoir. The second Bogoliubovapproximation was to keep only the largest contributions in the interacting part of theHamiltonian. The largest contributions are those that involve the condensate. Therefore,one keeps only the terms that involve a0 two times or more. Equation (8.12) becomes:


Fig. 8.1: The energy spectrum of liquid Helium II, showing phonons as k → 0 and rotons as the high-k dip.

H = N0V20 +

∑k,k =0

(E(k) + N0(Vk + V0)

)a†kak

+ N0

∑k

Vk

(a†ka†

−kaka−k

). (8.14)

The Bogoliubov approximations conserve off-diagonal coupling terms that inducethe appearance of new eigenmodes. Then, a change of basis is made through the trans-formations (5.37) that, according to this procedure, diagonalizes the Hamiltonian (8.12)into

H = N0V20 +

∑k =0

EBog(k)α†kαk , (8.15)

as a function of new operators αk, α†k, chosen to remain bosonic operators, and with, as

an all-important consequence,

EBog(k) =√

E(k)[E(k) + 2N0Vk] . (8.16)

This is eqn (8.16) that justifies the Landau spectrum. When Vk is given by the Fouriertransform of Coulomb potential, the spectrum assumes the shape displayed in Fig. 8.1.It is also called the Bogoliubov spectrum. The unperturbed dispersion is recovered if N0

vanishes. On the other hand, if one considers the existence of a condensate, there followsfrom a quadratic unperturbed dispersion

E(k) =2k2

2m, (8.17)

the renormalized spectrum near k = 0 equal to


EBog(k) ≈ k

√N0V0

m. (8.18)

This is the linear part of the Landau spectrum, which displays features of phononquasiparticles with sound velocity

vs =

√N0V0

m. (8.19)

Considering a large polariton population of N0 = 106 and a system size of 100 µm,the typical bogolon velocity in a GaAs microcavity would be 5×105 m s−1. In the sameway one can estimate the reciprocal-space region where the bogolon spectrum is linearas the set of k values such that

k <2

√mN0V0 . (8.20)

This corresponds to an angular width of about 3 degrees in the above-mentionedconditions.

8.3.3 Polariton superfluidity

Superfluidity is a property deeply associated with BEC and at first glance it seems thatone of these properties cannot exist without the other. This is not exactly true. BEC islinked with the appearance of a Dirac function at k = 0 in the distribution function ofbosons. The Fourier transform of this Dirac function gives the extension in the directspace of the condensate wavefunction, which is infinite and constant. BEC thereforemeans the appearance of a homogeneous phase in direct space. This homogeneity im-plies superfluidity. Particles can move throughout space along a phase-coherent, dissi-pationless path. Superfluidity means that statistically, two points in space are connectedby a phase-coherent path, even if the whole space is not covered by a phase-coherentwavefunction. As a conclusion, a superfluid state can occur without the existence ofstrict BEC. This is the kind of state that arises in two dimensions where a strict BECis forbidden. We now describe qualitatively how this phase transition takes place, andwe estimate the Kosterlitz–Thouless (KT) transition temperature TKT for the polaritoncase.

At temperatures higher than the critical temperature TKT, the superfluid numberdensity ns is zero, but local condensation can take place. Condensate droplets can havequite large sizes as we shall see later, but they are characterized by an exponentiallydecreasing correlation function and are not connected together. Free vortices preventlong-range ordering, i.e., percolation of the quasicondensate droplets. However, oncethe critical temperature TKT is reached, single vortices are no longer stable. They bind,forming pairs or clusters with the total winding number (or verticity) equal to zero, al-lowing for a sudden percolation of the quasicondensate droplets that therefore form asuperfluid. For temperatures slightly below TKT, the superfluid number density is pro-portional to TKT with a universal coefficient (see the discussion by Nelson and Koster-litz (1977)):


ns =2mkBTKT

π2, (8.21)

where m is the bare polariton mass at q = 0. The pairs of vortices remain well be-low TKT and the correlation function is not constant, but decreases as a power of thedistance. The superfluid wavefunction has thus a finite extension in reciprocal space, andconsequently is not a BEC wavefunction. Complete homogeneity and true long-rangeorder can be achieved only at T = 0, where vortices disappear. Below TKT normal andsuperfluid phases coexist. The normal fluid can be characterized by a density nn and avelocity vn, while the superfluid has a density ns and velocity vs. The total fluid densityis:

n = nn + ns , (8.22)

where nn can be calculated following, for instance, Koinov (2000). Despite the absenceof BEC in two dimensions, the energies of quasiparticles (bogolons) in the superfluidphase are still given by the Bogoliubov expression

EBog(k) =√

E(k)[E(k) + 2µ] . (8.23)

We need to know the chemical potential of interacting polaritons to calculate thequasiparticle dispersion. With the meaning previously given for the chemical potential,when one considers added particles going into the ground-state, the associated interac-tion energy yields:

µ = NV0 . (8.24)

Once the quasiparticle dispersion is known, one can use the famous Landau for-mula to calculate the normal mass density. This formula reads for a two-dimensionalsystem:104,105

nn =1

(2π)2

∫E(k, T, µ)

(− ∂fB(Ebog(k, T, 0))

∂Ebog(k)

)dk . (8.25)

Note that the Bose distribution function eqn (8.1) entering this expression is taken atzero bogolon chemical potential, while the nonzero polariton chemical potential is stillpresent in the bogolon dispersion relation (8.23). Equations (8.23)–(8.25) yield ns(T, n).Its substitution into eqn (8.22) allows one to obtain TKT(n) and therefore to plot a po-lariton phase diagram. Such a phase diagram is shown in Fig. 8.2.

Solid lines (a–d) show the critical concentration for a KT phase transition accord-ing to the above-mentioned procedure, and calculated for typical microcavity structures

104The derivation of the Landau formula requires a current-conservation law and therefore is exact forparticles with a parabolic spectrum only. In the case of cavity polaritons with a non-parabolic dispersion,eqn (8.25) remains a good approximation at low temperatures, where the exciton-like part of the low polaritondispersion branch is weakly populated. Keeling (2006) has shown that at higher temperatures, the criticaldensity for the KT transition is lower than is predicted by eqn (8.25). See Appendix C for more details.

105The integral in eqn (8.25) can be computed approximately in the case of a parabolic dispersionif µ/(kBT ) 1, as shown by Fisher and Hohenberg (1988). This approximation is for example well satis-fied for excitons having critical temperature in the Kelvin range. It is no longer the case for polaritons becauseof their small masses. Therefore, numerical integration has to be carried out.


Fig. 8.2: Phase diagrams for GaAs (a), CdTe (b), GaN (c), and ZnO (d) based microcavities at zero detuning.Vertical and horizontal dashed lines show the limits of the strong-coupling regime imposed by the excitonthermal broadening and screening, respectively. Solid lines show the critical concentration Nc versus tem-perature of the polariton KT phase transition. Dotted and dashed lines show the critical concentration Nc forquasicondensation in 100 µm and in one meter lateral size systems, respectively. The thin dashed line (upperright) symbolizes the limit between vertical-cavity surface-emitting laser (VCSEL) and light-emitting dioderegimes.

based on GaAs (a), CdTe (b), GaN (c), and ZnO (d). In all cases we assume zero de-tuning of the exciton resonance and the cavity photon mode. For GaAs- and CdTe-based microcavities we have used the parameters of the samples studied by Senellartand Bloch (1999), Senellart et al. (2000) and Le Si Dang et al. (1998). The parame-ters of model GaN and ZnO microcavity structures have been given by Malpuech et al.(2002a) and Zamfirescu et al. (2002). The latter structure has only hypothetical interest,since strong coupling has not yet been achieved experimentally in ZnO-based cavities.Vertical and horizontal dashed lines show the approximate limit of the strong-couplingregime in a microcavity that come from either exciton screening by a photoinducedelectron–hole plasma or from temperature-induced broadening of the exciton resonance.Below the critical density, if still in the strong-coupling regime, a microcavity operatesas a polariton diode emitting incoherent light, while in the weak-coupling regime thedevice behaves like a conventional light-emitting diode. Above the critical density, inthe weak-coupling regime, the microcavity acts as a conventional laser. Thin dottedlines in Fig. 8.2 indicate the limit between the latter two phases that cannot be found


in the framework of our formalism limited to the strong-coupling regime. One can notethat critical temperatures achieved are much higher than those that can be achieved inexciton systems. In existing GaAs- and CdTe-based cavities these temperatures are highenough for experimental observation of the KT phase transition under laboratory con-ditions, but do not allow one to produce devices working at room temperature. Recordcritical temperatures of TKT=450 K and 560 K for GaN- and ZnO-based model cavitiesare given by extremely high exciton dissociation energies in these semiconductors. It isinteresting to note that even above TKT(n), n − nn does not vanish, which reflects theexistence of isolated quasicondensate droplets. As we show below, these droplets canreach substantial size, even above the Kosterlitz–Thouless temperature or density. Theirproperties may dominate the behaviour of real systems in some cases.

At high excitonic densities the behaviour of the polariton liquid can be affectedby fermionic effects linked to the composite nature of the exciton. This may result inthe crossover from the quasicondensed or superfluid phase to the quantum-correlatedplasma similar to the gas of electronic pairs in a superconductor. This phase is com-monly referred to as BCS from the first letters in the names of Bardeen, Cooper andSchrieffer, who proposed a theoretical model to explain the superconductivity in termsof collective phenomena in a stongly correlated gas of composite bosons, called Cooperpairs. The group of Littlewood in Cambridge has published a series of papers basedon the Dicke model (see Section 5.6), that was generalized by Eastham and Littlewood(2001) and applied to the study of disorder and structural imperfections, for instanceby Marchetti et al. (2004), since the model is especially suited to that purpose. Keel-ing et al. (2005) have calculated the phase diagram displayed on Fig. 8.3 where theregimes of BEC, BCS and weak-coupling regimes (termed “BEC of photons” by theauthors) can be distinguished. The only parameter that govens the phase boundaries inthis model is the mean field, which characterises the strength of interparticle interac-tions in the electron/hole/exciton-polariton plasma. Experimentally, the BCS phase ofexciton-polaritons has not been observed so far.

8.3.4 Quasicondensation and local effects

In this section we define a rigorous criterion for boson quasicondensation in finite-sizesystems. For the sake of simplicity we neglect here all kinds of interactions betweenparticles. Let us consider a system of size R. The particle density is given by

n(T,R, µ) =N0

R2+

1

R2

∑k

k>2π/R

fB(k, T, µ) , (8.26)

where N0 is the ground-state population. We define the critical density as the maximumnumber of bosons that can be accommodated in all the states but the ground-state:

nc(R, T ) =1

R2

∑k

k>2π/R

fB(k, T, 0) . (8.27)

The quasicondensate density is thus given by n0 = n − nc. In this case, formally,the chemical potential µ is always strictly negative, but it approaches zero, allowing


Fig. 8.3: BCS–BEC crossover caused by the internal structure of polaritons, as studied by Keeling et al.(2005). The thermodynamics of composite polaritons is investigated through the critical temperature of con-densation. At low density there is no deviation from the internal structure. At higher density, the systemshifts to BCS-like behaviour with an associated critical temperature now comparable to the Rabi splitting.The phase diagram shown and obtained by Keeling et al. (2004) displays the phase boundary in a mean-fieldtheory (solid line) and phase boundaries taking into account fluctuation corrections for different values ofthe photon mass. Temperature is plotted in units of g

√n with g the coupling strength and n the density of

excitons.

one to put as many bosons as desired in the ground-state, while keeping the concen-tration of bosons in all other states finite and limited by nc. The concentration (8.27)can be considered as the critical concentration for local quasi-Bose condensation intwo-dimensional systems. Further, we shall refer to Tc defined in this way as the criti-cal temperature of Bose condensation in a finite two-dimensional system. On the otherhand, it appears possible, knowing the temperature and density, to deduce the typicalcoherent droplet size, which is given by the correlation length of the quasicondensate.

From a practical point of view, experiments are performed on samples having a lat-eral size of about 1 cm. Electron–hole pairs are generated by laser light with a spotarea of about 100 µm. These electron–hole pairs rapidly (typically on a timescale lessthan 10 ps) form excitons, which relax down to the optically active region, where theystrongly interact with the light field to form polaritons. Excitons that form polaritonshave a finite spatial extension in the plane of the structure, but they are all coupledto each other via light, as illustrated for instance in the applications developed byMalpuech and Kavokin (2001) or Kavokin et al. (2001). The polariton system thus cov-ers the whole surface where excitons are generated. If the KT critical conditions are notfulfilled, but if typical droplet sizes are larger than the light spot size, the whole polari-ton system can be transiently phase-coherent and thus exhibits local BEC. As we shallshow below, this situation is the most likely to happen in current optical experimentsperformed at low temperature.

Let us underline at this point an important advantage of polaritons with respect toexcitons weakly coupled to light for the purposes of BEC or superfluidity. Individualexcitons in real structures are subject to strong localization in inevitable potential fluc-tuations that prevent them from interacting and forming condensed droplets. Polaritons


are basically delocalized, even though the excitons forming them could be localized,a point emphasized in the work of Keeling et al. (2004) and Marchetti et al. (2006).This is why their interactions are expected to be more efficient and bosonic behaviourmore pronounced. The dotted and dashed lines in Figs. 8.2(a)–(d) show the criticalconcentration for local quasicondensation in microcavity systems of 100 µm and 1 mlateral size, respectively. In the high-temperature (high-concentration) limit, the criti-cal concentrations are very similar for both lateral sizes and they slightly exceed thecritical concentrations of the KT phase transition. This means that in this limit the KTtransition takes place before the droplet size reaches 100 µm. Conversely, in the low-temperature (low-concentration) limit the KT curve is between the transition curves ofthe 100 µm and 1m size systems. This shows that droplets at the KT transition arelarger than 100 µm but smaller than 1m. Since the typical laser spot size is of about100 µm, this means that local Bose condensation takes place before the KT transitionat low pumping. A detailed analysis could allow one to obtain the percolating dropletsize versus temperature, which is beyond the scope of our present discussion.

Note finally that inhomogeneous broadening of the exciton resonance leads to broad-ening of the polariton ground-state in the reciprocal space. Any broadening in the recip-rocal space is formally equivalent to localization in the real space. Such a localization,present even in an infinite microcavity, allows for quasicondensation, in principle. Agra-novich et al. (2003) have recently studied disordered organic semiconductors prone toexhibit such effects.

Experimental results such as those of Richard et al. (2005b) evidenced localizationof the condensate in real space due to the disorder, whereas no superfluid behaviour—like linearization of the excitation spectrum—was observed. The impact of localizationis therefore extremely strong and the following qualitative picture can be drawn. In thepresence of disorder, the lowest-energy states are localized states. In a non-interactingboson picture, only the lower energy state should be filled giving rise to a fully localizedcondensate. This picture changes drastically in presence of interactions. Indeed, once alocalized state starts to be filled, it blueshifts because of polariton–polariton interaction.The chemical potential increases and reaches the energy of another localized state that inturn starts to be populated and blueshifts. The system therefore assumes an assembly ofstrongly populated localized states, all with the same chemical potential. This situationwhich does not exhibit any superfluidity, is known as a Bose glass. See, for instance,the discussion by Fisher et al. (1989). Once the chemical potential reaches the valueof the localization energy, a delocalization of the condensate occurs and a standard KTphase transition can take place as shown by Berman et al. (2004). In realistic systemsone should therefore expect two successive bosonic phase transitions on increasing thedensity.

8.4 High-power microcavity emission

In Chapter 7 we have explored the optical non-linearities produced by strong polaritoninteractions when they are resonantly injected on the lower-polariton dispersion. How-ever, around the same period of research it became clear that peculiar effects are alsoobserved when the strong-coupling microcavities are pumped non-resonantly at higher

HIGH-POWER MICROCAVITY EMISSION 295

power. In this case, the pump photons inject electron–hole pairs at high energy intothe continuum absorption of the quantum wells, and these carriers rapidly bind and re-lax onto the exciton dispersion at high k. We have discussed above how these excitonscan approach easily to the edge of the lower-polariton trap near k = 0, but then thereis a bottleneck for their further relaxation. As the density of excitons is increased bypumping harder, suddenly an entirely new regime of microcavity emission appears.

Such possibilities were suggested by Imamoglu and Ram (1996) and Imamogluet al. (1996) for stimulated scattering of excitons into polaritons at k = 0, based onthe bosonic nature of the final state. However, most experiments in III-V microcavi-ties showed limited evidence for this effect due to the competing problem of excitonionization at high carrier densities. More recent measurements on III-V microcavitiesunder near-resonant pumping of the lower-polariton branch at high angles by Deng et al.(2003) showed how different the polariton-laser emission is from a conventional photonlaser.

Fig. 8.4: CdTe microcavity studied by Le Si Dang et al. (1998) and its (a) reflectivity and (b) PL for a rangeof excitation densities at 4 K.

The experiment by Le Si Dang et al. (1998) that shows this dramatic new regimemost clearly uses CdTe microcavities with large Rabi splittings (23 meV) and exci-tons stable to higher densities due to their larger binding energy (25 meV). As thepump power is increased the lower-polariton luminesence emerging at normal incidence(k = 0) shifts to higher energy and increases in magnitude faster than the pump power(Fig. 8.4). From our previous discussion about polariton-induced blueshifts to the dis-persion, this provides compelling evidence for the buildup of large polariton popula-tions, while the narrowing of the emission line and its beaming into a narrow angularrange indicates the coherent nature of this population. Unfortunately, these microcavi-ties are complex to fabricate and no further devices of this performance level have yetbeen grown.


Fig. 8.5: Far-field angular emission from Richard et al. (2005b) for (a,b) below and (c,d) above the stimulationthreshold. (a,d) display far-field emission in the θx,y Fourier plane, (b,e) display the emission measured inthe E, θx plane.

What remains unclear is the extent to which disorder on the tens of micrometre scaleaffects the process of polariton condensation into a macroscopic coherent state. Forinstance, observations of the angular emission of these CdTe microcavities by Richardet al. (2005b) show that for some positions on a sample, the coherent emission is in theform of an annular ring above the pump threshold (Fig. 8.5). Similar observations havebeen observed in III-V microcavities for the condition of near-resonant pumping of theexciton reservoir by Savvidis et al. (2002). However, also clearly visible in the II-VIexperiments is the increase in angular intentsity fluctuation in this condition, which hasalso been observed in III-V microcavities in the spatial domain. Hence, one of the keytheoretical questions is how a polariton condensate can emerge, and what its propertieswill be in a situation of weak spatial inhomogeneities.

Another interesting question has been whether the scattering of excitons from theirreservoir into the polariton trap can be enhanced to allow efficient polariton condensa-tion in III-V microcavities. One possible route has been to weakly dope the quantumwell with electrons to provide a lighter quasiparticle that absorbs more energy from re-laxing excitons, as proposed by Malpuech et al. (2002b) and Malpuech et al. (2002a).Experimentally, although some promising results have been achieved, for instance in thework of Lagoudakis et al. (2003) and Bajoni et al. (2006), it appears that the speed-upfrom the exciton-electron scattering is insufficient to surmount the polariton bottleneckbefore exciton ionization is also produced.

Experiments in wide-bandgap semiconductors look most promising at this time, butare developing as the technology for their fabrication matures. We should also mentionthe possibility of hybrid microcavities that contain organic semiconductors or dye lay-ers in the strong-coupling regime, such as studied by Lidzey et al. (1998). While they

THRESHOLDLESS POLARITON LASING 297

show typical strong coupling features in reflectivity, and even in photoluminescenceand electroluminescence, as reported by Tischler et al. (2005), their nonlinear proper-ties have proved to be dominated by other states in the system and as yet no polaritonscattering has been identified, see the discussion by Lidzey et al. (2002) and Savvidiset al. (2006). Currently, this is believed to be due to the localized nature of the excitonsin these systems that reduces the exciton–exciton scattering interaction. A number ofother realizations of strong coupling with organic semiconductors also look potentiallyinteresting, including plasmon–exciton modes on noble-metal surfaces, as discussed byBellessa et al. (2004). Thus, a wide variety of systems in which excitons are coupled toelectromagnetic modes can produce conditions suitable for polaritonic optical nonlin-earities.

8.5 Thresholdless polariton lasing

We have already introduced the polariton-laser as an optoelectronic device based onthe spontaneous emission of coherent light by a polariton Bose condensate in a micro-cavity. In order to motivate the full description of such a device, we present a simpleanalysis that allows a direct comparison with the standard Boltzmann equations (6.49b)describing conventional lasing (Chapter 6). As we have seen in discussions of para-metric scattering (Chapter 7), a key ingredient of scattering into the polariton trap ispolariton–polariton scattering. This is true not only for polaritons in the vicinity of thepolariton trap, but also for polaritons that are at high k, and thus predominantly excitonic(Fig. 8.6).

Fig. 8.6: Dispersion relations of polariton-laser, showing pair scattering of excitons at k1,2 feeding energyinto the polariton trap at k‖=0.

Our central premise to build this simple model is that the exciton and k = 0 po-lariton populations are mutually coupled only by Coulomb pair scattering. Acousticphonons have too small a velocity to couple polaritons to excitons (energy-momentumconservation is impossible) while, at the temperatures discussed here, both the opticalphonons and the exciton states ωLO ∼ 36 meV (in GaAs) above the k = 0 polaritons,are unpopulated. However, the exciton–exciton interaction is sufficiently strong to po-tentially allow the excitons to mutually thermalize (as will be discussed in the full modelwe develop later in this chapter).


0.50

(c) T=300K

0.50

Nx (×1018

cm-3

)

(b) T=100K

R S

1.0

0.5

0.0

Para

met

ric

Scat

teri

ng R

ates

(ps

-1)

0.50

(a) T=30K

Fig. 8.7: Exciton-density dependence of the integrated spontaneous (S) and stimulated (R) scattering ratesfor Ω=7 meV and InGaAs QW parameters.

Our model (using the same equations 6.49b)) couples N0 polaritons with an excitonreservoir of density Nx, assumed uniform over the polariton active mode volume V . Thestrong coupling splits the polariton states at k = 0 by the Rabi frequency Ω, producinga classical trap for polaritons in momentum space of depth Ω/2 and FWHM widthc∆k <

√6εΩEex (for effective sample dielectric constant ε). Pair scattering of two

excitons at k1, k2 can deposit a polariton into k0 = 0 and a higher-momentum excitoninto k3 (Fig. 8.6). For pair scattering to occur both energy and momentum must beconserved but, because of the quadratic shape of the exciton dispersion, this requiresexcitons with a sufficiently high k necessitating a large enough density or temperature.From k1 +k2 = k3 and E1 + E2 = ELP + E3, where Ei = Eex + 2|ki|2/(2M), andELP = Eex−Ω/2, these produce the constraint

2k1 ·k2/M = Ω , (8.28)

where M is the exciton mass. The total scattering rate depends on the occupation of thestates involved. Here, the stimulated scattering discussed in Chapter 7 shows that the N0

polaritons behave as bosons. The pair-scattering rate ΓPS = V [(1+N0)f1f2(1−f3)−N0f3(1−f1)(1−f2)] where the first term is the final-state enhanced scattering into theN0 polariton condensate and the second term is the re-ionization out of this trap. TheCoulomb coupling constant V ∝ e2/(4πε0εBaB), where aB is the exciton Bohr radius.The Fermi functions fi = 1/(1 + e(Ei−Ef )/kBT ) provide an approximate distributionfor the reservoir of excitons at temperature kBT , and Fermi level Ef . Extracting theN0-dependent and N0-independent parts, this can be written in the form

N0R(Nx) + S(Nx) =

∫dk1

∫dk2 ΓPS(k1, k2) . (8.29)

This integral is computationally tractable and its behaviour can be easily understood.For low temperatures compared to the trap potential depth (kBT <Ω/2), pair scattering


only occurs if Ef > Ω/2, implying that a sufficient number of excitons are needed topopulate states at high enough k for which energy–momentum conservation becomespossible. For small exciton densities, no pair scattering is possible until the tempera-ture is raised enough to populate these higher-k states. In the low-temperature regime,f3 =0 and so R=S implying that only pair scattering into the polariton trap can occur,this being stimulated by occupation of the k = 0 state. As the temperature increases,the greater range of populated k1 and k2 increases the pair scattering although ioniza-tion of polaritons out of the trap is also now possible from very high-k excitons, thusreducing R compared to S (Fig. 8.7(c)).

Full evaluation of the scattering integrals (eqn 8.29) allows the rate equations tobe solved for the polariton emission (Figs. 8.8(a) and (b)). At low densities the pairscattering behaves quadratically as expected from a two-body incoherent process, sothat

S = cN2x , (8.30)

R = c(N2x − d

Γ0

ΓnrNx) , (8.31)

where d allows for polariton ionization. Below the threshold,

Pth = ωV Γnr

√Γ0/c (8.32)

the output power Pout = ωV cP 2/Γ2nr corresponds to spontaneous pair scattering,

while above threshold, Pout = P − Pth (Fig. 8.8). A more complicated behaviouris seen if ionization of polaritons becomes significant, giving rise to an intermediateregime with a second threshold. It is thus clear that the L−I curve characteristic differsfrom normal lasers, and gives scope for judicious tailoring for practical applications.

The polariton-laser threshold is calculated for two different polaritons trap depths(Figs. 8.8(c) and (d)) corresponding to the conditions for InGaAs (Ω = 7 meV) andCdTe (Ω = 20 meV) QW-based microcavities. The different curves correspond to dif-ferent pair-scattering rates, with the solid line corresponding to rates measured in time-resolved experiments. The minimum laser threshold is found for a temperature kBT ∼Ω/2, which ensures enough excitons are available in the high-k states for strong pairscattering. Such simple estimates show that it should be possible to obtain thresholds atinput powers significantly below 1 mW. The lasing threshold is higher in II-VI micro-cavities with large Rabi splittings because the deeper trap potential requires a compa-rable increase in exciton Fermi energy. In contrast to previous discussions about the in-fluence of acoustic phonons in semiconductor microcavities, temperature plays a ratherdifferent role here: it modifies the occupation of the exciton dispersion and thus controlsthe pair scattering.

Observation of the expected quadratic dependence at low powers is experimentallycomplicated by changes in the emission angular width. A key question is the actualtemperature of the exciton distribution compared to the lattice temperature. For everyphoton emitted, the excitons are heated by Ω/2, however, this is generally much lessthan the energy provided to the lattice as photo- or electrically injected carriers relax


10-6

10-4

10-2

100

102

Out

put p

ower

(m

W)

102

101

100

10-1

10-2

Pump Power (mW)

(a) Ω=7meV

10K 30K 100K 300K

P

P2

102

101

100

10-1

10-2

Pump Power (mW)

30K 100K 200K 300K

(b) Ω=20meV

102 4 6 8

1002 4 6

Temperature (K)

(d) Ω=20meV

0.1

1

10

Thr

esho

ld p

ower

(m

W)

102 4 6 8

1002 4

Temperature (K)

(c) Ω=7meV

Fig. 8.8: Output power vs. power absorbed in the microcavity at different exciton temperatures for microcav-ities of (a) InGaAs QWs and (b) CdTe QWs. At higher temperatures a two-threshold behaviour is seen. Belowthreshold, a quadratic behaviour is seen. (c,d) Corresponding temperature dependence of the threshold, forrelative pair scattering rates of (from top to bottom) 0.01,0.1,1,10. The solid line is the equivalent temperatureof the trap potential Ω/2, while the dashed line is the exciton binding energy.

to form excitons at the bandedge. Further experiments are thus required to verify theexpected exciton density and temperature, and have recently been reported (see previoussection).

In a polariton-laser, absorption at the emission wavelength is extremely weak, evenat low exciton densities. Ionization of polaritons from their trap is prevented by thefast relaxation of high-energy excitons forming a “thermal lock” (Fig. 8.6). The strong-coupling regime allows the large reservoir of exciton states providing gain to exist at en-


Fig. 8.9: Comparison of the key features of VCSELs and polariton-lasers (“plasers”), showing the differ-ent contributions of the spontaneous and final-state stimulated scattering rates S, R as the reservoir densityis increased. In VCSELs, the emitted photons can be reabsorbed by the electronic states, while in plasers,polaritons excited in the trap simply decay again without being ionized into the exciton reservoir (for lowenough temperature).

ergies above the radiating polariton. Pair scattering separates the processes of absorptionand emission through control of the exciton k-distribution, in contrast to conventionallasers for which absorption and emission are intimately linked. Because of this scheme,gain does not require large carrier densities to invert the carrier population, and the strictcondition for inversion is avoided. VCSELs operate by stimulation of photons across theelectron–hole transition, while the polariton-laser operates by stimulation of polaritonsacross the inplane dispersion (rapidly leading to photon emission). Because of the largenumber of possible stimulated pair transitions, the “inversion” in a polariton-laser mustbe defined using the integrated exciton density Nx rather than the occupation at eachk. In spite of this, the exciton density at threshold is several times less than the trans-parency density Nt. Laser action can occur as soon as more than one polariton buildsup in the trap, and the pair-scattering rate becomes stimulated by this final state occupa-tion. This in turn depends on the pair scattering strength, the exciton density, the cavityfinesse and the polariton dispersion, all of which can be tuned by device design.

The polariton-laser works well until the buildup of carriers produces excessive broad-ening of the exciton transition resulting in absorption at the polariton energy, whichcollapses the strong coupling and destroys the polariton trap.

This simple model makes a number of assumptions for the sake of simplicity, whichcan only be checked in the full quantum approach that is detailed in the rest of thischapter. However, they provide a first intuition as to how the polariton-laser comparesto conventional VCSELs, and highlight the nature of the inversionless operation.


8.6 Kinetics of formation of polariton condensates: semiclassical picture

As pointed out in the previous section, the condensate of cavity polaritons is an equilib-rium state of the polaritonic system for a wide range of external parameters. However,polaritons have a finite lifetime in microcavities and are therefore non-equilibrium par-ticles. Relaxation of polaritons in the steepest zone of the polariton dispersion, andtherefore towards the ground-state, is a slow process compared to the polariton lifetime.The slow energy relaxation of polaritons combined with their fast radiative decay resultsin the bottleneck effect. The photoluminescence mainly comes from the “bottleneck re-gion” and the population of the ground-state remains much lower than what one couldexpect from the equilibrium distribution function (see Section 7.1.4). A suitable formal-ism to describe polariton population dynamics is the semiclassical Boltzmann equation.This formalism has been widely used, particularly by Tassone and Yamamoto (1999),Malpuech et al. (2002b), Malpuech et al. (2002a) and Porras et al. (2002). It has provensuccessful when its results have been compared with experimental data, for instance inthe work of Butte et al. (2002). We now discuss qualitatively the main features of po-lariton photoluminescence, and show how the semiclassical Boltzmann equations canbe used to describe polariton relaxation. A major weakness of this approach is, however,that it only allows calculation of the populations of polaritonic quantum states. All otherquantities of interest, such as the order parameter and various correlation functions, arebeyond its scope and a derivation involving quantum features of the system must beundertaken and will be presented later.

8.6.1 Qualitative features

Typical dispersion curves of bulk polaritons are shown in Fig. 4.14. Excitons created byan initial laser excitation relax along the lower-polariton dispersion, which is essentiallythe bare exciton dispersion, except near the exciton–photon resonance. In this region,the polariton density of states is strongly reduced and the excitonic contribution to thepolariton is decreased. One should note that strictly speaking, a photon with k = 0 doesnot exist, and that consequently, the k = 0 polaritonic state of the LPB does not existeither. The polariton dispersion has no minimum and polaritons accumulate in a largenumber of states in the bottleneck region, already mentioned, from where the light ismainly emitted. In this respect, PL experiments performed on the bulk can be viewedas being influenced by the polaritonic effect. More simply, the bulk bottleneck effect isinduced by the sharpness of the energy/wavevector region where an exciton can emita photon considering energy and wavevector conservation conditions. In microcavities,the polariton dispersion is completely different and the LPB has a minimum at k = 0.A bottleneck effect still arises because of the sharpness of this minimum, but light isclearly emitted from the whole polariton dispersion including the ground-state. The po-laritonic effect is from this point of view much clearer than in the bulk, as the PL signalcomes from polariton modes, which are easily distinguishable from bare exciton andphoton modes. States that emit light in a strongly coupled microcavity are polaritonstates, despite the localization effect. The consequences are twofold. First, PL gives di-rect access to the polariton dispersion as pointed out by Houdre et al. (1994). Secondly,a theoretical description of PL experiments should account for the polariton effect and

FORMATION OF POLARITON CONDENSATES: SEMICLASSICAL PICTURE 303

the particle relaxation should be described within the polariton basis.The initial process is a non-resonant optical excitation or an electrical excitation of

the semiconductor. At our level, the differences between the two kinds of excitation areonly qualitative. This excitation generates non-equilibrium electron–hole pairs that self-thermalize on a picosecond timescale. A typical temperature of this electron–hole gasis of the order of hundreds or even thousands of Kelvin. This electron–hole gas stronglyinteracts with optical phonons and is cooled to a temperature smaller than ωLO/kB ona picosecond timescale. During these few picoseconds excitons may form and populatethe exciton dispersion. The exact ratio between excitons and electron–hole pairs andtheir relative distribution in reciprocal space is still the subject of intense research activ-ity, for example from the work of Selbmann et al. (1996) or from Gurioli et al. (1998).For simplicity we choose to completely neglect these early-stage processes. Rather, wechoose to consider as an initial condition the direct injection of excitons in a particularregion of reciprocal space. We assume that the typical timescale needed to achieve sucha situation is much shorter than the typical relaxation time of polaritons within their dis-persion relation. Therefore, our objective is to describe the relaxation of particles (po-laritons) moving in a dispersion relation composed of two branches (the upper-polaritonmode and the lower-polariton mode), as shown in Fig. 8.10(a). We, moreover, assumefor simplicity that the upper branch plays only a minor role since it is degenerate withthe high-k LPB, and that polaritons only relax within the LPB. The peculiar shape ofthe dispersion relation plays a fundamental role in the polariton relaxation kinetics.

The LPB is composed of two distinct areas. In the central zone, excitons are coupledto the light. In the rest of reciprocal space, excitons have a wavevector larger than thelight wavevector in the vacuum and are therefore dark, as shown on Fig. 8.10(a). In theactive zone, the polariton lifetime is mainly associated with radiative decay and is of theorder of a few picoseconds. In the dark zone, polaritons only decay non-radiatively witha decay time of the order of hundreds of picoseconds. The dark zone has a parabolicdispersion associated with the heavy-hole exciton mass, which is of the order of thefree-electron mass. The optically active zone is strongly distorted by strong exciton–light coupling. The central part of this active zone can be associated with a very smalleffective mass (about 10−4m0). This mass rapidly increases with wavevector to reachthe exciton mass at the boundary between the optically active and dark zones.

We now list the physical processes involved in polariton relaxation towards lowerenergy states.

Polariton–acoustic phonon interaction: Interactions between excitons and acousticphonons are much less efficient than optical phonon–exciton interactions. Each relax-ation step needs about 10 ps and no more than 1 meV can be exchanged. About 100-200 ps are therefore needed for a polariton to dissipate 10-20 meV of excess kineticenergy and to reach the frontier zone between dark and active areas. This relaxationtime is shorter than the particle lifetime within the dark zone and some thermalizationcan take place in this region of reciprocal space. Once polaritons have reached the edgeof the active zone they still need to dissipate about 5-10 meV to reach the bottom of thepolariton trap. This process assisted by the acoustic phonon needs about 50 ps, which is


Fig. 8.10: The polariton dispersion re-lation and dynamics of relaxation. (a)Bare exciton and bare cavity photon dis-persions (dashed line), and polariton dis-persion (solid line) as a function of thewavector (×107m−1). In the “dark zone”,the exciton wavevector is larger than thelight wavevector in the media. (b) Sketchof the polariton relaxation within thelower-polariton branch showing the “bot-tleneck” where polaritons accumulate. Asketch of polaritons relaxation due to directpolariton–polariton interaction has alreadybeen shown in Fig. 8.6 in a special casewith population of the ground-state as a re-sult. (c) Sketch of polariton–electron scat-tering process (the curve on the right-handside displays the free-electron dispersion).

at least ten times longer than the polariton lifetime in this region. Therefore, polaritonscannot strongly populate the states of the trap. The distribution function takes largervalues in the dark zone and at the edge of the active zone than in the trap. It cannotachieve thermal equilibrium values because of the slow relaxation kinetics. This effecthas been called the bottleneck effect by Tassone et al. (1997), since it is induced by theexistence of a relaxation “neck” in the dispersion relation. Such a phenomenon doesnot take place in a single QW with a parabolic dispersion. The energy difference be-tween the dark-active zone frontier is only 0.05 meV and a dark exciton can reach theground-state by a single scattering event.


Polariton–polariton interaction: This elastic scattering mechanism is a dipole–dipoleinteraction with a typical timescale of a few ps. It is very likely to happen because ofthe non-parabolic shape of the dispersion relation. Each scattering event can provide anenergy exchange of a few meV. It is the main available process allowing population ofthe polariton trap and overcoming the “bottleneck effect”. All experimental results inthis field confirm that polariton–polariton interactions strongly affect polariton relax-ation. However, the polariton–polariton interaction does not dissipate energy and doesnot reduce the temperature of the polariton gas. If one considers the process sketchedin Fig. 8.10(b), one polariton drops into the active zone where it will rapidly decay,whereas the other polariton gains energy and stays in a long-living zone. Altogether,this process heats the polariton gas substantially and may generate a non-equilibriumdistribution function as we shall see later.

Polariton–free-carrier interaction This scattering is sketched in Fig. 8.10(c). As statedabove, optical pumping generates hot free carriers, which may interact with polaritons.Actually, the formation time of excitons or of strongly correlated electron–hole pairsis much shorter than the polariton lifetime. It is reasonable to assume that they do notplay a fundamental role in polariton relaxation. However, a free-carrier excess may existin modulation-doped structures, or may even be photoinduced if adapted structures areused, as has been done by Harel et al. (1996) and Rapaport et al. (2000). A large free-carrier excess destroys excitonic correlations. However, at moderate density it can keeppolaritons alive and provide a substantial relaxation mechanism. The polariton–electroninteraction is a dipole–charge interaction and the associated scattering process has asub-picosecond timescale. Electrons are, moreover, quite light particles in semiconduc-tors (typically 4–5 times lighter than heavy-hole excitons). An electron is therefore ableto exchange more energy by exchanging a given wavevector than an exciton. This as-pect is extremely helpful in providing polariton relaxation in the steepest zone of thepolariton dispersion. An electron–polariton scattering event is, moreover, a dissipativeprocess for the polariton gas. It may be argued that the electron system can be heated bysuch an interaction. This is only partially true, and we assume that the two-dimensionalelectron gas covering the entire sample represents a thermal reservoir. In this frame-work, an electron gas plays a role similar to acoustic phonons in polariton relaxation,but with a considerably enhanced efficiency. This efficiency allows, in some cases, theachievement of a thermal polariton distribution function, as proposed theoretically byMalpuech et al. (2002b) and later investigated experimentally by Qarry et al. (2003),Tartakovskii et al. (2003) and more recently by Perrin et al. (2005) and Bajoni et al.(2006).

8.6.2 The semiclassical Boltzmann equation

The classical Boltzmann equation describes the relaxation kinetics of classical particles.In reciprocal space this equation reads:

dnk

dt= Pk − Γknk − nk

∑k′

Wk→k′ +∑k′

Wk′→knk′ , (8.33)


Ludwig Boltzmann (1844–1906) developed the statistical theory ofmechanics and thermodynamics. As such he fathered Boltzmann’sconstant kB (see footnote 58 on page 121), the Maxwell–Boltzmanndistribution and the Boltzmann equation to describe the dynamics of anideal gas, among others.

Suffering from depression he committed suicide, some say be-cause of the unsatisfying response to his work. His dense and intricateproduction was later lightened and disseminated by Ehrenfest (who alsocommitted suicide).

where Pk is the generation term, due to optical pumping or to any other physical pro-cess, Γk is the particle decay rate, and W is the total scattering rate between the statesand due to any kind of physical process. Uhlenbeck and Gropper (1932) first proposedto include the quantum character of the particles by taking into account their fermionicor bosonic nature. Equation (8.33) written for fermions reads:

dnk


∑k′

Wk→k′(1− nk′) + (1− nk)∑k′

Wk′→kn′k , (8.34)

whereas for bosons it is:

dnk


∑k′

Wk→k′(1 + nk′) + (1 + nk)∑k′

Wk′→kn′k . (8.35)

Equations (8.34) and (8.35) are called the semiclassical Boltzmann equations. The maintask to describe the relaxation kinetics of particles in this framework is to calculate scat-tering rates. One should first identify the main physical processes that provoke scatteringof particles. Then, scattering rates can be calculated using the Fermi golden rule. Thisprocedure is usable only if the scattering processes involved are weak and can be treatedin a perturbative way. Interactions should provoke scattering of particles within their dis-persion relation and not provoke energy renormalization. For example, the coupling ofparticles with the light should be a weak coupling, only responsible for a radiative de-cay. In a strongly coupled microcavity one cannot describe relaxation of excitons usinga Boltzmann equation. One should first treat non-perturbatively the exciton–photon cou-pling giving rise to the polariton basis. Then, polaritons weakly interact with their envi-ronment. This weak interaction provokes scattering of polaritons within their dispersionrelation and eqn (8.35) can be used. The scattering rates can indeed be calculated in aperturbative way (Fermi golden rule) because they are induced by weak interactions.

In a semiconductor microcavity the main scattering mechanisms identified are:

• Polariton decay (mainly radiative)• Polariton–phonon interactions• Polariton–free-carrier interactions


• Polariton–polariton interactions• Polariton–structural-disorder interactions.

The calculations of the rates of these scattering mechanisms are presented in ap-pendix B.

8.6.3 Numerical solution of Boltzmann equations, practical aspects

Despite its very simple mathematical form, eqn (8.35) is very complicated to inte-grate106 and one must use numerical methods. We detail below one possible such ap-proach.

The phase space, in our case the reciprocal polariton space, must first be discretized.As mentioned previously, it is reasonable to assume cylindrical symmetry for the distri-bution function. The elementary cells of the chosen grid should reflect this cylindricalsymmetry, and therefore these cells should be annular. The cell number i, C(i) shouldcontain all states with wavevectors k satisfying k ∈ [ki, kk+1[. Various choices of scalefor the ki (linear, quadratic or other) have been used in the literature already quoted.The most important requirement being that the distribution function does not vary tooabruptly from cell to cell, one should use small cells in the steep zone of the polaritondispersion, whereas very large cells can be used in the flat excitonic area. The natureof the polariton dispersion makes, therefore, the choice of a nonlinear grid a much bet-ter candidate. In all cases, one state requires particular attention over all the others:the ground-state, especially if one wishes to describe “condensation-like phenomena”,namely a discontinuity of the polariton distribution function. If such a discontinuitytakes place the actual size of the cells plays a role. We cannot choose infinitely smallcells numerically. This means that one cannot solve numerically the Boltzmann equa-tions in the thermodynamic limit in the case of Bose condensation. What can be done isto account for a finite system size R. The spacing between states becomes finite (of theorder of 2π/R). The grid size plays a role, but it is no longer arbitrary but related to areal physical quantity. In such a case the cell size should follow the real state spacing inthe region where the polariton distribution function varies abruptly.

8.6.4 Effective scattering rates

The total scattering rate from a discrete state to another discrete state is the sum of allthe scattering rates:

Wk→k′ =wk→k′

S= W phon

k→k′ + W polk→k′ + W el

k→k′ . (8.36)

We now need to calculate two kinds of transition rate. The first is the transition ratebetween a discrete initial state k and all the states belonging to a cell of the grid, indexedby the integer i.

W outk→k′

k′∈C(i)

=∑k′∈C

wk→k′

S. (8.37)

We pass to the thermodynamic limit, changing the sum to an integral:

106Hilbert worked on the problem of integrating the Boltzmann equation but without success.


W outk→k′

k′∈C(i)

=S

(2π)2

∫k′∈C(i)

wk→k′

Sdk′ =

1

(2π)2

∫k′∈C(i)

wk→k′ dk′ . (8.38)

Here, the integration takes place over final states. The cells have cylindrical symme-try, which means that the scattering rate does not depend on the direction of k. The totalscattering rate towards cell i is therefore the same for any state belonging to cell j. So:

If k ∈ C(j), W outi→j = W out

k→k′

k′∈C(i)

. (8.39)

One also needs to calculate the number of particles reaching a state from the cell C(i):

W ink→k′

k′∈C(i)

= W ini→j =

1

(2π)2

∫k′∈C(i)

wk→k′ dk . (8.40)

Here, as opposed to eqn (8.38), the integration takes place over initial states k.If one wishes to describe condensation in a finite-size system, the ground-state cell

is constituted by a single state and no integration takes place when the final state is theground-state:

W outk→0 = W out

i→0 =wout

k→0

S. (8.41)

This scattering rate is inversely proportional to the system size.In this framework, eqn (8.35) becomes:

dni

dt= Pi − Γini − ni

∑j

Wi→j(1 + nj) + (1 + ni)∑

j

Wj→inj . (8.42)

It is really an ensemble of coupled first-order differential equations that can be easilysolved numerically. One should point out that despite the cylindrical symmetry hypoth-esis, and despite the one-dimensional nature of the final equation, all two-dimensionalscattering processes are correctly accounted for.

8.6.5 Numerical simulations

The bottleneck region of the LPB corresponds to the transition from the exciton-like tothe photon-like part of the dispersion. Exciton–exciton scattering has been proposed asan efficient relaxation process for polaritons. It remains, however, an elastic scatteringprocess that does not dissipate the total polariton energy. It may allow stimulated scat-tering to take place, but as we shall see below, it hardly allows the achievement of athermal distribution. Currently, we see two possible ways to suppress the bottleneck ef-fect and to achieve polariton lasing. First, in future GaN, ZnSe or ZnO-based cavities atroom temperature, if the strong-coupling regime still holds, acoustic phonon relaxationshould be much more efficient than in presently available cavities at helium temper-ature. The case of CdTe cavities is already much more favourable than that of GaAssamples. Theoretical calculations conducted by Porras et al. (2002) show that a thermaldistribution of cavity polaritons with large ground-state occupation can be achieved inthese systems, as shown in Fig. 8.11(a).


Fig. 8.11: a) Distribution function of polaritons calculated by Porras et al. (2002) solving the semi-classical Boltzman equation in a CdTe microcavity (Rabi splitting 10 meV). Polariton–phonon andpolariton–polariton interactions are taken into account. The lattice temperature is 10 K. The po-lariton reservoir is assumed to be thermalized with an excess temperature computed consistently.The pumping power is px = 1, 2, 5, 8, 15 × 1010 cm−2/100 ps, from bottom to top.b) distribution function of polaritons at 10 K calculated by Malpuech et al. (2002b) when non-resonantlypumped with a power of 4.2 W/cm2. Results are shown for (a) polariton-acoustic phonon scattering (dotted),(b) as (a) plus polariton–polariton scattering (dashed) , and (c) as (b) plus polariton-electron scattering (solid).The thin dotted line shows the equilibrium Bose distribution function with zero chemical potential.

This is confirmed by recent experimental achievements (see previous chapter). Analternative (complementary) way could be to use n-doping of microcavities, which isexpected to allow efficient electron-polariton scattering within the photon-like part ofthe dispersion. The advantage of this scattering mechanism with respect to previouslydiscussed ones is that the matrix element of electron–exciton scattering is quite large.Also, an electron has a much smaller mass than a heavy-hole exciton. Thus, the energyrelaxation of a polariton from the bottleneck region to the ground state requires fewerscattering events than for polariton–polariton or polariton–phonon scattering. These ad-vantages have been found theoretically to be strong enough to restore fast polaritonrelaxation and allow the polaritons to condense into their trapped state.

In order to illustrate this, we consider a GaAs-based microcavity containing a sin-gle quantum well. We also take into account a finite system size R that is assumedto be given by a 100 µm excitation spot size. This is practically done by consider-ing a spacing of 2π/R between the ground-state and the first excited states, whereasthe remaining reciprocal space states are assumed to vary continuously. Figure 8.11(b)shows polariton distributions calculated by Malpuech et al. (2002b) by solving the com-plete set of Boltzmann equations for all k states. Taking into account only the acous-tic phonon scattering (curve a in Fig. 8.11), a thermal distribution function is seenonly beyond k = 2 × 104 cm−1 (the bottleneck region) where polaritons accumulate.Equilibrium is reached 10 ns after the start of the non-resonant pumping, leaving anequilibrium polariton density of 2.5 × 1010 cm−2. Including both polariton–polaritonand polariton–acoustic-phonon scattering processes (curve b) shows partial relaxationof the bottleneck and a flat polariton distribution. However, the equilibrium polaritondensity in the cavity remains the same, close to the saturation density for excitons(about 5 × 1010 cm−2). The distribution function near the polariton ground-state ap-


proaches one. This result is in excellent agreement with experimental results obtained bySenellart and Bloch (1999), Butte et al. (2002), Tartakovskii et al. (1999), Tartakovskiiet al. (2000) and Emam-Ismail et al. (2000). This shows that the amplification thresholdfor the distribution function at the trap state is reached when the strong-coupling regimeis likely to be suppressed. To obtain an increase in the population of the lowest k stateone has to increase the population of a large number of states, requiring a large densityof excitons. This is due to the flat shape of the distribution function that comes from thenature of the polariton–polariton scattering process (each scattering event increases thepopulation of the high-k states; relaxation of polaritons from these states is then assistedonly by phonons and is slow).

The radiative efficiency, which we estimate as the ratio of the concentration of pho-tons leaving the cavity within a cone of less than 1 to the pumping intensity, is thusfound to be only 1.7%. When a small free-electron density of 10 × 10 cm−2 is takeninto account (curve c in Fig. 8.11(b)), a huge occupation number of the lowest-energystate of more than 104 is achieved. This system thus acts as a polariton-laser, in whichscattering of polaritons injected at high k by optical or electrical pumping is stimulatedby population of low-k states. In such a situation the light power emitted in a cone of1 is 3.3 W/cm2 and the efficiency of the energy transfer from pump to emitted light isabout 80%. The light emitted by the cavity is much more directional and comes froma smaller number of states than in case b. The equilibrium polariton density in the cav-ity is now 1.25 × 109 cm−2, i.e., 20 times lower than in cases (a, b). Pump powers atleast forty times stronger can be used before the strong to weak coupling threshold isreached. The thin dotted line in Fig. 8.11 shows the equilibrium polariton density froma Bose distribution function plotted for zero chemical potential. It follows quite closelycurve c, which clearly demonstrates that a thermodynamic equilibrium is practicallyachieved for this value of the chemical potential, which is a signature of Bose conden-sation of polaritons. Another interesting feature is found in Fig. 8.12, which shows theradiative efficiency of the cavity as a function of input power with, (a), and without, (b),polariton–electron scattering. In the first case, the emission rises quadratically up to thethreshold, while in the second case it is much larger and increases linearly. The dottedline on curve (b) marks the excitation conditions for which the strong-coupling regimecollapses because of the bleaching of the excitons. A ground-state population largerthan one is achievable within strong coupling, especially using microcavities made oflarger bandgap semiconductors, such as CdTe, as investigated by Le Si Dang et al.(1998), Richard et al. (2005a) and Kasprzak et al. (2006). Figure 8.12 shows the resultsof a simulation performed by Porras et al. (2002) for a CdTe structure. A thermal ex-citon bath of variable temperature was assumed. At low pumping a bottleneck effect isclearly visible that vanishes at higher pumping allowing a large ground-state populationbuilding up.

8.7 Kinetics of formation of polariton condensates: quantum picture in theBorn–Markov approximation

In this section we use standard procedures developed in quantum optics that we adaptto describe the relaxation kinetics and condensation kinetics of cavity polaritons. Our

FORMATION OF POLARITON CONDENSATES: QUANTUM PICTURE 311

Fig. 8.12: Radiative efficiency versus power absorbed in the microcavity considered at 10 K, for (a) a dopedcavity, ne = 1010 cm−2 and (b) an undoped cavity. The dotted part of the curve (b) corresponds to acalculated exciton density > 5 × 1010 cm−2.

goal is to provide a self-consistent microscopic derivation starting with the polaritonHamiltonian. Polaritons are modelled as weakly interacting bosons moving within thelower-polariton branch. They can in principle interact with phonons and free carriers(we will limit ourselves to the interaction with electrons). Furthermore, we include inthe Hamiltonian coupling to an external light field that allows for a self-consistent de-scription of pumping and radiative lifetimes. Then, the procedure used can be summa-rized as follows. We write the equation of motion for the density matrix of the system(von Neumann equation) and perform some approximations. In all cases we will allowfor the Markov approximation, cancelling memory effects. In a second step we performthe so-called Born approximation that allows us to decouple the density matrices of dif-ferent systems and eventually different polariton states. This Born approximation willbe applied in any case to decouple the polariton density matrix from the phonon den-sity matrix, the electron density matrix and the external free-photon density matrix. Allsystems except the polariton one are considered as reservoirs and we trace (average)over them. At the end of this procedure we will find a master equation for the densitymatrix of the whole polariton system (eqn 8.45) that will depend only on polariton op-erators and on semiclassical scattering rates. This equation will be our starting pointfor further approximations on two different levels. First, we describe the case wherewe fully apply the Born approximation to the polariton system. Namely, we decoupledensity matrices of all polariton states. Then, we trace over all polariton states. Thisobtains the semiclassical Boltzmann equations, which are therefore found to be rigor-ously justified. A partial trace applied only on excited states allows us to get a masterequation for the ground-state density matrix, depending only on semiclassical quantitiesthat in turn can be calculated, solving the Boltzmann equation. As we will see, this lastequation can be formally solved and we will discuss in detail the numerical results thatemerge from this model. However, we can already say that in this framework, spon-taneous coherence buildup cannot take place, which shows the need to relax some of


the approximation performed. This alternative approach is presented in the next sectionwhere we partially relax the Born approximation keeping the correlations between theground-state and the excited states. As a result we get a quantum Boltzmann masterequation that describes the dynamics of the ground-state statistics. The numerical so-lution of the coupled Boltzmann-Master equation is presented. It demonstrates that thespontaneous coherence buildup takes place in the polariton system.

8.7.1 Density matrix dynamics of the ground-state

The procedure we are going to outline is familiar in quantum optics in open systemswhere it has been used by numerous authors. The account by Carmichael (2002) isespecially readable and insightful. Shen (1967) and Zel’dovich et al. (1968) provide arather more historical approach.

However, our case presents additional features with no counterpart in pure quantumoptics. This is essentially because polaritons:

• are massive particles with a non-parabolic dispersion• are self-interacting• have a finite lifetime.

In what follows, we calculate the evolution of the density matrix of the system ρ(t)under the influence of the polariton Hamiltonian (5.163). The tools required to followthe derivation have been introduced in Chapter 3. First, we iterate the Liouville–vonNeumann eqn (3.33) to obtain

iρ = [H, ρ(−∞)] +

∫ t

−∞[H(t), [H(τ), ρ(τ)]] dτ . (8.43)

Then we perform a Born approximation between the polariton field and the phonon,electron and photon fields. The density matrix of the system therefore reads:

ρ(t) = ρpol(t)ρphonρelργ . (8.44)

The polariton density matrix evolves in time, while the phonon and electron sub-systems are considered as thermal baths at the lattice temperature. ργ describes thephoton vacuum of the electromagnetic field outside of the cavity (to which is coupledthe electromagnetic field from within the cavity). ρphon, ρel and ργ are kept equal totheir equilibrium value. We then trace over phonon, electron and photon states. Aftersome lengthy but straightforward algebra, this partial trace gives results in a Lindbladform for the dissipative processes, i.e., with terms of the kind LL†ρ + ρLL† − 2L†ρL,eqn (5.83), (we now write simply ρ instead of ρpol), multiplied by time-dependent co-efficients and with operators L depending only on polariton operators. It is at this stagethat the Markov approximation is invoked: the populations are assumed to vary slowlywith time and are taken out of the integral in eqn (8.43) at t = τ . The remaining prod-uct of exponentials is then integrated to give the energy-conserving delta functions. Theequation of motion for the polariton density matrix then becomes:

ρ = Lpumpρ + Llifetimeρ + Lphonρ + Lelρ + Lpolρ , (8.45)


where L are superoperators, that is, an expression constraining normal operators to acton the density matrix. For practical purposes they can be understood as notational con-venience. Because of the equation they derive from, they are sometimes called Liouvil-lian. This is a convenient name as following from the linearity of Liouville equation onecan associate to each part of the Hamiltonian a corresponding Liouvillian that affectsthe evolution of the density matrix. We define them now. The pump Liouvillian reads:

Lpumpρ =1

2

∑k

Pk

(2a†

kρak − aka†kρ− ρaka†

k

), (8.46)

where

Pk =2π

|g(k)|2|Kpump| . (8.47)

The lifetime Liouvillian reads

Llifetimeρ =1

2

∑k

Γk

(2akρa†

k − a†kakρ− ρa†

kak

), (8.48)

where

Γk =2π

|γ(k)|2 (8.49)

is the radiative coupling constant of the state to the external photon field. This result isobtained assuming that the photon modes are empty and therefore unable to replenishthe corresponding polariton mode. Only their quantum fluctuations are playing a role,namely bringing a perturbation at the origin of the transition.

There is a clear symmetry between eqns. (8.46) and (8.48), reversing the orderingof a and a†. Note that each expression is equal to its hermitian conjugate.

The Liouvillian of interaction with phonons reads

Lphon = − 1

2

∑k′

∑k′ =k

W phonk′→k(2a†

kak′ρa†k′ak + a†

k′aka†kak′ρ + ρa†

k′aka†kak′)

− 1

2

∑k′

∑k =k′

W phonk→k′(2a†

k′akρa†kak′ + a†

kak′a†k′akρ + ρa†

kak′a†k′ak) ,

(8.50)

where

W phonk′→k =

2π

∑q

|U(q)|2(1 + nq)δ(E(k′)− E(k)∓ ωq) , (8.51a)

W phonk→k′ =

2π

∑q

|U(q)|2(ξ± + nq)δ(E(k′)− E(k)∓ ωq) , (8.51b)

where nq = 〈b†qbq〉 is the phonon distribution function given by the Bose distribution,and ξ+ = 1 (with matching + sign in the delta function) corresponds to emission of aphonon, while the case ξ− = 0 corresponds to absorption (if E(k) < E(k′)).


The electronic Liouvillian Lel is the same as Lphon but for the transition rates thatget replaced with

W elk→k′ =

2π

∑q

|U el(q,k,k′)|2neq(1− ne

q+k′−k)

× δ(E(k′)− E(k) +

2

2me(q2 − |q + k− k′|2)) , (8.52)

with neq the Fermi distribution function.

Exercise 8.1 (∗) Derive the Liouvillian Lpol for polariton–polariton scattering in theapproximation where it can be dealt with perturbatively. Observe the similarity withexpressions eqns (8.46), (8.48) and (8.50). Relate it to the Lindblad form, i.e., what isthe operator L of eqn (5.83) in this case?

At this level, making the trace on polariton states in eqn (8.45) produces fourth-order correlators for the phonon and electron terms and even eighth-order correlatorsfor the polariton–polariton terms. These correlators have now to be decoupled in orderto get a closed set of kinetic equations.

For simplicity we consider only polariton scattering with acoustic phonons. Othercases can be related easily to the following derivation. We now perform the Born ap-proximation on the polariton system itself. This means that we factorize the densitymatrix of the polariton system as ρ(t) = ρ0(t)ρk1

(t) · · · ρk(t) · · · . This implies thatcorrelators can be decoupled in the following way:

〈a†kaka†

k′ak′〉 = 〈a†kak〉〈a†

k′ak′〉 , (8.53)

if k = k′. This quantity is of course the product of polariton populations. We shall notethese with uppercase letters, to separate them from phonon populations that we notewith lower case:

〈a†kak〉 = Nk, 〈b†kbk〉 = nk . (8.54)

In this framework, we make the trace over all polariton states except the ground-state. Using this procedure, we get as foretold the semiclassical Boltzmann equationsthat describe the dynamics of the polariton distribution function:

dNk

dt= Pk − ΓkNk −Nk

∑k′

Wk→k′(1 + Nk′) + (1 + Nk)∑k′

Wk′→kN ′k . (8.55)

The dynamics of the ground-state population is also governed by this equation butwe also end up with a master equation for the ground-state density matrix that reads:

ρ0 =1

2Win(t)(2a†

0ρ0a0 − a0a†0ρ0 − ρ0a0a

†0)

+1

2(Γ0 + Wout(t))(2a0ρ0a

†0 − a†

0a0ρ0 − ρ0a†0a0) , (8.56)


where

W in(t) =∑k

Wk→0Nk and W out(t) =∑k

W0→k(1 + Nk) , (8.57)

the former is the total scattering rate toward the ground-state and the latter the totalscattering rate from the ground-state toward excited states. This equation is completelysimilar to the one usually accepted to describe a single-mode linear amplifier—an ex-haustive description of which is given by Mandel and Wolf (1995)—except for the timedependence of the transition rates eqns. (8.57). Equation (8.56) can nevertheless besolved using the Glauber–Sudarshan representation of the density matrix. The deriva-tion and the discussion of this solution have been given by Rubo et al. (2003) and Laussyet al. (2004c). The result appears below:

Exercise 8.2 (∗∗) Show that the solution of eqn (8.56) in Glauber representation withthe initial condition P (α, α∗, 0) = δ(α− α0) is

P (α, α∗, t) =1

πm(t)exp

(− |α−G(t)α0|2m(t)

)(8.58)

in terms of time-dependent parameters:

G(t) = exp

[1

2

∫ t

0

(W in(τ)−W out(τ)) dτ

], m(t) = G(t)2

∫ t

0

W in(τ)G(τ)2 dτ .

(8.59)

The exact solution, eqns. (8.58) and (8.59), shows that in the Born–Markov approx-imation, the state of the condensate is that of a thermalized coherent state, parame-ters G and m relating to the relative importance of the coherent and thermal fractions,respectively. The quantities of interest that we can derive directly from the solutionare the occupation number N0 (ground-state population), the order parameter 〈a0〉, theground-state statistics p0(n) = 〈n| ρ |n〉 and the second-order coherence g(2). Thesecan be obtained from the complete solution by integrating their equations of motion(derived from eqn (8.56)). First, equations for the scalar quantities (we use for conve-nience η = 2− g(2) rather than g(2) directly):

N0 = (Win −Wout − Γ0)N0 + Win , (8.60a)

˙〈a0〉 =1

2(Win − (Wout + Γ0))〈a0〉 , (8.60b)

∂t(ηN20 ) = 2(Win − (Wout + Γ0))ηN2

0 . (8.60c)

Secondly, the set of coupled differential equations for the statistics:

p0(n) = (n + 1)(Wout + Γ0)p0(n + 1)

− [n(Wout + Γ0) + (n + 1)Win]p0(n0) + nWinp0(n− 1) . (8.61)

The quantities 〈a0〉4 and ηN20 are both extensive and proportional to N2

0 . They aresurprisingly described by the same equation. Whereas 〈a0〉 depends on off-diagonal el-ements of the density matrix, ηN2

0 depends only on diagonal elements. The equation of


motion for the normalized (intensive) quantities η that describe the diagonal coherenceand 〈a0〉2/N0 that describes the off-diagonal coherence are given by:

η = −2Winη

N0, (8.62)

∂

∂t

( 〈a0〉2N0

)= −Win

N0

〈a0〉2N0

. (8.63)

8.7.2 Discussion

The set of equations (8.60)–(8.62) is particularly simple and the meaning of the variousterms is transparent. Equation (8.60) is inhomogeneous, it is composed of a spontaneousscattering term and of a stimulated scattering term. The equations for the order parame-ter and for the quantity η are both homogeneous and governed by the stimulated terms.This means that the ground-state initially empty or in a thermal state will stay thermalforever even if a large number of particles comes in. However, a coherent seed in theground-state can be amplified if stimulated scattering takes place. As we will see in thenext section, this set of equations allows us to describe the transfer of a large number ofincoherent reservoir particles within a ground state having a high degree of coherence.This means that the total coherence of the system can increase. However, the coherencedegree of the ground-state itself can only decay in time as one can see on eqns. (8.62),except if the ground-state population becomes infinite, which can be the case only ininfinite systems.

In the steady-state regime the equilibrium value for N0 is:

N0(∞) =Win

Wout(∞) + Γ0 −Win(∞), (8.64)

and the system is in a thermal state with zero diagonal and off-diagonal coherence.The characteristic decay constant of the order parameter is called the phase diffusioncoefficient that is equal to the emission linewidth of the ground state:

D =1

2

[Win(∞)−Wout(∞)− Γ0

]a =

Win

2N0(∞). (8.65)

The energy broadening of the condensate is no longer given by the radiative life-time but by the balance between incoming and outgoing scattering rates. In the low-temperature limit Wout is small and Win ≈ Γ0 that allows recovery of the well-knowndiffusion coefficient from laser theory

D ≈ Γ0

2N0. (8.66)

However, one should notice that the polariton–polariton interaction that is not in-cluded in the former equations plays an important role in the phase diffusion coefficient,as discussed by Laussy et al. (2006b) and in more detail later.


8.7.3 Coherence dynamics

Before presenting numerical results, we give a qualitative analysis of the polariton con-densate formation. The kinetics are characterized by a transient regime, during whichthe polaritons come to the condensate, after being excited in some k = 0 state at t = 0.Their relaxation rate depends nonlinearly on the pumping intensity. For strong enoughpumping the stimulated scattering of polaritons into the condensate flares up at a timet > 0, so that the in-scattering rate increases drastically and becomes much greaterthan the out-scattering rate. In the time domain, where Win(t) > Wout(t) + Γ0, thesolution becomes unstable. This instability allows the condensation to happen in a co-herent quantum state. The formation of the condensate with t = 0 implies breakingthe symmetry of the system, which cannot happen spontaneously in the framework ofthe formalism used. Therefore, to study the possibility of coherence buildup we intro-duce an initial seed (a coherent state with small average number of polaritons). Thisinitial coherence can survive and be amplified for the high relaxation rates, as long as atime window exists in which Win(t) > Wout(t) + Γ0. After the steady-state regime isreached the point 〈a0〉 becomes stable again, since the rates reach the time-independentvalues Win(∞) and Wout(∞), with Wout(∞) + Γ0 > Win(∞). However, the differ-ence between the stationary rates is very small, inversely proportional to the systemarea, which corresponds to a large stationary number of condensed polaritons. If thecoherence is formed its decrease due to spontaneous scattering is slow in large cavities.

As before, we consider the cavity containing 1010 electrons/cm−2. The coefficientsWin(t) and Wout(t) are extracted from the Boltzmann equation. We model the follow-ing experiment: at t = 0 an ultrashort laser pulse generates a coherent ground-statecontaining a variable polariton number (the seed). At the same time, an incoherent non-resonant cw pumping is turned on. Three pumping densities (0.8, 8 and 160 W/cm2) areconsidered. In all cases the strong-coupling regime is maintained. Figure 8.13 showsthe evolution of the ground-state population for different pumping densities and a seedwith N0 = 102 particles. Table 8.2 gives the parameters obtained in the steady-stateregime, namely ground-state populations, the ratio of populations of the ground-stateand the first excited state, the ratio of the ground-state population and the total popula-tions, and the chemical potential µ = kBT ln(1 − 1/N0). The steady-state distributionfunctions are found to be very close to the Bose distribution function.

Table 8.1 Parameters used for numerical computations.

Cavity photon Non-radiative Heavy-hole Electronlifetime lifetime exciton mass mass

8 ps 1 ns 0.5m0 0.07m0

Figure 8.14 shows the evolution of two second-order coherence parameters of thesystem. η has already been defined as the ratio of coherent polaritons in the ground-stateover the total number of polaritons in the entire system, and χ is the ratio of coherentpolaritons in the ground-state over the total number of polaritons in the ground-state.They are linked by η = χn0/

∑q nq. Ground-state coherence χ is maximum at initial


Fig. 8.13: Time dependence of the Bose-condensate occupation number N0. The pumping densities are 0.8,8, and 160 W/cm2 for curves (a), (b) and (c), respectively. The evolution of the seed at the initial stage isshown in the inset.

Fig. 8.14: Ground-state coherence η (dashed lines)and total system coherence χ (solid line) for pumpingpower of 0.8, 8 and 160 W/cm2, respectively.

time, which corresponds to the artificial introduction of a coherent seed. The relevantphysical quantity at t = 0 is η, which is vanishingly small. Figure 8.14(a) is the regimebelow threshold for coherent amplification, where the seed coherence is rapidly washedout and that of the entire system remains zero. Figure 8.14(c) in contrast displays theregime where the coherent seed is coherently amplified: a macroscopic number of par-ticles populate the coherent fraction of the initially small seed. Figure 8.14(b) displaysan intermediate regime.


Table 8.2 Main equilibrium values obtained numerically.

Pumping density N0 N0/N1 N0/N −µ(W/cm2) (µeV)

0.8 7.7×103 23.5 0.04 568 2.7×105 510 0.59 1.6160 5.8×106 5800 0.95 0.07

Figure 8.14(c) which shows quasicondensation of polaritons, is characterized bythe buildup of the order parameter and by the amplification of the initial seed coherentstate. The whole system coherence also strongly increases from 0 to more than 90%. Atintermediate pumping densities the values of the steady-state coherence degree dependnoticeably on the seed’s characteristics.

8.8 Kinetics of formation of polariton condensates: quantum picture beyond theBorn–Markov approximation

8.8.1 Two-oscillator toy theory

To gain insight into the mechanisms at work, we first revert to a toy model that reducesall the relevant physics to its bare minimum. Later we give the full picture suitable todescribe a realistic microcavity.

Since dimensionality is not an issue because it is not the accommodation of a pop-ulation in phase space but dynamical effects that are responsible for populating theground-state, we describe the system by a zero-dimensional two-oscillator model, oneoscillator representing the ground-state, the other an excited state (or assembly of ex-cited states combined as a whole). We also neglect interparticle interactions, which willclearly show that efficient relaxation is required (conserving particle number), but thatintrinsic interparticle interactions are not necessary. The number of polaritons in theentire system fluctuates, but we shall see that the correlations implied by conservationof polaritons in their relaxation are at the heart of our mechanism. One can reconcilethe conservation of particles with a fluctuation in their total number through an inter-pretation in terms of a pulsed experiment, where a laser injects periodically in time afluctuating number of particles in the system. Each relaxation taken separately involvesan exact and constant number of particles, while observed results are averaged overpulses and thus echo an overall fluctuating population.

We label the states as 1 and 2. There is only one parameter to distinguish them thatis the ratio ξ of the rate of transitions w1→2 and w2→1 between these states:

ξ ≡ w2→1

w1→2. (8.67)

These w are constants, especially they have no time dependence coming from pop-ulations included in these scattering terms. We assume ξ > 1 , which identifies state 1as the ground-state (i.e., state of lower energy), since from elementary statistics:

w2→1

w1→2= e(E2−E1)/kT , (8.68)


with Ei the energy of state i (by definition of the ground state, E2 > E1) and T is thetemperature of the system once it has reached equilibrium.

The Hamiltonian for the two-oscillator system coupled through an intermediary os-cillator (depicting a phonon) in the rotating wave approximation reads in the interactionpicture

H = V ei(ω1−ω2+ω)ta0a†1b + h.c. , (8.69)

where a1, a2 and b are (time-independent) annihilation operators with bosonic alge-bra for first oscillator (ground-state), second oscillator (excited state) and the “phonon”respectively, with free propagation energy ω1, ω2, ω and coupling strength V . Car-rying the same procedure as previously for , i.e., evaluating the double commutatorgives

∂tρ = − 12 [w1→2(a

†1a1a2a

†2ρ + ρa†

1a1a2a†2 − 2a1a

†2ρa†

1a2)

+w2→1(a1a†1a

†2a2ρ + ρa1a

†1a

†2a2 − 2a†

1a2ρa1a†2)] , (8.70)

after the Markov approximation ((τ) ≈ (t)) and factorization of the entire systemdensity matrix into ρρph, with ρ, ρph the density matrices describing the two oscilla-tors and the phonons, respectively. Of course at this stage the whole construct is veryclose to our previous considerations, but note that no Born approximation is made on ρso that correlations between the two oscillators are fully taken into account. The transi-tion rates are given by:

w1→2 = 2π|V |2〈b†b〉/(ω2 − ω1) , (8.71a)

w2→1 = 2π|V |2(1 + 〈b†b〉)/(ω2 − ω1) . (8.71b)

We now obtain from eqn (8.70) the equation for diagonal elements

p(n,m) ≡ 〈n,m|ρ|n,m〉 , (8.72)

where p(n,m) is the joint probability distribution to have n particles in state 1 and min 2. This equation reads

∂tp(n,m) = (n + 1)m[w1→2p(n + 1,m− 1)− w2→1p(n,m)]

+ n(m + 1)[w2→1p(n− 1,m + 1)− w1→2p(n,m)] . (8.73)

This equation for a probability distribution parallelling the Boltzmann equation is thequantum Boltzmann master equation (QBME) for the two-oscillator model.

The QBME can be derived rigorously from a microscopic Hamiltonian, as was doneby Gardiner who pioneered this approach in a series of papers on quantum kineticsstarted by Gardiner and Zoller (1997). However, in the two-oscillator toy model, thephysical picture is so straightforward that one hardly needs this approach (that is inves-tigated below). In a similar spirit, Scully (1999) applied a fixed number of particles tothe case of Bose condensation of atoms. Coming back to our two modes, and focusing,


for instance, on the first term on the right-hand side, one sees that it expresses how theprobability can be increased to have (n,m) particles in states (1, 2) through the processwhere starting from (n + 1,m − 1) configuration, one reaches (n,m) by transfer ofone particle from state 1 to the other state. This is proportional to n + 1, the numberof particles in state 1 and is stimulated by m − 1 the number of particles in state 2 towhich we add one for spontaneous emission, whence the factor (n + 1)m. We repeatthat w1→2 and w2→1 are constants and should not be confused with the bosonic transi-tion rate defined as w1→2(1 + m) and w2→1(1 + n) to account in a transparent way forstimulation. Our present discussion will be clarified by being explicit.

For all quantities Ω that pertain to a single state only, say the ground-state (so wecan write Ω(n)), it suffices to know the reduced probability distribution for this state,i.e., for the ground-state:

p1(n) ≡∞∑

m=0

p(n,m) , (8.74)

and vice versa, i.e., for excited state p2(m) ≡∑∞n=0 p(n,m). So that indeed

〈Ω(n)〉 =∞∑

n=0

∞∑m=0

Ω(n)p(n,m) =∞∑

n=0

Ω(n)p1(n) . (8.75)

We will soon undertake to solve exactly eqn (8.73) but to delineate its quality inexplaining how coherence arises in the system we first show that if we make the ap-proximation to neglect correlations between the two states, i.e., if we assume the factor-ization

p(n,m) = p1(n)p2(m) , (8.76)

then the system at equilibrium will never display any coherence, i.e., in accord with ourprevious discussion, both states will be in a thermal state no matter the initial condi-tions, the transition rates or any other parameters describing the system. Indeed puttingeqn (8.76) into eqn (8.73) and summing over m, we obtain:

∂tp1(n) = p1(n + 1)w1→2(n + 1)(〈m〉+ 1)

− p1(n)(w2→1(n + 1)〈m〉+ w1→2n(〈m〉+ 1)

)+ p1(n− 1)w2→1n〈m〉 ,

(8.77)

with 〈m〉 ≡ ∑m mp2(m) the average number of bosons in state 2. At equilibrium the

detailed balance of these two states gives the solution

p1(n + 1) =〈m〉

〈m〉+ 1

w2→1

w1→2p1(n) . (8.78)

The same procedure for state 2 yields likewise

p2(m + 1) =〈n〉

〈n〉+ 1

w1→2

w2→1p2(m) , (8.79)


0

0.5

1

0 5 10 15 20

<n>

θ=<n>/(1+<n>)

0

1

2

3

0 1 2 3 4kBT/−hω

<n>=θ/(1-θ)=(e−hω/kBT-1)-1

θ=e-−hω/kBT

Fig. 8.15: “Thermal parameter” θ as a function of 〈n〉 (left figure) and temperature (right figure, dashed line).θ varies in the interval [0, 1[ and tends towards unity with increasing particle number. The average number ofparticles is also displayed as a function of the temperature on the right figure (solid line).

with the notational shortcuts

θ ≡ 〈n〉〈n〉+ 1

, ν ≡ 〈m〉〈m〉+ 1

, (8.80)

eqns (8.78) and (8.79) read after normalization

p1(n) = (1− νξ)(νξ)n , (8.81a)

p2(m) = (1− θ/ξ)(θ/ξ)m , (8.81b)

so that 〈n〉 ≡∑np1(n) = νξ/(1− νξ), which inserted back into eqn (8.80) yields

ξ =θ

ν, (8.82)

or, written back in terms of occupancy numbers and transition rates:

w2→1

w1→2=

〈n〉〈n〉+ 1

〈m〉+ 1

〈m〉 , (8.83)

which give in eqns. (8.78) and (8.79):

p1(n + 1) =〈n〉

〈n〉+ 1p1(n) and p2(m + 1) =

〈m〉〈m〉+ 1

p2(m) , (8.84)

achieving the proof that both states are (exact) thermal states under the hypothesis (8.76)that we will now relax. This will give rise to a likewise regime where both states arethermal states, but also to another regime where the excited state (state 2) is still ina thermal state, but the ground-state (state 1) is non-thermal (and in some limit, hasthe statistics of a coherent state). This is possible if one takes into account correlationsbetween states. In our case these correlations come from the conservation of particlenumber, so that the knowledge of particle number in one state determines the number


in the other state. In fact, observe how the QBME connects elements of p(n,m) that lieon antidiagonals of the plane (n,m). One such antidiagonal obeys the equation

n + m = N, (8.85)

where N is a constant, namely, the distance of the antidiagonal to the origin from thegeometrical point of view, and the number of particles from the physical point of view.One such antidiagonal is sketched in Fig. 8.16. The equation can be readily solved ifonly one antidiagonal is concerned, i.e., if there are exactly N particles in the system. Inthe case where the particle number in the system is only known with some probability,one can still decouple the equation onto its antidiagonal projections, solve for themindividually and add up afterwards weighting each antidiagonal with the probability tohave the corresponding particle number. We hence focus on such an antidiagonal Nwhose conditional probability distribution is given by d(n|N) ≡ p(n,N − n), with anequation of motion given by eqn (8.73) as:

∂td(n|N) = (n + 1)(N − n)[w1→2d(n + 1|N)− w2→1d(n|N)]

+n(N − n + 1)[w2→1d(n− 1|N)− w1→2d(n|N)] .(8.86)

The equation is well behaved within its domain of definition 0 ≤ n ≤ N since itsecures that d(n|N) = 0 for n > N . This also ensures unicity of solution despite therecurrence solution being of order 2, for d(1|N) is determined uniquely by d(0|N),itself determined by normalization.

The stationary solution is obtained in this way (or from detailed balance):

d(n + 1|N) =w2→1

w1→2d(n|N), (8.87)

with solution

d(n|N) = d(0|N)

(w2→1

w1→2

)n

, (8.88)

where d(0|N) is defined for normalization as

d(0|N) =ξ − 1

ξN+1 − 1, ξ ≡ w2→1

w1→2. (8.89)

Technically solving for d resembles the procedure already encountered to solve theequation under assumption (8.76). However, we are now paying full attention to corre-lations between the two states, which turns detailed balancing of eqns (8.78) and (8.79)into one of an altogether different type, eqn (8.88). This gives by weighting eqn (8.88)the solution to the QBME:

p(n,m) =ξ − 1

ξn+m+1 − 1ξnP (n + m) , (8.90)

whereP (N) ≡

∑n+m=N

p(n,m) (8.91)

is the distribution of total particle number, i.e., the probability to have N particles in theentire system. P (N) is time independent since the microscopic mechanism involved


n

m

p(n,m)

antidiagonaln+m=7

Fig. 8.16: p(n, m) steady state solution from Laussy et al. (2004a) in the case where the distribution functionfor the number of particles in the entire system P (N) is a Gaussian of mean (and variance) 15 and ξ = 1.2.One “antidiagonal”, n + m = 7, is shown for illustration. The projection on the n-axis displays a coherentstate, whereas the projection on the m-axis displays a thermal state.

conserves particle number for any transition (one can also check that ∂tP (N) = 0).This allows us to derive the statistics of separate states:

p1(n) = ξn∞∑

N=n

ξ − 1

ξN+1 − 1P (N) ,

p2(n) = ξ−n∞∑

N=n

ξ − 1

ξN+1 − 1ξNP (N) .

(8.92)

Observe how the n dependence of the sum index prevents a trivial relationship be-tween p1 and p2 of the kind p1(n) = p2(N − n). Also, the asymmetry between groundand excited state is obvious from eqn (8.90). It is this feature that allows two states withdrastically different characteristics, typically a thermal and a coherent state. Indeed, p1

(resp. p2) is the product of a sum with an exponentially diverging (resp. converging tozero) function of ξ. In both cases, the sum of positive terms is a decreasing functionof n, so that clearly no coherence can ever survive in the excited state whose fate isalways thermal equilibrium, or at least, in accord with our definitions,

p2(n) > p2(n + n0) for all n, n0 in N. (8.93)

For p1, however, ξn diverges with n, what leaves open the possibility of a peak notcentred about zero in this distribution, while it can still be a decreasing function if the


sum converges faster still. It is to P (N) to settle this issue, which as a constant of motionis completely determined by the initial condition. The solution for the case where P (N)is a Gaussian of mean (and average) 15 is displayed in Fig. 8.16. p(n,m) is in this casemanifestly not of the type p1(n)p2(m) and there is always coherence in the system. Inthe next section we investigate the more interesting situation where coherence does notexist a priori in the system.

8.8.1.1 Growth of the condensate at equilibrium By growth at equilibrium we meanthat, still in the approximation of infinite lifetime, coherence can arise when one lowersthe temperature, i.e., increases ξ, in a system where initially all states are thermal states.In this case the initial condition for the system is the thermal equilibrium

p(n,m) = (1− θ)(1− ν)θnνm , (8.94)

where θ, ν are the thermal parameters for ground and excited states, respectively. Theylink to 〈n〉, the mean number of particles in the ground-state, through

〈n〉 =θ

1− θ, (8.95)

or, the other way around,

θ =〈n〉

1 + 〈n〉 . (8.96)

Similar relations hold for ν and m. This is one possible steady solution of eqn (8.73)and we discuss how it arises from eqn (8.90) below. The thermal state is wildly fluctu-ating. Once in a while, thermal kicks transfer in the state one or many particles, which,however, do not stay for long before the state is emptied again or replaced by other,unrelated particles. This accounts for the chaotic, or incoherent, properties of such astate. This essentially empty but greatly fluctuating statistics brings no conceptual prob-lem for small populations, but one might enquire whether it is conceivable to have athermal distribution with high mean number. This is possible for a single state but notfor the system as a whole. A macroscopic population can distribute itself in a vast col-lection of states so that each has thermal statistics, constantly exchanging particles withother states and displaying great fluctuations, but as expected from physical grounds,the whole system does not fluctuate greatly in its number of particles. Therefore, weexpect P (N), the distribution of particles in the entire system, to be peaked about anonzero value, typically to be a Gaussian of mean and variance equal to N . In the pureBoltzmann case, this results from the central limit theorem since the total number ofparticles is the sum of a large number of uncorrelated random variables, and thus is it-self a Gaussian random variable. Remembering our previous definitions, this, however,does not qualify the system as a coherent emitter, since the statistics must refer to a sin-gle state, not to a vast assembly of differing emitters. Thus, not surprisingly, coherencearises when a single quantum mode models or copies features of a macroscopic system,typically its population distribution. The two-oscillator system, which is a rather coarseapproximation to a macroscopic system, will, however, display very clearly this mecha-nism. In the limit where ξ 1 it is already clear from eqn (8.92) that p1(n) ≈ P (n), so


Fig. 8.17: Left: ground-state distribution p1(n) for ξ = θν≈ 1.26 with 3 particles in the excited state and 17

in the ground-state, both in thermal equilibrium, g(2)(0) = 2. Centre: same with ξ raised to 1.5. Distributionis non-thermal, especially p1(1) > p1(0), though the distribution is then always decreasing. g(2)(0) ≈ 1.89.Right: with ξ → 0. The distribution is that for the entire system, p1(n) = P (N), cf. eqn (8.101). Yet it isfar from coherent in this model, as g(2)(0) → 1.745.

that the statistics of the entire system indeed serves as a blueprint for the ground-state(and it alone, excited states being always decreasing as already shown). At equilibrium,with two thermal states, the distribution for the whole system reads:

P (N) =∑

n+m=N

p(n,m) = (1− θ)(1− ν)θN+1 − νN+1

θ − ν. (8.97)

This exhibits a peak at a nonzero value provided that

ν + θ > 1 . (8.98)

This criterion refers to a first necessary condition: there must be enough particles inthe system. The fewest particles available so that eqn (8.98) is fulfilled, is two. Thisminimum required to grow coherence fits nicely with the Bose–Einstein condensationpicture (one needs at least two bosons to condense). It is not a necessary condition,though; also the dynamical aspect is important as shown by the key role of ξ. Indeed, ifthe system is steady in configuration (8.94), ξ is not a free parameter but is related to θand ν by:

ξ =θ

ν, (8.99)

and in this case the distribution of the ground-state

p1(n) = (1− θ)(1− ν)ξn∞∑

N=n

θN+1 − νN+1

ξN+1 − 1

ξ − 1

θ − ν, (8.100)

reduces by straightforward algebra to p1(n) = (1 − θ)θn, i.e., as should be for con-sistency, the ground-state is in a thermal state, independently of the value of θ (i.e.,no matter what is the number of particles in the ground-state). This can come as asurprise, but it must be borne in mind that this two-oscillator model is an extreme sim-plification that cannot dispense with some pathological features, namely, the ability tosustain a thermal macroscopic population, an ability that we understand easily sincethe ground-state accounts for half of the system! It is expected that with increasing


number of states, dimensionality will forbid such an artifact. Also, the shape of P (N)hardly resembles a Gaussian (see Fig. 8.17, right panel) but already in this limitingcase it is able to display a peak at a nonzero value provided there are enough particles.With increasing number of states, the central limit theorem will turn this distributioninto an actual Gaussian. Once again, P (N) is time independent because the relaxationmechanism conserves particle number, which results in correlations between the twostates. By increasing ξ to ξ′, one might search new values of θ, ν, say θ′, ν′, so thatθ/(1− θ) + ν/(1− ν) = θ′/(1− θ′) + ν′/(1− ν′) (conservation of particle number)and ξ′ = θ′/ν′. This is possible if one allows P (N) to change, in which case the twonew states are also thermal states. If P (N) is constrained by correlations induced bystrict conservation of particle number—so that the uncertainty is not shifted as the sys-tem evolves—then eqn (8.99) breaks down and this allows eqn (8.90) to grow a coherentstate in the ground-state. This process is illustrated in Fig. 8.17, starting from thermalequilibrium and lowering temperatures (increasing ξ). In the two-oscillator model, co-herence grown out of thermal states cannot come much closer to a Gaussian than illus-trated in Fig. 8.17, right panel, where the limiting case ξ → 0 for which the ground-statedistribution reduces to P (N) is displayed, cf. eqn (8.97),

p1(n) = (1− θ)(1− ν)θn+1 − νn+1

θ − ν, (8.101)

which is obvious on physical grounds (one particular realization is the one for whichw1→2 = 0 and thus with all particles eventually reaching the ground-state) and rein-forces our understanding of Bose condensation as the ground-state distribution functioncoming close to the macroscopic distribution, with complete condensation correspond-ing to identification of p1(n) with P (N).

8.8.1.2 Growth of the condensate out of equilibrium The previous case holds inan equilibrium picture and for that matter refers to coherence buildup in systems likecold atom BEC. To address the polariton-laser case, it is necessary to extend the two-oscillator model with the additional complication of finite lifetime τ of particles instate 1, with a balance in the total population provided by a pump that injects parti-cles in state 2 at a rate Γ. (We will not crucially need a finite lifetime in the excited stateand thus neglect it, which is a good approximation in a typical microcavity where theradiative lifetime drops by a factor of ten to a hundred in the photon-like part of thedispersion.) Although the QBME can be readily extended phenomenologically to takethese into account,

p(n,m) = (n + 1)m[w1→2p(n + 1,m− 1)− w2→1p(n,m)]

+ n(m + 1)[w2→1p(n− 1,m + 1)− w1→2p(n,m)]

+1

τ(n + 1)p(n + 1,m)− 1

τnp(n,m)

+ Γp(n,m− 1)− Γp(n,m) ,

(8.102)

the couplings between different particle numbers forbid solving this new equation alongthe same analytical lines as previously, though the numerical solution can be obtained


straightforwardly. Introducing 〈m〉n the mean number of polaritons in the excited stategiven that there are n in the ground-state, i.e.,

〈m〉n p0(n) =

∞∑m=0

mp(n,m) , (8.103)

with p0(n) ≡∑m p(n,m) is the reduced ground-state statistics, we obtain, by averag-

ing eqn (8.73) over excited states, an equation for the ground-state statistics only:

∂tp1(n) = (n + 1)(w1→2(〈m〉n+1 + 1) + 1/τ)p1(n + 1)

−

n((〈m〉n + 1)w1→2 + 1/τ

)+ (n + 1)〈m〉nw2→1

p1(n)

+ n〈m〉n−1w2→1p1(n− 1) . (8.104)

However, in this out-of-equilibrium regime, the excited state is not as important asin the equilibrium case where it must be thermal and whose configuration is of utmostconsequence on the ground-state. Thus, we can dispense with the actual distributionof the excited state and simply use the mean 〈m〉n obtained from

∑m mp(n,m) =

〈m〉np2(n). In this case eqn (8.102) can be decoupled to give an equation for p1(n)alone, and in the “dynamical” steady-state, the detailed balance reads:

p1(n + 1) =w2→1〈m〉n

w1→2(〈m〉n+1 + 1) + 1/τp1(n) . (8.105)

Up to this point it is still exact, and also in the out-of-equilibrium regime we grantthe conservation of particle number as the origin of correlations between the two states,but because of lifetime and pumping, it can now be secured only in the mean, leadingus to the following approximation for 〈m〉n:

〈m〉n = N − n . (8.106)

The pump, which has quantitatively disappeared from the formula, is implicitly takeninto account through this assumption, since even though particles have a finite lifetime,their number is constant on average. In the coherent case, p1(n) is a Poisson distributionwith maximum at N − Nc, so that this dependency of N on the pump is in this caseN = τP + Nc.

When the population has stabilized in the ground-state by equilibrium of radiativelifetime and pumping, it is found in a coherent state if N > Nc with Nc the criticalpopulation defined by:

Nc =1

τ(w2→1 − w1→2), (8.107)

obtained from eqns (8.105) and (8.106) with the requirement that p1(1) > p1(0). If thispopulation is exceeded, coherence builds up in the system along with the population,which stabilizes at an average given by the maximum of the Gaussian-like distribution:


〈n〉 = nmax = N −Nc , (8.108)

obtained from p1(n) = p1(n + 1); so that effectively if N < Nc there is no such Gaus-sian and coherence remains low with a thermal-like state whose maximum is for zerooccupancy. If N > Nc the state is a Gaussian whose mean increases with increasingdeparture of population from the critical population. Thus, the higher the number of par-ticles, the less the particle number fluctuations of the state, and the better its coherence.

In Fig. 8.18 is displayed the numerical solution of eqn (8.102) for p1(n, t), whereparameters (see legend) have been chosen so that N exceeds eqn (8.107) and thereforegrow some coherence from an initially empty ground-state (cf. Fig. 8.18(a)). The co-herence is maintained for infinite times and the statistics for the ground-state occupancytends towards a Gaussian-like function neatly peaked about a high value (Fig. 8.18(b)).We define a coherence degree equal to 2 − g(2)(0), so chosen to be 0 for a genuinethermal state and 1 for a genuine coherent state. In the case where (8.107) is exceeded,the coherence degree of ground-state quickly reaches unity (Fig. 8.18(c)). In Fig. 8.19 isdisplayed the counterpart situation where parameters (see legend) result in a subcriticalpopulation so that the steady-state is thermal, as shown in Fig. 8.19(b). The dynamicsof p(n), starting from vacuum, is merely to grow this thermal state (cf. Fig. 8.19(a)) andthe coherence degree remains low (cf. Fig. 8.19(c)).

We have varied the temperature (through w2→1) for simplicity but kept all other pa-rameters constant. This is not very convenient experimentally as this requires adjustingthe pumping. The total number of particles N can be expressed as a function of otherparameters as follows: the rate equation of particle numbers in the ground and excitedstate are

∂tn = −γn− w1→2n(N − n + 1) + w2→1(n + 1)(N − n) , (8.109a)

∂t(N − n) = P − (n + 1)(N − n)w2→1 + n(N − n + 1)w1→2 . (8.109b)

where γ = 1/τ . Going to the steady-state yields the relation

N =P (γ2 + w2→1(γ − P ) + w1→2(γ + P ))

γ(w1→2(γ + P )− w2→1P ), (8.110)

which provides an equivalent set of parameters to cross the threshold as a function of Por γ, which is more relevant experimentally.

8.8.2 Coherence of polariton-laser emission

The extension from the two-oscillator model to a realistic microcavity is straightforwardas far as the formalism is concerned, but the resulting equations cannot be tackled di-rectly as previously. As before, the pumping and finite lifetime should not be neglected,but for the relaxation, one can still consider phonon-mediated scattering only (neglect-ing polariton–polariton scatterings). Since all these processes are dissipative, we seek amaster equation of the Lindblad type for the density matrix ρ of polariton states in thereciprocal space:

∂tρ = (Lpol−ph + Lτ + Lpump)ρ . (8.111)


Fig. 8.18: System configuration suitable forcoherence buildup: N = 350, w2→1 =10−5 (arb. units), w1→2 = 0.75 × 10−5

(arb. units) and 1/τ = 20×10−5 (arb. units).All units have the same dimension of an in-verse time. (a) is a density plot for the timeevolution of p(n), starting from vacuum itquickly evolves towards a coherent state. (b) isthe projection of p(n) in the steady-state. (c)is the time evolution of the normalized coher-ence degree η = 2 − g(2)(0): full coherenceis quickly attained.

Fig. 8.19: System configuration unable to de-velop coherence. Parameters are the same asfor Fig. 8.18 except w2→1 = 0.95 × 10−5

(arb. units) corresponding to a higher temper-ature. (a) Starting from vacuum the ground-state steadies in a thermal state for which aprojection (b) is shown. (c) The normalizedcoherence degree remains low.


Here, L are Liouville superoperators that describe, respectively, scattering (throughphonons), lifetime and pumping. The polariton–polariton scattering would add a uni-tary (non-dissipative) contribution Lpol−pol = − i

[Hpol−pol, ρ]. In the following, we

undertake the derivation of Lpol−ph from the microscopic Hamiltonian Hpol−ph forpolariton–phonon scattering. Exactly the same procedures can be carried out for Lτ

and Lpump to yield:

Lτρ = −∑k

1

2τk(a†

kakρ + ρaka†k − 2akρa†

k) , (8.112a)

Lpumpρ = −∑k

Pk

2(aka†

kρ + ρa†kak − 2a†

kρak) , (8.112b)

with τk the lifetime, Pk the pump intensity in the state with momentum k and ak theBose annihilation operator for a polariton in this state. The expression for the lifetimecomes from the quasimode coupling of polaritons with the photon field outside thecavity in the vacuum state (thereby linking spontaneous emission with the perturbationfrom vacuum fluctuations). We later neglect finite lifetime elsewhere than in the ground-state, where it is typically several orders of magnitude shorter because of the dominantphoton fraction. In our simulations, pumping injects excitons 10 meV above the bottomof the bare exciton band that we model by nonzero values of Pk for a collection of k-states normally distributed about a high momentum mean value. Expression (8.112b)describes an incoherent pumping provided by a reservoir that pours particles in thesystem but does not allow their return. Its effect is thus merely to populate the systemwith incoherent polaritons, which will relax towards the ground-state where they mightjoin in a coherent phase before escaping the cavity by spontaneous emission (the lightthus emitted retaining this coherence).

We pay special attention to Lpol−ph that contains the key ingredients of our results.We repeat here the interaction picture polariton–phonon scattering term from the polari-ton Hamiltonian:

Hpol−ph =∑

k,q =0

Vqei(Epol(k+q)−Epol(k)−ωq)tak+qa†

kb†q + h.c. , (8.113)

with Vq the interaction strength, Epol the lower-polariton-branch dispersion, ωq thephonon dispersion and aq, resp. bq, the Bose annihilation operator for a polariton,resp. a phonon, in state q.

As previously, we compute Lpol−ph starting with the Liouville–von Neumann equa-tion for polariton–phonon scattering,

∂t = − i

[Hpol−ph, ] , (8.114)

where is the density matrix for polaritons and phonons. Its useful part, namely thepolariton density matrix, is obtained by tracing over phonons, ρ ≡ Trph. The densitymatrix for phonons is combined as a reservoir in equilibrium with no phase coherence


nor correlations with ρ. To dispense from this reservoir we write the equation for toorder two in the commutator and trace over phonons,

∂tρ(t) = − 1

2

∫ t

−∞Trph[Hpol−ph(t), [Hpol−ph(τ), ]] dτ . (8.115)

We defineEk,q(t) ≡ Vqe

i(Epol(k+q)−Epol(k)−ωq)t , (8.116)

and for convenience we write

H(t) ≡∑k,q

Ek,q(t)ak+qa†kb†q , (8.117)

so that Hpol−ph = H+H†. Operators are time independent. Because the phonon densitymatrix is diagonal, [Hpol−ph(t), [Hpol−ph(τ), ]] reduces to [H(t), [H†(τ), (τ)]]+h.c.,which halves the algebra. Also, the conjugate hermitian follows straightforwardly, so weare left only with explicit computation of two terms, of which the first reads:

[H(t),H†(τ)] =∑

k,q =0

∑l,r =0

[Ek,q(t)ak+qa†kb†q, E∗l,r(τ)ala

†l+rbr(τ)] , (8.118)

which, taking the trace over phonons and calling θqρ ≡ Trph(b†qbq), becomes:

Trph[H(t),H†(τ)(τ)]

=∑

k,l,q =0

Ek,q(t)E∗l,q(τ)θq(ak+qa†kala

†l+qρ(τ)− ala

†l+qρ(τ)ak+qa†

k) . (8.119)

Solving numerically this equation is a considerable task, which, however, has alreadybeen carried out for a similar equation by Jaksch et al. (1997), using quantum Monte-Carlo simulations. We prefer to make further approximations to reduce its simulationto a level of complexity of the same order as for the Boltzmann equations: we takeinto account correlations between ground-state and excited states only, neglecting allcorrelations between excited states. This is legitimized by the fast particle redistribu-tion between excited states and their rapid loss of phase correlations. Physically thismeans that if a particle reaches the ground-state, its absence is felt to some extent inthe collection of excited states in a way that ensures particle number conservation.On the contrary, redistribution of particles between excited states will be seen to obeythe usual Boltzmann equations that pertain to averages only. Formally, we thus neglectterms like 〈ak1

a†k2

ak3a†k4〉 if ki involve nondiagonal elements in the excited state. For

nonvanishing terms, we further allow 〈ak1a†k1

ak2a†k2〉 = 〈ak1

a†k1〉〈ak2

a†k2〉 if neither

k1 nor k2 equal 0, while otherwise we retain the unfactored expression. Terms fromeqn (8.119) featuring the ground-state are:∑

k =0

Ek,−k(t)Ek,−k(τ)∗θk(a0a†0a†

kakρ(τ)− aka†0ρ(τ)a0a†

k) (8.120a)

+∑k =0

E0,k(t)E0,k(τ)∗θk(a†0a0aka†

kρ(τ)− a0a†kρ(τ)a†

0ak) . (8.120b)


Recall this expression (8.120) is one part of the term inside the time integral thatgives ∂tρ(t) evolution. Since

Ek,q(t)Ek,q(τ)∗ = |Vq|2 exp(− i

(Epol(k + q)− Epol(k)− ωq)(t− τ)

),

the time integration would yield a delta function of energy if ρ in eqn (8.120) wasτ -independent. This delta would itself provide selection rules for allowed scatteringprocesses through the sum over k. That ρ(τ) time evolution is slow enough as comparedto this exponential to mandate this (Markov) approximation can be checked throughevaluation of the phonon reservoir correlation time, which, when the reservoir has abroadband spectrum as in our case, is short enough to allow the approximation of ρ(τ)by ρ(t). In this case, eqn (8.120b) vanishes as a non-conserving energy term. Gatheringother terms similar to eqn (8.119) eventually gives (from now on we no longer write ρtime dependence, which is t everywhere):

∂tρ = − 1

2

∑k =0

W0→k(a†0a0aka†

kρ + ρa†0a0aka†

k − 2a0a†kρa†

0ak) (8.121a)

− 1

2

∑k =0

Wk→0(a0a†0a†

kakρ + ρa0a†0a†

kak − 2a†0akρa0a†

k) , (8.121b)

where

W0→k ≡ 2π

|Vk|2θkδ(Epol(k)− Epol(0)− ωk) (8.122a)

Wk→0 ≡ 2π

|Vk|2(1 + θk)δ(Epol(k)− Epol(0)− ωk) . (8.122b)

We call p(nk) the diagonal of the polariton density matrix, i.e., the dotting of ρwith |nk〉 = |n0, nk1

, · · · , nki, · · · 〉 the Fock state with nki

polaritons in state ki:

p(nk) ≡ 〈· · · , nki, · · · , nk1

, n0|ρ|n0, nk1, · · · , nki

, · · · 〉 . (8.123)

This is the probability that the system be found in configuration nk whose equationof motion is the master equation obtained from eqn (8.121) as

p(nk) = −∑k

(W0→kn0(nk + 1) + Wk→0(n0 + 1)nk)p(nk)

+∑k

W0→k(n0 + 1)nkp(n0 + 1, . . . , nk − 1, . . .) (8.124)

+∑k

Wk→0n0(nk + 1)p(n0 − 1, . . . , nk + 1, . . .) .

This is the counterpart to eqn (8.73), which also parallels closely the Boltzmann equa-tion with which it shares the same transition rates (8.122) given by Fermi’s goldenrule, and so it represents the QBME for polariton lasers. We substitute eqn (8.112) in


eqn (8.121) and, following the same spirit, we do not solve it for the entire joint prob-ability p(nk) but average over all excited states to retain the statistical character forthe ground-state only. Excited states will be described with a Boltzmann equation, thuswith thermal statistics. Calling

p0(n0) ≡∞∑

i=1

∞∑nki

=0

p(n0, nk1, nk2

, · · · , nkj, · · · ) , (8.125)

the ground-state reduced probability (the sum is over all states but the ground-state, cf.eqn (8.74)), and

〈nk〉n0p0(n0) ≡

∑nk1

,nk2,···

nkp(nk) , (8.126)

cf. eqn (8.103), we get the ground-state QBME equation:

p0(n0) = (n0 + 1)(Wn0+1out + 1/τ0)p0(n0 + 1)

−(n0(Wn0

out + 1/τ0) + (n0 + 1)Wn0

in

)p0(n0) (8.127)

+ n0Wn0−1in p0(n0 − 1) ,

with rate transitions now a function of the ground-state population number n0:

Wn0

in (t) ≡∑k

Wk→0〈nk(t)〉n0, (8.128a)

Wn0

out(t) ≡∑k

W0→k(1 + 〈nk(t)〉n0) , (8.128b)

while for excited states, in the Born–Markov approximation, we indeed recover theBoltzmann equations:

˙〈nk〉 = 〈nk〉∑q =0

Wk→q(〈nq〉+1)−(〈nk〉+1)∑q =0

Wq→k〈nq〉, k = 0 . (8.129)

Inclusion of Lτ and Lpump for the above adds −〈nk〉/τk + Pk to this expression. Ob-serve that in this case, the transition rates are constants.

Cast in this form, eqn (8.127) has the same transparent physical meaning in termsof a rate equation for the probability of a given configuration, much like the usual rateequations for occupation numbers in the framework of the Boltzmann equations. Thedifference is that transitions from one configuration to a neighbouring one occur at ratesthat depend on the configuration itself, through the population of the state. 〈nk〉n0

is afunction of n0 that we estimate through a first-order expansion about 〈n0〉. This impliesthat fluctuations of excited states are proportional (with opposite sign) to fluctuations ofthe ground-state:

〈nk〉n0≈ 〈nk〉〈n0〉 +

∂〈nk〉n0

∂n0

∣∣〈n0〉(n0 − 〈n0〉) , (8.130)

where 〈nk〉〈n0〉 is given by a Boltzmann equation. Since the derivative does not dependon n0 (it is evaluated at 〈n0〉), we compute it by evaluation of both sides at a known


value, for instance n0 = N with N the total particle number in the entire system, groundand excited states together. This gives

∂〈nk〉n0

∂n0

∣∣∣∣∣〈n0〉

=〈nk〉

〈n0〉 −N, (8.131)

since 〈nk〉N = 0 (no particles are left in excited states when they are all in the ground-state). With the knowledge of eqn (8.130) and (8.131), which are known from semiclas-sical Boltzmann equations, this is now only a matter of numerical simulations.

8.8.3 Numerical simulations

Parameters used are for a CdTe microcavity of 10 µm lateral size with one QW and aRabi splitting of 7 meV at zero detuning. The size corresponds to the light spot radiusreported by Deng et al. (2002). This is an important parameter as correlations increasewith decreasing size of the system. Scattering is mediated by a bath of phonons at atemperature of 6 K and with a thermal gas of electrons of density 1011 cm−2 accountedfor to speed up relaxation. This is below the exciton bleaching density. The cavity isinitially empty and pumped non-resonantly from t = 0 onwards.

Fig. 8.20: Time evolution of normalized ground-state population 〈n0(t)〉/neq (dotted line) andof normalized coherence ratio η = 2 − g(2)

(solid line), starting from the vacuum (no coher-ence): both population and coherence buildup areobtained in the timescale of hundreds of ps.

Fig. 8.21: Density plot of p0(n0) (ground-statepolaritons distribution) as a function of time fora realistic microcavity (with parameters given inthe text). Darker colors correspond to smaller val-ues. Initial state is the vacuum. A coherent stateis obtained in the timescale of hundreds of ps.

Figure 8.20 displays the ground-state population normalized to its steady-state valueneq and the coherent fraction χ. Starting from zero for the vacuum state, coherencesteadily rises in the system as more polaritons enter the ground-state. Interestingly, eventhough the dynamics can give rise to a temporary decrease in the number of ground-statepolaritons, the coherence does not drop in echo but continues its ascent. This is betterunderstood with Fig. 8.21 where a density plot for p0(n0) is shown and where onecan observe how the coherent statistics is reached through a tightening of the numberof states with high probability of occupancy about the average. The function p0(n0),


Fig. 8.22: Projections of p0(n0) of previousfigure at three various times: still in a thermalstate (exponential decay) close to initial times,≈ 75 ps; in a fully grown coherent state (Gaus-sian) in the steady-state and in the intermediateregion where coherence is building up, ≈ 100 ps.

Fig. 8.23: Ground-state population normalized to pumppower 〈n0〉/P (solid line) and normalized coherence de-gree η in the steady-state (dashed) as a function of thereduced pumping P/Pthreshold; we find Pthreshold =26 W/cm−2.

which at first varies wildly, quickly flattens with a large number of particles in a ther-mal state, then a nonzero maximum appears and the statistics evolves as Gaussian-liketowards a Poisson distribution of mean 〈n0〉 in the steady-state. The polariton densityof 1010 cm−2 achieved in the simulation of Laussy et al. (2004b) is more than one or-der of magnitude smaller than the strong/weak coupling transition density in CdTe. InFig. 8.23 is displayed the steady-state coherence ratio and population as a function ofpumping. It exhibits a clear threshold behaviour.

8.8.4 Order parameter and phase diffusion coefficient

8.8.4.1 Formalism and discussion The phase diffusion coefficient D is the decayconstant of the order parameter 〈a0(t)〉 = 〈a0(0)〉eiω0te−Dt where ω0 is the free prop-agation energy of the considered field. This quantity has been introduced in Chapter 6and discussed in section 8.7.2. It is directly related to the laser linewidth and inverselyproportional to its coherence length. For normal lasers, this phase diffusion coefficientis due to the finite ratio that exists between spontaneous and stimulated emission oflight. D is therefore inversely proportional to the number of photons in the lasing mode.This number is finite in any finite size system and in any realistic system.

For polariton-lasers, as we have seen in Sections 8.3.1 and 8.3.2, this decay constanthas, for non-interacting particles, the same physical origin as for normal lasers. It isdue to the spontaneous scattering toward the ground-state that provokes a dephasing.This dephasing is responsible for a broadening of the emission linewidth. The emissionspectrum can indeed be calculated through:

I(ω, t) =1

π∫ +∞

0

〈a†0(t + τ)a0(t)〉eiωτ dτ . (8.132)

This quantity becomes independent of t in the steady-state to give


I(ω, t) = ∫ +∞

0

〈a†0(τ)a0(0)〉eiωτ dτ =

DN0/π

(ω − ω0)2 + D2. (8.133)

This is the classical Lorentzian spectrum. This aspect has been widely discussed inlaser theory. However, polaritons are interacting particles and this may lead to a substan-tial shift of the emission energy and to an increase of the linewidth. Formally, this shiftand broadening originates from the terms

∑k Vka†

0a0a†kak of the Hamiltonian. These

do not invoke any real scattering and they do not modify either the populations or thestatistics. They are called dephasing terms. Their impact has been analysed by Porrasand Tejedor (2003) in the steady-state and by Laussy et al. (2006b) on the polarization.Porras and Tejedor considered the broadening of a thermal state. Figure 8.24 showsthe full-width at half-maximum of the ground-state line versus pumping power. Thewidth first decays above the threshold because of the decrease of dephasing inducedby spontaneous scattering. However, the dephasing induced by interactions increasesproportionally to the ground-state population and becomes dominant for large pumpingpowers. In what follows we give an intuitive derivation that gives the same results as thequantum formalism developed by Laussy et al. (2006b).

Fig. 8.24: Ground-state occupancy of a polariton-laser versus pumping power (dashed line) and full-width athalf-maximum of the ground-state emission (solid line), as predicted by Porras and Tejedor (2003).

We consider again a polariton system with N particles in all and N0 in the ground-state. We also define N1 = N − N0 the number of particles in the excited states.The interparticle interaction constant is supposed to be constant and is given by V .Therefore, the average interaction energy of the ground-state is simply given by V NN0.The energy per particle is equal to the energy shift of the line:

Eshift =V NN0

N0= V N . (8.134)

This shift is a rigid shift of the complete dispersion, if one neglects the dependenceof V on the wavevector. N , however, is a fluctuating quantity governed by a statisticaldistribution. The uncertainty of the number of particles leads to an uncertainty in theenergy. We are going to separate the uncertainty from N1 and the uncertainty from N0.If one considers a large ensemble of independent states, all having thermal statistics and


a small average number of particles, the statistics of the total number of particles in thissystem is Poissonian. This is a consequence of the central limit theorem. Therefore, thenumber of particles in excited states follows a Poissonian distribution P1(n1). The rootmean square of this distribution is σ1 =

√N1. The emission line resulting from the

interaction between the ground-states and the excited states is therefore Poissonian witha width at half-maximum of V 2

√2 ln 2

√N1. On the other hand, the uncertainty of the

number of particles in the condensate is sensitive to the statistics of exciton-polaritonsin this state. For a state having the thermal statistics, σ0 =

√N2

0 + N0 ≈ N0. Theassociated line has the shape of a thermal distribution, namely exponential on the high-energy tail, and with an abrupt cutoff corresponding to the case n0 = 0. Such statisticsyields an asymmetric broadening as shown in Fig. 8.24. The total line is in this case asuperposition of two very different functions. If, however, the ground-state is coherentthe uncertainty in the number of particles for the coherent statistics is only σ0 =

√N0.

In a general case, the condensate is in a mixed state having both coherent and thermalfractions. Its statistics is described by the second-order coherence function g2(0), whichvaries from 2, for the thermal state, to 1 for the coherent state. The root mean square ofsuch statistics is given by σ0 =

√N0 + N2

0 (g2(0)− 1). If g2(0) is close enough to 1,the associated statistics is well described by a Gaussian. Making the assumption that thetwo random variables N0 and N1 are not correlated, the statistics of their sum is givenby a Gaussian of mean N0 + N1 and of root mean square σ =

√σ2

0 + σ21 . Now, if we

neglect the spontaneous broadening that is very small if N0 is large, the linewidth of theemission line is given by:

∆E ≈ V 2√

2 ln 2√

N1 + N0 + N20 (g2(0)− 1) . (8.135)

Now, we suppose that N0 can be measured experimentally. We use eqn (8.134)to get N1 as a function of N0 and Eshift. The knowledge of these two quantities—Vcombined with a measurement of ∆E—finally allows one to deduce the second-ordercoherence as:

g2(0) = 1 +1

8 ln 2

(∆E

V N0

)2

− Eshift

V N20

. (8.136)

This formula has of course a weak precision if σ1 is larger than σ0. In the oppositelimit it is probably very accurate and represents a clear and direct way to measure g2(0).This technique can only be applied to the case of interacting particles. It is specific tothe polariton system with respect to a photon system.

8.9 Semiconductor luminescence equations

In this section we discuss an alternative view of polariton lasing based on the considera-tion of fermion-like electrons and holes instead of the boson-like exciton-polaritons. Theobservations of Pau et al. (1996) claiming experimental evidence for the boser action,i.e., polariton lasing, have been reproduced by the group of Khitrova and Gibbs and pub-lished with a theoretical model by Kira et al. (1997) in conceptual opposition with theboson approach started with the so-called “boser” of Imamoglu et al. (1996). These au-thors criticized the bosonic approach for approximating “the full electron–hole Coulomb

SEMICONDUCTOR LUMINESCENCE EQUATIONS 339

Fig. 8.25: PL spectra reported by Kira et al. (1997), experimental (left column) and theoretical as computedwith eqn (8.142) solving numerically eqns (8.137)–(8.139) (right column). The dashed curves were obtainedby discarding line (8.137b), which is reponsible for strong coupling, in the absence of which the emittedspectra are clearly seen to emit at the bare cavity and “exciton” modes. The cavity detuning of 3 meV isultimately responsible for the observed splitting in the high-pumping regime (top of figure) where the upper-polariton branch is emitting superlinearly. The shaded curves are the excitonic absorption spectra, the excitonemission is not sensitive to the dressing of the exciton, even when the exciton resonance is bleached.

interaction in the many-body Hamiltonian [. . . ] by including only some aspects of theinterband attractive part” and proposed instead a full quantum picture starting from atwo-band Hamiltonian that includes consistently Coulomb interactions and correlationsbetween electrons and holes. They obtained a complex set of coupled equations for thepolarization Pk ≡ c†v,kcc,k, the population operators c†c,kcc,k and the photon annihila-tion operators Bq for photons with zero inplane wavevector (q‖ = 0, q). These are aspecial case of a general theory to describe light–matter interaction in semiconductorsdeveloped by the authors, which they call the “semiconductor luminescence equations”.With the dynamical decoupling scheme (explained in details, e.g., by Binder and Koch(1995))—retaining correlators 〈B†

qPk〉 and 〈B†qbq′〉 and electron fe

k ≡ 〈cc,k†cc,k〉, hole


fhk ≡ 1 − 〈cv,k

†cv,k〉 populations—a closed set of equations is obtained, which, how-ever, remains very complicated and requires essentially numerical treatment:

i∂t〈B†qPk〉 =(−Ec(k)− Ev(k)− ωq + EG)〈B†

qPk〉 (8.137a)

+ (fek + fh

k − 1)Ω(k, q) (8.137b)

+ fekfh

kΩSE(k, q) , (8.137c)

i∂t〈B†qBq′〉 =(ωq′ − ωq)〈B†

qBq′〉 (8.138a)

+ iEquq〈Bq′PH〉+ iEq′ u∗q′〈B†

qPH〉 , (8.138b)

i∂tfe(h)k = 2i[µ∗

cv(k)〈PkD〉] . (8.139)

The detailed expressions for the various quantities involved need not concern us here,but let us instead list the physical meaning of each quantity, which yields the gist of thismodel: Ec,v(k) are the dispersions for conduction/valence electrons (renormalized byinteractions and structure details like the QW confinement factor, in a way we do notreproduce), EG the gap energy, Eq the radiation field vacuum amplitude, u(r) the ef-fective cavity mode wavefunction and µcv the dipole matrix element (Kira et al. (1997)provide a full discussion). More importantly,

Ω(k, q) ≡ µcv(k)〈B†qE〉+

∑k′

V (k′ − k)〈B†qPk′〉 , (8.140)

ΩSE(k, q) ≡ iµcv(k)Equq , (8.141)

which involve the Coulomb matrix element V and the quantized radiation field E ∝∑q Bquq, and provide the source for field-particle correlations (cf. eqn (8.137c)). Since

fekfh

kΩSE is nonzero in the presence of excitations (electrons and holes), it triggers〈B†

qPk〉 even if they are initially zero. We note that Pk is the amplitude for an interbandtransition of the electron, so that 〈B†

qPk〉 is the photon-mediated polarization. Thus,ΩSE, which drives the emission through electron–hole recombination, is interpreted asa spontaneous emission term. In the same way, Ω on line (8.137b) provides stimulation(the second term in eqn (8.140) being the renormalization of stimulation to the first“bare” stimulation term).

From these equations, one can compute the normal-incidence luminescence spec-trum as the time variation of the light field intensity:

I(k) ∝ ∂t〈B†kBk〉 . (8.142)

This, however, demands numerical simulations, which the authors have performedonly in 1D and found to be in agreement with experimental observations. Two appeal-ing features of such numerical approaches are the possibility to artificially neutralizesome contributions, e.g., the stimulation term (8.137b), and to gain access to some ex-perimentally awkward or poorly defined quantities, such as for instance 〈[X,X†]〉. For

CLAIMS OF EXCITON AND POLARITON BOSE–EINSTEIN CONDENSATION 341

a true boson, this quantity is exactly one, cf. eqn (3.39). The computations have shown,however, that this quantity varies from 0.7 down to 0.3 as a function of increasing den-sities (from ≈ .5 to ≈ 3× 1011 cm−2), leading to the conclusion that the boson pictureis not valid, and that Pau et al.’s (1996) explanation in terms of “close-to-boson” polari-tons was mistaken. The more convincing result from this work, however, comes fromthe other numerical latitude, namely the artificial switch-off of stimulation, which ineffect amounts to discarding strong coupling. In doing so, they observed the persistenceof the splitting, which was previously claimed as evidence of strong coupling. The cor-rection of the semiconductor luminescence equation and the bosonic picture remains anunresolved issue.

8.10 Claims of exciton and polariton Bose–Einstein condensation

Finally, we crown this chapter with the state-of-the-art of the experimental findings, re-porting a positive outcome to the quest for solid state BEC. Indeed, very recently a num-

Fig. 8.26: Far-field angular emission below and above threshold showing the formation of a polariton Bose–Einstein condensate, as claimed by Kasprzak et al. (2006). The cover illustration is based on the k-spacecounterpart of this figure.


ber of groups have found that thermalization of polaritons on the lower-polariton branchis possible in conditions of positive cavity detuning. In II-VI microcavities, Kasprzaket al. (2006) could obtain a direct fitting of the experimentally observed polariton oc-cupations using the Bose–Einstein distribution above and below threshold, as seen inFig. 8.26, and show that this is different from Maxwell–Boltzmann statistics, i.e., strongevidence of the bosonic nature of the polaritons. In addition, it is clear that despite thestrong spatial fluctuations and disorder-induced localization of the emission, the differ-ent emitting “hot” spots are coherently locked in phase, i.e., they are part of the samestate. The transition to this polariton BEC state is found around 20 K, where the polari-ton temperature is close to the lattice temperature. Similarly, in III-V microcavities newevidence is emerging that polariton BECs are indeed thermalized at low temperatures,whether for positive cavity detuning when excited at resonant energies of the excitons,as done by Deng et al. (2006), or using localized stress to form a real-space trap forpolaritons, as done by Balili et al. (2006). All these new experiments clearly show theappearance of bosonic-induced coherence in the semiconductor microcavity system inthe strong coupling regime, and pave the way to coherent matter-wave monolithic de-vices.

8.11 Further reading

Griffin et al. (1995) edited a proceedings volume for the first Levico conference onBEC in 1993, held before its experimental realization two years later and announced“hot off the press” at the following conference. Its lucky timing and contributions fromeminences of the time made it an important publication in the field. The review papersgive insightful and personal accounts by pioneers of the field, while brief reports providean interesting historical snapshot of the time. It also includes some unusual systems forthe condensed-matter physicist, such as BEC of mesons or Cooper pairing in nuclei.Griffin remembers the conference, now an historical one, and the story of this book isin J. Phys. B: At. Mol. Opt. Phys. 37 (2004).

More pedagogical and unified texts have flourished since. Especially notable is theexcellent text by Pitaevskii and Stringari (2003). Pethick and Smith (2001) provide agood introductory description of the atomic case that became the “hero” of BEC asits first experimental realization, and a source of inspiration for the condensed-mattercommunity. Another useful proceedings is the Enrico Fermi’s Varenna summer schoolvolume edited by Inguscio et al. (1999).

Moskalenko and Snoke (2000) provide a good overview of theoretical work (withmany discussions of experimental results) related to BEC in semiconductors. The firstchapter opens with “Many people seem to have trouble with the concept of an exci-ton. . . ” and gives a gentle introduction to an otherwise involved exposition. The contentindeed borrows a lot from research papers of the Russian literature. It is therefore a use-ful window to the non-Russian-speaking reader into the extensive amount of theoreticalwork made by the Soviet school on this topic.

An enduring classic dealing with superfluidity is the textbook by Khalatnikov (1965)which covers all the main characteristics of Bose gas and liquids, especially the excita-tion spectrum, hydrodynamics and kinetics of the problem. The discussion of the Lan-

FURTHER READING 343

dau spectrum and its two-liquid model is one of the best expositions. The Bogoliubovmodel is reviewed in full details in the review by Zagrebnov and Bru (2001).

We also recommend readers interested in this controversial subject to the recentpaper by M. Combescot et al., “polariton-polariton scattering: Exact results through anovel approach”, Europhysics Letters, 79, 17001-17006 (2007), where interactions ofexciton-polaritons in microcavities are described beyond the bosonic approximation.This paper contains a list of references to the previous papers by Combescot and co-authors on interacting systems of composite bosons, where the so-called “Shiva dia-grams” were proposed and a number of interesting conclusions of general relevancefor the many-body physics have been drawn. Being perfectly aware that the bosonic ap-proximation for composite particles—unlike exciton-polaritons—has its limitations, weuse it largely throughout this book for two reasons: because it allows a qualitative un-derstanding of most of the new nonlinear optical effects observed in microcavities, andbecause the full theory of cavity polaritons treated as composite particles is still underconstruction. In particular, the important issue of spin and polarization-related effectshas only been treated within the bosonic approximation, as shown in the next chapter.

9

SPIN AND POLARIZATION

In this chapter we consider a complex set of optical phenomena linkedto the spin dynamics of exciton-polaritons in semiconductormicrocavities. We review a few important experiments that reveal themain mechanisms of the exciton-polariton spin dynamics and presentthe theoretical model of polariton spin relaxation based on the densitymatrix formalism. We also discuss the polarization properties of thecondensate and the superfluid phase transitions for polarizedexciton-polaritons. Finally, theoretically predicted optical spin-Hall andspin-Meissner effects are described.107

107We acknowledge enlightening discussions with I. Shelykh, Yu. Rubo and K. Kavokin who contributedsubstantially to the content of this chapter.

346 SPIN AND POLARIZATION

When optically created, polaritons inherit their spin and dipole moment from the ex-citing light. Their polarization properties can be fully characterized by a Stokes vectoror—using the language of quantum physics—a pseudospin accounting for both spin anddipole moment orientation of a polariton. From the very beginning of their creation ina microcavity, polaritons start changing their pseudospin state under the effect of ef-fective magnetic fields of various natures and due to scattering with acoustic phonons,defects, and other polaritons. This makes pseudospin dynamics of exciton-polaritonsrich and complex. It manifests itself in non-trivial changes in the polarization of lightemitted by the cavity versus time, pumping energy, pumping intensity and polariza-tion. Experimental studies of the polarization properties of microcavities have givenevidence of a set of unusual effects (giant Faraday rotation, polarization beats in photo-luminescence, polarization inversion in the parametric oscillation regime, etc.). Linearoptical effects (like Faraday rotation) have been interpreted in terms of the theory ofnon-local dielectric response (developed in Chapter 3). The analysis of the data of non-linear optical experiments require more substantial theoretical effort. The experimentsunambiguously indicate that energy and momentum-relaxation of exciton-polaritons arespin dependent, in general. This is typically the case in the regime of stimulated scat-tering when the spin polarizations of initial and final polariton states have a huge ef-fect on the scattering rate between these states. It appears that critical conditions forpolariton Bose-condensation are also polarization dependent. In particular, the stimula-tion threshold (i.e. the pumping power needed to have a population exceeding 1 at theground-state of the lower-polariton branch) has been experimentally shown to be lowerunder linear than under circular pumping. These experimental observations have stim-ulated theoretical research toward understanding the mutually dependent polarization-and energy-relaxation mechanisms in microcavities.

9.1 Spin relaxation of electrons, holes and excitons in semiconductors

The concept of spin was introduced by Dutch-born US physicists Samuel AbrahamGoudsmit and George Eugene Uhlenbeck in 1925. In the same year, the Austrian theo-rist Wolfgang Pauli proposed his exclusion principle that states that two electrons cannotoccupy the same quantum state. Later, it was understood that this principle applies to allparticles and quasiparticles with a semi-integer spin (fermions) including electrons andholes in semiconductors. On the other hand, it is not valid for particles (quasiparticles)having an integer spin, in particular for excitons.

Spins of electrons and holes govern the polarization of light generated due to theirrecombination. The conservation of spin in photoabsorbtion allows for spin-orientationof excitons by polarized light beams (optical orientation), an effect that manifests itselfalso in the polarization of photoluminescence. σ+ and σ− circularly polarized light ex-cites J = +1 and −1 excitons, respectively. Linearly polarized light excites a linearcombination of +1 and −1 exciton states, so that the total exciton spin projection onthe structure axis is zero in this case. Optical orientation of carrier spins in bulk semi-conductors was discovered by a French physicist George Lampel in 1968. In quantumwells, this has been extensively studied since the 1980s by many groups. For good re-

SPIN RELAXATION IN SEMICONDUCTORS 347

Georges Uhlenbeck (1900–1988), H. A. Kramers and S. A. Goudsmit (1902–1978) on the left. WolfgangPauli (1900–1958) on the right.

Uhlenbeck is mainly known for his introduction (with Goudsmit) of the concept of spin of the elec-tron, although he favoured and was most active in statistical physics, where lies his work on quantum kinetics(see on page 306). Known to prefer rigour to originality, one of his student comments about him that “He feltthat something really original one did only once—like the electron-spin—the rest of one’s time one spent onclarifying the basics.”

Pauli’s first talk in Sommerfeld’s “Wednesday Colloquium” impressed the venerable professor so much thathe entrusted the young student to prepare an article on relativity for the “Encyklopadie der mathematischenWissenschaften.” He produced a 237-pages article that impressed Einstein himself who reviewed the work as“grandly conceived.” He remained widely praised for the mastery and clarity of his expositions, with his tworeview articles on quantum mechanics in the Handbuch der Physik known as the “Old and New Testament.”Oppenheimer called the later “the only adult introduction to quantum mechanics.” However, he publishedlittle as compared to his actual scientific production, especially as he aimed for a thorough understanding.He onced commented “The fact that the author thinks slowly is not serious, but the fact that he publishesfaster than he thinks is inexcusable.” Many of his own results were confined to private correspondence, likethe equivalence of Heisenberg and Schrodinger pictures (in a letter to Jordan), the time–energy uncertaintyrelation (to Heisenberg), or the neutrino to rescue energy conservation in radioactivity. He almost never caredabout recognition. He developed the “Ausschliessungsprinzip” or exclusion principle in 1924, already knownas the Pauli principle when it earned him the Nobel prize in 1945. Despite a strong opposition to the initialidea of the spin of the electron (which Kronig never published after his idea was ridiculed by Pauli), heformalized the concept of spin following the young theory of Heisenberg, culminating with Pauli matrices.He later laid the foundations of quantum field theory with the spin-statistics theorem, linking bosons andfermions to integer and half-integer spins, respectively. After his first marriage to a dancer broke down in lessthan a year and when his mother committed suicide, he suffered a deep personal crisis and started to drink.He consulted the psychologist Carl Jung (who first delegated a female assistant thinking that Pauli’s problemwere linked to women) for whom he detailed over a thousand dreams and established a deep relationship asa strong believer in psychology. He wrote that “in the science of the future reality will neither be ‘psychic’nor ‘physical’ but somehow both and somehow neither.” Numerous anecdotes are in circulation about him,like the “Pauli effect” dooming an experiment if he was in its vicinity. He was also known for his severityand scathing comments, but also for his wit, illustrated by his famous remark “This isn’t right. This isn’teven wrong.” One of the most brilliant theorists of all times, Albert Einstein described him as his intellectualsuccessor during Pauli’s Nobel celebration in Princeton.


views we address the reader to the famous volume “Optical orientation” edited by Meierand Zakharchenia (1984).

Fig. 9.1: Polarization of optical transitions in zincblende semiconductor quantum wells. Dark, grey and lightgrey lines show σ+, σ− and linearly polarized transitions, respectively.

An exciton is formed by an electron and hole, i.e., by two fermions having projec-tions of the angular momenta on a given axis equal to Je

z = Sez = ± 1

2 for an electronin the conduction band with S-symmetry and Jh

z = Shz + Mh

z = ± 12 ,± 3

2 for a hole inthe valence band with P-symmetry (in zincblende semiconductor crystals). The stateswith Jh

z = ± 12 are formed if the spin projection of the hole Sh

z is antiparallel to the pro-jection of its mechanical momentum Mh

z . These states are called light holes. If the spinand mechanical momentum are parallel, the heavy holes with Jh

z = ± 32 are formed (see

Section 4.2). In bulk samples, at k = 0 the light- and heavy-hole states are degenerate.However, in quantum wells the quantum confinement in the direction of the structuregrowth axis lifts this degeneracy so that energy levels of the heavy holes lie typicallycloser to the bottom of the well than the light-hole levels (see Fig. 9.1).

The ground-state exciton is thus formed by an electron and a heavy hole. The to-tal exciton angular momentum108 J has projections ±1 and ±2 on the structure axis.Bearing in mind that the photon spin is 0 or ±1 and that the spin is conserved in theprocess of photoabsorbtion, excitons with spin projections ±2 cannot be optically ex-cited. These are spin-forbidden states that, since they are not coupled to light, are alsocalled dark states. This vivid terminology extends to states that couple to light (withspin projection ±1) calling them bright states. Since, in quantum microcavities, J = 2states are not coupled to the photonic modes, we shall neglect them in the followingconsideration but it should be borne in mind, however, that in some cases these dark

108In this text, as is common practice in the field, the total exciton angular momentum is referred to forconvenience as the exciton spin.

SPIN RELAXATION IN SEMICONDUCTORS 349

states still come into play. They can, for instance, be mixed with the bright states by aninplane magnetic field or be populated due to the polariton–polariton scattering.

Georges Lampel Professor at the Ecole Polytechnique, Paris and Boris P. Zakharchenia (1928–2005).

Lampel is the father of “optical orientation”, an important branch of crystal optics studying the opticallyinduced spin effects in solids. One of Lampel’s PhD students was Claude Weisbuch, who was the first toobserve the strong light–matter coupling in semiconductor microcavities. At present, Lampel and Weisbuchwork in the same laboratory of the famous Ecole Polytechnique in Paris.

Zakharchenia was a PhD Student of Evgenii Gross and participated in the early studies on the Wannier–Mottexcitons in CuO2. Later, he headed a laboratory of the Ioffe Institute in Saint Petersburg that published thepioneering works on what is now called “spintronics”. Zakharchenia is also known for his theory on thePushkin’s duel place.

The polarization of exciting light cannot be retained infinitely long by excitons.Sooner or later they lose polarization due to inevitable spin and dipole moment relax-ations. Excitons, being composed of electrons and holes, are subject to mechanisms ofspin-relaxation for free carriers in semiconductors. The main important ones being:

1. Elliot (1954)–Yaffet mechanism involving the mixing of the different spin wave-functions with k = 0 as a result of the kp interaction with other bands. In quantumwells this effect plays a major role in the spin relaxation of holes and can inducetransitions between the optically active and dark exciton states |+1〉 → |−2〉 and|−1〉 → |+2〉.

2. D’yakonov and Perel (1971b) mechanism caused by the spin-orbit interaction in-duced spin splitting of the conduction band in non-centrosymmetric crystals (likezincblende crystals) and asymmetric quantum wells (Dresselhaus and Rashbaterms, respectively) at k = 0. This mechanism is predominant for the electronsand also leads to transitions between the optically active and dark exciton states,|+1〉 → |+2〉 and |−1〉 → |−2〉.

3. The Bir–Aronov–Pikus (BAP) mechanism first published by Pikus and Bir (1971),involving the spin-flip exchange interaction of electrons and holes. For excitonsthe efficiency of this mechanism is enhanced, as the electron and the hole forma bound state. The exchange interaction consists of so-called “short-range” and


“long-range” parts of the Coulomb interaction109. The short-range part leads tothe coupling between heavy-hole (hhe) and light-hole excitons (lhe) and thus issuppressed in the quantum wells where the degeneracy of lhe and hhe is lifted.The long-range part leads to transitions within the optically active exciton dou-blet110.

4. Spin-flip scattering between carriers and magnetic ions in diluted magnetic semi-conductors. As an example, paramagnetic semiconductors containing Mn2+ ions(spin 5/2) allow for efficient spin relaxation of electrons, heavy and light holesflipping their spins with the magnetic ion spins.111

Vladimir Idelevich Perel (b. 1922) and Mikhail Igorevich Dyakonov (b. 1939) in 1976.

Dyakonov and Perel personalize the excellence of the Soviet theoretical physics. Their names are associatedwith numerous physical effects and theories, including the non-radiative recombination and spin-relaxationmodels, spin-Hall effect, among others. Their elegant and seemingly simple analytical models helped theunderstanding of extremely complex phenomena of modern solid-state physics. The famous “Tea seminar”created by Dyakonov and Perel still run at the Ioffe institue of Saint Petersburg every Thursday at 5 pm.

109For details see E.L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures, Alpha, Harrow(2005), pages 252–253.

110The ability of the long-range part of the exchange interaction to couple the exciton doublet can lead tothe inversion of the circular polarization in time-resolved polarization measurements, as shown by Kavokinet al. (2004a).

111The spin-flip scattering is discussed at length by P. A. Wolff, in Semiconductors and Semimetals, editedby J. K. Furdyna and J. Kossut, Diluted Magnetic Semiconductors, Vol. 25. (Academic Press, Boston, 1988).

MICROCAVITIES IN THE PRESENCE OF A MAGNETIC FIELD 351

In a key paper, Maialle et al. (1993) have shown that the third (BAP) mechanism ispredominant for the quantum-confined excitons in non-magnetic semiconductors. Thelong-range electron–hole interaction leads to the longitudinal-transverse splitting of ex-citon states, i.e., energy splitting between excitons having a dipole moment parallel andperpendicular to the wavevector. This splitting is responsible for rapid spin-relaxationof excitons in quantum wells. For description of exciton-polaritons in microcavities ithas a very important consequence: the dark states can be neglected, which allows usto consider an exciton as a two-level system and use the well-developed pseudospinformalism for its description. From the formal point of view the exciton can be thusdescribed by a 2×2 spin density matrix that is completely analogous to the spin densitymatrix for electrons.

Exercise 9.1 (∗) In 1924, British physicists Wood and Ellett (1924) reported an amaz-ing polarization effect: they measured the circular polarization degree of fluorescenceof the mercury vapour resonantly excited by a circularly polarized light. In their initialexperiments a high degree of circular polarization was observed, while it significantlydiminished in later experiments. They wrote: “It was then observed that the apparatuswas oriented in a different direction from that which obtained in the earlier work, andon turning the table on which everything was mounted through ninety degrees, bringingthe observation direction East and West, we at once obtained a much higher value ofthe polarization.” Explain this effect.

9.2 Microcavities in the presence of a magnetic field

A magnetic field strongly affects excitons in quantum wells, and thus it also affectsexciton-polaritons in microcavities. One can distinguish between three kinds of linearmagneto-exciton effects in cavities:

• The energies of electron and hole quantum-confined levels change as a functionof the magnetic field, following the so-called Landau fan diagram. As a result,the energies of electron–hole transitions increase by

δE = (l + 1/2)ωc , (9.1)

where the cyclotron frequency ωc = eB/(µc) depends on the reduced effectivemass of electron–hole motion µ = mem

‖h/(me + m

‖h) and magnetic field B. m

‖h

is the inplane hole mass, me is the electron effective mass and l = 0, 1, 2, · · · isthe Landau quantum index. Landau quantization of exciton energies takes placedue to the magnetic field effect on the orbital motion of electrons and holes. Thiseffect is polarization independent. In microcavities, Landau quantization resultsin the appearance of a fine structure of polariton eigenstates and gives the possi-bility of tuning of different Landau levels into resonance with the cavity mode, asobserved by Tignon et al. (1995).

• Zeeman splitting of the exciton resonance. Excitons with spins parallel or antipar-allel to the magnetic field have different energies. The splitting is given by:

∆E = µBgB, (9.2)


where µB ≈ 0.062 meV/T is the Bohr magneton and g is the exciton g-factorthat depends on the materials composing the quantum well and the magnetic-field orientation with respect to the QW. In the following we shall consider theso-called Faraday geometry, i.e. the magnetic field parallel to the wavevectorof incident light and, consequently, normal to the QW plane (normal-incidencecase). Note that g can be positive or negative; in most semiconductor materials itvaries between −2 and 2, and in GaAs/AlGaAs QWs it changes sign as the QWwidth changes. Exciton Zeeman splitting leads to an effect known as resonantFaraday rotation, i.e., rotation of the polarization plane of linearly polarized lightpassing through a QW in the vicinity of the exciton resonance. In microcavities,this effect is strongly amplified due to the fact that light makes a series of roundtrips, each time crossing the QW before escaping from the cavity. This is to bediscussed in detail below.

• An increase of the exciton binding energy and oscillator strength due to theshrinkage of the exciton wavefunction in a magnetic field. This effect is importantfor strong enough magnetic fields, for which the magnetic length L =

√/(eB)

is comparable to the exciton Bohr diameter (typically about 100 A). An increaseof the exciton oscillator strength in a magnetic field enhances the vacuum-fieldRabi splitting.

Figure 9.2 shows the relative exciton radiative broadening, Rabi splitting and periodof Rabi oscillations (i.e. oscillations in time-resolved reflection due to beats betweentwo exciton-polariton modes in the cavity) measured experimentally and calculated witha variational approach of Berger et al. (1996). One can see that the oscillator strengthincreases by about 80% with the magnetic field changing from 0 to 12 T, which resultsin the Rabi splitting increasing by more than 30% and a corresponding decrease in theperiod of Rabi oscillations. Note that due to the exciton diamagnetic shift δE − EB

(with δE given by eqn (9.1) and EB being the exciton binding energy, the energy ofthe polariton ground-state shifts up. This shift is parabolically dependent on the mag-netic field at low fields. On the other hand, due to the increase of the exciton oscillatorstrength and resulting increase of the Rabi splitting, the polariton ground-state energyis pushed down. In realistic cavities, the latter effect dominates at low fields and theground-state energy shifts down, while in the limit of strong fields, when the magneticlengths become comparable to the exciton Bohr radius, the ground state is expectedto start moving up. Moreover, due to the increase of the Rabi splitting, the exciton-polariton dispersion gets slightly steeper near the ground-state, so that the polaritoneffective mass decreases with the increase of the magnetic field. The variational calcu-lation of the change of the Rabi splitting in presence of a magnetic field applied to amicrocavity is given in Appendix D.

9.3 Resonant Faraday rotation

Faraday rotation is rotation of the electric field vector of a linearly polarized light wavepropagating in a medium in the presence of a magnetic field oriented along the lightpropagation direction (see Chapter 2). Consider the propagation of linearly (X) polar-ized light, whose initial polarization is described by a Jones vector

RESONANT FARADAY ROTATION 353

Fig. 9.2: Relative exciton radiative broadening, Rabi splitting and period of Rabi oscillations, i.e., oscilla-tions in time-resolved reflection due to beats between two exciton-polariton modes in the cavity, measuredexperimentally from a GaAs-based microcavity with InGaAs/GaAs QWs (points), and calculated (lines), fromBerger et al. (1996). (

10

)=

1

2

(1−i

)+

1

2

(1−i

), (9.3)

here the upper and lower components of the vectors correspond to the electric-fieldprojections in the x- and y-directions, respectively, and the two terms on the right-handside describe σ+ (right-circular) and σ− (left-circular) polarized waves, respectively. Ifthe structure is placed in a magnetic field, the transmission coefficients for σ+ and σ−

polarized waves become different, so that the transmitted light can be represented as

t =1

2A exp(iϕ+)

(1i

)+

1

2C exp(iϕ−)

(1−i

), (9.4)


where the amplitudes A and C and phases ϕ+ and ϕ− coincide in the absence of thefield, but may be different in the presence of the field. t is the polarization vector ofelliptically polarized light, which can be conveniently represented as a sum of waveslinearly polarized along the main axes of the ellipse:

t =i

2(B − C) exp(iψ)

(sin φ− cos φ

)+

1

2(A + C) exp(iψ)

(cos φsinφ

), (9.5)

where ψ = (ϕ+ + ϕ−)/2 and φ = (ϕ− − ϕ+)/2 is the Faraday rotation angle.The transmission coefficient of the structure for light detected in X-polarization is

Tx =1

4|A exp(iϕ+) + C exp(iϕ−)|2 , (9.6)

and in Y -polarization it is

Ty =1

4|A exp(iϕ+)− C exp(iϕ−)|2 (9.7)

In σ+ and σ− polarization the amplitude of light transmitted across the quantumwell at the exciton resonance frequency is given by

tσ+,σ− = 1 +iΓ0

ωσ+,σ−

0 − ω − i(γ + Γ0), (9.8)

where ωσ+,σ−

0 is the exciton resonance frequency in the two circular polarizations,whose splitting in a magnetic field is referred to as exciton Zeeman splitting, Γ0 isthe exciton radiative decay rate and γ is the exciton non-radiative decay rate.

Hereafter, we shall neglect the exciton inhomogeneous broadening. The polarizationplane of linearly polarized light passing through the QW rotates by the angle

φ =1

2

[arctan

(ω−0 − ω)Γ0

(ω−0 − ω)2 + (γ + Γ0)2

− arctan(ω+

0 − ω)Γ0

(ω+0 − ω)2 + (γ + Γ0)2

],

≈ (ωσ−

0 ωσ+

0 )Γ0

(γ + Γ0)2. (9.9)

In the case of a microcavity, the Faraday rotation can be greatly amplified. Let us firstanalyse the expected effects in the framework of ray optics. Consider a cavity-polaritonmode as a ray of light travelling backwards and forwards inside the cavity within itslifetime. At the anticrossing point of the exciton and photon modes the lifetime of cavitypolaritons τ is

τ =1

2(κ + γ), (9.10)

where κ is the cavity decay rate, dependent on the reflectivity of the Bragg mirrors. Theaverage number of round trips of light inside the cavity is

N =τc

2ncLc=

Q

2, (9.11)

where nc is the cavity refractive index and Lc is its length, Q is the quality factor of thecavity (see Chapter 2). In standard-quality GaAs-based microcavities this factor reaches

SPIN RELAXATION OF EXCITON-POLARITONS: EXPERIMENT 355

a few hundred. While circulating between the mirrors the light accumulates a rotation,before escaping the cavity. The amplitude of the emitted light can be found as

E = tB + tBrBeiφ + tBr2Be2iφ + · · · = tB

1− rBeiφ, (9.12)

where rB and tB are the amplitude reflection and transmission coefficients of the Braggmirror, respectively. The angle of the resulting rotation of the linear polarization is

θ = arg(E) = arctanrB sin φ

1− rB cos φ. (9.13)

Note that this consideration neglects reflection of light by a QW exciton, since theamplitude of the QW reflection coefficient is more than an order of magnitude smallerthan the transmission amplitude.

To observe a giant Faraday rotation a peculiar experimental configuration is needed.In the reflection geometry only a very small rotation of the polarization plane can be ob-served (Kerr effect). Actually, the reflection signal is dominated by a surface reflectionfrom the upper Bragg mirror that does not experience any polarization rotation. To ob-serve the giant effect predicted by eqn (9.13), either the measurement should be carriedout in the transmission geometry, which would imply etching any absorbing substrate,or a pump-probe technique should be used to introduce the light into the cavity at anoblique angle and then to probe emission at the normal angle. The Faraday rotationdescribed above is a linear optical effect induced by an external magnetic field appliedto the cavity. There exists also an optically induced Faraday rotation in microcavitiesthat is a nonlinear effect having a strong influence on the polarization of emission of thecavities. It will be considered in detail below.

Exercise 9.2 (∗∗∗) Describe the resonant Faraday rotation of TE-polarized light inci-dent on a QW at oblique angle. A magnetic field is oriented normally to the QW plane.

9.4 Spin relaxation of exciton-polaritons in microcavities: experiment

The spin dynamics of cavity polaritons has been experimentally studied by measure-ment of time-resolved polarization from quantum microcavities in the strong-couplingregime. At the beginning of the twenty-first century, a series of experimental works ap-peared in this field, which reported unexpected results. Let us briefly summarize themost important of them.

In the experimental work by Martin et al. (2002), the dynamics of the circular po-larization degree c of the photoemission from the ground-state of a CdTe-based micro-cavity was measured. c was determined as

c =I+ − I−

I+ + I−=

N+k=0(t)−N−

k=0(t)

Nk=0(t), (9.14)

where I± denotes the measured circularly polarized intensities, N±k=0 represent the pop-

ulation of the ground-state (k = 0) with polaritons having spin projections on the struc-ture axis equal to±1, respectively, Nk=0(t) = N+

k=0(t)+N−k=0(t). The pump pulse was


circularly polarized and centred on the energy above the stop-band of the Bragg mirrorscomposing the microcavity (non-resonant excitation). In the linear regime, when thestimulated scattering of the exciton-polaritons did not play any role, an exponential de-cay of the circular polarization degree of photoemission was observed. However, abovethe stimulation threshold, ρc exhibited a non-monotonic temporal dependence. At pos-itive detuning, the initial polarization of ≈ 30% first increased up to ≈ 90% and thenshowed damped oscillations (see Fig. 9.3). For negative detuning, the polarization de-gree started from a positive value ≈ 50%, rapidly decreased down to strongly negativevalues and then increased showing attenuated oscillations with a period of about 50 ps.As we show in the next section this effect is caused by the TE–TM splitting of polaritonstates with k = 0.

Fig. 9.3: Experimentally measured temporal evolution of the photoluminescence of a CdTe-based microcavityexcited by circularly polarized light at the positive detuning, upper polariton branch (a, δ = 10 meV) andnegative detuning, lower polariton branch (b, δ = −10 meV). The filled circles/solid line (open circles/dashedline) denote the σ+ (σ−) emission. The deduced time evolution of the circular polarization degree for positiveand negative detunings is shown in (c) and (d), respectively. The inset shows the maximum value of thepolarization degree at 20 ps in the negative detuning case. From Martin et al. (2002).

In the experiments carried out by the group of Y. Yamamoto and published by She-lykh et al. (2004), the microcavity was pumped resonantly at an oblique angle (≈50).


The dependence of the intensity and of the polarization of light emitted by the ground-state on pumping polarization and power was studied, with all experiments carried outin the continuous-pumping regime. The pump intensity corresponding to the stimula-tion threshold appeared to be almost twice as high in the case of circular pumping asin the case of linear pumping as shown in Figs. 9.4(a) and (b). Figure 9.4(c) shows thedependence of the circular polarization degree of the emitted light on the pumping in-tensity for different pumping polarizations. For both polarizations, c ≈ 0 is observedfar below the threshold and c ≈ 0.9 near threshold. This indicates that spin relax-ation is complete at low excitation density, and that stimulated scattering into a definitespin component (say, spin-up) of the ground-state takes place at high densities. Abovethreshold, c remains large in the case of a circular pump. In the case of a linear pump,however, c decreases sharply at high pumping intensities. This effect was interpretedin terms of the interplay between ultrafast scattering and spin-relaxation of exciton-polaritons due to TE–TM splitting (see also below).

The polarization dynamics of polariton parametric amplifiers was the focus of ex-perimental research carried out by several groups. The parametric amplifier is realizedif polaritons are created by resonant optical pumping close to the inflection point ofthe lower dispersion branch at the “magic angle” (see Chapter 7). In this configurationthe resonant scattering of two polaritons excited by the pump pulse toward the signal(k = 0) and the idler state is the dominant relaxation process. The scattering can bestimulated by an additional probe pulse used to create the seed of polaritons in the sig-nal state, or it can be strong enough to be self-stimulated.

In the experiments carried out by Lagoudakis et al. (2002), the polarization of theprobe pulse was kept right-circular, whereas the pump polarization was changed fromright- to left-circular passing through elliptical and linear polarization. Consequently,the spin-up and spin-down populations of the pump-injected polaritons were variedwhile the pump intensity was kept constant. The two circular components of polariza-tion of light emitted by the ground-state and four inplane linear components (vertical,horizontal and the two diagonal ones) were detected as functions of the circular polariza-tion degree of the excitation. To briefly summarize the results of these measurements—shown in Fig. 9.5(a), (b) and (c): in the case of a linearly polarized pump pulse andcircularly polarized probe pulse the observed signal was linearly polarized, but with aplane of polarization rotated by 45 degrees with respect to the pump polarization. Inthe case of elliptically polarized pump pulses the signal also became elliptical, whilethe direction of the main axis of the ellipse rotated as a function of the circular polar-ization degree of the pump. In the case of a purely circular pump, the polarization ofthe signal was also circular, but its intensity was half that found for a linear pump. Thepolarization of the idler emission emerging at roughly twice the magic angle showed asimilar behaviour, although in the case of a linearly polarized pump the idler polariza-tion was rotated by 90 degrees with respect to the pump polarization. As we show below,the rotation of the polarization plane in this experiment is a manifestation of the opti-cally induced Faraday effect: the imbalance of populations of spin-up and spin-downpolariton states produces a spin-splitting of the polariton eigenstates. This imbalancehas been introduced by the elliptically polarized pump. The whole effect, referred to as


Fig. 9.4: a) Emission from k‖ ≈ 0 polaritons vs. pump power under the circular pump. The two circular-polarization components of the emission and their total intensity are plotted. b) Same as a) but for the linearpump. c) Circular polarization degree c vs. pump power with circular (triangles) and linear (circles) pumps.From Shelykh et al. (2004).

“self-induced Larmor precession” will be discussed in the next section.The polarization dynamics in a parametric oscillator without a probe has been con-

sidered in the experimental work by the Toulouse group (Renucci et al. 2005b). Quitesurprisingly, it was observed that for a linear pump, the polarization of the signal is alsolinear, but rotated by 90 (see Fig. 9.6). This effect is connected with the anisotropy ofthe spin-dependent polariton–polariton scattering.

An interesting polarization effect in a microcavity was observed in 2002 by Wolf-gang Langbein (Fig. 9.7) who illuminated a microcavity with a spot of linearly polarizedlight (say, along the x-axis) and detected spatially and temporally resolved emissionfrom the sample in co- and crosslinear polarizations. In the crosspolarization he ob-served a characteristic cross (Langbein cross), showing that the conversion of co- to


Fig. 9.5: Experimental (a)–(c) and theoretical intensities of circularly (a)–(d) and linearly (b), (c), (e) and (f),polarized components of the light emitted by a microcavity ground-state as a function of the circular polar-ization degree of the pumping light. From Kavokin et al. (2003).


Fig. 9.6: Experimentally measured degree of linear polarization of emission from the ground exciton-polariton state in a microcavity measured under linearly polarized pumping at the magic angle. Differentcurves correspond to different pumping intensities. One can see that the linear polarization degree quicklybecomes negative, which corresponds to 90 of rotation of the polarization plane with respect to the pumpinglight polarization. From Renucci et al. (2005a).

crosslinear polarization is efficient within the quarters of the reciprocal space separatedby the x- and y-axes, but not in the vicinity of the axes. These beautiful images appeardue to precession of the pseudospin of propagating exciton-polaritons in the effectivemagnetic field induced by TE–TM splitting (see next section).

9.5 Spin relaxation of exciton-polaritons in microcavities: theory

As we mentioned in Section 9.1, the main mechanisms for spin relaxation of exciton-polaritons in the linear regime are the transitions within the optically active doubletdue to the longitudinal transverse TE–TM splitting of exciton-polaritons. Consequently,the dark states can be neglected in most cases and an exciton-polariton state with agiven inplane wavevector k can be treated as a two-level system. It can be described bythe 2 × 2 density matrix ρk that is completely analogous to the spin density matrix ofan electron.

It is convenient to decompose the polariton pseudospin density matrix as:

ρk =Nk

21 + Sk ·σσσ , (9.15)

where 1 is the identity matrix, σσσ is the Pauli-matrix vector,112 Sk is the mean pseu-dospin of the polariton state characterized by the wavevector k. It describes both theexciton spin state and its dipole moment orientation (see Fig. 9.8).

The pseudospin is the quantum analogue of the Stokes vector (see Section 2.2): themean value of the pseudospin operator coincides with the Stokes vector of partially po-larized light in our notation. Note that here and further we use the basis of circularly

112The Pauli matrix vector is a vector σσσ = (σx, σy , σz) whose components σi are matrices, namely thePauli matrices (3.12) on page 81.

SPIN RELAXATION OF EXCITON-POLARITONS IN MICROCAVITIES: THEORY 361

Fig. 9.7: Spatial imaging of polariton propagation (linear greyscale, all images from W. Langbein in Pro-ceedings of 26th International Conference on Physics of Semiconductors, (Institute of Physics, Bristol, 2003),p.112.). Top: colinearly polarized polaritons. Bottom: crosslinearly polarized polaritons.

polarized states, i.e., associate the states having definite Sz with the polariton radiativestates with their spin projection on the structure axis equal to ±1. Their linear com-binations correspond to eigenstates of Sx and Sy yielding linearly polarized emission.The pseudospin parallel to the x-axis corresponds to X-polarized light, the pseudospinantiparallel to the x-axis corresponds to Y -polarized light and the pseudospin orientedalong the y-axis describes diagonal linear polarizations.

For non-interacting polaritons the temporal evolution of each density matrix (9.15)is governed by its Liouville–von Neumann equation (3.30)

i∂tρk = [Hk, ρk] , (9.16)

where the Hamiltonian Hk reads in terms of the pseudospin:

Hk = En(k)− gµBBeff,k ·Sk . (9.17)

Here, En(k) is the energy of the nth polariton branch, g is the effective polariton g-factor, µB is the Bohr magneton and Beff,k is an effective magnetic field. Unlike realmagnetic fields the effective field only applies to the radiatively active doublet and doesnot mix optically active and dark states. We do not consider here the effects of a realmagnetic field on the polariton spin dynamics.


Fig. 9.8: Poincare sphere with pseudospin (identical to the Stokes vector in this case, cf. Fig. 2.2). Theequator of the sphere corresponds to different linear polarizations, while the poles correspond to two circularpolarizations.

The x- and y-components of Beff,k are nonzero if the exciton states whose dipolemoments are oriented in, say, x- and y-directions have different energies. This alwayshappens for excitons having nonzero inplane wavevectors. The splitting of exciton stateswith dipole moments parallel and perpendicular to the exciton inplane wavevector iscalled longitudinal-transverse splitting (LT-splitting, see Chapter 4 for the bulk polari-ton LT-splitting). The longitudinal-transverse splitting of excitons in quantum wells isa result of the long-range exchange interaction. It can be described by the reduced spinHamiltonian of Pikus and Bir (1971):

Hex =3

16

|Φex(0)||Φbulk

ex (0)|2 ωLTf(k)

k

[k2 (kx − iky)2

(kx + iky) k2

], (9.18)

where ωLT is the longitudinal-transverse splitting of a bulk exciton (see Chapter 4)and Φex(0), Φbulk

ex (0) are bulk and QW exciton envelope functions, respectively, takenwith equal electron and hole coordinates. The form-factor f(k) is given by

f(k) =

∫∫Ue(z)Uhh(z)e−k|z−z′|Ue(z

′)Uhh(z′) dzdz′ , (9.19)

with Ue(z), Uhh(z) being electron and heavy-hole envelope functions normal to theQW plane direction. Off-diagonal terms of the Hamiltonian (9.17) lead to polaritonspin flips and thus create an effective inplane magnetic field Beff,k referred to belowas the Maialle field. If no other fields are present Beff,k = BLT,k. The Maialle fieldis in general not parallel to k but makes with the x-axis twice the angle as k. Theeffective magnetic field is zero for k = 0 and increases as a function of k followinga square root law at large k. For more details on the longitudinal-transverse splitting

SPIN RELAXATION OF EXCITON-POLARITONS IN MICROCAVITIES: THEORY 363

the reader can refer to the work of Tassone et al. (1992). In microcavities, splitting oflongitudinal and transverse polariton states is amplified due to the exciton coupling withthe cavity mode. Note that the cavity mode frequency is also split in TE- and TM- lightpolarizations (see Chapter 2). The resulting polariton splitting strongly depends on thedetuning between the cavity mode and the exciton resonance and, in general, dependsnon-monotically on k. Figure 9.9 shows the TE–TM polariton splitting calculated for aCdTe-based microcavity sample for different detunings between the bare cavity modeand the exciton resonance. For these calculations, polariton eigenfrequencies in twolinear polarizations have been found numerically by the transfer matrix method. Onecan see that the splitting is very sensitive to the detuning, and may achieve 1 meV,which exceeds by an order of magnitude the bare-exciton LT splitting.

Fig. 9.9: Longitudinal-transverse polariton splitting calculated for detunings: +10 meV, 0 meV, −10 meV,−19 meV. Solid lines: lower polariton branch, dashed lines: upper-polariton branch. From Kavokin et al.(2004b).

Using the pseudospin representation, one can rewrite the Liouville–von Neumannequation for the density matrix as

∂tSk =gµB

Sk ×Beff,k . (9.20)

The Maialle field thus induces precession of the pseudospin of the ensemble ofcircularly polarized polaritons and can cause oscillations of the circular polarization de-gree of the emitted light in time as was observed experimentally by Martin et al. (2002)(Fig. 9.3) and conversion from linear to circular polarization observed by the Yamamotogroup (Fig. 9.4). In the latter case, the pseudospin of exciton-polaritons excited at theoblique angle rotates about the Maialle field, while in the ground state (k = 0) the ro-tation vanishes. Thus, linear to circular polarization conversion depends on the ratio ofthe rotation period at the oblique angle and energy relaxation time of exciton-polaritons,i.e., the time needed for a polariton to relax to the ground-state. At low pumping inten-sity, the relaxation time is longer than the rotation period, at some intermediate pumpingthe two times coincide and at high pumping the relaxation time becomes shorter than


the period of pseudospin rotation. This explains the non-monotonic dependence of thecircular-polarization degree of emission on the pumping intensity in the linear pumpingcase (Fig. 9.4).

The precession of the polariton pseudospin about the Maialle field resembles theprecession of electron spins about the Rashba effective magnetic field.113 The physi-cal origin of these two fields is, however, different. The Rashba field is created by thespin-orbit interaction in asymmetric quantum wells, whereas the Maialle field appearsbecause of the long-range electron–hole interaction and TE–TM splitting of the cavitymodes. The orientation of these two fields is also different. The Rashba field is perpen-dicular to the electron wavevector, whereas the Maialle field is neither perpendicularnor parallel to the exciton (polariton) wavevector, in general.

9.6 Optical spin Hall effect

In high-quality microcavities, exciton-polaritons can propagate ballistically, i.e., with-out scattering, over a few picoseconds. This presents an opportunity of observing thepseudospin precession of individual exciton-polaritons under the effect of the Maiallefield. The pseudospin precession in the ballistic regime leads to the optical spin Halleffect detailed by Kavokin et al. (2005).

The spin Hall effect is the appearance of a spin flux due to the direct current flow in asemiconductor. This effect predicted by D’yakonov and Perel (1971a) found its experi-mental proof only recently in the work of Kato et al. (2004). It has a remarkable analogyin semiconductor optics, namely, in Rayleigh scattering of light in microcavities. Thespin polarization in a scattered state can be positive or negative, dependent on the ori-entation of the linear polarization of the initial state and on the angle of rotation of thepolariton wavevector during the act of scattering. Very surprisingly, spin polarizationsof the polaritons scattered clockwise and anticlockwise have different signs.

Consider a semiconductor microcavity in the strong-coupling regime. We supposethat one of the k = 0 states of the lower-polariton dispersion branch is resonantly ex-cited by linearly polarized light (here, k = (kx, ky) is the inplane polariton wavevector).The Rayleigh-scattered signal comes from the quantum states whose wavevector is ro-tated with respect to the initial k by some nonzero angle θ. We study polarization ofthe scattered light as a function of θ and k taking into account the pseudospin rotationunder the effect of the Maialle field (9.20).

Let us assume that light is incident in the (x, z)-plane, and is polarized along thex-axis (TM-polarization). It excites the polariton state with a pseudospin parallel tothe x-axis. As the pseudospin is parallel to the effective field, it does not experienceany precession at the initial point (see Fig. 9.10(b)). Consider now the scattering actthat brings our polariton into the state (k′

x, k′y) with k′

x = k cos θ and k′y = k sin θ.

Following the classical theory of Rayleigh scattering, we assume that the polarizationdoes not change during the scattering act, so that at the beginning the pseudospin ofthe scattered state remains oriented in the x-direction (Fig. 9.10(b)). As the effective

113The Rashba effect predicted in 1984 (Y.A. Bychkov, E.I. Rashba, JETP Lett., 39, 78 (1984)) has stimu-lated development of spintronics, a new area of semiconductor physics studying propagation of spin-polarizedelectrons.

OPTICAL SPIN HALL EFFECT 365

Fig. 9.10: (a) experimental configuration allowing for observation of the optical spin Hall effect: linearlypolarized light is incident at an oblique angle, circular polarization of the scattered light is analysed, (b) ori-entations of the effective magnetic field induced by the LT splitting of the polaritons for different orientationsof the inplane wavevector, (c) pseudospin of polaritons created by the TM-polarized pump pulse at the pointshown by the large circle, arrows show the pseudospins of polaritons just after the Rayleigh-scattering act andthe effective field orientation.

field is no longer parallel to the pseudospin, it starts precessing. Due to precession, thepseudospin acquires a z-component and circular polarization emerges. The polarizationof emission depends on the ratio between the period of precession (given by the TE–TMsplitting of the polariton state) and the polariton lifetime. If both quantities are of thesame order, a peculiar angle dependence of the circular polarization of scattered lightappears.

Figure 9.11 shows schematically the dependence of the circular polarization degreeof light scattered by the cavity in different directions. Dark areas correspond to the right(left) circularly polarized light. One can notice an inequivalence of clockwise and anti-clockwise scattering: if the spin-up majority of polaritons (right-circular polarization)dominate scattering at the angle θ, the signal at angle−θ is mostly emitted by spin-down


Fig. 9.11: Schematic diagram of the angular dependence of circular polarization of emission of the micro-cavity. Dark areas correspond to the right-(left-) circularly polarized light.

polaritons (left-circular polarization), and vice versa. In order to obtain the polarizationdistribution in scattering of TE-polarized light (incident electric field in the yz-plane)one should simply interchange areas in Fig. 9.11.

Rotation of the polariton pseudospin around the Maialle field is responsible for theappearance of the Langbein cross (Fig. 9.7). In his experiment, in x-, −x-, y−, and−y-directions, X-polarized light corresponds to one of the eigenstates of the system(TE- or TM-polarized). The effective field in these points is parallel (antiparallel) to thepseudospin of the exciting light, thus no precession takes place and the polarization isconserved. On the other hand, within the quarters, the effective field is inclined at someangle to the pseudospin, so that the precession takes place, and the emission signal incrosslinear polarization appears.

9.7 Optically induced Faraday rotation

The z-component of Beff,k splits J = +1 and J = −1 exciton states. It would be zeroin the absence of polariton–polariton interactions. However, in the nonlinear regime itcan arise due to the difference of concentrations of spin-up and spin-down polaritons,which leads to the optically induced Faraday rotation in microcavities. To show this, letus first consider the connection of the pseudospin formalism with the second quantiza-tion representation.

If the polariton concentration is lower than the saturation density, the polaritonsbehave as good bosons, and thus a pair of bosonic annihilation operators ak,↑, ak,↓can be introduced to describe the polariton quantum states having a wavevector k. Theoccupation numbers of spin-up, spin-down polariton states and the z-component of thecorresponding pseudospin can be found from

OPTICALLY INDUCED FARADAY ROTATION 367

Nk↑ = Tr[ρka†

k,↑ak,↑]

= 〈a†k,↑ak,↑〉 , (9.21a)

Nk↓ = Tr[ρka†

k,↓ak,↓]

= 〈a†k,↓ak,↓〉 , (9.21b)

Sk,z =1

2

[〈a†k,↑ak,↑〉 − 〈a†

k,↓ak,↓〉] . (9.21c)

To find the dynamics of inplane-components of the pseudospin one should introducethe bosonic operators for linear-polarized polaritons as follows:

ak,x = 1√2

(ak,↑ + ak,↓

), ak,−x =

1√2

(ak,↑ − ak,↓

), (9.22a)

ak,y = 1√2

(ak,↑ + iak,↓

), ak,−y =

1√2

(ak,↑ − iak,↓

). (9.22b)

Knowing the dynamics of these operators, Sk,x and Sk,y are expressed as:

Sk,x =1

2

[〈a†

k,xak,x〉 − 〈a†k,−xak,−x〉

]= 〈a†

k,↓ak,↑〉 , (9.23a)

Sk,y =1

2

[〈a†

k,yak,y〉 − 〈a†k,−yak,−y〉

]= 〈a†

k,↓ak,↑〉 . (9.23b)

We shall consider the polariton pseudospin dynamics in the nonlinear regime. Let usstart from the simplest case where all the polaritons are in the same quantum state, i.e.,form a “condensate”, so that the scattering to the other states is completely suppressed.Of course, this is true only at zero temperature. The general form of the interactionHamiltonian reads

H = ε(a†↑a↑ + a†

↓a↓) + V1(a†↑a

†↑a↑a↑ + a†

↓a†↓a↓a↓) + 2V2(a

†↑a↑a

†↓a↓) (9.24)

where the index corresponding to the polariton state k in the reciprocal space is omit-ted.114 In eqn (9.24), ε is the free polariton energy, while the matrix elements V1 and V2

correspond to the forward scattering of the polaritons in the triplet configuration (paral-lel spins) and in the singlet configuration (antiparallel spins). If the polariton–polaritoninteractions were spin-isotropic, i.e., the Hamiltonian (9.24) were covariant with re-spect to the linear transformation of the operators (9.22a), the matrix elements wouldbe interdependent, so that

V1 = V2 . (9.25)

However, this situation is not realized in semiconductor microcavities, where the majorcontribution to polariton–polariton interaction is given by the exchange term, so that thepolariton–polariton interaction is in fact anisotropic (dependent on the mutual orienta-tion of spins of interacting polaritons) and |V1| |V2|. This anisotropy manifests itself

114The Hamiltonian without notation shortcut reads

H = ε(a†k,↑ak,↑+a†

k,↓ak,↓)+V1(a†k,↑a†

k,↑ak,↑ak,↑+a†k,↓a†

k,↓ak,↓ak,↓)+2V2(a†k,↑ak,↑a†

k,↓ak,↓) .


experimentally in the optically induced splitting of the spin-up and spin-down polaritonstates, which has been experimentally measured as a function of the circular polariza-tion degree of the excitation (directly linked to the imbalance of populations of spin-upand spin-down states).

The Hamiltonian (9.24) conserves N↑ and N↓ but not the inplane components ofthe pseudospin. It is straightforward to directly evaluate the commutator of this Hamil-tonian with the operators governing linear polarization. This commutator is zero only ifthe condition (9.25) is satisfied, which is not the case experimentally. The equation ofmotion for 〈a†

↓a↑〉 can be obtained from eqns (9.16) and (9.24) to read

∂t〈a†↓a↑〉 =

i

2(V1 − V2)

[〈a†

↓a↓a†↓a↑〉 − 〈a†

↑a↑a↑a↓〉]

. (9.26)

In the mean-field approximation, the fourth-order correlators in the right side ofeqn (9.26) can be decoupled and eqn (9.26) can be transformed into an equation of pre-cession of the pseudospin in an effective magnetic field Bint oriented along the structuregrowth axis. The absolute value of the field is determined by the difference between thepopulations of spin-up and spin-down polaritons:

gµB|Bint| = 2(V1 − V2)(N↓ −N↑) . (9.27)

Experimentally, the effect manifests itself as a rotation of the polarization plane of emis-sion if the σ+ and σ− populations are imbalanced, i.e., in the case of optically inducedFaraday rotation.

To summarize, in general the effective magnetic field acting on the polariton pseu-dospin has two components, Beff,k = BLT,k + Bint,k. The inplane component is gov-erned by the TE–TM splitting of the cavity modes and the long-range electron–hole in-teraction in the exciton. It is concentration independent and leads to the beats betweenthe circularly polarized components of the photoemission. The component parallel tothe structure growth axis arises because of the anisotropy of the polariton–polariton in-teraction and depends on the imbalance between spin-up and spin-down polaritons. Itleads to the Faraday rotation of the polarization plane of light propagating through thecavity.

9.8 Interplay between spin and energy relaxation of exciton-polaritons

The linear model developed above neglects the inelastic scattering of polaritons lead-ing to their relaxation in reciprocal space and their energy relaxation. Interpretation ofnonlinear polarization effects in microcavities requires its generalization. Formally, ki-netic equations for the occupation numbers and pseudospins can be decoupled only inthe linear regime. Thus, a theoretical description of nonlinear processes requires theself-consistent accounting of both energy and spin relaxation processes.

Our starting point is the Hamiltonian of the system in the interaction representation.

H = Hshift + Hscat . (9.28)

INTERPLAY BETWEEN SPIN AND ENERGY RELAXATION 369

Polariton interaction with acoustic phonons and polariton–polariton scattering are takeninto account here. Only the lower-polariton branch is considered, coupling with the up-per branch and dark exciton states are neglected. The “shift” term describes interactionof exciton-polaritons having the same momentum but possibly different spins:

Hshift =∑

k,σ=↑,↓

(gBµBBLTa†

σ,ka−σ,k + V(1)k,k,0(a

†σ,kaσ,k)2 + V

(2)k,k,0(a

†σ,ka−σ,k)2

)+

∑k,k′ =kσ=↑,↓

(V

(1)k,k′,0a

†σ,ka†

σ,k′aσ,kaσ,k′ + V(2)k,k′,0a

†σ,ka†

−σ,k′aσ,ka−σ,k′

). (9.29)

Here, a↑,k, a↓,k are annihilation operators of the spin-up and spin-down polaritons,BLT,k = BLT,k,x + iBLT,k,y (BLT,k,x and BLT,k,y are the x- and y-projections ofthe effective magnetic field). The “scattering term” Hscat describes scattering betweenstates with different momenta:

Hscat =1

4

∑k,k′ =kσ=↑,↓

exp( i

(E(k) + E(k′)− E(k + q)− E(k′ − q))t

)

× (V

(1)k,k′,qa†

σ,k+qa†σ,k′−qaσ,kaσ,k′ + 2V

(2)k,k′,qa†

σ,k+qa†−σ,k′−qaσ,ka−σ,k′ + h.c.

)

+1

2

∑k,q =0σ=↑,↓

exp( i

(E(k) + ωq − E(k + q))

)Uk,qa†

σ,k+qaσ,kbq + h.c. (9.30)

In eqn (9.30), bq is an acoustic phonon operator, Uk,q is the polariton–phononcoupling constant, E(k) is the dispersion of the low polariton branch. The matrix ele-ments V

(1)k,k′,q and V

(2)k,k′,q describe scattering of two polaritons in the triplet and the sin-

glet configurations, respectively. As has been discussed above, in real microcavities thepolariton–polariton interaction is strongly anisotropic: the triplet scattering is usuallymuch stronger than the singlet one. Moreover, the matrix elements V

(1)k,k′,q and V

(2)k,k′,q

can have opposite signs, as shown by Kavokin et al. (2005). Indeed, the interactionbetween two polaritons with parallel spins is always repulsive, while polaritons withopposite spins are characterized by an attractive interaction and can even form a boundstate (bipolariton), as discussed by Ivanov et al. (2004).

We use the Hamiltonian (9.28) to write the Liouville equation for the total densitymatrix of the system:

idρ

dt= [H(t), ρ(t)] = [Hshift + Hscat(t), ρ(t)] . (9.31)

We solve eqn (9.31) within the Born–Markov approximation already used in previ-ous chapters. The Markov approximation means physically that the system is assumedto have no phase memory. This is, in general, not true for the coherent processes de-scribed by the Hshift part of the Hamiltonian but is a reasonable approximation for the


scattering processes involving the momentum transfer. We apply the Markov approxi-mation to the scattering part of eqn (9.31), which reduces to:

dρ

dt= − i

[Hshift, ρ]− 1

2

∫ t

−∞[Hscat(t), [Hscat(τ), ρ(τ)]] dτ . (9.32)

The next step is to perform the Born approximation ρ = ρphon ⊗∏

k ρk. Thephonons are then traced out with their occupation numbers being treated as fixed param-eters determined by the temperature. The density matrices are given by eqn (9.15). Theycontain information about both occupation numbers and pseudospin components of allstates in the reciprocal space. Equations. (9.28–9.32) together with formulas (9.21a)for occupation numbers and pseudospins are sufficient to derive a closed set of dy-namics equations for N↑,k, N↓,k and Sk,⊥ = exSk,x + eySk,y, as has been done byShelykh et al. (2005). Maialle spin-relxation because of the TE–TM splitting and self-induced Larmor precession are reduced in this model to the precession of the polaritonpseudospin about an effective magnetic field Beff,k that arises from the Hamiltonianterm Hshift. Once polariton populations and pseudospins are known, the intensities ofthe circular and linear components of photoemission are given by:

I+k = N↑,k , (9.33a)

I−k = N↓,k , (9.33b)

Ixk =

N↑,k + N↓,k

2+ Sx,k , (9.33c)

Iyk =

N↑,k + N↓,k

2− Sx,k . (9.33d)

Note that light polarizations parallel to the x- and y-axes correspond to the pseudospinparallel and antiparallel to the x-axis, respectively.

The general form of the kinetic equations (9.28)–(9.32) is complicated and theirsolution requires heavy numerical calculations. However, some particular cases can beconsidered analytically. If the pump intensity is weak, the polariton–polariton scatteringis dominated by the scattering with acoustic phonons. The polariton–polariton interac-tion terms can be thus neglected and the system of kinetic equations becomes muchsimpler. For the occupation numbers and pseudospins we have in this case:

dNk

dt= − 1

τkNk +

∑k′

[(Wk→k′ −Wk′→k)(

1

2NkNk′ + 2Sk ·Sk′)

+ (Wk→k′Nk′ −Wk′→kNk)],

(9.34a)

dSk

dt= − 1

τskSk +

∑k′

[(Wk→k′ −Wk′→k)(NkSk′ + Nk′Sk)

+ (Wk→k′Sk′ −Wk′→kSk)]

+gsµB

[Sk ×BLT,k] ,

(9.34b)

INTERPLAY BETWEEN SPIN AND ENERGY RELAXATION 371

where the transition rates are

Wk→k′ =

⎧⎪⎨⎪⎩

2π

|Uk,k′−k|2nphon,k′−kδ(E(k′)− E(k)− ωk′−k)

2π

|Uk,k′−k|2(nphon,k′−k + 1)δ(E(k′)− E(k)− ωk′−k) .

(9.35)

nphon are acoustic phonon occupation numbers. The Dirac delta functions accountfor energy conservation during the scattering processes. Mathematically, they appearfrom the integration of the time-dependent exponents in the second term of eqn (9.32).Writing the energy-conserving delta functions in eqn (9.35) we assume that the polari-ton longitudinal transverse splitting does not modify strongly the polariton dispersioncurve. In numerical calculations, the delta functions may be replaced by resonant func-tions, e.g., Lorentzians, having a finite amplitude that can be estimated as an inverseenergy broadening of the polariton state. The polariton lifetime has been introduced ineqn (9.34) to take into account the radiative decay of polaritons. The pseudospin life-time is in general less than τk, it can be estimated as τ−1

sk = τ−1k + τ−1

sl , where τsl is thecharacteristic time of the spin-lattice relaxation (which accounts for all the processesof relaxation within the polariton spin doublet apart from one due to the LT splitting).The last term in eqn (9.34b) is the same as in eqn (9.20). It describes the pseudospinprecession about an effective inplane magnetic field given by the polariton LT splitting.This term is responsible for oscillations of the circular polarization degree of the emittedlight.

Qualitatively, the spin relaxation of exciton-polaritons can be understood from thefollowing arguments. Below the threshold, the spin system is in the collision-dominatedregime, i.e., relaxation of polaritons down to the ground-state goes through a huge num-ber of random passes each corresponding to the scattering process with an acousticphonon. The polarization degree displays a monotonic decay in this case. This is easyto understand, as in each scattering event the direction of the effective magnetic fieldchanges randomly, so that on average no oscillation can be observed. The situation iscompletely analogous to one for electrons undergoing spin relaxation while moving inan effective Rashba field. Spin relaxation of spin-up and spin-down polaritons proceedswith the same rate, in general.

The situation changes dramatically above the stimulation threshold. In this case,the relaxation rates for spin-up and spin-down polaritons become different, in general.Once the ground-state is populated preferentially by polaritons having a given spin ori-entation, the relaxation rate of polaritons having this spin orientation is enhanced. Thisstimulated scattering process makes the circular polarization degree of the emission in-crease in time, as was experimentally observed. Also, the polarization degree is foundto oscillate with a period sensitive to the pumping power and the detuning. The detuning(difference between bare exciton and cavity mode energies) has an important effect onthe polariton spin relaxation. First, the LT splitting at a given value of the wavevectorstrongly depends on the detuning, as one can see from Fig. 9.9. Also, the energy relax-ation is extremely sensitive to the detuning. At positive detuning, polaritons relax to theground-state with more random steps than at negative detuning.


Exercise 9.3 (∗∗) Polaritons linearly polarized in the y-direction propagate ballisti-cally under the effect of an effective magnetic field oriented (a) in the x-direction, (b)in the y-direction, (c) along the z-axis (normal to the cavity plane). Find the time-dependent intensity of light emitted by the cavity in X,Y and circular polarizations if thepolariton lifetime is τ .

9.9 Polarization of Bose condensates and polariton superfluidity

In this section we address an important issue of polarization of the condensates ofexciton-polaritons in microcavities. Of course, the condensate polarization is cruciallydependent on the dynamics of its formation, i.e., pumping polarization, polariton relax-ation mechanisms, etc. A realistic polariton system is always out of equilibrium, strictlyspeaking, just because the polaritons have a finite lifetime and the system should bepumped from outside to compensate the losses of polaritons due to the lifetime. Thethermodynamic limit where the polariton distribution is given by an equilibrium Bose–Einstein distribution function is hardly achievable in reality.115 Nevertheless, consider-ation of this limit is very important for understanding the fundamental effects of Bosecondensation of polaritons. The characteristics of polariton condensates out of equilib-rium deviate from the parameters predicted assuming thermal equilibrium, but the mainqualitative tendencies remain valid.

Here, we assume that the Bose condensation of exciton-polaritons has already takenplace and the polariton system is fully thermalized. The condensate is in a purely co-herent state that can be characterized by a wavefunction. The lifetime of polaritons isassumed infinite and there is no pumping. Moreover, in the first part of this Section weshall assume zero temperature. We take into account the spin structure of the conden-sate, however. An exciton-polariton can have a +1 or−1 spin projection on the structureaxis, thus we have a two-component (or spinor) Bose condensate. This is an importantpeculiarity of the polariton system with respect to various known 0-spin bosonic sys-tems (atoms, Cooper pairs, He4) and 1-spin systems (He3). We consider the heavy-holeexciton-polaritons polarized in the plane of the cavity. In this case the condensate canbe described by a spinor:

φφφ =

(φx

φy

), (9.36)

where φx, φy are complex functions of time and coordinates describing projections ofthe polarization of the condensate on two corresponding inplane axes. In this basis, thefree energy of the system can be represented as:

F = φφφ∗T (−i∇)φφφ−µ(φφφ ·φφφ∗)2 +1

4(V1 +V2)(φφφ ·φφφ∗)2− 1

4(V1−V2)|φφφ×φφφ∗|2 , (9.37)

where T (−i∇) is the kinetic energy operator, µ is the chemical potential, which cor-responds to the experimentally measurable blueshift of the photoluminescence line of

115Recent experiments of the group of Le Si Dang (Grenoble) show, however, that under sufficiently strongpumping the polariton gas quickly thermalizes, so that its distribution function becomes very close to theBose–Einstein function.

POLARIZATION OF BOSE CONDENSATES AND POLARITON SUPERFLUIDITY 373

the microcavity due to formation of the condensate, V1 and V2 are interaction constantsof polaritons with parallel and antiparallel spins, respectively. The two last terms ofeqn (9.37) describe contributions of polariton–polariton interactions to the free energy.It can be shown that no other terms of this order can exist in the free energy if the cavityis isotropic in the xy-plane.

At zero temperature, only the ground-state is occupied, so that substitutions can bemade in eqn (9.37):

n = φφφ ·φφφ∗ , (9.38a)

Sz =i

2|φφφ×φφφ∗| , (9.38b)

where n is the occupation number of the condensate, and Sz is the normal-to-planecomponent of the pseudospin of the condensate, which characterizes the imbalance ofpopulations of spin-up, n↑, and spin-down, n↓ polaritons:

n = n↑ + n↓ , (9.39a)

Sz =1

2(n↑ − n↓) , (9.39b)

and is related to the circular polarization degree of the emitted light ρ by:

ρ =2Sz

n. (9.40)

The free energy therefore reads:

F = −µn +1

4(V1 + V2)n

2 + (V1 − V2)S2z . (9.41)

In the experimentally studied semiconductor microcavities

V1 > 0 > V2 > −V1 . (9.42)

That is why, at zero magnetic field, the minimum of the free energy corresponds to Sz =0 (linearly polarized condensate). In perfectly isotropic microcavities, the orientation oflinear polarization would be chosen spontaneously by the system and would randomlychange from one experiment to another. In real structures it is generally pinned to oneof the crystallographic axes. The pinning of linear polarization can be caused by vari-ous factors, including the exchange splitting of the exciton state and the photon modesplitting at k = 0. The second scenario is the most likely one, as the slightest birefrin-gence in the cavity (say, inplane variation of the refractive index by 0.01%) would yieldthe splitting of the cavity mode by about 0.1 meV, which is quite sufficient for pinningof the condensate polarization. Here, and further, we shall assume the condensate isx-polarized in the absence of any magnetic field.


The chemical potential of the condensate can be obtained by minimization of thefree energy over n, which yields:

µ =V1 + V2

2n . (9.43)

As we mentioned above, this value corresponds to the blueshift of the photolumi-nescence line due to formation of the condensate. Such a blueshift, linearly dependenton the occupation number of the condensate, has been experimentally observed by thegroup of Le Si Dang (Fig. 9.12).

Fig. 9.12: Angle-dependent photoluminescence (PL) from a microcavity in the strong-coupling regime atdifferent pumping intensities (in units of the threshold pumping P0) (a, b, c) by Richard et al. (2005a). Onecan clearly see formation of the condensate at k = 0 point (c) as well as the blueshift of the ground-stateenergy by about 1 meV between (a) and (c).

9.10 Magnetic-field effect and superfluidity

We consider an exciton-polariton condensate in a microcavity subject to a magneticfield normal to the plane (B ‖ z). We neglect the field effect on electron–hole relativemotion and the resulting diamagnetic shift of the condensate and only consider the spinsplitting of exciton-polaritons resulting from the Zeeman effect. The diamagnetic shiftwould result in a parabolic dependence of the energy of the condensate on the amplitudeof the applied field, but would not influence the superfluidity and spin structure of thepolaritons.

The free energy of the system can be represented as:

FΩ = F − iΩ|φφφ×φφφ∗| , (9.44)

where F is the free energy in the absence of the magnetic field given by eqn (9.37), Ω =µBgB/2 with gB being the exciton-polariton g-factor, µB being the Bohr magneton, Bbeing the amplitude of the applied magnetic field.

MAGNETIC-FIELD EFFECT AND SUPERFLUIDITY 375

At zero temperature, one can use eqn (9.41) for F so that eqn (9.44) yields

FΩ = −µn− 2ΩSz +1

4(V1 + V2)n

2 + (V1 − V2)S2z . (9.45)

If a weak magnetic field is applied to the cavity (we shall define later what “weak”means in this case), the pseudospin projection and the number of polaritons n can still beconsidered as independent variables. In this regime, further referred to as the weak-fieldregime, the free-energy minimization over Sz yields

Sz =Ω

V1 − V2. (9.46)

The condensate in this case emits elliptically polarized light with

ρ =µBgB

(V1 − V2)n. (9.47)

Interestingly, in this regime the chemical potential is still given by eqn (9.43) as imme-diately follows from minimization of the free energy. Thus, the condensate emits lightat the same energy. The red shift of the polariton energy due to the Zeeman effect is ex-actly compensated by an increase of the blueshift due to polarization of the condensate.The minimum free energy of the system decreases quadratically with the field:

FΩ,min = − Ω2

V1 − V2− V1 + V2

4n2 . (9.48)

At the critical magnetic field

Bc =(V1 − V2)n

µBg, (9.49)

the condensate becomes fully circularly polarized (ρ = 1).Beyond this point, Sz and n are no longer independent parameters, and Sz = n/2.

Therefore, minimization of the free energy eqn (9.44) over n yields a different result:

µ = −Ω + V1n , (9.50)

FΩ,min = −V1

2n2 . (9.51)

In this regime, further referred to as the strong-field regime, the emission energy ofthe condensate decreases linearly with the field, so that the normal Zeeman effect canbe observed in photoluminescence. The magnetic susceptibility of the condensate isdiscontinuous at the critical field.

In order to reveal the superfluid properties of exciton-polaritons we consider theGross–Pitaevskii equation:

i∂φj

∂t=

∂FΩ

∂φ∗j

, (9.52)


Landau and his group in Moscow (1956).Back row: S.S. Gershtein, L.P. Pitaevskii, L.A. Vainshtein, R.G. Arkhipov, I.E. Dzyaloshinskii.Front row: L.A. Prozorova, A.A. Abrikosov, I.M. Khalatnikov, L.D. Landau, E.M. Lifshitz.

here the free energy F is given by eqn (9.37), j = x, y. The dynamics of the inplanecomponents of the polarization of the system is given by:

i∂φx

∂t= Txy(−i∇)φx − µφx + iΩφy +

1

2(V1 + V2)(φφφ ·φφφ∗)φx

+1

2(V1 − V2)|φφφ×φφφ∗|φy ,

(9.53a)

i∂φy

∂t= Tyx(−i∇)φy − µφy − iΩφx +

1

2(V1 + V2)(φφφ ·φφφ∗)φy

− 1

2(V1 − V2)|φφφ×φφφ∗|φx .

(9.53b)

The last terms in eqns (9.53) describe the self-induced Larmor precession of the polari-ton pseudospin that exactly compensates precession induced by the external magneticfield weaker than the critical field (9.49). This follows from eqns (9.38b) and (9.46).The exact compensation of the Larmor precession is one of the manifestations of thefull paramagnetic screening or spin Meissner effect in polariton condensates (discussedin detail below).

Following Lifshitz and Pitaevskii (1980, problem of §30) one can obtain the disper-sion of excited states of the system from eqns (9.53) by substitution:

φφφ(r, t) =√

ne + Aei(k · r−ωt) + C∗e−i(k · r−ωt) , (9.54)

MAGNETIC-FIELD EFFECT AND SUPERFLUIDITY 377

where

e =

⎧⎨⎩

x cos θ + iy sin θ , if B < Bc ,

1√2(x + iy) , if B > Bc ,

(9.55)

and θ = 12 arcsin

(Ω/[n(V1 − V2)]

).

Retaining only the terms linear in A and C, and separating the terms with differentcomplex exponential functions, one can obtain a system of four linear algebraic equa-tions for Ax, Ay , Cx and Cy . The condition for existence of a non-trivial solution of thissystem allows one to obtain the spectra of excited polariton states. At magnetic fieldsbelow Bc it reads:

ω2 = ω20 + nV1ω0 ± ω0

√(nV2)2 + (V 2

1 − V 22 )n2

Ω2

Ω2c

, (9.56)

where ω0(k) is the energy of the lower-branch polariton state as a function of the in-plane wavevector k in the absence of the condensate, ω0(k = 0) = 0, Ωc = µBgBc/2.One can see that both branches start at the same point at k = 0, which corresponds tothe bottom of the lowest polariton band blueshifted by µ (see eqn (9.43)). This showsthat the ground state of the system remains two-fold degenerate in the presence of theexternal magnetic field below Bc, i.e., the Zeeman effect is fully suppressed (see in-set in Fig. 9.12). This suppression results from the full paramagnetic screening of themagnetic field. The polarization plane of light going through the microcavity does notexperience any Faraday rotation in this case. The polariton spins orient along the field,so that the energy of the system decreases as in any paramagnetic material, but thisdecrease is exactly compensated by the increase of polariton–polariton interaction en-ergy. This effect can be considered as a spin analogue of the Meissner effect familiar insuperconductors.

Fig. 9.13: Sound velocities of two branches of the excitations of a polariton condensate in a microcavity asa function of applied magnetic field, as predicted by Rubo et al. (2006). The inset shows Zeeman splitting ofthe polariton ground state, µ0 = n(V1 + V2)/2.


Both dispersion branches described by eqn (9.54) have a linear part characteristic ofa superfluid. The peculiarity of our two-component superfluid consists in the differenceof the sound velocities for two branches:

v± =∂ω

∂k

∣∣∣∣k=0

=

√√√√ V1n

2m∗ ±1

2m∗

√(nV2)2 + (V 2

1 − V 22 )n2

Ω2

Ω2c

, (9.57)

where m∗ is the effective mass of the lowest polariton band. Note that at zero magneticfield the difference of two sound velocities persists, it is given by

√n/(2m∗)(

√V1 + V2

−√V1 − V2). The sound velocity of the branch polarized with the condensate increaseswith magnetic field and achieves

√V1n/m∗ at the critical field (Fig. 9.13). On the other

hand, the sound velocity of the other branch decreases with increasing field and vanishesat B = Bc.

At strong fields, the ground-state of the lowest polariton band is split into the right-circularly and left-circularly polarized doublet. The lowest (right-circular, if g > 0)branch remains superfluid. Its dispersion is given by:

ω2 = ω20 + 2nV1ω0 . (9.58)

The sound velocity is independent of the field and always equals√

V1n/m∗ in thisregime. On the contrary, the higher (left-circular) branch has the same shape as thebare-polariton band:

ω = ω0 + 2(Ω− Ωc) . (9.59)

The second term in the right part of eqn (9.58) describes the Zeeman splitting of theground-state.

The superfluidity of exciton-polaritons at zero temperature exists below and abovethe critical field: at B < Bc both branches of excitations have a linear dispersion, andthey touch each other at k = 0 (zero Zeeman splitting). At B = Bc the Zeeman splittingis still zero, but one of two branches of excitations no longer has linear component inthe dispersion. The condensate of polaritons exists only at T = 0 in two-dimensionalsystems with a parabolic spectrum, thus at any finite temperature the polariton superflu-idity disappears at the critical field. Interestingly, at B > Bc the superfluidity reappearsagain due to the Zeeman gap that opens between the two branches of excitations: the po-lariton dispersion in the vicinity of the lowest-energy state again becomes linear (see thephase diagram in Fig. 9.14). With further increase of the field the critical temperature ofthe superfluid transition is expected to increase and approach the Kosterlitz–Thoulesstransition temperature, i.e., the critical temperature of the superfluid transition in a one-component interacting Bose gas. One can see that the two-component (spinor) nature ofthe polariton condensate leads to its very peculiar behaviour in a magnetic field, in par-ticular, to the full paramagnetic screening (or spin Meissner effect) and the appearanceof a bicritical point in the phase diagram at B = Bc.

9.11 Finite-temperature case

At nonzero temperature the excited polariton states are not empty. The temperatureincrease leads to depletion of the condensate, i.e., departure of the polaritons from the

FINITE-TEMPERATURE CASE 379

Fig. 9.14: Phase diagram from Rubo et al. (2006) of the polariton superfluidity: at zero field the superfluid islinearly polarized, in the weak-field regime it is elliptically polarized and Zeeman splitting is equal to zero,at the critical field no superfluidity exists, the condensate disappears at any finite temperature, at strong fieldsthe superfluidity exists again, the condensate is circularly polarized.

condensate to the states having a nonzero inplane wavevector) and its depolarization(as some polaritons can flip their spins). The depletion results in the decrease of nwith increasing temperature. Let us analyse the depolarization effect assuming a givenoccupation number of the condensate n and finite temperature T .

Fig. 9.15: Zeeman splitting of a polariton condensate at zero temperature (dashed lines) and finite temper-atures increasing from right to left. One can see that the spin Meissner effect is relaxed as the temperatureincreases.

We shall consider only the k = 0 state assuming that all other states are weakly oc-cupied and do not contribute to the polarization of emission. We take into account all theexcited states of the condensate characterized by different projections of its pseudospinon the magnetic field. The chemical potential of the system can be found from:

µ =∂

∂n〈E〉 , (9.60)

where the average energy of the condensate


〈E〉 = −2Ω〈Sz〉+1

4(V1 + V2)n

2 + (V1 − V2)〈S2z 〉 , (9.61)

with

〈Sz〉 =

∑n/2j=−n/2 j exp

(− Ej/(kBT ))

∑n/2j=−n/2 exp

(− Ej/(kBT )) , (9.62a)

〈S2z 〉 =

∑n/2j=−n/2 j2 exp

(−Ej/(kBT ))

∑n/2j=−n/2 exp

(− Ej/(kBT )) , (9.62b)

where Ej = −2Ωj + (V1 − V2)j2. One can see that with T → 0, Sz → Ω/(V1 − V2)

if Ω < (V1 − V2)n/2 and Sz → n/2 if Ω > (V1 − V2)n/2, which yields the resultsobtained above (eqns (9.46) and (9.49)). In order to obtain a compact expression for thechemical potential we substitute the sums (9.62a) by integrals. This is perfectly validfor large enough occupation numbers n. Now, the derivatives are easily calculated andthe chemical potential reads

µ =1

2(V1+V2)n−2Ω

exp(−(V1 − V2)n2/4kBT )

Σn

(n

2sinh

Ωn

kBT− 〈Sz〉 cosh

Ωn

kBT

)

+ (V1 − V2)exp(−(V1 − V2)n

2/4kBT )

Σncosh

Ωn

kBT

(n2

4− 〈S2

z 〉)

, (9.63)

where

Σn =

∫ n2

−n2

exp

(−E(x)

kBT

)dx , E(x) = −2Ωx + (V1 − V2)x

2 . (9.64)

Figure 9.15 shows schematically the behaviour of the chemical potential of a con-densate as a function of the magnetic field at different temperatures (the lower branchesin the figure show the chemical potential behaviour as a function of the magnetic field).At nonzero temperature, the chemical potential decreases with the field increase even ifthe field is lower than the critical one. The condensate is never fully circularly polarizedbut its magnetic susceptibility χ = ∂µ/∂B remains strongly field dependent, however.The upper set of curves shows the excitation energy of the polariton having its spinopposite to the magnetic field, i.e., the upper component of the Zeeman doublet observ-able in reflection or transmission experiments. One can see that at finite temperatures,the Zeeman splitting is never exactly zero. It is, however, strongly suppressed below thecritical field due to the paramagnetic screening (or spin Meissner effect).

Exercise 9.4 (∗∗) Obtain the Bogoliubov spectrum of excitations (dispersion relation)from the Gross–Pitaevskii equation for a scalar wavefunction.

SPIN DYNAMICS IN PARAMETRIC OSCILLATORS 381

9.12 Spin dynamics in parametric oscillators

In microcavity-based optical parametric oscillators (OPOs), the redistribution of po-lariton population between the initial (“pump”) state and two final states (“signal” and“idler”) takes place:

pump↔ signal + idler ,

so that both the energy and the total inplane wavevector of the polariton system are con-served (see Section 9.7). Frequently, such OPOs are studied in the amplification regimewhen a weak probe pulse is used to seed the signal state and stimulate the scattering ofpolaritons from the pump state. The backscattering of polaritons to the pump state canoften be neglected in this regime. Experimentally, the polarization properties of such anamplification process have been studied by Lagoudakis et al. (2002) (see Section 7.2.1).In terms of the classical nonlinear optics the scattering of polaritons from the pump tosignal state stimulated by a probe pulse can be understood as a resonant four-wavemixing process. In this section we first consider phenomenologically the amplificationprocess and describe the experiment of Lagoudakis in terms of the four-wave mixing,then present the quantum theory of polarization-dependent OPO based on the Liouvilleequations.

9.13 Classical nonlinear optics consideration

We consider an experimental configuration of Lagoudakis et al.: a polarized pump ex-cites the cavity at the magic angle and generates a coherent polariton population at theinflection point of the lower-polariton branch (see Fig. 9.16). Its polarization is changedfrom right-circular to left-circular passing through linear polarization in a series of ex-periments. A circularly polarized probe generates polaritons in the ground-state (k = 0)that stimulates the resonant scattering of polaritons created by the pump pulse to the

Fig. 9.16: In-plane dispersion of uncoupled ex-citon and photon (dashed) and polariton (solid)modes in a microcavity in the strong-couplingregime. Arrows show scattering of exciton-polaritons from the pump state into signal andidler states.


ground-state. The signal polarization dependence on the pump polarization is studied(see Fig. 9.5(a)–(c)).

To model this experiment, one can represent an electric field of an electromagneticwave propagating in the structure as a Jones vector:

E =

(Ex

Ey

), (9.65)

having the inplane components Ex, Ey . The electric field of the four-wave mixing signalis given by:

Esigα = TαβγδPβP ∗

γ Sδ , (9.66)

where α, β, γ and δ take the values x or y, Pα are the components of the pump pulse,Sα are the components of the probe pulse.

In order to obtain the tensor let us analyse the two-component matrix

Mαδ = TαβγδPβP ∗γ . (9.67)

By reasons of symmetry, it can be represented generally in the following way:

Mαβ = A(PxP ∗x + PyP ∗

y )δα,β + B(PαP ∗β + PβP ∗

α) + C(PαP ∗β − PβP ∗

α) , (9.68)

where A, B, C are constants and δ is the Kronecker symbol. The first term in the rightpart of eqn (9.68) describes the isotropic optical response of the system, the second termyields the inplane anisotropy induced by the pump pulse, and the third term describesthe pump-induced gyrotropy.

The gyrotropy comes from the spin-splitting of the exciton resonance in the caseof circular or elliptical pumping. The splitting of the exciton resonance in σ+ and σ−

polarizations influences the linear optical response of the quantum well (optically in-duced Faraday rotation results from this effect). We take this effect into account whilecalculating the linear propagation of light in the cavity. Putting A = 1 we reduce thenumber of unknown parameters of the problem to two. The best fit to the experimentaldata (Fig. 9.5(d)–(f) is obtained with the following set of parameters: B = (i − 1)/2and C = i/8 for the signal, B = −1/2 and C = 2i/3 for the idler.

Note that if the pump pulse is linearly polarized, the third term in the right partof eqn (9.68) vanishes, so that the polarization of the signal is governed by the onlyparameter B. If it were zero (no optical anisotropy case) the signal polarization wouldbe the same as the probe polarization. In the experiment of Lagoudakis, however, thesignal polarization is purely linear at circular probe. This indicates that optically inducedanisotropy (Kerr effect) takes place in microcavities: if the sample is illuminated bylinearly polarized light, the polariton ground state (signal state) splits into a doubletco- and crosspolarized with respect to the incident light. Most likely, this effect has itsorigins in the anisotropy of polariton–polariton interactions, as we shall see below.

To understand the origin of constants B and C a microscopic (quantum) considera-tion is needed.

POLARIZED OPO: QUANTUM MODEL 383

9.14 Polarized OPO: quantum model

In Section 9.7 we have given a set of equations (9.34) that describe polariton relaxationassisted by acoustic phonons, polariton–polariton scattering and spin-relaxation becauseof the Maialle field. It represents a general formalism that can be applied for numericalmodelling of any particular experimental situation. To determine the qualitative ten-dencies analytically, this formalism can be simplified by neglecting different spin orenergy relaxation mechanisms. In OPO case, polariton–polariton scattering is the dom-inant mechanism of the polariton redistribution in the reciprocal space. If we retain ineqns (9.34) only polariton–polariton scattering and neglect all other terms (except theradiative lifetime term responsible for emission of light), the system of kinetic equationsfor polariton occupation numbers and and inplane pseudospins can be derived:

dNk,↑dt

= Tr(a†k↑ak,↑

dρ

dt) = − 1

τkNk↑ +

∑k′,q

W(1)k,k′,q

[(Nk↑ + Nk′↑ + 1)Nk+q↑Nk′−q↑ − (Nk+q↑ + Nk′−q↑ + 1)Nk↑Nk′↑]

+ W(2)k,k′,q

[(Nk↑ + Nk′↓ + 1)Nk+q↑Nk′−q↑ + Nk+q↓Nk′−q↑ + 2S⊥,k+q ·S⊥,k′−q

−(Nk↑Nk′↓+S⊥,k+q ·S⊥,k′)(Nk+q↑+Nk′−q↓+Nk+q↓+Nk′−q↑+2)]

+W(12)k,k′,q

[Nk′↑S⊥,k′ ·S⊥,k′−q+Nk′−q↑S⊥,k′ ·S⊥,k+q+Nk′↑S⊥,k · (S⊥,k′−q+S⊥,k+q)

+ S⊥,k′ ·S⊥,k+q(Nk′−q↑ + Nk′−q↓ + Nk′↑ −Nk′↓)

+ S⊥,k ·S⊥,k+q(Nk′−q↑ + Nk+q↓ −Nk′↑ −Nk′↓)]

, (9.69)

dS⊥,k

dt= − 1

τskS⊥,k +

∑k′,qW

(1)k′,k′,q

2S⊥,k

[Nk+q↑Nk′−q↑ + Nk+q↓Nk′−q↓

−Nk′↑(Nk+q↑ + Nk′−q↑ + 1)−Nk′↓(Nk+q↓ + Nk′−q↓ + 1)

+ 2S⊥,k+q(S⊥,k′ ·S⊥,k′−q) + 2S⊥,k−q(S⊥,k′ ·S⊥,k+q)

− 2S⊥,k′(S⊥,k′+q ·S⊥,k′−q)]

+W

(2)k′,k′,q

2

[2(S⊥,k + S⊥,k′)(Nk+q↑Nk′−q↑ + Nk+q↓Nk′−q↑ + 2S⊥,k ·S⊥,k′)

− (S⊥,k(Nk′↑ + Nk′↓ + Nk↑ + Nk↓))(Nk+q↑ + Nk′−q↑ + Nk+q↓ + Nk′−q↓ + 2)

− 4S⊥,k(S⊥,k′ ·S⊥,k+q + S⊥,k′ ·S⊥,k′−q)]

+W

(12)k′,k′,q

2

[S⊥,k′−q2((Nk′↑ + 1)Nk+q↑ + (Nk′↓ + 1)Nk+q↓)

+ (Nk+q↓ + Nk+q↓ −Nk′↑ −Nk′↓)(Nk↑ + Nk↓)+ S⊥,k′−q2((Nk′↑ + 1)Nk+q↑ + (Nk′↓ + 1)Nk+q↓)

+ (Nk+q↓ + Nk+q↓ −Nk′↑ −Nk′↓)(Nk↑ + Nk↓)]

, (9.70)


where ez is a unitary vector in the direction of the structure growth axis. The equationfor spin-down occupation numbers can be obtained from eqn (9.69) by changing thespin indices. The transition rates are as follows:

W(1)k′,k′,q =

2π

|V (1)

k,k′,q|2δ(E(k) + E(k′)− E(k + q)− E(k′ − q)

),

W(2)k′,k′,q =

2π

|V (2)

k,k′,q|2δ(E(k) + E(k′)− E(k + q)− E(k′ − q)

), (9.71)

W(12)k′,k′,q =

2π

(V (1)

k,k′,qV(2)∗k,k′,q

)δ(E(k) + E(k′)− E(k + q)− E(k′ − q)

).

As usual, the delta functions ensure energy conservation. The signs of the transi-tion rates W

(1)k′,k′,q, W

(2)k′,k′,q and W

(12)k′,k′,q can differ. Although W

(1)k′,k′,q and W

(2)k′,k′,q

are always positive, W(12)k′,k′,q can be positive or negative depending on the phase shift

between the matrix elements of the singlet and triplet scattering V (2) and V (1), respec-tively. In particular, it is negative if these matrix elements are real and have oppositesigns, which is typically the case for microcavity polaritons.

Fig. 9.17: Scattering of two X-polarized polaritons (schematic): both linearly polarized polariton statesare linear combinations of circularly polarized (spin-up and spin-down) states. Due to different signs of thescattering constants in triplet (parallel spin) and singlet (antiparallel spin) configurations the pair of polaritonsafter the act of scattering will have Y -polarization. If V (1) = V (2) the pair of polaritons would lose theirpolarization after the scattering process.

Modelling the polarization dynamics in the OPO we can restrict our consideration toonly three states in reciprocal space corresponding to the pump, signal and idler (furthermarked as p, s, i states).

Let us consider the OPO pumped at the magic angle (p-state) using TE-polarizedlight. No probe pulse is sent, and the polarization emitted by the ground state (s) is stud-ied. This corresponds to the experimental configuration used by Renucci et al. (2005b).Their experiment showed that the signal polarization is rotated by 90 with respect topump polarization. Let us see how this surprising observation can be interpreted withinthe quantum kinetic formalism.

In the spontaneous scattering regime, when Ns↑ = Ns↓ = 0, S⊥,s the system ofkinetic equations (9.69) and (9.70) strongly simplifies and for the signal state we have:

CONCLUSIONS 385

dNs↑dt

= −dNs↑τs

+ W (1)N2p↑ + 2W (2)(Np↑Np↓ + S⊥,p ·S⊥,p) , (9.72a)

dNs↓dt

= −dNs↓τs

+ W (1)N2p↓ + 2W (2)(Np↑Np↓ + S⊥,p ·S⊥,p) , (9.72b)

dS⊥,s

dt= −dS⊥,s

τs+ 2W (12)S⊥p(Np↑ + Np↓) . (9.72c)

Equations (9.72a) show that the orientation of the inplane pseudospin of the signalstate is governed by the sign of W (12). For W (12) < 0 (which is fulfilled if the signsof V (1) and V (2) are different) the pseudospin of the signal is opposite to the pseu-dospin of the pump. The inversion of the pseudospin corresponds to the 90 rotation ofthe polarization plane of photoemission (see the scheme in Fig. 9.17). This rotation hasbeen observed experimentally by several groups, the above-mentioned one but also byKrizhanovskii et al. (2006). The linear polarization degree of the signal in the sponta-neous scattering regime can be estimated as

L =4W (12)

W (1) + 4W (2). (9.73)

As the triplet scattering amplitude V (1) is usually about 20 times higher than the singletscattering amplitude V (2), |W (12)| W (1) and thus L is very small (typically, a fewper cent as one can see from Fig. 9.6. However, enhancement of the pump power caninduce a considerable increase of the linear polarization degree of the emission. Indeed,due to the polarized spontaneous scattering a seed population of polaritons is createdin the signal state. These polaritons have a weak linear polarization perpendicular tothe pump-pulse polarization. The negative polarization degree of this seed can then beenhanced by bosonic stimulation if the intensity of the pump exceeds the stimulationthreshold value (in Fig. 9.6 it achieves −70% for high pumping).

To conclude this section, polariton–polariton scattering is polarization dependentand the final-state polarization does not coincide with the initial-state polarization, ingeneral. Inversion of linear polarization between initial and final state is caused by dif-ferent signs of polariton–polariton interaction constants in singlet and triplet geome-tries. When the final state is populated by a seed of circularly polarized polaritons(Lagoudakis experiment) rotation of the polarization plane by 45 instead of 90 degreeshas been observed, which indicates that light-induced optical anisotropy (birefringence)may take place in the microcavities. In general, the TE–TM splitting of polariton statesand self-induced Faraday rotation should also be accounted for to describe the compli-cated polarization dynamics of cavity OPOs. The kinetic equations derived using thepseudospin density matrix approach and Born–Markov approximation describe all theabove-mentioned effects. Their simplified version retaining only polariton–polariton in-teraction allows for qualitative understanding of the experimental data in many cases.

9.15 Conclusions

Progress in polarized optics of microcavities paves the way to realization of “spin-optronic” devices that would transform light on a nanometre and picosecond scale.


Nowadays, optical communication technologies are based on intensity and frequencymodulation. The light polarization modulation is rarely used. On the other hand, polar-ization of light represents an additional degree of freedom that could be efficiently usedfor encoding of information. Polarization-modulated signals can not be used for long-range telecommunication as polarization is quickly lost via polarization scrambling inoptical fibres. However, short-range optical information transfer, or even optical infor-mation transfer on a processor scale are excellent application areas for “spin-optronic”devices. For this purpose amplifiers and switches sensitive to polarization are neededas well as polarization converters, polarization modulators, and stable sources of po-larized light. Ideally, these several functionalities should be embedded within a singlesystem. A spin-optronic device should be extremely low power consuming as it is sup-posed to be integrated on a chip together with classical electronic functions. The deviceshould be electronically controlled in order to provide an efficient interface with theinformation-processing unities.

Future polariton devices based on microcavities have a potentiality to fulfil all theserequirements, which is why they seem to be well adapted for the purposes of spin-optronics. They are expected to consume little power, have a nanosize and allow forultrafast manipulations with the polarization of light. Experiments performed in theresonant-pumping geometry show that microcavities efficiently convert linear to circu-lar polarization and vice versa. They also act as switches sensitive to the polarizationof a reference beam. The main obstacle on the way towards realization of spin-optronicdevices based on microcavities is linked with the need of having an electrically pumpedpolariton-laser operational at room temperature. If this problem is solved, polariton de-vices would respond to the actual technological needs of spin-optronics. A new gen-eration of optoelectronic devices is therefore potentially to be based on microcavitiesoperating in the strong-coupling regime.

9.16 Further reading

Hanle (1924) provides excellent additional reading on spin-relaxation. We already men-tionned the volume “Optical orientation” edited by Meyer and Zakharchenia (1984),which contains excellent experimental and theoretical chapters on the polarized opticsof semiconductors. For those interested in spintronics, we propose to compare the opti-mistic review of Wolf et al. (2001) with the skeptical one of D’yakonov (2004).

GLOSSARY

This glossary serves two purposes: it can first be used as an index thatlocates where in the book the key is introduced, discussed or used. It canalso be used as a dictionary that provides a succinct definition thatmight otherwise not be found elsewhere.

388 GLOSSARY

AAbsorption of light in a crystal is a measurement that can be made

by calorimetres or deduced from the reflectivity and thetransmission spectra. It characterizes the efficiency of thelight–matter coupling. See Section 2.2.2 on page 31.

Acoustic phonon is a phonon that can be excited by a photon. In-teraction of exciton-polaritons with acoustic phonons isone of the most important mechanisms of the polaritonenergy relaxation in microcavities. See on page 138, 146,257, 297, and Section 8.6.1 on page 303.

Active layer In semiconductor lasers, this is the layer of a semi-conductor material, e.g., a quantum well, where the in-version of electronic population between the energy lev-els in the valence and conduction bands is achieved. Thestimulated emission of light dominates its absorption inthis layer at some frequency. See on page 4, 10, 14, 234,Fig. 6.10 on page 235, page 235, 240.

Anticrossing is a signature of the strong-coupling regime in a sys-tem of two coupled oscillators. Due to the interactionsbetween the oscillators, the eigenfrequencies of the sys-tem remain splitted (and this splitting is in fact maximum)when the normal frequencies of the two individual oscil-lators coincide. See the discussion around eqn (4.101) onpage 152.

Antinode (of the light field in a cavity) is the maximum of inten-sity of the electric field of a standing light mode. Typ-ically an active element (quantum well, wire or dot) issought to be placed at the antinode of the field as thisprovides the largest exciton–light coupling strength in thecavity. See on page 56, 237.

BBandgap is the region of forbidden states in the band diagram of

a semiconductor. The bandgap energy is the energy dif-ference between the conduction and the valence band of asemiconductor, that is, the energy required or released tobring one electron from one to the other. See on page 65,Fig. 4.4 on page 126, page 310.

BCS is a famous theoretical model explaining low-temperature su-perconductivity. The term comes from the abbreviationof the surnames of its three authors: Bardeen, Cooper,and Schrieffer. BCS assumes the formation of so-calledCooper pairs of electrons, which have bosonic propertiesin a metal. The Cooper pair’s size strongly exceeds theaverage distance between pairs in contrast with the situa-tion in diluted Bose gases (like exciton gases) where thesize of a boson is, in general, much less than the distancebetween bosons. See on page 282 and on page 292.

Bloch theorem defines the expression of the electron wavefunc-tion in a periodic-crystal potential and introduces the quasi-wavevector as its quantum number. It also applies to pe-riodic photonic structures (where it gives rise to photoniccrystals). See on page 52 and on page 124.

Bloch equations describe a driven two-level quantum system andcan be generalized to describe optical properties of semi-conductors (becoming the Maxwell-Bloch equations). SeeSection 5.3 on page 173.

Boltzmann equations are differential equations that describe thekinetics of the occupation numbers of the eigenstates ofan infinite system. They operate with ensemble-averagedpopulations at a classical level and as such do not describequantum correlations. Boltzmann equations are a power-ful tool for the description of relaxation processes in in-teracting gases, like an exciton-polariton gas. See Section8.6.2 on page 305.

Born approximation consists in decoupling the dynamics of cou-pled quantum systems such as, e.g., excitons and phonons.This allows typically to factorize the total density matrixinto products of density matrices for each subsystem. Seeon page 188.

Bose–Einstein statistics describes the energy distribution of quan-tum particles with integer spin, called bosons (e.g., pho-tons). It reduces to the Maxwell–Boltzmann statistics ofan ideal classical gas in the high-temperature and high-energy limit. At low temperatures, it predicts the accumu-lation of bosons in the lowest energy state (Bose–Einsteincondensation). See eqn (8.1) on page 281.

Bottleneck The (phonon) bottleneck effect is a slowing down ofthe rapid polariton relaxation along the lower dispersionbranch that is rapid in the exciton-like part due to scat-tering with acoustic phonons, but then becomes slowerin the vicinity of the anticrossing point of the excitonand photon modes because of kinetic blocking of polari-ton relaxation. The main obstacle for polariton relaxationin the bottleneck region comes from the lack of acousticphonons that are able to scatter with polaritons of verylow effective mass. The bottleneck effect prevents polari-tons from relaxing down to their ground-state at k = 0,which represents a major problem for the realization ofpolariton lasers. The bottleneck effect exists also for bulkor quantum well exciton-polaritons. See on page 257, 258,285, 295, Fig. 8.10 on page 304, pages 308—310.

Bose–Einstein condensation (BEC), also simply “Bose conden-sation”, is a phase transition for bosons leading to the for-mation of a coherent multiparticle quantum state charac-terized by a wavefunction. The Bose condensate occupiesthe lowest energy level of the system that coincides withthe chemical potential. Strictly speaking, BEC can onlytake place in infinite systems with dimensionality higherthan 2. In finite size and/or low-dimensional systems onecan speak of quasi-BEC and Kosterlitz–Thouless phasetransition. See on page 276 and Chapter 8.

Bra (see ket first) In quantum mechanics, the dual state 〈ψ| ofa ket |ψ〉. The product of a bra 〈ψ| with any ket |φ〉gives the braket 〈ψ|φ〉 that is the inner product of theirassociated Hilbert space, whence the name. See note 21on page 76.

Bragg mirror is a mirror formed by alternating layers of semicon-ductors with different refractive index. Each layer bound-ary partially reflects the incoming wave and through theeffect of constructive interferences, very high reflectiv-ity is achieved. To obtain the strongest interference, thethicknesses of these layers must be chosen equal to aquarter of the wavelength of light in the correspondingmaterial at some frequency referred to as the Bragg fre-quency. The reflection spectrum of a Bragg mirror ex-hibits a plateau of very high reflectivity centred on theBragg frequency. This plateau is referred to as a stop-band, which represents a one-dimensional photonic band-gap. See on page 2, 8, 9, 45, 53, Fig. 2.8 on page 55, page56, Fig. 2.9 on page 58, Fig. 4.20 on page 150, page 236.

Broadening refers to the fact that a spectral line is never exactly adelta function but always has a width that is due to variousmechanisms that blurs the energy definition at which thetransition is expected to take place. See “homogeneous”and “inhomogeneous broadenings” in the glossary.

Bright mode also known as a superradiant mode, is a collectivestate of a few oscillators (atoms, excitons, polaritons, etc.)which has a higher radiative decay rate than any of theseoscillators taken in isolation. See Section 5.6 on page 198.

Bulk microcavities are microcavities for which the cavity consistsof a bulk semiconductor without any embedded quantumobjects, like quantum wells, wires or dots.

GLOSSARY 389

CCoherence is one of the basic characteristics of light. According

to the Glauber classification, different orders of the co-herence can be defined. The first-order coherence is de-pendent on temporal correlations of the amplitude of thelight field, the second-order coherence is dependent onthe intensity correlations, etc. Fully coherent light is fullycorrelated to all orders. The coherence time and coher-ence length of light are linked to the first-order coherence.See on page 10, Section 2.3 on page 32, page 42, Sec-tion 3.3.2 on page 101, page 106, 108. The term is alsowildly applied to other concepts, such as quantum coher-ence (e.g., page 96) or condensate coherence (Chapter 8).Often these other meanings themselves split further intomore definitions for unrelated concepts.

Coherent state in classical optics is a state of light characterizedby a fixed phase. In quantum optics, it is a state that mini-mizes the position–momentum uncertainty in an harmonicpotential, spreading equally these uncertainties (otherwiseit becomes a squeezed coherent state, or “squeezed state”for short. It has a Poisson distribution as its diagonal el-ements in a density matrix representation (it is a purestate). It contrasts with the thermal state where the vac-uum is found with the highest probability. See Section3.3.2 on page 101.

Collapse The “process” that a quantum state undergoes upon mea-surement to become the eigenstate |ωi0

〉 associated tothe eigenvalue ωi0

measured in the course of the exper-iment. This postulate has been made to match the exper-imental fact that (immediately) repeated measures of thesame observable on a quantum system always return thesame result. This assumption is one of the pillars of theCopenhagen interpretation and is also known as the re-duction of the wavefunction or quantum jump. The exactprocess responsible for it is as yet debated but is describedby the theory of decoherence. See on pages 80—82, 84.

Copenhagen interpretation One early interpretation of quantummechanics issued by the joint efforts of Bohr and Heisen-berg c. 1927 while collaborating in the capital of Den-mark (whence the name). See on page 76.

Coupled cavities are cavities that have a common mirror, usually.

DDark exciton also referred to as an optically inactive exciton is an

exciton that cannot be created by resonant absorption ofa photon. Examples include: indirect in real or reciprocalspace excitons, excitons with a spin projection on a givenaxis equal to ±2, excitons having wavevectors exceed-ing the wavevector of light in vacuum at their resonancefrequency. See on page 284, 348. By analogy, a so-calledbright exciton is directly coupled to light.

Density matrix is the extension of the concept of the wavefunc-tion into the statistical realm. A quantum state which isdescribed by the wavefunction |ψ〉 is also described bythe density matrix |ψ〉〈ψ|, but not all states that aredescribed by a density matrix can also be described by awavefunction. A state that admits a wavefunction descrip-tion is called a pure state. A state that does not is called amixed state. The latter lacks a full knowledge of the sys-tem that is supplemented by classical probabilities. SeeSection 3.2.1 on page 95.

Detuning refers to the difference in energy between two coupledmodes. “Changing the detuning” means bringing the twomodes in and out of resonance (equal energy). For in-stance, the detuning between photon and exciton modes

in a microcavity is the difference between the eigenfre-quencies of the bare cavity mode and the exciton reso-nance at zero inplane wavevector. One speaks of posi-tive detuning if the cavity mode is above the exciton res-onance and of negative detuning if it is below. See onpage 62, 157, Fig. 4.23 on page 158, 172.

Dispersion in optics and in quantum mechanics describes the fre-quency (energy) dependence on the wavevector. If a waveequation is dispersive, the profile of a wavepacket usuallyget distorted as it propagates (if combinations of disper-sion and nonlinearity concur in maintaining its shape, asoliton is formed).

EElastic circle is a circle in a two-dimensional reciprocal space cen-

tred at k = 0 which radius is given by the absolute valueof the wavevector of a particle. All states on the elasticcircle have the same kinetic energy and are therefore pos-sible final states for the elastic (Rayleigh) scattering ofthat particle.

Etching is a process to remove unwanted parts of a semiconductordevice during its fabrication. Many techniques exist, e.g.,wet etching using acid chemicals and dry etching vaporiz-ing the material, selective or anisotropic etchings allow toshape the structure by etching different parts at differentrates. See on pages 10—14, 237 and 355.

Exchange interaction is a quantum-mechanical interaction mech-anism based on the indistinguishability of quantum parti-cles. For example, if a pair of electrons from quantumstates i and j scatters to quantum states m and k, it is im-possible to say which one went to the state m and whichone to the state k, so that the two scenari (i going to m, jgoing to k and vice-versa) should be taken into account.If i = m and j = k, the first scenario corresponds tothe direct interaction while the second one corresponds tothe exchange interaction.

Exciton is a Coulomb correlated electron–hole pair in a semicon-ductor. One can distinguish between Frenkel and Wannier–Mott excitons, the former having much larger binding en-ergies and much smaller Bohr radius than the latter. Seeon page 117, Section 4.3 on page 127, Fig. 4.7 on page 129,Section 5.7.3 on page 202.

Exciton-polariton is a quasiparticle formed by a photon propa-gating in the crystal and an exciton resonantly excited bylight. Exciton-polaritons are true eigenstates of light incrystals in the vicinity of the resonant frequencies of ex-citonic transitions. See on page 5, Fig. 1.3 on page 6, page117, 137, footnote 69 on page 138, page 143, 146, Sec-tion 4.4.4 on page 150, Fig. 4.23 on page 158, page 208,Section 5.8.1 on page 210, Section 5.8.2 on page 211,Chapters 7 and 8.

FFabry–Perot resonator is a kind of cavity formed by a dielectric

layer sandwiched between two mirrors. Its eigenmodesare standing waves whose wavelength is related to thesize of the resonator. If the mirrors are ideal, the integernumber of half-wavelengths of an eigenmode should beequal to the thickness of the cavity. See on page 6, 59, 62,239, 247.

Faraday rotation is a rotation of the polarization plane of linearlypolarized light passing through a media subject to a mag-netic field parallel to the light propagation direction. Un-like natural optical activity, the Faraday effect can be ac-cumulated in optical resonators and microcavities due tothe multiple round-trips of light. See on page 45, Section9.2 on page 351, Section 9.3 on page 352, Section 9.7 onpage 366.

390 GLOSSARY

Fermi–Dirac statistics applies to particles with half-integer spin,called fermions (e.g., electrons and holes in semiconduc-tors). It is based on the Pauli exclusion principle, and in-troduces a so-called Fermi energy below which all energylevels are occupied and above which all energy levels areempty at zero temperature. At high temperatures, it re-duces to the Maxwell-Boltzmann ditribution.

Fock state also referred to as a number state, is a quantum statecharacterized by a fixed number of photons. It has no clas-sical countepart and is highly sought for quantum infor-mation processing. See Section 3.3.1 on page 100.

HHanbury Brown–Twiss setup is a photon-counting optical setup

that allows one to measure the intensity–intensity corre-lations in a light beam and extract from them the second-order coherence of light g2 . See Exercise 2.9 on page 73and Section 3.3.6 on page 106.

Hermitian operator In mathematics, an operator Ω that is self-adjoint, i.e., such that Ω† = Ω. As a consequence itseigenvalues are real. Such an operator is typically used todefine an observable. See the second postulate of quan-tum mechanics on page 80.

Heterostructure The superposition of several thin layers of differ-ent (hetero) types of semiconductors that together form astructure whose bandgap varies with position. A junctionbetween two semiconductors is the simplest heterostruc-ture. In the celebrated double heterostructure, two semi-conductors sandwich a lower-bandgap semiconductor soas to create a potential trap for both electrons and holes.Such a region is the core for semiconductor lasing. Seeon page 131, 234, 235, Fig. 6.10 on page 235.

Hilbert space In quantum mechanics, mainly used as a synonymfor “the set of quantum states” for a given system. Inmathematics, a separable complete vector space that isthe foundation for the mathematical formulation of thetheory. See the first postulate of quantum mechanics onpage 76 and appendix A.

Homogeneous broadening is the broadening of a transition due tothe lifetime of the particle. It has the shape of a Gaussian.See Section 2.3.3 on page 38.

IInhomogeneous broadening is the broadening of a transition due

to the potential disorder that randomly shifts the transitionenergies up or down in different regions of the space. Ittypically has the shape of a Lorentzian. See Section 2.3.3on page 38.

JJaynes–Cummings model describes the coupling of a single atom

with the quantized optical field. It is a rare instance ofa fully-integrable fully-quantum Hamiltonian. See Sec-tion 5.5 on page 192.

Jones vector is a two-component complex vector that describesthe polarization of light. Its components correspond to theamplitudes of two orthogonal linear polarizations. See onpage 29, 3.1.2 and on page 352.

KKet In quantum mechanics, a vector noted |ψ〉 (by Dirac) part

of a Hilbert space that describes the state of a quantumsystem. See “Bra” on page 388 and note 21 on page 76.

Kosterlitz–Thouless phase transition is a transition towards a su-perfluid phase in two-dimensional bosonic systems. It wasdescribed for the first time by J.M. Kosterlitz and D.J.Thouless in 1973. In infinite two-dimensional systems,Bose condensation is impossible, while a superfluid canbe formed. A superfluid is a collective bosonic state, inwhich the particles can move throughout space along aphase-coherent, dissipationless path. In ideal infinite mi-crocavities, exciton-polaritons may undergo the Kosterlitz–Thouless-like transition and form a superfluid if a criti-cal condition linking the concentration of polaritons withtemperature is fulfilled. However, strictly speaking, theKosterlitz–Thouless theory cannot be directly applied toexciton-polaritons in microcavities as it ignores the two-component nature of a polariton superfluid coming fromthe specific spin structure of the exciton-polaritons. Seeon page 285.

LLamb shift is the shift of the emission spectrum due to reabsorp-

tion of light. See on page 216.

Landau levels are the quantization levels for the electron’s orbitalmotion in a magnetic field. See Section 9.2 on page 351.

Laser is historically the acronym for light amplification by stimu-lated emission of radiation but is now a generic term torefer to a device emitting a coherent output with someor all the features of the original laser, even though itsspecifics can differ (case of the polariton-laser, for in-stance, or the atom laser that has nothing to do with light).In its original acceptance the laser generates light (its pre-decessor the maser generates microwaves) from stimu-lated emission of photons by an inverted population ofemitters (atoms, excitons. . . ) with oscillations of the radi-ation. The oscillation—or positive feedback—is providedby the cavity. See the second half of Chapter 6.

Leaky modes are escape channels in a structure designed to con-fine. In a planar cavity, they are the light modes with suchfrequencies and wavevectors that the Bragg mirrors donot confine them in the cavity. In other words, the leakymodes propagate within the transparency ranges of theBragg mirrors. Exciton-polaritons scattered to the leakymodes easily escape from the cavity. See the term f ineqn (1.5) on page 5, on page 64 and Section 6.1.5 onpage 223.

Light cone is the cone limited by the maximal angle at which lightemitted by a spherical source inside the dielectric mediumcan go out to vacuum. This angle is dependent on the di-electric constant of the medium. See on page 140, 240.

Liouville equation also known as Liouville–von Neumann equa-tion is a linear differential equation that describes the dy-namics of a density matrix. It is the counterpart of theSchrodinger equation for a pure state (wavefunction). SeeSection 3.1.4 on page 87.

Locality is a property of a response function: a local response at agiven point in space depends only on the argument of thefunction at the same point in space. See 2.2.3 on page 32.

Longitudinal–transverse splitting, abbreviated as TE–TM split-ting, is the splitting between optical modes that have theirpolarization vector parallel and perpendicular to the wave-vector, respectively. It also applies to exciton-polaritons.See eqn (2.153) on page 64, on page 159 and Section 9.5on page 360.

GLOSSARY 391

MMagic angle is the incidence angle of light that allows one to ex-

cite the polariton state close to the inflection point of thelowest polariton branch so that the energy and wavevec-tor conservation conditions are fulfilled for the polariton–polariton scattering from this state into the ground-state(k = 0) and some higher-energy state belonging tothe lower-polariton branch (called “idler”). The resonantpolariton–polariton scattering plays a central role in micro-cavity-based optical parametric amplifiers and oscillators.Excitation at the magic angle allows one to populate qua-sidirectly the polariton ground state, thus transferring thecoherence of the exciting laser pulse to light emitted bythe cavity normally to its surface. In typical GaAs-basedcavities the magic angle varies between 15 and 20 degreesat detunings close to zero. See Fig. 7.9 on page 260, pages261—263, page 265, 266, 357, 381, 384.

Markov approximation consists in the assumption that the timeevolution of the quantum state of a system at time t de-pends on its state and on the external conditions at thesame time t only and as such, does not have any memoryof its previous dynamics. See on page 190 and Section 8.7on page 310.

Master equation is an equation of motion for a density matrix. Ittherefore describes the dynamics of the probabilities ofthe occupation numbers of a quantum system. See Sec-tion 3.1.4 on page 87, eqns (5.82) on page 190 (for theharmonic oscillator) and (8.45) on page 312 (for a fullpolariton system).

Maxwell-Boltzmann distribution describes the probability to finda particle with a given velocity in an ideal classical gas.

Metallic reflection is the reflection of light by materials that havea nonzero imaginary part in their refractive index. See onpage 57.

Microsphere, microdisk are semiconductor or dielectric spheres(disks) with a radius comparable to the wavelength of vis-ible light in the media. See Section 2.8.2 on page 69.

Motional narrowing is a quantum effect that consists in the nar-rowing of a distribution function of a quantum particlepropagating in a disordered medium due to averaging ofthe disorder potential on the size of the wavefunction of aparticle. In other words, quantum particles that are neverlocalized at a given point of the space, but always occupysome nonzero volume, have a potential energy that is theaverage of the potential within this volume. This is why,in a random fluctuation potential, the energy distributionfunction of a quantum particle is always narrower thanthe potential distribution function. See Section 4.4.5 onpage 160, Section 7.1.3 on page 256

OOrder parameter of a phase transition is a characteristic of the

system that is zero above the critical temperature of thetransition and nonzero below. In the case of the Bose–Einstein condensation, the wavefunction of the conden-sate is an order parameter. Within the second quantiza-tion formalism, the expectation value of the boson cre-ation (annihilation) operator in the condensate plays thesame role. From the point of view of experimental obser-vation of the superfluid phase transition in the system ofspin-degenerate exciton-polaritons, the spontaneous lin-ear polarization of the polariton condensate provides anorder parameter. See on page 275, 282, 284, 315, 316,Section 8.8.4 on page 336.

Optical parametric amplifier (OPA) is a process of resonant scat-tering of two particles (like photons or polaritons) of fre-quency ω0 into two particles of frequency ω0 + ω1and ω0 − ω1 that are called idler and signal, respec-tively. In terms of classical optics this is a nonlinear pro-cess governed by a χ3 susceptibility. If a nonlinear me-dia generating the parametric amplification is placed ina resonator, the corresponding device can be referred toas an optical parametric oscillator (OPO). In microcavi-ties such a parametric amplification process is extremelyefficient if one pumps at the magic angle. In this case,the scattering of two pumped polaritons into a ground-state (signal) and an excited state (idler) is resonant (con-serves both energy and wavevector). The driving force ofthe scattering is the Coulomb interaction between polari-tons. The parametric amplification can be stimulated bya probe pulse that seeds the ground-state, injecting a po-lariton population larger than one. The process can alsobe strong enough to be self-stimulated. See Chapter 7.

Optical phonon is a crystal-lattice vibration mode characterizedby the relative motion of cations and anions.

Optical spin Hall effect is the angle-dependent conversion of lin-ear to circular polarized light in microcavities. It is basedon the resonant Rayleigh scattering of exciton-polaritonsand is governed by their longitudinal-transverse splitting.See Section 9.6 on page 364.

Organic materials are made of organic molecules, containing car-bon, oxygen and hydrogen atoms, usually.

Oscillator strength characterizes the strength of the coupling be-tween light and an oscillating dipole (e.g., an exciton).See eqn (4.50) on page 139.

PPauli principle forbids two fermions to occupy the same quantum

state. See Section 3.1.3 on page 85.

Phase diagram shows the functional dependence between the crit-ical parameters (temperature, density, magnetic field, etc.)separating different phases of a system at thermal equilib-rium.

Phase transition is a transition between different phases (e.g., thesolid and the liquid phase).

Phonon is a quantized mode (longitudinal or transverse) of vibra-tion in a crystal lattice. As such they are the counterpartfor sound of what the photon is for light. Phonons cancontrol thermal and electrical conductivities. In particu-lar, long-wavelength phonons transport sound in a solid,whence the name (voice in Greek). There are two types ofphonons, acoustic phonons and optical phonons. See onpage 125, 212, 269, Fig. 8.1 on page 288.

Photoluminescence is a powerful method of optical spectroscopythat studies the light emission from a sample illuminatedby light of a higher frequency than that that is emitted.

Photonic crystal is a periodic dielectric structure characterized byphotonic bands (including allowed bands and gaps). Cav-ities in photonic crystals allow study of fully localizeddiscrete photonic states. See on page 2, 6, Section 1.7 onpage 19, Fig. 1.19 on page 20, page 64.

Pillar microcavity is a pillar etched from a planar microcavity struc-ture. Its diameter is comparable to the wavelength of lightat the frequency of the planar cavity mode. It allows oneto obtain full (three-dimensional) photonic confinement.See Section 1.5 on page 12, Fig. 1.10 on page 13, Sec-tion 2.8 on page 64, Fig. 2.14(c) on page 67.

392 GLOSSARY

Plasmon is a quantized light mode of propagation on a metal or ahighly doped semiconductor. Plasmons can be longitudi-nal or transverse, localized at the surface or freely prop-agating in the bulk crystal. In metallic microspheres orother microstructures confined plasmon-polaritons can beformed. See on page 20, 297.

Poincare sphere is a sphere each point of which surface corre-sponds to a given polarization of light. Points of the vol-ume within the sphere describe partially polarized state.The Poincare sphere can also be used to describe the quan-tum state of a two-level system, in which case it is knownas the Bloch sphere. See on page 31, 175 and Fig. 9.8 onpage 362.

Polariton is a mixed quasiparticle formed by a photon and a crys-tal excitation (phonon, magnon, plasmon or exciton). Po-laritons can be formed in bulk crystals, at their surfaces,in quantum-confined structures and microcavities. In thisbook we mostly consider the exciton-polaritons (see itsentry in the glossary).

Polariton-laser A polariton-laser is a coherent light source basedon Bose–Einstein condensation of exciton-polaritons. Con-trary to VCSELs polariton-lasers have no threshold linkedto the population inversion. Amplification of light in pola-riton-lasers is governed by the ratio between the lifetimeof exciton-polaritons and their relaxation time towards thecondensate. See on page 21, 280, 285, 295, Section 8.5 onpage 297, Table 8.9 on page 301, page 310, 327, Section8.8.2 on page 329, 333, 336.

Polarization of light is an important characteristic of a light modethat describes the geometrical orientation and dynamicsof the electric field vector. See Chapter 9.

Pseudospin is a complex vector describing the quantum state ofa two-level quantum system in the same manner than theJones vector describes the polarization of light.

Purcell effect consists in the modification of the radiative decayrate of an emitter (typically an atom or an exciton) due tothe changes in the density of photonic states of the sur-rounding media. If the density of final state is reduced,emission is inhibited, if it is increased, emission is en-hanced. The Purcell effect is the landmark of the weak-coupling regime. See Section 6.1 on page 216.

QQuantum computation refers to the application of quantum in-

formation to process qubits to undergo useful computa-tional procedures (or algorithms) that have been foundin some cases to outclass their classical equivalents. Forinstance, the Shor algorithm factorizes large integers inpolynomial time and the Grover algorithm speeds up quer-ries in unstructured spaces. The possibility to use micro-cavities to do quantum computation is in a prehistoric re-search stage. See on page 79.

quantum cryptography Application of quantum information tocommunicate a message securely, taking advantage, e.g.,of conjugate bases for measurement of a qubit or of EPRcorrelations. The possibility to use microcavities to doquantum computation is, like quantum computation, in aprehistoric research stage. See Exercise 3.23 on page 114.

quantum dot (QD) is a semiconductor nanocrystal that confinesexcitation in all three dimensions. It is the ultimate ex-tension of the concept of the reduced dimensionality of aquantum well. See below and Fig. 4.8 on page 132, Sec-tion 5.7.3 on page 202.

quantum information The formulation of (classical) informationtheory with quantum systems as the carriers of informa-tions, which proved to be a worthwile extension, yieldingas subbranches quantum cryptography and quantum com-putation.

quantum state A vector in a Hilbert space that fully describes aquantum-mechanical system according to the postulatesof quantum mechanics. See the first postulate on page 76.

quantum well (QW) is a semiconductor heterostructure having aprofile of conduction and/or valence band edges in theform of a potential well where the free carriers or excitonscan be trapped in one-dimensional sheets and propagatefreely in the others two (the so-called plane of the quan-tum well.) See Fig. 4.8 on page 132. Multiple quantumwells are a system of parallel quantum wells separated bybarriers. If the barriers are thin enough to allow for effi-cient tunnelling between the wells, a system of multiplequantum wells is then called a superlattice.

quantum wire is an electrically conducting wire whose dimen-sions are so small as to impose quantum confinement inthe directions normal to the axis. It extends the conceptof the reduced dimensionality of a quantum well one stepfurther. See Fig. 4.8 on page 132.

Quasi-Bose–Einstein condensation is a term frequently used todescribe the accumulation of a macroscopic quantity ofbosons in the same quantum state in a finite-size quan-tum system, in obvious analogy to Bose–Einstein con-densation that strictly speaking is a phase transition forinfinite-size systems. See Chapter 8.

Qubit A quantum two-level system. The term appears in connec-tion with quantum information where it is the elementaryunit of information carried by a quantum system, and isthe support for related effects, like dense coding. In mi-crocavities, any two-level system such as the pseudospinof a polariton in principle qualifies as a qubit, providedthat the coherence time and control of the state are goodenough, which are still open questions. The term qubit hasbeen introduced by Schumacher (1995). See on page 79.

RRabi splitting is the splitting of an energy level due to the cou-

pling to a cavity mode. The term came to microcavityphysics from atomic physics where an atomic resonanceis split in energy. The appearance of Rabi splitting is a sig-nature of the strong-coupling regime in microcavities. Insemiconductor microcavities, this term is frequently usedinstead of exciton-polariton splitting. It can be detectedby anticrossing of exciton and cavity-photon resonancesin reflection spectra taken at different incidence angles. Itshould be noted, however, that two dips in reflection canbe seen even in the weak-coupling regime, if the exci-ton inhomogeneous broadening exceeds the cavity modewidth. Thus, the dip positions in reflection spectra do notcoincide, in general, with the eigenmodes of a microcav-ity. Typical values of the Rabi splitting are from a fewmeV in GaAs-based microcavities with a single QW orin bulk microcavities, to more than 100 meV in organiccavities with Frenkel excitons. See on page 152, 154, 171,Fig. 5.3 on page 183.

Radiative lifetime is the characteristic time of the depopulation ofa given quantum state caused by the emission of photons.In semiconductors, radiative recombination follows fromthe spontaneous recombination of an electron–hole pair.

Rate equations characterize the dynamics of the occupation num-bers in a finite quantum system. The system of rate equa-tions for an infinite system is described by the Boltzmannequations.

Reflectivity measures the intensity of light reflected by a sampledivided by the intensity of the incident light.

Relaxation is a statistical process of the reduction of a physicalparameter (energy, wavevector, etc.)

GLOSSARY 393

Resonant excitation in optics, is the generation of a quasiparticleor of an excited electronic state by absorption of photonswhich energy is equal to the energy of the created quasi-particle (or electronic state).

Resonant Rayleigh scattering of light is an elastic scattering wherethe wavevector of light changes but not its frequency. It isan important tool of optical spectroscopy of semiconduc-tors. See on page 66, 149, 157, 161, 364, 364.

SScattering is a process which changes the wavevector of an in-

cident particle. The elastic scattering conserves energy,while the inelastic scattering does not.

Schrodinger equation is the time evolution equation for a non-relativistic quantum state. It is the basis for other dynami-cal equations of quantum systems. See Section 3.1.2. Theequation itself appears as eqn (3.1) on page 76 (in theSchrodinger picture.)

Screening is the attenuation of a force or of some influence due tothe presence of a surrounding media.

Second quantization is a mathematical formalism largely used inquantum field theory and quantum optics where physicalprocesses are described in terms of creation and annihila-tion operators. See Section 3.2.2 on page 97.

Single-photon emitter is a device that ideally emits a single pho-ton on demand. It has application in quantum cryptogra-phy. They are currently usually based on single quantumdots. Due to the Pauli exclusion principle, the dot can-not host more than one electron and one hole at the low-est energy level and in a given spin configuration. Oncesuch an electron–hole pair recombines, emitting a photon,some time is needed to recreate it again in the dot. There-fore photons are emitted one by one. The same effect canbe realized using single atoms, molecules, or defects. Seefootnote 43 on page 101.

Spatial dispersion refers to the dependence of the dielectric con-stant on the wavevector of light.

Spin is the intrinsic angular momentum of a particle or quasipar-ticle, as opposed to the orbital angular momentum whichis defined as for the case of a classical particle. Spin isa quantum number which can take only specified valuesand cannot be known simultaneously along all axes. Spinis linked to the statistics of the particles: bosons have in-teger spin and fermion half-integer spin. The spin of thephoton is related to its polarization.

Spin dynamics studies the evolution of the spin in time.

Spin Meissner effect is the same as full paramagnetic screeningin exciton (exciton-polariton) BEC. No Zeeman splittingof the condensate takes place until some critical magneticfield dependent on the occupation number of the conden-sate and polariton–polariton interaction constants. See onpage 376.

Spontaneous symmetry breaking is a signature of any phase tran-sition according to the Landau theory. In microcavities,BEC or the superfluid phase transition of exciton-pola-ritons require symmetry breaking and the appearance ofan order parameter (wavefunction) in the polariton con-densate. The signature of spontaneous symmetry break-ing in isotropic planar microcavities is the buildup of lin-ear polarization of light emitted by a polariton conden-sate. The orientation of the polarization plane is randomlychosen by the system. See Section 7.4.1 on page 274,page 284, 317.

Squeezing refers to the reduction of the quantum uncertainty ofan observable by increasing the uncertainty in the conju-gate variable, so that the Heisenberg uncertainty relationremains satisfied. It leads to characteristic modificationsof the particle-number statistics.

Stimulated scattering is a scattering that is enhanced by the Bosestatistics. The probability of scattering becomes propor-tional to the occupation number (plus one) of the finalstate to which the bosons are scattered. The term +1 isthe term independent of the statistics (providing the scat-tering rate in absence of stimulation). If the final state ismacroscopically populated, i.e., forms a Bose condensate,scattering towards such a state is strongly amplified andbecomes extremely rapid. Exciton polaritons (which aregood bosons) are subject to stimulated scattering that pro-vides the underlying action of polariton lasing. See Sec-tion 7.2.1 on page 258, page 260, Fig. 7.10 on page 261,Fig. 7.11 on page 262, page 268, 270, 274, 280, Sec-tion 8.6.5 on page 308, Section 8.7.2 on page 316, page346, 356, 357, 371, 381.

Stokes shift is a redshift between the absorption and the emissionpeaks, usually caused by some relaxation or localizationprocesses. See on page 30.

Stokes vector is a 4-component vector that fully characterizes thepolarization of light. See eqn (2.29) on page 30.

Strong coupling between two systems refers to a regime wherethe quantum Hamiltonian dynamics predominates overthe dissipation of the system. The dynamics cannot bedealt with perturbatively and new quantum states of thesystem emerge. In a cavity, the strong coupling refers tosuch a coupling between exciton and light giving rise topolaritons. It manifests itself in the appearance of a split-ting between the real parts of the eigenfrequencies of po-lariton modes, which is maximum at the resonance be-tween bare exciton and photon modes. In this regime, theimaginary parts of two polariton eigenfrequencies coin-cide at the resonance. The signature of strong couplingis a characteristic anticrossing observed in the reflection(transmission) spectra when the light mode crosses theexciton resonance or vice versa. It requires domination ofthe exciton–photon coupling strength over different damp-ing factors (acoustic phonon broadening, inhomogeneousbroadening etc.) It is the opposite of the weak-couplingregime. See Section 1.1.7 on page 5, Fig. 4.21 on page 152,page 154, 210, 216, 223, Chapters 7 and 8, page 364,Fig. 9.16 on page 381

Superlattice is a multiple quantum well system where the wellsare so close to each other that electrons can easily tunnelbetween them. As a result, minibands are formed.

Superradiance is the enhancement of the radiative decay rate in asystem of coupled oscillators (atoms, excitons, polaritons,etc.)

Superfluidity is a specific property of bosonic liquids at ultra-low temperatures. The liquid propagates with zero viscos-ity and has a linear dependence of the kinetic energy onwavevector. The appearance of superfluidity is a conse-quence of the repulsive interaction between bosons. Ac-cording to recent theories, exciton-polaritons in micro-cavities may become superfluid under certain conditions.See on page 271, 282—286, Section 8.3.3 on page 289,page 294, 342.

TThermal state is a the state of a system that has reached thermal

equilibrium with its surroundings. It is a mixed state char-acterized by exponentially decreasing probabilities of theoccupation numbers. See Section 3.3.4 on page 103.

394 GLOSSARY

Transfer matrix is a mathematical technique that allows to solvethe Schrodinger or Maxwell equations in multilayer sys-tems. See Section 2.5 on page 45.

Transmission of light is an optical spectroscopy technique that al-lows to detect the intensity of light passing through a sam-ple.

VVacuum Rabi splitting refers to the linear optical regime where

the interaction of a single photon with a single atom isimplied in the case of atomic cavities. Nonlinear vacuumRabi splitting has not been evidenced so far so that thedistinction has not yet gained importance and “Rabi split-ting” is often used to mean “vacuum Rabi splitting”. Seeabove.

VCSEL is an acronym for vertical cavity surface emitting laser. Itis a device based on a microcavity in the weak-couplingregime. Stimulated emission of light by an active elementinside the cavity (typically quantum wells, where the in-version of population of electron levels of the conductionand valence bands is achieved due to electrical injectionof charge carriers) pumps one of the confined light modesof the cavity. The light emitted by this laser goes out atright angles to the surface of the mirror, contrary to “hor-izontal” lasers, where the generated light propagates inthe plane of the laser cavity. See Fig. 1.11 on page 14,Fig. 6.10 on page 235, Section 6.2.3 on page 236.

WWeak coupling between two systems refers to the regime opposed

to strong coupling (see above) where dissipation dom-inates over the system interaction so that the couplingbetween the modes can be dealt with pertubatively andboth modes retain essentially their uncoupled properties.The weak coupling between exciton and light manifestsitself in the appearance of the splitting between the imag-inary parts of the eigenfrequencies of exciton-polaritonmodes at the resonance between bare exciton and pho-ton modes. In this regime the real parts of two polaritoneigenfrequencies coincide at the resonance, and two po-lariton resonances in the reflection or transmission spec-tra usually coincide (while in the case of a strong im-balance between the widths of the exciton and photonmodes the doublet structure in reflection and transmis-sion can be seen even in the weak-coupling regime). SeeSection 1.1.7 on page 5, Chapter 6.

Whispering-gallery modes are standing light modes localized atthe equator of a sphere. Contrary to the “breathing modes”,the whispering-gallery modes are characterized by highorbital quantum numbers. See on page 6, Section 1.6 onpage 15, Fig. 2.16 on page 69.

ZZeeman splitting is the magnetic-field-induced energy splitting of

a quantum state into a couple of states characterized bydifferent spin projections onto the magnetic-field direc-tion. See on page 351, 352, 354, Fig. 9.13 on page 377,Fig. 9.15 on page 379, page 380.

APPENDIX A

LINEAR ALGEBRA

This appendix reviews briefly the essential notions of linear algebra required for a goodunderstanding of the postulates and concepts of quantum mechanics discussed in Chap-ter 3. It also settles notations.

A complex vector space is a non-empty setH with linear stability, i.e., such that

∀ |u〉 , |v〉 ∈ H, ∀α, β ∈ C, |u〉 , |v〉 ∈ H ⇒ α |u〉+ β |v〉 ∈ H (A.1)

(we use Dirac’s notations introduced in Chapter 3). Axioms for this structure are i)associativity, |u〉 + (|v〉 + |w〉) = (|u〉 + |v〉) + |w〉, and ii) commutativity, |u〉 +|v〉 = |v〉 + |u〉, of the vector addition, iii) existence of a neutral element 0 (such that|u〉+0 = |u〉), iv) existence for all |v〉 of |w〉 such that |v〉+|w〉 = 0, v) associativity ofscalar multiplication, α(β |u〉) = (αβ) |u〉, vi) equality 1 |v〉 = |v〉 with 1 the identityof C, and distributivities vii) over vector addition, α(|u〉 + |v〉) = α |u〉 + α |v〉, andviii) over scalar addition, (α + β) |u〉 = α |u〉+ β |u〉.

All these axioms are trivial for a physicist and we shall pass quickly over such as-pects of the theory. The mathematical structure withstanding quantum theory is put tothe front because this is largely an abstract theory whose connections to physical “real-ity” or to measurements is not intrinsic to the “quantum state”, whose better definitionremains to be part of a vector space. Note that we write the null vector 0 rather than |0〉,which often will more conveniently refer to a nonzero vector.

This space is further endowed with a norm that defines an inner product for whichthe space is complete and this qualifies it as a Hilbert space. A norm is an applicationNfromH to R∗

+ such that for all α ∈ C and for all |u〉, |v〉 ∈ H,

i N (|u〉) ≥ 0, positivity.ii N (α |u〉) = |α|N (|u〉), scalability.

iii N (|u〉+ |v〉) ≤ N (|u〉) +N (|v〉), triangle inequality.iv N (|u〉) = 0 if and only if |u〉 = 0, positive-definiteness.

The norm provides a notion of length of a vector. The inner product extends to the notionof angles and projections; it is defined as the application from H2 to C, satisfying thefollowing axioms:

i 〈u|v + w〉 = 〈u|v〉+ 〈u|w〉, 〈u + v|w〉 = 〈u|w〉+ 〈v|w〉, additivity.ii 〈u|u〉 ≥ 0, non-negativity.

iii 〈u|u〉 = 0 iff |u〉 = 0, non-degeneracy.iv 〈u|v〉 = 〈v|u〉∗, conjugate symmetry.v 〈u|αv〉 = α〈u|v〉, sesquilinearity.

395

396 LINEAR ALGEBRA

We have contracted an expression that should read⟨ |u〉 , |v〉 + |w〉 ⟩ to 〈u|v + w〉.

This demonstrates the considerable simplifications afforded by Dirac’s notation. Subse-quently, N (|u〉) will be written |〈u|u〉|2.

Completeness is a topological notion of “no-missing point” in the space, it demandsformally that all sequences of points un in the space that converge in the sense of Cauchy(such that whatever is ε > 0, there exists Nε for which N (un, vm) < ε for n, m > N ),have their limit also in the space.

Although infinite-dimensional spaces of functions are also of prime importance,as is the case for instance to describe the harmonic oscillator, we can safely rely onintuition gained from finite-dimensional linear algebra and not study the full theory,known as spectral theory. We cannot spare a few elementary facts of linear algebra,though. We list them now for later reference (if needs be of further details, cf. Halmos,op. cit.)

Let H be a complex Hilbert space of dimension n. There exists a basis of states|φi〉 , i ∈ [1, n], such that any |ψ〉 ∈ H can be written as:

|ψ〉 =

n∑i=1

ci |φi〉 , with ci ∈ C . (A.2)

The basis is said to be orthogonal if 〈φi|φj〉 ∝ δi,j and orthonormal if each state isnormed to unity, |〈φi|φi〉|2 = 1.

Operators on H are linear applications (endomorphisms) from H to H. A generaloperator Ω can be decomposed along the same lines as (A.2) as

Ω =n∑i

n∑j

Ωi,j |φi〉〈φj | , (A.3)

where Ωi,j = 〈φi|Ω |φj〉. In particular, the unity 1 defined such that 1 |ψ〉 = |ψ〉 forall |ψ〉 can be decomposed on basis states as follows (this is an important propertyknown as closure of the identity that we write in the case of a continuum basis also):

1 =n∑

i=1

|φi〉〈φi| , 1 =

∫|φ〉〈φ| dφ , (A.4)

from which one can recover eqn (A.3) by applying it on each side of Ω.The set of applications on the Hilbert space itself forms a vector space. The compo-

sition of applications, denoted with , is defined as follows: let Λ and Ω be two linearapplications onH (endomorphisms), then for every |u〉 ∈ H, the new application Λ Ωis given by:

(Λ Ω) |u〉 = Λ(Ω |u〉) , (A.5)

where Ω |u〉 is another vector |v〉 ∈ H on which Λ is evaluated. Dropping the unneces-sary parentheses on the right-hand side of eqn (A.5), the result reads ΛΩ |u〉 so that thesymbol is dropped for convenience and the composition is noted as a product. When

LINEAR ALGEBRA 397

the applications in a finite-dimensional space are written as matrices, the compositionis effectively the product of these matrices.

Operators are usually not commutative, i.e., ΛΩ = ΩΛ in general. The commutatoris the operator defined as:

[Λ,Ω] = ΛΩ− ΩΛ , (A.6)

and is an important quantity in quantum mechanics. Sometimes the anticommutator isrequired, it is noted with curly brackets:

Λ,Ω = ΛΩ + ΩΛ . (A.7)

For a good and more detailed coverage of the mathematics involved, cf., e.g., Finite-Dimensional Vector Spaces, P. R. Halmos, Springer (1993).

APPENDIX B

SCATTERING RATES OF POLARITON RELAXATION

This appendix complements Chapter 8 with detailed calculations of the rates of polari-ton scattering with phonons, electrons and other polaritons in microcavities.

B.1 Polariton–phonon interaction

The theoretical description of carrier–phonon or of exciton–phonon interaction has re-ceived considerable attention throughout the history of semiconductor heterostructures.Here we present a simplified picture that is, however, well suited to our problem. Cavitypolaritons are two-dimensional particles with only an inplane dispersion. They are scat-tered by phonons that are in the QWs we consider, mainly three-dimensional (acousticphonons) or two-dimensional (optical phonons). Scattering events should conserve thewavevector in the plane. We call q the phonon wavevector and q‖, qz the inplane andz-component of q:

q = (q‖, qz) . (B.1)

ωq will denote the phonon energy.Using Fermi’s golden rule, the scattering rate between two discrete polariton states

of wavevector k and k′ reads:

W phonk→k′ =

2π

∑q

|M(q)|2(θ± + Nq=k−k′+qz)δ(E(k′)− E(k)± ωq) , (B.2)

where Nq is the phonon distribution function and θ± is a quantity whose sign matchesthe one in the delta function corresponding to phonon emission and absorption, and isdefined as θ+ = 1 and θ− = 0. In the case of an equilibrium phonon distribution, Nq

follows the Bose distribution. The sum of eqn (B.2) is over phonon states.M is the matrix element of interaction between phonons and polaritons. If one con-

siders polariton states with a finite energy width γk, the function can be replaced by aLorentzian and eqn (B.2) becomes:

W phonk→k′ =

2π

∑q

|M(q)|2(θ± + Nq=k−k′+qz)

γk′/π

(E(k′)− E(k)± ωq)2 + γ′k/π2

.

(B.3)Wavevector conservation in the plane actually limits the sum of eqn (B.3) to the z-

direction. In the framework of the Born approximation the matrix element of interactionreads:

|M(q)|2 =∣∣∣⟨ψpol

k

∣∣∣Hpol−phonq

∣∣∣ψpolk′

⟩∣∣∣2 = |Xk|2|Xk′ |2 ∣∣〈ψexk |Hex−phon

q |ψexk′ 〉

∣∣2 ,

(B.4)

399

400 SCATTERING RATES OF POLARITON RELAXATION

where∣∣∣ψpol

k

⟩is the polariton wavefunction and |ψex

k 〉 the exciton wavefunction, and Xk

is the exciton Hopfield coefficient (which squares to the exciton fraction). The excitonwavefunction reads:

ψexk (re, rh) = Ue(ze)Uh(zh)

1√S

eik(βere+βhrh)

√2

π

1

a2DB

exp(− |re − rh|

a2DB

), (B.5)

where ze, zh are electron and hole coordinates along the growth axis and re and rh

their coordinates in the plane, Ue and Uh are the electron and hole wavefunctions in thegrowth direction, a2D

B is the two-dimensional exciton Bohr radius, βe,h = me,h/(me +mh).

B.1.1 Interaction with longitudinal optical phonons

This interaction is mainly mediated by the Frolich interaction (Frohlich 1937). In threedimensions the exciton–LO-phonon matrix element reads:

MLO(q) = −e

q

√4πωLO

SLε0

(1

ε∞− 1

εs

)=

MLO0

q√

SL, (B.6)

where ωLO is the energy for creation of a LO-phonon, ε∞ is the optical dielectricconstant, εs the static dielectric constant, L is the dimension along the growth axisand S the normalization area. In two dimensions one should consider confined opti-cal phonons with quantized wavevector in the z-direction. L becomes the QW widthand qm

z = mπ/L with m an integer. Moreover, the overlap integral between excitonand phonon wavefunctions quickly vanishes while m increases. Therefore, we consideronly the first confined phonon state and the matrix element (B.6) becomes:

MLO(q) =MLO

0√|q‖|2 + (π/L)2√

SL. (B.7)

The wavevectors exchanged in the plane are typically much smaller than π/L andeqn (B.7) can be approximated by:

MLO(q) =MLO

0

π

√L

S. (B.8)

Considering a dispersionless phonon dispersion for LO-phonons, the LO-phononcontribution to Eq. (B.3) reads:

W phon−LOk→k′ =

2L

π2S|Xk|2|Xk′ |2|MLO

0 |2×(θ± +

1

exp(−ωLO/kBT )− 1

)γk′

(E(k′)− E(k)± ωLO)2 + γ′2k

. (B.9)

Optical phonons interact very strongly with carriers. They allow fast exciton formation.Their energy of formation ωLO is, however, of the order of 20 to 90 meV, depending

POLARITON–PHONON INTERACTION 401

on the nature of the semiconductor involved. An exciton with a kinetic energy smallerthan 20 meV can no longer emit an optical phonon. The probability of absorbing anoptical phonon remains extremely small at low temperature. This implies that an ex-citon gas cannot cool down to temperature lower than 100–200 K by interacting onlywith optical phonons. Optical phonons are therefore extremely efficient at relaxing ahot-carrier gas (optically or electrically created) towards an exciton gas with a tempera-ture 100–300 K in a few picoseconds. The final cooling of this exciton gas towards thelattice temperature should, however, be assisted by acoustical phonons or other scatter-ing mechanisms. The semiconductor currently used to grow microcavities, and whereoptical phonons play the largest role, is CdTe. In such a material, ωLO is only 21 meV,namely larger than the exciton binding energy in CdTe-based QWs. Moreover, the Rabisplitting is of the order of 10–20 meV in CdTe-based cavities. This means that the directscattering of a reservoir exciton towards the polariton ground-state is a possible processthat may play an important role.

B.1.2 Interaction with acoustic phonons

This interaction is mainly mediated by the deformation potential. The exciton–acousticphonon matrix element reads:

Mac(q) =

√q

µ2ρcsSLG(q‖, qz) , (B.10)

where µ is the reduced mass of electron–hole relative motion, ρ is the density and cs isthe speed of sound in the medium. Assuming isotropic bands, G reads

G(q‖, qz) = DeI⊥e (qz)I

‖e (q‖)DhI⊥h (qz)I

‖h(q‖) ≈ DeI

‖e (q‖)DhI

‖h(q‖) . (B.11)

De, Dh are the deformation coefficients of the conduction and valence band, respec-tively, and I

⊥(‖)e(h) are the overlap integrals between the exciton and phonon mode in the

growth direction and in the plane, respectively:

I‖e(h)(q‖) =

(√2

π

1

a2DB

)2 ∫exp

(− 2r

a2DB

)exp

(i

mh(e)

me + mhq‖ · r

)dr

=

(1 +

(mh(e)q‖a2D

B

2Mx

))2

, (B.12a)

I⊥e(h)(qz) =

∫|fe(h)(z)|2eiqzz dz ≈ 1 . (B.12b)

Using this matrix element and moving to the thermodynamic limit in the growthdirection, the scattering rate (B.3) becomes:

W phonk→k′ =

2π

L

2π|Xk|2|Xk′ |2

∫qz

q

2ρcsSL|G(k− k′)|2(θ± + Nphon

k−k′+qz)

γk′/π

(E(k′)−E(k)± ωk−k′+qz)2 + γ2

k′

. (B.13)


Moving to the thermodynamic limit means that we let the system size in a given di-rection (here the z-direction) go to infinity, substituting the summation with an integral,using the formula

∑qz→ (L/2π)

∫dqz .

Equation (B.13) can be easily simplified:

W phonk→k′ =

|G(k− k′|22πZρcs

|Xk|2|Xk′ |2∫

qz

|k− k′ + qz|(θ± + Nphonk−k′+qz

)

γk′/π

(E(k′)− E(k)± ωk−k′+qz)2 + γ2

k′

. (B.14)

B.2 Polariton–electron interaction

The polariton–electron scattering rate is calculated using Fermi’s golden rule as

W elk→k′ =

2π

∑q

|M elq,k,k′ |2|Xk|2|Xk′ |2N e

q(1−N eq+k′−k)

γk′(E(k′)− E(k) + 2

2me(q2 − |q + k− k′|2)

)2

+ γ2k′

, (B.15)

where N eq is the electron distribution function and me the electron mass. If one considers

electrons at thermal equilibrium, it is given by the Fermi–Dirac electron distributionfunction with a chemical potential

µe = kBT ln(

exp( 2ne

πkBTme

)− 1

), (B.16)

where ne is the electron concentration. M el is the matrix element of interaction betweenan electron and an exciton. A detailed calculation of the electron–exciton matrix elementhas been given by Ramon et al. (2003). M el is composed of a direct contribution and ofan exchange contribution:

M el = M eldir ±M el

exc . (B.17)

The + sign corresponds to a triplet configuration (parallel electron spins) and the −to a singlet configuration (antiparallel electron spins). If both electrons have the samespin, the total exciton spin is conserved through the exchange process. However, if bothelectron spins are opposite, an active exciton state of spin +1, for example, will bescattered towards a dark state of spin +2 through the exchange process. Here, and inwhat follows, we shall consider only the triplet configuration for simplicity.

In order to calculate M el we adopt the Born approximation and obtain:

POLARITON–ELECTRON INTERACTION 403

M eldir =

∫∫∫ψ∗

k(re, rh)f∗q(r′e)

[V(|re − r′e|

)− V

(|rh − r′e|)]

ψk′(re, rh)f∗q+k−k′(r′e)dredrhdre′ ,

(B.18a)

M elexc =

∫∫∫ψ∗

k(re, rh)f∗q(r′e)

[V(|re − r′e|

)− V(|rh − re|

)− V

(|rh − r′e|)]

ψk′(re, rh)f∗q+k−k′(r′e)dredrhdre′ ,

(B.18b)

with Coulomb potential V (r) = e2/(4πεε0r) with ε the dielectric susceptibility ofthe QW.

The free-electron wavefunction f is given by:

fq(r′e) =1√S

eiq · r′e . (B.19)

Integrals (B.18) can be calculated analytically. One finds:

M eldir =

e2

2Sεε0|k− k′|[(1 + ξ2

h)−3/2 − (1 + ξ2e )−3/2

], (B.20)

Mdirdir =

2e2

Sεε0

[(1 + ξ2

c )−3/2 − (1 + 4ξ2h)−3/2(

a−2 + |q− βek′|2)1/2

− (1 + 4ξ2h)−3/2(

a−2 + |k′ − k− q− βek′|2)1/2

],

(B.21)

where

ξe,h =1

2βe,h|k′ − k|a2D

B and ξc = |βek + k′ − k− q|a2DB . (B.22)

Passing to the thermodynamic limit, eqn (B.15) becomes

W elk→k′ =

S

2π

∫q

|M eldir + M el

exc|2|Xk|2|Xk′ |2N eq(1−N e

q+k′−k)

× γk′(E(k′)− E(k) + 2

2me(q2 − |q + k− k′|2)

)2

+ γ2k′

. (B.23)

The polariton–electron interaction is a dipole-charge interaction that takes place on apicosecond time scale. An equilibrium electron gas can thermalize a polariton gas quiteefficiently. A more complex effect may, however, take place such as trion formation orexciton dephasing.


B.3 Polariton–polariton interaction

The polariton–polariton scattering rate reads:

W polk→k′ =

2π

∑q

|M ex|2|Xk|2|Xk′ |2|Xq|2|Xq+k′−k|2Npolq (1 + Npol

q+k′−k)

× γk′(E(k′)− E(k) + E(q + k′ − k)− E(q)

)2

+ γ2k′

. (B.24)

The exciton–exciton matrix element of interaction is also composed of a direct andan exchange term. It has been investigated and calculated by Ciuti et al. (1998) andrecently by Combescot et al. (2007). Here, and in what follows, we use a numericalestimate provided by Tassone and Yamamoto (1999) that we further assume constantover the whole reciprocal space:

Mex ≈ 6(a2D

B )2

SEb =

1

SM0

exc , (B.25)

where Eb is the exciton binding energy. Passing to the thermodynamic limit in the plane,eqn (B.24) becomes:

W polk→k′ =

1

2πS

∫|M0

ex|2|Xk|2|Xk′ |2|Xq|2|Xq+k′−k|2Npolq (1 + Npol

q+k′−k)

× γk′(E(k′)− E(k) + E(q + k′ − k)− E(q)

)2

+ γ2k′

. (B.26)

As one can see, the a priori unknown polariton distribution function is needed to cal-culate scattering rates. This means that in any simulation these scattering rates shouldbe updated dynamically throughout the simulation time, which can be extremely timeconsuming.

Polariton–polariton scattering has been shown to be extremely efficient when a mi-crocavity is resonantly excited. It also plays a fundamental role in the case of non-resonant excitation. Depending on the excitation condition and on the nature of the semi-conductor used, the exciton–exciton interaction may be strong enough to self-thermalizethe exciton reservoir at a given temperature.

B.3.1 Polariton decay

There are three different regions in reciprocal space: [0, ksc], ]ksc, kL] and ]kL,∞[.

• [0, ksc] is the region where the exciton–photon anticrossing takes place. Cavitymirrors reflect the light only within a finite angular cone, which corresponds toan inplane wavevector ksc that depends on the detuning. In this central region thepolariton decay is mainly due to the finite cavity photon lifetime Γk = |Ck|2/τc,where Ck is the photon Hopfield coefficient (which squares to the photon fractionof the polariton) and τc is the cavity photon lifetime. ksc values are typically ofthe order of 4 to 8× 106 m−1 and τc is in the range between 1 and 10 ps.

POLARITON–STRUCTURAL-DISORDER INTERACTION 405

• ]ksc, kL] where kL is the wavevector of light in the medium. In this region excitonsare only weakly coupled to the light and polariton decay is Γk = Γ0, which is theradiative decay rate of QW excitons.

• ]kL,∞[. Beyond kL excitons are no longer coupled to light. They only decaynon-radiatively with a decay rate Γnr. We do not wish to enter into the detailsof the mechanism involved in this decay, which we consider as constant in thewhole reciprocal space. This quantity is given by the decay time measured intime-resolved luminescence experiments, and is typically in the range between100 ps and 1 ns.

B.4 Polariton–structural-disorder interaction

This scattering process is mainly associated with the excitonic part of polaritons. Struc-tural disorder induces coherent elastic (Rayleigh) scattering with a typical timescale ofabout 1ps. It couples very efficiently all polaritons situated on the same “elastic circle”in reciprocal space (see the results of Freixanet et al. (1999) and Langbein and Hvam(2002)). This allows us to simplify the description of polariton relaxation by assum-ing cylindrical symmetry of the polariton distribution function. Also, disorder induces abroadening of the polariton states, which should be accounted for when scattering ratesare calculated.

APPENDIX C

DERIVATION OF THE LANDAU CRITERION OFSUPERFLUIDITY AND LANDAU FORMULA

This appendix presents the derivation of the important “Landau formula” used to esti-mate the critical temperature of the superfluid phase transition.

Let us consider a uniform fluid at zero temperature flowing along a capillary at aconstant velocity v. The only dissipative process assumed is the creation of elementaryexcitations due to the interaction between the fluid and the boundaries of the capillary.If this process is allowed by the conservation laws and Galilean invariance, the flow willdemonstrate viscosity; otherwise it will be superfluid, i.e., dissipativeless. The basic ideaof the derivation is to calculate energy and momentum in the reference frame movingwith the fluid and in the static frame, then making the link between the two frames by aGalilean transformation. If a single excitation with momentum p = k appears, the totalenergy in the moving frame is E = E0 + ε(k), where E0 is the energy of the ground-state and ε(k) is the dispersion of the fluid excitations. In the static frame however, theenergy and momentum of the fluid read:

E = E0 + ε(k) + k ·v +1

2Mv2 , (C.1a)

P = p + Mv , (C.1b)

where M is the total mass of the fluid.Equation (C.1a) shows that the energy of the elementary excitations in the static

system is ε(k) + k ·v. Dissipation is possible, only if the creation of elementary exci-tations is profitable energetically, which means:

ε(k) + k ·v < 0 . (C.2)

Dissipation can therefore take place only if v > ε(k)/(k). On the other hand the fluxis stable if the velocity is smaller than:

vc = min

(ε(k)

k

). (C.3)

Formula (C.3) is the Landau criterion of superfluidity. In the case of a parabolic disper-sion, vc is zero and there is no superfluid motion. In the opposite case of a Bogoliubovdispersion (eqn (8.16)), vc is simply the speed of sound, which means that the fluid canmove without dissipation with any velocity less than the speed of sound.

We now consider the finite-temperature case. In such a situation it is natural to as-sume that part of the thermally excited particles does not remain superfluid. We there-fore consider the coexistence of a superfluid component of velocity vs and a normal

407

408 LANDAU CRITERION OF SUPERFLUIDITY AND FORMULA

component of velocity vn. In the frame moving with the normal fluid, the energy of anelementary excitation reads ε(k)+k · (vs−vn). The occupation number of elementaryexcitations is

fB

(ε(k) + k · (vs − vn)

), (C.4)

where fB is the Bose distribution function, eqn (8.1). The total mass density of the fluidcan be written as ρ = ρn + ρs, where ρn and ρs are the mass densities of the normalfluid and superfluid. At this stage, we will assume that all particles of the fluid have thesame mass m, which is not correct, in principle, for polaritons. In the static frame themass current of the liquid reads:

mj = ρsvs + ρnvn . (C.5)

Following eqn (C.1b), the total momentum of the fluid in the static frame can alsobe written as P = Mvs +

∑i ki where the sum is taken over all the excitations. The

mass current therefore reads:

mj = ρvs +1

S

∑i

ki = ρvs +

(2π)2

∫fB

(ε(k) + k · (vs − vn)

)k dk . (C.6)

Comparing eqns (C.5) and (C.6), one gets:

ρs(vs − vn) =

(2π)2

∫fB

(ε(k) + k · (vs − vn)

)k dk . (C.7)

We are now going to assume that |vs − vn| is small with respect to the speed ofsound and develop eqn (C.4) in power series of vs − vn, which gives:

fB(ε(k) + k · (vs − vn)) ≈ fB(ε(k)) + k · (vs − vn)dfB(ε(k))

dε. (C.8)

We can now insert eqn (C.8) in eqn (C.7) to get:

ρn = − 2

(2π)2

∫dfB(ε(k))

dεk2dk . (C.9)

This formula is known as the Landau formula of superfluidity. It is here written fora 2D system. It is valid only for the case of a parabolic dispersion. In Chapter 8, weuse this result for polaritons, replacing 2k2/(2m) by the polariton dispersion. The for-mula used remains, however, fundamentally inexact since the hypothesis of the constantparticle mass is needed in order to derive eqn (C.9).

APPENDIX D

LANDAU QUANTIZATION AND RENORMALIZATION OF RABISPLITTING

This appendix addresses the renormalization of the exciton binding energy and oscilla-tor strength in a quantum well subjected to an external magnetic field, resulting in thevariation of the Rabi splitting in a microcavity.

Consider an exciton confined in a QW and subject to a magnetic field normal tothe QW plane. Separating the exciton centre of mass motion and relative electron–holemotion in the QW plane, and assuming that the exciton does not move as a whole in theQW plane, we obtain the following exciton Hamiltonian:

H = He + Hh + Hex , (D.1)

where

Hν = − 2

2mν

∂2

∂z2ν

+ Vν(zν)− µBgνsνB + (lν + 1/2)ωνc , ν = e, h,

Hex = − 2

2µ

[1

ρ

∂

∂ρ

(ρ

∂

∂ρ

)− ρ2

4L4

]− e2

4πε0ε√

ρ2 + (ze − zh)2, (D.2)

ρ is the coordinate of electron–hole relative motion in the QW plane, L =√

/(eB)is the so-called magnetic length, B is the magnetic field, se(h) is the electron (hole)spin, me(h) is the electron (hole) effective mass in normal to the plane direction, µ is thereduced mass of electron–hole motion in the QW plane, Ve(h) is the QW potential for an

electron (hole), ge(h) and ωe(h)c are the electron (hole) g-factor and cyclotron frequency,

respectively, l = 0, 1, 2, · · · and ε is the dielectric constant. Hereafter, we neglect theheavy–light hole mixing.

The excitonic Hamiltonian (D.2) was first derived by Russian theorists Gor’kov andDzialoshinskii (1967). It contains a parabolic term dependent on the magnetic field. Ifthe field increases, the magnetic length L decreases, which leads to the shrinkage of thewavefunction of the electron–hole relative motion. Thus, the probability of finding theelectron and hole at the same point increases, leading to an increase of the exciton oscil-lator strength. In order to estimate this effect, let us solve the Schrodinger equation (3.1)variationally for the wavefunction Ψexc(ze, zh, ρ) chosen as an approximate (or trial)expression equal to

Ψexc(ze, zh, ρ) = Ue(ze)Uh(zh)f(ρ); (D.3)

where ze(h) is the electron (hole) coordinate in the direction normal to the plane and ρ isthe coordinate of electron–hole inplane relative motion. If the conduction-band offsets

409

410 LANDAU QUANTIZATION AND RENORMALIZATION OF RABI SPLITTING

are large in comparison to the exciton binding energy, which is the case in conventionalGaAs/AlGaAs QWs, we find Ue,h(ze,h) as a solution of single-particle problems ina rectangular QW. We separate variables in the excitonic Schrodinger equation, andchoose

f(ρ) =

√2

π

1

a⊥e−ρ/a⊥ , (D.4)

where a⊥ is a variational parameter. Substituting this trial function into the Schrodingerequation for electron–hole relative motion with Hamiltonian (D.1), we obtain the exci-ton binding energy

EB = − 3

16

2a2⊥

µL4− 2

2µa2⊥

+4

a2⊥

∫ ∞

0

ρdρe−2ρ/a⊥V (ρ)−(le+1/2)ωec−(lh+1/2)ωh

c ,

(D.5)where

V (ρ) =e2

4πε0ε

∫ ∞

−∞

∫ ∞

−∞dzedzh

U2e (ze)U

2h(zh)√

ρ2 + (ze − zh)2. (D.6)

The parameter a⊥ should maximize the binding energy. Differentiating eqn (D.5)with respect to a⊥ we obtain:

Fig. D.1: Typical “fan-diagram” of an InGaAs/GaAs QW. Circles show the resonances in transmission spectraof the sample associated with the heavy-hole exciton transition. In the limit of strong fields Landau quanti-zation dominates over the Coulomb interaction of electron and hole, and the energies of excitonic transitionsincrease linearly with field. Square and diamond correspond to the light-hole exciton transitions. From Seisyanet al. (2001).

LANDAU QUANTIZATION AND RENORMALIZATION OF RABI SPLITTING 411

2

8µ

[1− a⊥

2L

4]=

∫ ∞

0

ρdρ(1− ρ

a⊥

)e−2ρ/a⊥V (ρ) . (D.7)

The exciton radiative damping rate Γ0, defined in Chapter 3, can be expressed interms of exciton parameters as

Γ0 =ω0

cωLT

√εa2

Ba−2⊥ J2

eh . (D.8)

Here, Jeh =∫

Ue(z)Uh(z) dz, ω0 is the exciton resonance frequency and ωLT

and aB are the longitudinal-transverse splitting and Bohr radius of the bulk exciton,respectively.116 The vacuum-field Rabi splitting in a microcavity is

Ω ∝√

Γ01

a⊥. (D.9)

Shrinkage of the wavefunction of electron–hole relative motion in the magnetic fieldbecomes essential if the magnetic length L is comparable to the exciton Bohr radius aB,i.e., for magnetic fields of about 3 T and more in the case of GaAs QWs. Taking intoaccount the fact that L ≈ 70 A at B = 10 T, the exciton Bohr radius can be realisticallyreduced by a factor of two.

116Typical parameters for GaAs are ωLT = 0.08 meV and aB = 14 nm.

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AUTHORS

Prof. Alexey Kavokin

Chair of Nanophysics and Photonics at the University of Southampton and Marie-CurieChair of Excellence at the University of Rome II. Coordinator of the EU Research-Training network “Physics of microcavities.” One of the world-leading theoreticiansspecializing in the optics of semiconductors and spintronics. Ioffe institute prizes forthe Best scientific work of the year in 1995 and 1999. Previously worked in Russia,France, Italy. Married with two children, his hobbies include writing (published fairytale “Saladine the Cat”, in Russian), chess, drawing.

Prof. Jeremy Baumberg

Director of NanoScience and NanoTechnology at the University of Southampton andProfessor in the Schools of Physics and of Electronics & Computer Science. Estab-lished innovator in NanoPhotonics, opening new areas for exploitation. 2004 Royal So-ciety Mullard Prize, 2004 Mott Lectureship of the Institute of Physics and the CharlesVernon Boys Medal in 2000. Previously worked for Hitachi and IBM and recently spun-out his research into a company, Mesophotonics Ltd. Wide range of research interestsincluding ultrafast coherent control, magnetic semiconductors, ultrafast phonon prop-agation, photonic crystals, single semiconductor quantum dots, semiconductor micro-cavities, and self-assembled photonic and plasmonic nanostructures. Married with twochildren, his hobbies include playing the piano, tennis and making kinetic sculptures.

Dr. Guillaume Malpuech

Researcher at the Centre National de la Recherche Scientifique (CNRS) since 2002.Postdoctoral researcher in Rome and Southampton in 2001 and 2002. Presently head ofthe Nanophotonic group of LASMEA, joint laboratory of CNRS and Universite BlaisePascal (Clermont–Ferrand, France). Authors of about 80 research papers on opticalproperties of semiconductors and of the monograph “cavity polaritons” (with AlexeyKavokin). Among other contributions, he developed with A. Kavokin, F. Laussy andI. Shelykh the theory of Bose–Einstein condensation of cavity polaritons and proposedthe concept of spin-optronic devices. Hobbies include skiing, alpinism and hiking.

Dr. Fabrice P. Laussy

Postdoctoral researcher at the Universidad Autonoma de Madrid (Spain), known forsignificant contributions to the quantum theory of polariton lasers and Bose condensa-tion of exciton-polaritons. PhD of Universite Blaise Pascal, Clermont-Ferrand, 2005,research associate at the University of Sheffield (United Kingdom), 2006. Hobbies in-clude smoking pipe, theatre, poetry.

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430 AUTHORS

The Authors (2006)Alexey KAVOKIN, Jeremy BAUMBERG, Guillaume MALPUECH and Fabrice LAUSSY

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