Atom-PhotonInteractionsBasic Processesand Applications

Claude Cohen-Tannoudj iJacques Dupont-Ro c

Gilbert Grynberg

Preface xxi

Introduction 1

ITRANSITION AMPLITUDES IN ELECTRODYNAMICS

Introduction 5

A. Probability Amplitude Associated with a Physical Process 7

B. Time Dependence of Transition Amplitudes 91. Coupling between Discrete Isolated States 92. Resonant Coupling between a Discrete Level and a Contin -

uum

1 03. Couplings inside a Continuum or between Continua 1 2

C. Application to Electrodynamics 1 51. Coulomb Gauge Hamiltonian 1 52. Expansion in Powers of the Charges q a 163. Expansion in Powers of the Interaction with the Transvers e

Field 1 74. Advantages of Including the Coulomb Interaction in the

Particle Hamiltonian 1 85. Diagrammatic Representation of Transition Amplitudes 1 9

COMPLEMENT A I -PERTURBATIVE CALCULATION O FTRANSITION AMPLITUDES -SOME USEFUL RELATIONS

Introduction.

, , . :,, : . , ,..: .

: 23

1. Interaction Representation

,- : ., : ;~ : 23

2. Perturbative Expansion of Transition Amplitudes-a . Pertur-bative Expansion of the Evolution Operator. b. First-Orde rTransition Amplitude. c. Second-Order Transition Amplitude . . ,

253. Transition Probability-a. Calculation of the Transition Proba-

bility to a Final State Different from the Initial State. b. Transi-tion Probability between Two Discrete States. Lowest-Orde rCalculation. c. Case Where the Final State Belongs to an En-ergy Continuum. Density of States. d. Transition Rate toward aContinuum of Final States. e. Case Where both the Initial an dFinal States Belong to a Continuum

3 1

COMPLEMENT B 1 -DESCRIPTION OF THE EFFECT OF APERTURBATION BY AN EFFECTIVE HAMILTONIA N

1. Introduction-Motivation 382. Principle of the Method

4 13. Determination of the Effective Hamiltonian-a . Iterative Cal-

culation of S. b. Expression of the Second-Order Effectiv eHamiltonian . c. Higher-Order Terms 43

4. Case of Two Interacting Systems 46

COMPLEMENT C 1 -DISCRETE LEVEL COUPLED TO A BROA DCONTINUUM: A SIMPLE MODE L

Introduction 49

1. Description of the Model-a . The Discrete State and the Con-tinuum. b. Discretization of the Continuum. c. SimplifyingAssumptions 50

2. Stationary States of the System . Traces of the Discrete State inthe New Continuum-a. The Eigenvalue Equation. b. GraphicDetermination of the New Eigenvalues. c. Probability Density ofthe Discrete State in the New Continuum 5 1

3. A Few Applications of This Simple Model-a . Decay of theDiscrete Level. b. Excitation of the System in the Discrete Leve lfrom Another State . c. Resonant Scattering through a Discret eLevel. d. Fano Profiles

564. Generalization to More Realistic Continua . Diagonalization of

the Hamiltonian without Discretization 64

IIA SURVEY OF SOME INTERACTION PROCESSE S

BETWEEN PHOTONS AND ATOM S

Introduction : :,k : ;

; , : 67

A. Emission Process: A New Photon Appears 691. Spontaneous Emission between Two Discrete Atomic Lev-

els . Radiative Decay of an Excited Atomic State-a . Dia-grammatic Representation . b. Spontaneous Emission Rate.c. Nonperturbative Results 69

2. Spontaneous Emission between a Continuum State anda Discrete State-a. First Example: Radiative Capture.b . Second' Example: Radiative Dissociation of a Molecule 73

3. Spontaneous Emission between Two States of the Ioniza-tion Continuum-Bremsstrahlung 76

B. Absorption Process : A Photon Disappears 781. Absorption between Two Discrete States 782. Absorption between a Discrete State and a Continuum

State-a. First Example: Photoionization. b. Second Exam-ple : Photodissociation 79

3. Absorption between Two States of the Ionization Contin-uum : Inverse Bremsstrahlung 82

4. Influence of the Initial State of the Field on the Dynamicsof the Absorption Process 83

C. Scattering Process : A Photon Disappears and Another Photo nAppears 861. Scattering Amplitude-Diagrammatic Representation 862. Different Types of Photon Scattering by an Atomic or

Molecular System-a. Low-Energy Elastic Scattering:Rayleigh Scattering. b. Low-Energy Inelastic Scattering:Raman Scattering. c. High-Energy Elastic Scattering : Thom -son Scattering. d. High-Energy Inelastic Scattering with th eFinal Atomic State in the Ionization Continuum: Compto nScattering 88

3. Resonant Scattering 93

D. Multiphoton Processes : Several Photons Appear or Disappear

981. Spontaneous Emission of Two Photons 982. Multiphoton Absorption (and Stimulated Emission) be -

tween Two Discrete Atomic States 100

3. Multiphoton Ionization 1024. Harmonic Generation 1045. Multiphoton Processes and Quasi-Resonant Scattering

.

10 6

E. Radiative Corrections : Photons Are Emitted and Reabsorbe d(or Absorbed and Reemitted) 1091. Spontaneous Radiative Corrections-a . Case of a Free Elec -

tron: Mass Correction. b. Case of an Atomic Electron: Natu-ral Width and Radiative Shift 109

2. Stimulated Radiative Corrections 114

F. Interaction by Photon Exchange 11 81. Exchange of Transverse Photons between Two Charge d

Particles : First Correction to the Coulomb Interaction 11 82. Van der Waals Interaction between Two Neutral Atoms -

a . Small Distance : D « A ab . b . Large Distance Aab « D

12 1

COMPLEMENT A II -PHOTODETECTION SIGNALS AND

CORRELATION FUNCTIONS

Introduction 127

1. Simple Models of Atomic Photodetectors-a . Broadband Pho-todetector. b. Narrow-Band Photodetector 128

2.. Excitation Probability and Correlation Functions- -a . Hamilto-nian. Evolution Operator. b. Calculation of the Probability Tha tthe Atom Has Left the Ground State after a Time dt.c . Atomic Dipole Correlation Function. d. Field Correlatio nFunction 129

3. Broadband Photodetection-a . Condition on the CorrelationFunctions. b. Photoionization Rate 137

4. Narrow-Band Photodetection-a . Conditions on the Inciden tRadiation and on the Detector. b. Excitation by a Broadban dSpectrum. c. Influence of the Natural Width of the ExcitedAtomic Level 139

5. Double Photodetection Signals-a. Correlation between TwoPhotodetector Signals. b. Sketch of the Calculation of w1, 143

COMPLEMENT B 11 -RADIATIVE CORRECTIONS IN THE

PAULI-FIERZ REPRESENTATIO N

Introduction 147

1. The Pauli-Fierz Transformation-a . Simplifying Assumptions.b. Transverse Field Tied to a Classical Particle. c . Determinationof the Pauli-Fierz Transformation 148

2 . The Observables in the New Picture-a . Transformation of theTransverse Fields. b. Transformation of the Particle Dynamica lVariables. c. Expression for the New Hamiltonian 15 2

3 . Physical Discussion-a . Mass Correction. b. New InteractionHamiltonian between the Particle and the Transverse Field .c. Advantages of the New Representation . d. Inadequacy of theConcept ofa Field Tied to a Particle 15 7

IIINONPERTURBATIVE CALCULATION OF TRANSITIO N

AMPLITUDES

Introduction 165

A. Evolution Operator and Resolvent 16 71. Integral Equation Satisfied by the Evolution Operator 16 72. Green 's Functions-Propagators 16 73. Resolvent of the Hamiltonian 17 0

B. Formal Resummation of the Perturbation Series 17 21. Diagrammatic Method Explained on a Simple Model 17 22. Algebraic Method Using Projection Operators-a . Projector

onto a Subspace g'o of the Space of States. b. Calculation ofthe Projection of the Resolvent in the Subspace X0. c. Calcu-lation of Other Projections of G(z) . d. Interpretation of theLevel-Shift Operator 174

3. Introduction of Some Approximations-a. Perturbative Cal-culation of the Level-Shift Operator. Partial Resummation ofthe Perturbation Series. b. Approximation Consisting of Ne-glecting the Energy Dependence of the Level-Shift Operator . . .

17 9

C . Study of a Few Examples 18 31 . Evolution of an Excited Atomic State-a . Nonperturbative

Calculation of the Probability Amplitude That the Atom Re-

mains Excited. b. Radiative Lifetime and Radiative Level

Shift. c. Conditions of Validity for the Treatment of the TwoPreceding Subsections 18 3

2. Spectral Distribution of Photons Spontaneously Emitted byan Excited Atom-a . Relevant Matrix Element of the Resol-vent Operator. b. Generalization to a Radiative Cascade.c. Natural Width and Shift of the Emitted Lines 18 9

3 . Indirect Coupling between a Discrete Level and a Contin-uum . Example of the Lamb Transition-a . Introducingthe Problem. b. Nonperturbative Calculation of the Transi-tion Amplitude. c. Weak Coupling Limit. Bethe Formula .d. Strong Coupling Limit. Rabi Oscillation 197

4 . Indirect Coupling between Two Discrete States . Multi -photon Transitions-a . Physical Process and Subspace 8'0of Relevant States. b. Nonperturbative Calculation of theTransition Amplitude. c. Weak Coupling Case. Two-Photo nExcitation Rate. d. Strong Coupling Limit. Two-PhotonRabi Oscillation . e. Higher-Order Multiphoton Transitions .f. Limitations of the Foregoing Treatment 205

COMPLEMENT A III-ANALYTIC PROPERTIES OF THERESOLVEN T

Introduction 213

1. Analyticity of the Resolvent outside the Real Axis 2132. Singularities on the Real Axis 2153. Unstable States and Poles of the Analytic Continuation of th e

Resolvent 2174. Contour Integral and Corrections to the Exponential Decay 220

COMPLEMENT B III -NONPERTURBATIVE EXPRESSIONS FO R

THE SCATTERING AMPLITUDES OF A PHOTON BY AN ATO M

Introduction 222

1 . Transition Amplitudes between Unperturbed States-a . Usingthe Resolvent. b. Transition Matrix. c. Application to Reso -nant Scattering. d. Inadequacy of Such an Approach 222

2, Introducing Exact Asymptotic States-a. The Atom in the Ab -

sence of Free Photons. b. The Atom in the Presence of a Free

Photon 229

3 . Transition Amplitude between Exact Asymptotic States -

a . New Definition of the S-Matrix. b. New Expression for th e

Transition Matrix . Physical Discussion

, .. .

. : 233

COMPLEMENT CIII-DISCRETE STATE COUPLED TO A

FINITE-WIDTH CONTINUUM : FROM THE WEISSKOPF -WIGNE R

EXPONENTIAL DECAY TO THE RABI OSCILLATIO N

1. Introduction-Overview 239

2. Description of the Model-a . Unperturbed States. b. Assump -

tions concerning the Coupling. c. Calculation of the Resolvent

and of the Propagators. d. Fourier Transform of the Amplitude

Ub (T) 240

3. The Important Physical Parameters-a. The Function Eb(E) .

b . The Parameter f2 1 Characterizing the Coupling of the DiscreteState with the Whole Continuum. c. The Function d b(E) . . . . 244

4. Graphical Discussion-a . Construction of the Curve 2/b(E) .

b . Graphical Determination of the Maxima of 2/b(E) . Classifica -

tion of the Various Regimes 246

5. Weak Coupling Limit-a . Weisskopf-Wigner Exponential De -

cay. b . Corrections to the Exponential Decay 249

6. Intermediate Coupling. Critical Coupling-a . Power Expansion

of Fib ( E) near a Maximum. b. Physical Meaning of the Critica l

Coupling : . .. . 25 1

7. Strong Coupling 25 3

IVRADIATION CONSIDERED AS A RESERVOIR: MASTER

EQUATION FOR THE PARTICLE S

A. Introduction-Overview 257

B. Derivation of the Master Equation for a Small System .sadInteracting with a Reservoir .9? 26 2

1 . Equation Describing the Evolution of the Small System i nthe Interaction Representation 262

2. Assumptions Concerning the Reservoir-a . State of theReservoir. b. One-Time and Two-Time Averages for theReservoir Observables 26 3

3. Perturbative Calculation of the Coarse-Grained Rate o fVariation of the Small System 266

4. Master Equation in the Energy-State Basis 269

C. Physical Content of the Master Equation 2721. Evolution of Populations 2722. Evolution of Coherences 274

D. Discussion of the Approximations 27 81. Order of Magnitude of the Evolution Time for d 27 82. Condition for Having Two Time Scales 27 83. Validity Condition for the Perturbative Expansion 2794. Factorization of the Total Density Operator at Time t 2805. Summary 28 1

E. Application to a Two-Level Atom Coupled to the Radiatio nField 2821. Evolution of Internal Degrees of Freedom-a. Master

Equation Describing Spontaneous Emission for a Two-LevelAtom. b. Additional Terms Describing the Absorption an dInduced Emission of a Weak Broadband Radiation 282

2. Evolution of Atomic Velocities-a . Taking into Accoun tthe Translational Degrees of Freedom in the Master Equation .b . Fokker-Planck Equation for the Atomic Velocity Distribu-tion Function. c. Evolutions of the Momentum Mean Valueand Variance. d. Steady-State Distribution . ThermodynamicEquilibrium 289

COMPLEMENT A I v-FLUCTUATIONS AND LINEAR RESPONS EAPPLICATION TO RADIATIVE PROCESSES

Introduction 30 2

1 . Statistical Functions and Physical Interpretation of the Maste rEquation-a . Symmetric Correlation Function. b. Linear Sus-

ceptibility. c. Polarization Energy and Dissipation. d. Physica lInterpretation of the Level Shifts. e. Physical Interpretation ofthe Energy Exchanges 302

2 . Applications to Radiative Processes-a . Calculation of the Sta-tistical Functions. b. Physical Discussion. c. Level Shifts du eto the Fluctuations of the Radiation Field. d. Level Shifts due toRadiation Reaction . e. Energy Exchanges between the Atom andthe Radiation 31 2

COMPLEMENT B 1v-MASTER EQUATION FOR A DAMPED

HARMONIC OSCILLATO R

1. The Physical System 3222. Operator Form of the Master Equation 3233. Master Equation in the Basis of the Eigenstates of HA-

a. Evolution of the Populations. b. Evolution of a Few AverageValues 326

4. Master Equation in a Coherent State Basis-a . Brief Review ofCoherent States and the Representation P N of the Density Opera -tor. b. Evolution Equation for PN(ß, ß*, t). c. Physical Dis-cussion

329

COMPLEMENT Cw-QUANTUM LANGEVIN EQUATION SFOR A SIMPLE PHYSICAL SYSTEM

Introduction 33 4

1. Review of the Classical Theory of Brownian Motion-a. Langevin Equation. b. Interpretation of the Coefficient D.Connection between Fluctuations and Dissipation. c. A FewCorrelation Functions 33 4

2. Heisenberg-Langevin Equations for a Damped Harmonic Os-cillator-a . Coupled Heisenberg Equations. b. The QuantumLangevin Equation and Quantum Langevin Forces. c. Connec-tion between Fluctuations and Dissipation . d. Mixed Two-TimeAverages Involving Langevin Forces and Operators of d.e. Rate of Variation of the Variances %/N and YA . f General-ization of Einstein's Relation . g. Calculation of Two-Time Aver-ages for Operators of .i. Quantum Regression Theorem 340

VOPTICAL BLOCH EQUATION S

Introduction 35 3

A. Optical Bloch Equations for a Two-Level Atom 35 51. Description of the Incident Field 35 52. Approximation of Independent Rates of Variation 35 63. Rotating-Wave Approximation-a . Elimination of Antireso -

nant Terms. b. Time-Independent Form of the Optical BlochEquations. c. Other Forms of the Optical Bloch Equations . . . 357

4. Geometric Representation in Terms of a Fictitious Spin . . . 36 1

B. Physical Discussion-Differences with Other Evolution Equa -tions 36 41. Differences with Relaxation Equations . Couplings between

Populations and Coherences

3642. Differences with Hamiltonian Evolution Equations 3643. Differences with Heisenberg-Langevin Equations 36 5

C. First Application-Evolution of Atomic Average Values 36 71. Internal Degrees of Freedom-a . Transient Regime.

b. Steady-State Regime. c. Energy Balance. Mean Number ofIncident Photons Absorbed per Unit Time 36 7

2. External Degrees of Freedom . Mean Radiative Forces-a. Equation of Motion of the Center of the Atomic WavePacket. b. The Two Types of Forces for an Atom Initially a tRest. c. Dissipative Force . Radiation Pressure. d. ReactiveForce . Dipole Force 37 0

D. Properties of the Light Emitted by the Atom 37 91. Photodetection Signals . One- and Two-Time Averages of

the Emitting Dipole Moment-a . Connection between theRadiated Field and the Emitting Dipole Moment . b. Expres-sion of Photodetection Signals 37 9

2. Total Intensity of the Emitted Light-a . Proportionality tothe Population of the Atomic Excited State. b. CoherentScattering and Incoherent Scattering. c. Respective Contribu-tions of Coherent and Incoherent Scattering to the Tota lIntensity Emitted in Steady State 38 2

3. Spectral Distribution of the Emitted Light in Steady

State-a. Respective Contributions of Coherent and Incoher-ent Scattering. Elastic and Inelastic Spectra . b. Outline ofthe Calculation of the Inelastic Spectrum. c. Inelastic Spec-trum in a Few Limiting Cases 38 4

COMPLEMENT Av-BLOCH-LANGEVIN EQUATIONS AN D

QUANTUM REGRESSION THEOREM

Introduction 38 8

1. Coupled Heisenberg Equations for the Atom and the Field-a. Hamiltonian and Operator Basis for the System. b . EvolutionEquations for the Atomic and Field Observables . c. Rotating-Wave Approximation. Change of Variables . d. Comparison withthe Harmonic Oscillator Case 38 8

2. Derivation of the Heisenberg-Langevin Equations-a . Choiceof the Normal Order. b. Contribution of the Source Field.c . Summary. Physical Discussion 394

3. Properties of Langevin Forces-a . Commutation Relationsbetween the Atomic Dipole Moment and the Free Field. b. Cal-culation of the Correlation Functions of Langevin Forces .c . Quantum Regression Theorem. d. Generalized EinsteinRelations 39 8

VITHE DRESSED ATOM APPROACH

A. Introduction: The Dressed Atom 407

B. Energy Levels of the Dressed Atom 4101. Model of the Laser Beam 41 02. Uncoupled States of the Atom + Laser Photons System 41 23. Atom-Laser Photons Coupling-a . Interaction Hamiltonian.

b. Resonant and Nonresonant Couplings . c. Local Periodic-ity of the Energy Diagram. d. Introduction of the RabiFrequency 413

4. Dressed States-a. Energy Levels and Wave Functions.b. Energy Diagram versus hco L 41 5

5. Physical Effects Associated with Absorption and Induce dEmission 417

C. Resonance Fluorescence Interpreted as a Radiative Cascade ofthe Dressed Atom .

4191. The Relevant Time Scales 4192. Radiative Cascade in the Uncoupled Basis-a . Time Evolu-

tion of the System. b. Photon Antibunching. c. Time Inter-vals between Two Successive Spontaneous Emissions 420

3. Radiative Cascade in the Dressed State Basis-a. AllowedTransitions between Dressed States. b. Fluorescence Triplet.c . Time Correlations between Frequency Filtered FluorescencePhotons 423

D. Master Equation for the Dressed Atom 4271. General Form of the Master Equation-a . Approximation of

Independent Rates of Variation. b. Comparison with Optica lBloch Equations 42 7

2. Master Equation in the Dressed State Basis in the SecularLimit-a . Advantages of the Coupled Basis in the SecularLimit. b. Evolution of Populations. c. Evolution of Coher-ences-Transfer of Coherences. d. Reduced Populations an dReduced Coherences

4293. Quasi-Steady State for the Radiative Cascade-a . Initial Den-

sity Matrix. b. Transient Regime and Quasi-Steady State 435

E. Discussion of a Few Applications 4371. Widths and Weights of the Various Components of th e

Fluorescence Triplet-a . Evolution of the Mean Dipole Mo-ment. b. Widths and Weights of the Sidebands . c. Structureof the Central Line

4372. Absorption Spectrum of a Weak Probe Beam-a . Physical

Problem. b. Case Where the Two Lasers Are Coupled to th eSame Transition. c. Probing on a Transition to a ThirdLevel. The Autler-Townes Effect 442

3. Photon Correlations-a. Calculation of the Photon-Correla-tion Signal. b. Physical Discussion. c. Generalization to aThree-Level System : Intermittent Fluorescence 446

4. Dipole Forces-a. Energy Levels of the Dressed Atom in a

Spatially Inhomogeneous Laser Wave . b. Interpretation of theMean Dipole Force. c. Fluctuations of the Dipole Force . . . . 45 4

COMPLEMENT Av I -THE DRESSED ATOM IN THERADIO-FREQUENCY DOMAI N

Introduction 460

1. Resonance Associated with a Level Crossing or Anti-crossing-a . Anticrossing for a Two-Level System. b. Higher-Order Anticrossing. c. Level Crossing. Coherence Resonance . . . 46 1

2. Spin Dressed by Radio-Frequency Photons-a. Description ofthe System. b. Interaction Hamiltonian between the Atom an dthe Radio-Frequency Field. c. Preparation and Detection 468

3. The Simple Case of Circularly Polarized Photons-a . EnergyDiagram. b. Magnetic Resonance Interpreted as a Level-Anti-crossing Resonance of the Dressed Atom. c. Dressed StateLevel-Crossing Resonances 473

4. Linearly Polarized Radio-Frequency Photons-a . Survey of theNew Effects. b. Bloch-Siegert Shift. c. The Odd Spectrum ofLevel Anticrossing Resonances. d. The Even Spectrum ofLevel-Crossing Resonances . e. A Nonperturbative Calculation:The Lande Factor of the Dressed Atom . f. Qualitative Evolu-tion of the Energy Diagram at High Intensity 479

COMPLEMENT BvI -COLLISIONAL PROCESSES IN TH EPRESENCE OF LASER IRRADIATION

Introduction , 49 0

1. Collisional Relaxation in the Absence of Laser Irradiation-a . Simplifying Assumptions. b. Master Equation Describing theEffect of Collisions on the Emitting Atom 49 1

2. Collisional Relaxation in the Presence of Laser Irradiation-a . The Dressed Atom Approach . b. Evolution of Populations :Collisional Transfers between Dressed States. c. Evolution ofCoherences . Collisional Damping and Collisional Shift. d. Ex-plicit Form of the Master Equation in the Impact Limit 494

3. Collision-Induced Modifications of the Emission and Absorp-tion of Light by the Atom. Collisional Redistribution-a. Tak-ing into Account Spontaneous Emission. b. Reduced Steady-State Populations. c. Intensity of the Three Components of theFluorescence Triplet. d. Physical Discussion in the Limit ,(l, <<145 [, 1

T,--A 50 14. Sketch of the Calculation of the Collisional Transfer Rate-

a. Expression of the Transfer Rate as a Function of the Collisio nS-Matrix. b. Case Where the Laser Frequency Becomes Resonan tduring the Collision. Limit of Large Detunings 51 0

EXERCISES

1. Calculation of the Radiative Lifetime of an Excited Atomi cLevel . Comparison with the Damping Time of a Classica lDipole Moment 515

2. Spontaneous Emission of Photons by a Trapped Ion .Lamb-Dicke Effect 51 8

3. Rayleigh Scattering 5244. Thomson Scattering 5275. Resonant Scattering 5306. Optical Detection of a Level Crossing between Two Excite d

Atomic States 5337. Radiative Shift of an Atomic Level . Bethe Formula for the

Lamb Shift 5378. Bremsstrahlung . Radiative Corrections to Elastic Scattering b y

a Potential 5489. Low-Frequency Bremsstrahlung . Nonperturbative Treatment

of the Infrared Catastrophe 55 710. Modification of the Cyclotron Frequency of a Particle due t o

Its Interactions with the Radiation Field 56411. Magnetic Interactions between Spins 57 112. Modification of an Atomic Magnetic Moment due to Its Cou -

pling with Magnetic Field Vacuum Fluctuations 57613. Excitation of an Atom by a Wave Packet : Broadband Excita -

tion and Narrow-Band Excitation 58014. Spontaneous Emission by a System of Two Neighboring Atoms .

Superradiant and Subradiant States 58 515. Radiative Cascade of a Harmonic Oscillator 58 916. Principle of the Detailed Balance 596

17. Equivalence between a Quantum Field in a Coherent Stat eand an External Field 597

18. Adiabatic Elimination of Coherences and Transformation o fOptical Bloch Equations into Relaxation Equations 601.

19. Nonlinear Susceptibility for an Ensemble of Two-Level Atoms .A Few Applications 60 4

20. Absorption of a Probe Beam by Atoms Interacting with a nIntense Beam. Application to Saturated Absorption 608

APPENDIXQUANTUM ELECTRODYNAMICS IN THE COULOMBGAUGE-SUMMARY OF THE ESSENTIAL RESULT S

1. Description of the Electromagnetic Field-a . Electric Field Eand Magnetic Field B . b. Vector Potential A and Scalar Poten-tial U. c. Coulomb Gauge. d. Normal Variables. e. Principleof Canonical Quantization in the Coulomb Gauge. f QuantumFields in the Coulomb Gauge 62 1

2. Particles 62 83. Hamiltonian and Dynamics in the Coulomb Gauge-a . Hamil-

tonian. b . Unperturbed Hamiltonian and Interaction Hamilto -nian. c . Equations of Motion 629

4. State Space 63 35. The Long-Wavelength Approximation and the Electric Dipole

Representation-a. The Unitary Transformation. b. The Phys -ical Variables in the Electric Dipole Representation . c. TheDisplacement Field. d. Electric Dipole Hamiltonian 63 5

References

, 64 1

Index

. :~ - 645