Fleischhauer EIT Review Rmp

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Electromagnetically induced transparency: Optics in coherent media Michael Fleischhauer Fachbereich Physik, Technische Universität Kaiserslautern, D-67663 Kaiserslautern, Germany Atac Imamoglu Institute of Quantum Electronics, ETH-Hönggerberg, HPT G12, CH-8093 Zürich, Switzerland Jonathan P. Marangos Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College, London SW7 2BW, United Kingdom Published 12 July 2005 Coherent preparation by laser light of quantum states of atoms and molecules can lead to quantum interference in the amplitudes of optical transitions. In this way the optical properties of a medium can be dramatically modified, leading to electromagnetically induced transparency and related effects, which have placed gas-phase systems at the center of recent advances in the development of media with radically new optical properties. This article reviews these advances and the new possibilities they offer for nonlinear optics and quantum information science. As a basis for the theory of electromagnetically induced transparency the authors consider the atomic dynamics and the optical response of the medium to a continuous-wave laser. They then discuss pulse propagation and the adiabatic evolution of field-coupled states and show how coherently prepared media can be used to improve frequency conversion in nonlinear optical mixing experiments. The extension of these concepts to very weak optical fields in the few-photon limit is then examined. The review concludes with a discussion of future prospects and potential new applications. CONTENTS I. Introduction 633 II. Physical Concepts Underlying Electromagnetically Induced Transparency 635 A. Interference between excitation pathways 635 B. Dark state of the three-level -type atom 637 III. Atomic Dynamics and Optical Response 638 A. Master equation and linear susceptibility 638 B. Effective Hamiltonian and dressed-state picture 642 C. Transmission through a medium with EIT 643 D. Dark-state preparation and Raman adiabatic passage 644 E. EIT inside optical resonators 645 F. Enhancement of refractive index, magnetometry, and lasing without inversion 646 IV. EIT and Pulse Propagation 647 A. Linear response: Slow and ultraslow light 647 B. “Stopping of light” and quantum memories for photons 650 C. Nonlinear response: Adiabatic pulse propagation and adiabatons 652 V. Enhanced Frequency Conversion 654 A. Overview of atomic coherence-enhanced nonlinear optics 654 B. Nonlinear mixing and frequency up-conversion with electromagnetically induced transparency 657 C. Nonlinear optics with maximal coherence 659 D. Four-wave mixing in double- systems 660 VI. EIT with Few Photons 661 A. Giant Kerr effect 661 B. Cross-phase modulation using single-photon pulses with matched group velocities 664 C. Few-photon four-wave mixing 665 D. Few-photon cavity EIT 666 VII. Summary and Perspectives 667 Acknowledgments 670 References 670 I. INTRODUCTION Advances in optics have frequently arisen through the development of new materials with optimized optical properties. For instance, the introduction of new optical crystals in the 1970s and 1980s led to substantial in- creases in nonlinear optical conversion efficiencies to the ultraviolet UV. Another example has been the devel- opment of periodically poled crystals that permit qua- siphase matching in otherwise poorly phase-matched nonlinear optical processes, which greatly enhance non- linear frequency-mixing efficiencies. As we shall see, co- herent preparation is a new avenue that produces re- markable changes in the optical properties of a gas- phase atomic or molecular medium. The cause of the modified optical response of an atomic medium in this case is the laser-induced coher- ence of atomic states, which leads to quantum interfer- ence between the excitation pathways that control the optical response. We can in this way eliminate the ab- sorption and refraction linear susceptibility at the reso- nant frequency of a transition. This was termed electro- magnetically induced transparency EIT by Harris and REVIEWS OF MODERN PHYSICS, VOLUME 77, APRIL 2005 0034-6861/2005/772/63341/$50.00 ©2005 The American Physical Society 633

Transcript of Fleischhauer EIT Review Rmp

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Electromagnetically induced transparency: Optics in coherent media

Michael Fleischhauer

Fachbereich Physik, Technische Universität Kaiserslautern, D-67663 Kaiserslautern,Germany

Atac Imamoglu

Institute of Quantum Electronics, ETH-Hönggerberg, HPT G12, CH-8093 Zürich,Switzerland

Jonathan P. Marangos

Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College, LondonSW7 2BW, United Kingdom

�Published 12 July 2005�

Coherent preparation by laser light of quantum states of atoms and molecules can lead to quantuminterference in the amplitudes of optical transitions. In this way the optical properties of a medium canbe dramatically modified, leading to electromagnetically induced transparency and related effects,which have placed gas-phase systems at the center of recent advances in the development of mediawith radically new optical properties. This article reviews these advances and the new possibilities theyoffer for nonlinear optics and quantum information science. As a basis for the theory ofelectromagnetically induced transparency the authors consider the atomic dynamics and the opticalresponse of the medium to a continuous-wave laser. They then discuss pulse propagation and theadiabatic evolution of field-coupled states and show how coherently prepared media can be used toimprove frequency conversion in nonlinear optical mixing experiments. The extension of theseconcepts to very weak optical fields in the few-photon limit is then examined. The review concludeswith a discussion of future prospects and potential new applications.

CONTENTS

I. Introduction 633

II. Physical Concepts Underlying Electromagnetically

Induced Transparency 635

A. Interference between excitation pathways 635

B. Dark state of the three-level �-type atom 637

III. Atomic Dynamics and Optical Response 638

A. Master equation and linear susceptibility 638

B. Effective Hamiltonian and dressed-state picture 642

C. Transmission through a medium with EIT 643

D. Dark-state preparation and Raman adiabatic passage 644

E. EIT inside optical resonators 645

F. Enhancement of refractive index, magnetometry,

and lasing without inversion 646

IV. EIT and Pulse Propagation 647

A. Linear response: Slow and ultraslow light 647

B. “Stopping of light” and quantum memories for

photons 650

C. Nonlinear response: Adiabatic pulse propagation

and adiabatons 652

V. Enhanced Frequency Conversion 654

A. Overview of atomic coherence-enhanced nonlinear

optics 654

B. Nonlinear mixing and frequency up-conversion with

electromagnetically induced transparency 657

C. Nonlinear optics with maximal coherence 659

D. Four-wave mixing in double-� systems 660

VI. EIT with Few Photons 661

A. Giant Kerr effect 661

B. Cross-phase modulation using single-photon pulseswith matched group velocities 664

C. Few-photon four-wave mixing 665D. Few-photon cavity EIT 666

VII. Summary and Perspectives 667Acknowledgments 670References 670

I. INTRODUCTION

Advances in optics have frequently arisen through thedevelopment of new materials with optimized opticalproperties. For instance, the introduction of new opticalcrystals in the 1970s and 1980s led to substantial in-creases in nonlinear optical conversion efficiencies to theultraviolet �UV�. Another example has been the devel-opment of periodically poled crystals that permit qua-siphase matching in otherwise poorly phase-matchednonlinear optical processes, which greatly enhance non-linear frequency-mixing efficiencies. As we shall see, co-herent preparation is a new avenue that produces re-markable changes in the optical properties of a gas-phase atomic or molecular medium.

The cause of the modified optical response of anatomic medium in this case is the laser-induced coher-ence of atomic states, which leads to quantum interfer-ence between the excitation pathways that control theoptical response. We can in this way eliminate the ab-sorption and refraction �linear susceptibility� at the reso-nant frequency of a transition. This was termed electro-magnetically induced transparency �EIT� by Harris and

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co-workers �Harris et al., 1990�. The importance of EITstems from the fact that it gives rise to greatly enhancednonlinear susceptibility in the spectral region of inducedtransparency of the medium and is associated with steepdispersion. For some readable general accounts of ear-lier work in the field, see, for example, Harris �1997� orScully �1992�. Other more recent reviews on specific as-pects of EIT and its applications can be found in thearticles of Lukin, Hemmer, and Scully �2000�; Matsko,Kocharovskaya, et al. �2001�; Vitanov et al. �2001�, aswell as in the topical Colloquium by Lukin �2003�. Itshould be emphasized that the modification of atomicproperties due to quantum interference has been studiedextensively for 25 years; see, for example, Arimondo�1996�. In particular, the phenomenon of coherent popu-lation trapping �CPT� observed by Alzetta et al. �1976� isclosely related to EIT. In contrast to CPT which is a“spectroscopic” phenomenon that involves only modifi-cations to the material states in an optically thin sample,EIT is a phenomenon specific to optically thick media inwhich both the optical fields and the material states aremodified.

The optical properties of atomic and molecular gasesare fundamentally tied to their intrinsic energy-levelstructure. The linear response of an atom to resonantlight is described by the first-order susceptibility ��1�.The imaginary part of this susceptibility Im���1�� deter-mines the dissipation of the field by the atomic gas �ab-sorption�, while the real part Re���1�� determines the re-fractive index. The form of Im���1�� at a dipole-allowedtransition as a function of frequency is that of a Lorent-zian function with a width set by the damping. The re-fractive index Re���1�� follows the familiar dispersionprofile, with anomalous dispersion �decrease in Re���1��with field frequency� in the central part of the absorp-tion profile within the linewidth. Figure 1 illustrates boththe conventional form of ��1� and the modified form thatresults from EIT, as will be discussed shortly.

In the case of laser excitation where the magnitude ofthe electric field can be very large we reach the situationwhere the interaction energy of the laser coupling di-vided by � exceeds the characteristic linewidth of thebare atom. In this case the evolution of the atom-fieldsystem requires a description in terms of state-amplitudeor density-matrix equations. In such a description wemust retain the phase information associated with theevolution of the atomic-state amplitudes, and it is in thissense that we refer to atomic coherence and coherentpreparation. This is of course in contrast to the rate-equation treatment of the state populations often appro-priate when the damping is large or the coupling isweak, for which the coherence of the states can be ig-nored. For a full account of the coherent excitation ofatoms the reader is referred to Shore �1990�.

For a two-level system, the result of coherent evolu-tion is characterized by oscillatory population transfer�Rabi flopping�. The generalization of this coherent situ-ation to driven three-level atoms leads to many newphenomena, some of which, such as Autler-Townes split-

ting �Autler and Townes, 1955�, dark states, and EIT,will be the subject of this review. These phenomena canbe understood within the basis of either bare atomicstates or new eigenstates, which diagonalize the com-plete atom-field interaction Hamiltonian. In both caseswe shall see that interference between alternative exci-tation pathways between atomic states leads to modifiedoptical response.

The linear and nonlinear susceptibilities of a �-typethree-level system driven by a coherent coupling fieldwill be derived in Sec. III. Figure 1 shows the imaginaryand real parts of the linear susceptibility for the case ofa resonant coupling field as a function of the probe fielddetuning from resonance. Figure 2 shows the corre-sponding third-order nonlinear susceptibility. Inspectionof these frequency-dependent dressed susceptibilities re-veals immediately several important features. One rec-ognizes that Im���1�� undergoes destructive interference

FIG. 1. Susceptibility as a function of the frequency �p of theapplied field relative to the atomic resonance frequency �31,for a radiatively broadened two-level system with radiativewidth �31 �dashed line� and an EIT system with resonant cou-pling field �solid line�: top, imaginary part of ��1� characterizingabsorption; bottom, real part of ��1� determining the refractiveproperties of the medium.

FIG. 2. Absolute value of nonlinear susceptibility for sum-frequency generation ���3�� as a function of �p, in arbitraryunits. The parameters are identical to those used in Fig. 1.

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in the region of resonance, i.e., the coherently drivenmedium is transparent to the probe field. The fact thattransparency of the sample is attained at resonance isnot in itself of great importance, as the same degree oftransparency can be obtained simply by tuning suffi-ciently away from resonance. What is important is thatin the same spectral region where there is a high degreeof transmission the nonlinear response ��3� displays con-structive interference, i.e., its value at resonance is largerthan expected from a sum of two split Lorentzian lines.Furthermore, the dispersion variation in the vicinity ofthe resonance differs markedly from the steep anoma-lous dispersion familiar at an undressed resonance �seeFig. 1�. Instead there is a normal dispersion in a regionof low absorption, the steepness of which is controlledby the coupling-laser strength �i.e., very steep for lowvalues of the drive laser coupling�. Thus despite thetransparency the transmitted laser pulse can still experi-ence strong dispersive and nonlinear effects. It is mostsignificant that the refractive index passes through thevacuum value and the dispersion is steep and linear ex-actly where absorption is small. This gives rise to effectssuch as ultraslow group velocities, longitudinal pulsecompression, and storage of light. Furthermore, throughthe destructive �constructive� interference in Im��1� ��3�and the elimination of the effect of resonance upon therefractive index, the conditions for efficient nonlinearmixing are met.

Coherent preparation techniques are most effective—and have been most studied—in atomic and molecularsamples that are in the gas phase. The reason for this isthat the coherent evolution of states will in general beinhibited by dephasing of the complex-state amplitudes,but the coherence dephasing rates for gases are rela-tively small when compared to those in solid-state me-dia. Nevertheless, several workers have recently madeprogress in applying these techniques to a variety ofsolid-state systems �Faist et al., 1997; Ham, Hemmer, andShahiar, 1997; Schmidt et al., 1997; Zhao et al., 1997; Weiand Manson, 1999; Serapiglia et al., 2000�. Atomic gasesremain very important in optical applications for severalkey reasons; for example, they are often transparent inthe vacuum UV and infrared, and there exist techniquesto cool them to ultracold temperatures, thus eliminatinginhomogeneous broadening. They can also tolerate highoptical intensities. Gas-phase media are, for instance,necessary for frequency conversion beyond the high-and low-frequency cutoff of solid-state materials.

The dramatic modifications of the optical propertiesthat are gained through coherent preparation of the at-oms or molecules in a medium have ushered in a renais-sance in activity in the area of gas-phase nonlinear op-tics. New opportunities also arise due to the extremelyspectrally narrow features that can result from coherentpreparation; in particular, this gives ultralow group ve-locities and ultralarge nonlinearities. These are of poten-tial importance to new techniques in optical informationstorage and also in nonlinear optics at the few-photonlevel, both of which may be important to quantum infor-mation processing.

Throughout this review we shall try to point out theimportant applications that arise as a result of EIT anddescribe in outline some of the seminal experiments. Weattempt to cover comprehensively the alteration of lin-ear and nonlinear optical response due to electromag-netically induced transparency and related phenomena.We do not, however, cover the related topic of lasingwithout inversion �Kocharovskaya and Khanin, 1988;Gornyi et al., 1989; Harris, 1989; Scully et al., 1989� andthe interested reader is advised to look elsewhere forreviews on that subject �Knight, 1990; Kocharovskaya,1992; Mandel, 1993; Scully and Zubairy, 1997�. In thefollowing section we shall introduce the underlyingphysical concepts of EIT through the coupling of near-resonant laser fields with the states of a three-level sys-tem. In Sec. III we discuss in detail the dynamics inthree-level atoms coupled to the applied laser fields anddetermine the optical response. In Sec. IV we treat thetopic of pulse propagation in EIT and review the devel-opments that have culminated in the demonstration ofultraslow group velocities down to a few meters per sec-ond and even the “stopping” and “storing” of a pulse oflight. The utility of coherent preparation in enhancingthe efficiency of frequency conversion processes is dis-cussed in Sec. V. Then in Sec. VI we shall turn to thetreatment of EIT in the few-photon limit, where it isnecessary to apply a fully quantum treatment of thefields. Finally, we draw conclusions and discuss futureprospects in quantum optics and atom optics as a resultof EIT.

II. PHYSICAL CONCEPTS UNDERLYINGELECTROMAGNETICALLY INDUCED TRANSPARENCY

A. Interference between excitation pathways

To understand how quantum interference may modifythe optical properties of an atomic system it is informa-tive to follow the historical approach and turn first to thesubject of photoionization of multielectron atoms. Dueto the existence of doubly excited states the photoion-ization spectrum of multielectron atoms can show a richstructure of resonances. These resonances are broad-ened due to relatively rapid decay �caused by the inter-action between the excited electrons of these states� todegenerate continuum states, with lifetimes in the pico-second to subpicosecond range. For the reason that theydecay naturally to the continuum, these states are calledautoionizing states. Fano introduced to atomic physicsthe concept of the interference between the excitationchannels to the continuum that exist for a single au-toionizing resonance coupled to a flat continuum �Fano,1961; Fig. 3�a��. In the vicinity of the autoionizing reso-nance the final continuum state can be reached via ei-ther direct excitation �channel 1� or through resonancewith the configuration interaction, leading to the decaythat provides a second channel to the final continuumstate. The interference between these channels can beconstructive or destructive and leads to frequency-

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dependent suppression or enhancement in the photoion-ization cross section.

Experiments by Madden and Codling �1965� showedthe resulting transparency windows in the autoionizingspectrum of helium. Sometime later the autoionizing in-terference structures in strontium were used by Arm-strong and Wynne to enhance sum-frequency mixing ina frequency up-conversion experiment to generate lightin the vacuum UV �Armstrong and Wynne, 1974�.1 Inthis wave-mixing experiment the absorption was elimi-nated in spectral regions where the nonlinear responseremained large, hence an improved efficiency for fre-quency conversion was reported.

During the 1960s autoionizing spectra were muchstudied �see, for example, Garton, 1966�. Several authorsaddressed the issue of interaction between two or morespectral series in the same frequency range, where theinterference between closely spaced resonances needs tobe considered �Fano and Cooper, 1965; Shore, 1967�.Shapiro provided the first explicit analysis of the case ofinterference between two or more resonances coupledto a single continuum �Shapiro, 1970�. In this case theinterference is mainly between the two transition path-ways from the ground state to the final state via each ofthe two resonances �Fig. 3�b��. Naturally interferencewill be significant only if the spacing between these reso-nances is comparable to or less than their widths.

Hahn, King, and Harris �1990� showed how this situa-tion could be used to enhance four-wave mixing. In ex-periments in zinc vapor that showed significantly in-creased nonlinear mixing, one of the fields was tuned toa transition between an excited bound state and a pair ofclosely spaced autoionizing states that had a frequencyseparation much less than their decay widths.

The case of interference between two closely spacedlifetime-broadened resonances, decaying to the samecontinuum, was further analyzed by Harris �1989�. Hepointed out that this will lead to lasing without inver-sion, since the interference between the two decay chan-nels eliminates absorption while leaving stimulatedemission from the states unchanged. Although we shallnot discuss the subject of lasing without inversion fur-ther in this review, this work was an important step inthe story with which we are concerned. A breakthroughwas then made in the work of Imamoglu and Harris�1989� when it was realized that the pair of closely

spaced lifetime-broadened resonances were equivalentto dressed states created by coupling a pair of well-separated atomic bound levels with a resonant laser field�Fig. 4�. They thus proposed that the energy-level struc-ture required for quantum interference could be engi-neered by use of an external laser field. Harris et al.�1990� then showed how this same situation could beextended to frequency conversion in a four-wave mixingscheme among atomic bound states with the frequencyconversion hugely enhanced. This is achieved throughthe cancellation of linear susceptibility at resonance asshown in Fig. 1, while the nonlinear susceptibility is en-hanced through constructive interference. The latter pa-per was the first appearance of the term electromagneti-cally induced transparency �EIT�, which was used todescribe this cancellation of the linear response by de-structive interference in a laser-dressed medium.

Boller et al. �1991�, in discussing the first experimentalobservation of EIT in Sr vapor, pointed out that thereare two physically informative ways that we can viewEIT. In the first we use the picture that arises from thework of Imamoglu and Harris �1989�, in which thedressed states can be viewed as simply comprising twoclosely spaced resonances effectively decaying to thesame continuum �Boller et al., 1991; Zhang et al., 1995�.If the probe field is tuned exactly to the zero-field reso-nance frequency, then the contributions to the linear sus-ceptibility due to the two resonances, which are equallyspaced but with opposite signs of detuning, will be equaland opposite and thus lead to the cancellation of theresponse at this frequency due to a Fano-like interfer-ence of the decay channels. An alternative and equiva-lent picture is to consider the bare rather than thedressed atomic states. In this view EIT can be seen asarising through different pathways between the barestates. The effect of the fields is to transfer a small butfinite amplitude into state �2�. The amplitude for �3�,which is assumed to be the only decaying state and thusthe only way to absorption, is thus driven by tworoutes—directly via the �1�-�3� pathway, or indirectly viathe �1�-�3�-�2�-�3� pathway �or by higher-order variants�.Because the coupling field is much more intense thanthe probe, this indirect pathway has a probability ampli-tude that is in fact of equal magnitude to the direct path-1See the improved ��3� fit of Armstrong and Beers, 1975.

FIG. 3. Fano interferences of excitation channels into a con-tinuum: �a� for a single autoionizing resonance; �b� for twoautoionizing states. FIG. 4. Interference generated by coherent coupling: left, co-

herent coupling of a metastable state �2� to an excited state �3�by the dressing laser generates �right� interference of excita-tion pathways through the doublet of dressed states �a±��Autler-Townes doublet� provided the decay out of state �2� isnegligible compared to that of state �3�.

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way, but for resonant fields it is of opposite sign.

B. Dark state of the three-level �-type atom

The use in laser spectroscopy of externally appliedelectromagnetic fields to change the system Hamiltonianof course predates the idea of using this in nonlinearoptics or in lasing without inversion. We must mentionthe enormous body of work treating the effects of staticmagnetic fields �Zeeman effect� and static electric fields�Stark effect�. The case of strong optical fields applied toan atom began to be extensively studied following theinvention of the laser in the early 1960s. Hänsch andToschek �1970� recognized the existence of these typesof interference processes for three-level atoms coupledto strong laser fields in computing the system suscepti-bility from a density-matrix treatment of the response.They identified terms in the off-diagonal density-matrixelements indicative of the interference, although theydid not explicitly consider the optical and nonlinear op-tical effects in a dense medium.

Our interest is in the case of electromagnetic fields inthe optical frequency range, applied in resonance to thestates of a three-level atom. We illustrate the three pos-sible coupling schemes in Figs. 5 and 6. For consistencystates are labeled so that the �1�-�2� transition is alwaysthe dipole-forbidden transition. Of these prototypeschemes we shall be most concerned with the lambdaconfiguration in Fig. 5, since the ladder and vee configu-rations illustrated in Fig. 6 are of more limited utility forthe applications that will be discussed later.

The physics underlying the cancellation of absorptionin EIT is identical to that involved in the phenomena ofdark-state and coherent population trapping �Lounisand Cohen-Tannoudji, 1992�. We shall therefore reviewbriefly the concept of dark states. Alzetta et al. �1976�made the earliest observation of the phenomenon of co-herent population trapping �CPT� followed shortly bytheoretical studies of Whitley and Stroud �1976�. Ari-mondo and Orriols �1976� and Gray et al. �1978� ex-plained these observations using the notion of coherentpopulation trapping in a dark eigenstate of a three-levellambda medium �see Fig. 5�. In this process a pair ofnear-resonant fields are coupled to the lambda systemand result in the Hamiltonian H=H0+Hint, where theHamiltonian for the bare atom is H0 and that for theinteraction with the fields is Hint. The Hamiltonian Hhas a new set of eigenstates when viewed in a properrotating frame �see below�, one of which has the form�a0�=��1�−��2�, which contains no amplitude of the barestate �3� and has amplitudes � and � proportional to thefields such that it is effectively decoupled from the lightfields. In the experiments of Alzetta, population waspumped into this state via spontaneous decay from theexcited states and then remained there since the excita-tion probability of this dark state is canceled via inter-ference. An early account on the effect of CPT on thepropagation of laser fields was given by Kocharovskayaand Khanin �1986�. A very informative review of theapplications of dark states and the coherent populationtrapping that accompanies them in spectroscopy hasbeen provided by Arimondo �1996�.

We would now like to look a bit more closely at thestructure of the laser-dressed eigenstates of a three-levelatom illustrated in Fig. 5. This discussion is intended toprovide a simple physical picture that establishes theconnection between the key ideas of EIT and that ofmaximal coherence.

Within the dipole approximation the atom-laser inter-action Hint=� ·E is often expressed in terms of the Rabicoupling �or Rabi frequency� �=� ·E0 /�, with E0 beingthe amplitude of the electric field E, and � the transitionelectronic dipole moment. After introducing therotating-wave approximation, we can represent theHamiltonian of the three-level atom interacting with acoupling laser with real Rabi frequency �c and a probelaser with Rabi frequency �p �Fig. 5� in a rotating frameas

Hint = −�

2� 0 0 �p

0 − 2�1 − 2� �c

�p �c − 21� . �1�

Here 1=�31−�p and 2=�32−�c are the detunings ofthe probe and coupling laser frequencies �p and �c fromthe corresponding atomic transitions.

A succinct way of expressing the eigenstates of theinteraction Hamiltonian �1� is in terms of the “mixingangles” and � that are dependent in a simple wayupon the Rabi couplings as well as the single-photon

FIG. 5. Generic system for EIT: lambda-type scheme withprobe field of frequency �p and coupling field of frequency �c.1=�31−�p and 2=�32−�c denote field detunings fromatomic resonances and �ik radiative decay rates from state �i�to state �k�.

FIG. 6. Ladder �left� and vee-type �right� three-level schemes.These do not show EIT in the strict sense because of the ab-sence of a �meta�stable dark state.

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�1=� and two-photon � =1−2� detunings �see Fig.5�. For two-photon resonance �2=1, or =0� the mix-ing angles are given by

tan =�p

�c, �2�

tan 2� =�p

2 + �c2

. �3�

The eigenstates can then be written in terms of the bareatom states:

�a+� = sin sin ��1� + cos ��3� + cos sin ��2� , �4�

�a0� = cos �1� − sin �2� , �5�

�a−� = sin cos ��1� − sin ��3� + cos cos ��2� . �6�

The reader’s attention is drawn to the following features:While the state �a0� remains at zero energy, the pair ofstates �a+� and �a−� are shifted up and down by anamount ��±,

��± =�

2� ± 2 + �p

2 + �c2� . �7�

The states �a±� retain a component of all of the bareatomic states, but in contrast state �a0� has no contribu-tion from �3� and is therefore the dark state, since if theatom is formed in this state there is no possibility ofexcitation to �3� and subsequent spontaneous emission.

It should be noted that the dark state will always beone of the possible states of the dressed system, but thatthe details of the evolution of the fields can determinewhether the atom is in this state �a0�, in �a±�, or an ad-mixture. Evolution into the dark state via optical pump-ing �through spontaneous decay from �3�� is one way totrap population in this state that is well known in laserspectroscopy and laser-atom manipulation. EIT offersan alternative, adiabatic, and much more rapid route toevolve into this state.

To see the origin of EIT using the dressed-state pic-ture above, consider the case of a weak probe, that is,�p��c. In this case sin →0 and cos →1, which re-sults in the dressed eigenstates shown in Fig. 4. Theground state becomes identical to the dark state �a0�= �1� from which excitation cannot occur. Furthermore,when the probe is on resonance �=0� then tan �→1�i.e., �=� /2�, and then �a+�= �1/2���2�+ �3�� and �a−�= �1/2���2�− �3��; these are the usual dressed states rel-evant to EIT in the limit of a strong-coupling field and aweak probe. Conversely the picture of dark states showus that the EIT interference will survive in a lambdasystem in the limit of a pair of strong fields and so isrelevant to the situation of maximal coherence.

To ensure that a dark state is formed, an adiabaticevolution of the field is often adopted �Oreg et al., 1984�.The physically most interesting example of this involvesa so-called counterintuitive pulse sequence. This was

first demonstrated by Bergmann and co-workers �Kuk-linski et al., 1989; Gaubatz et al., 1990�, who termed thisstimulated Raman adiabatic passage �STIRAP�. With thesystem starting with all the amplitude in the ground state�1� the laser field at �c is first applied. This is similar tothe EIT situation since �p��c and the dark state in-deed corresponds exactly to �1� with negligible popula-tion in either �2� or �3�. But now �p is gradually �i.e.,adiabatically� increased and at the same time �c isgradually decreased so that eventually �c��p and thedark state becomes �a0�=−�2�. This population transfer isattained via the “maximally coherent” dark state when�p=�c, i.e., =� /4 in which the state takes the form�a0�= �1/2���1�− �2��. A full description of this processand the conditions for adiabaticity are given in the nextsection.

III. ATOMIC DYNAMICS AND OPTICAL RESPONSE

The essential features of EIT and many of its applica-tions can be quantitatively described using a semiclassi-cal analysis. This will be the focus of this section, wherewe shall assume continuous-wave �cw� classical fields in-teracting with a single atom that can be modeled as alambda system �Fig. 5�. For the derivation of linear andnonlinear susceptibilities, we concentrate on the steady-state solution of the atomic master equation. While amaster equation analysis is general and can be used forthe treatment of nonperturbative field effects, a single-atom wave-function approach is simpler and more illus-trative; we shall therefore use the latter to discuss thedressed-state interpretation of EIT. As many of the cen-tral features of EIT are related to optically thick media,we shall discuss the light transmission of such a medium.The process of establishing EIT will then be examinedand the connection to Raman adiabatic passage will beestablished. Finally, we shall discuss EIT inside opticalresonators and briefly review some applications of EITthat will not be expanded upon in the following sectionsof the review.

A. Master equation and linear susceptibility

We consider an ensemble of identical atoms whosedynamics can be described by taking into account onlythree of its eigenstates. In the absence of electromag-netic fields, all atoms are assumed to be in the lowest-energy state �1� �Fig. 5�. State �2� has the same parity as�1� and is assumed to have a very long coherence time.The highest-energy state �3� is of opposite parity and hasa nonzero electric dipole coupling to both �1� and �2�. A�near� resonant nonperturbative electromagnetic field offrequency �c, termed the coupling field, is applied on the�2�-�3� transition. A probe field of frequency �p is appliedon the �1�-�3� transition. EIT is primarily concerned withthe modification of the linear and nonlinear opticalproperties of this—typically perturbative—probe field.We emphasize that most atomic systems have a subspace

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of their state space that can mimic this simplified picturewhen driven by near-resonant polarized electromagneticfields.

The time-dependent interaction Hamiltonian in theinteraction picture that describes the atom-laser cou-pling is

Hint = −�

2��p�t��31e

i1t + �c�t��32ei2t + H.c.� , �8�

where �c�t� and �p�t� denote the Rabi frequency asso-ciated with the coupling and probe fields. �ij= �i�j� is theatomic projection operator �i , j=1,2,3�. This semiclassi-cal Hamiltonian can be obtained from the fully quan-tized interaction-picture interaction Hamiltonian by re-placing annihilation and creation operators by cnumbers.

The dynamics of laser-driven atomic systems are gov-erned by the master equation for the atomic density op-erator:

d�

dt=

1

i��Hint,�� +

�31

2�2�13��31 − �33� − ��33�

+�32

2�2�23��32 − �33� − ��33�

+�2deph

2�2�22��22 − �22� − ��22�

+�3deph

2�2�33��33 − �33� − ��33� , �9�

where the second and third terms on the right-hand sidedescribe spontaneous emission from state �3� to states �1�and �2�, with rates �31 and �32, respectively. We assumehere that the thermal occupancy of the relevant radia-tion field modes is completely negligible. For optical fre-quencies this approximation is easily justified. A de-tailed derivation of this master equation is given byCohen-Tannoudji et al. �1992�. We have also introducedenergy-conserving dephasing processes with rates �3dephand �2deph; the latter determines one of the fundamentaltime scales for practical EIT systems, as we shall seeshortly. For convenience, we define the total spontane-ous emission rate out of state �3� as �3=�31+�32. Thecoherence decay rates are defined as �31=�3+�3deph,�32=�3+�3deph+�2deph, and �21=�2deph.

The polarization generated in the atomic medium bythe applied fields is of primary interest, since it acts as asource term in Maxwell’s equations and determines theelectromagnetic field dynamics. The expectation valueof the atomic polarization is

P� �t� = − �i

eri�/V

=Natom

V��13�31e

−i�31t + �23�32e−i�32t + c.c.� . �10�

To obtain the expression in the second line, we assumedthat all Natom atoms contained in the volume V coupleidentically to the electromagnetic fields. The time-

dependent exponentials arise from the conversion to theSchrödinger picture. Assuming �13=�13z and �23=�23z,we let �=Natom/V and obtain Pz�t�=P�t� as

P�t� = ���13�31e−i�31t + �23�32e

−i�32t + c.c.� . �11�

We now focus on the perturbative regime in the probefield and evaluate the off-diagonal density-matrix ele-ments �31�t�, �32�t�, and �12�t� to obtain P�t�, or, equiva-lently, the linear susceptibility ��1��−�p ,�p�. Taking �11�1 and using a rotating frame to eliminate fast expo-nential time dependences, we find

�32 =i�ce

i1t

��32 + i22��12,

�12 = −i�ce

i2t

�21 + i2�2 − 1��13,

�31 =i�pei1t

��31 + i21�+

i�cei2t

��31 + i21��21. �12�

As in Sec. I we define the single-photon detuning as =1=�31−�p and two-photon detuning as =1−2=�21− ��p−�−c�. Keeping track of the terms that oscil-late with exp�−i�pt�, we obtain

��1��− �p,�p� =��13�2��0�

� 4 ���c�2 − 4 � − 4�212

���c�2 + ��31 + i2���21 + i2 ��2

+ i8 2�31 + 2�21���c�2 + �21�31�

���c�2 + ��31 + i2���21 + i2 ��2� .

�13�

The linear susceptibility given in Eq. �13� containsmany of the important features of EIT. First, it must bementioned that Eq. �13� predicts Autler-Townes splittingof an atomic resonance due to the presence of a nonper-turbative field. There is more, however, in this expres-sion than the modification of absorption due to the ap-pearance of dressed atomic states: in particular, for two-photon Raman resonance � =0�, both real andimaginary parts of the linear susceptibility vanish in theideal limit of �21=0. As depicted in Fig. 7, this result isindependent of the strength of the coherent couplingfield. It should be noted that changing the Rabi fre-quency of the coupling laser only changes the spectralprofile of absorption, the integral of Im��� over is con-served as �c is varied. In the limit ��c���31, the absorp-tion profile carries the signatures of an Autler-Townesdoublet. Important features in absorption, such as van-ishing loss at =0 and enhanced absorption on the low-and high-energy sides of the doublet, become evident ona closer inspection. For ��c���31, we obtain a sharptransmission window with a linewidth much narrowerthan �31. In this latter case, it becomes apparent that themodifications in the linear susceptibility call for an ex-planation based on quantum interference phenomena

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rather than a simple line splitting. Before proceedingwith this discussion, we highlight some of the other im-portant properties that emerge in Eq. �13�.

First and foremost it must be noted that the possibilityof eliminating absorption in an otherwise optically thickmedium has been demonstrated in several laboratoriesin systems ranging from hot atomic vapor cells to mag-netically trapped Bose-condensed atoms. Figure 8 showsthe absorption profile obtained in the first experiment todemonstrate the effect, carried out by Boller and co-workers �Boller et al., 1991�. The application of acoupling field opens up a transparency window in anotherwise completely opaque atomic cell. This experi-ment was carried out using an autonionizing state �3�of strontium with �31�2�1011 s−1. The decay rate of thelower state coherence �21�4�109 s−1 is determinedby the collisional broadening for an atomic density of��5�1015 cm−3.

The transparency obtained at two-photon resonanceis independent of the detuning of the probe field fromthe bare �1�-�3� transition ��. As increases, however,the distance between the frequencies where one obtainstransparency and maximum absorption becomes smaller,thereby limiting the width of the transparency window.At the same time, the transmission profile and the asso-ciated dispersion become highly asymmetric for ��31.Figure 9 shows contour plots of Im���1�� as a function ofthe detunings 1 and 2 of the two fields as well as thesingle-photon �� and two-photon � � photon detunings.It is evident that for large detuning 2 of the couplingfield the absorption spectrum is essentially that of a two-level system with an additional narrow Raman peakclose to the two-photon resonance. Exactly at the two-photon resonance point 1=2 the absorption vanishesindependent of the single-photon detuning.

In the limit 2=0 and for �c��31, the real part of thesusceptibility seen by a probe field varies rapidly at reso-nance ��0�. In contrast to the well-known �single�atomic resonances, the enhanced dispersion in the EITsystem is associated with a vanishing absorption coeffi-

cient, implying that ultraslow group velocities for lightpulses can be obtained in transparent media, which willbe discussed extensively in the following section.

Typically, observation of coherent phenomena inatomic gases is hindered by dephasing due to collisionsand laser fluctuations as well as inhomogeneous broad-ening due to the Doppler effect. While recent EIT ex-periments have been carried out using ultracold atomicgases driven by highly coherent laser fields, where Eq.�9� provides a perfect description of the atomic dynam-ics, it is important to analyze the robustness of EITagainst these nonideal effects.

In actual atomic systems, the dephasing rate of theforbidden �1�-�2� transition is nonzero due to atomic col-lisions. All the important features of EIT remain observ-able even when �21�0, provided that the coupling fieldRabi frequency satisfies

��c�2 � �31�21. �14�

Figure 10 shows the imaginary part of ��1��−�p ,�p� for�21=0 �Fig. 10�a��, �21=0.1�31 �Fig. 10�b��, and �21=10�31 �Fig. 10�c��. In all plots, we take �c=0.5�31. Wefirst note from Fig. 10�b� that the absorption profile ap-pears virtually unchanged with respect to the ideal case,provided that the inequality �14� is satisfied. The trans-parency is no longer perfect, however, and the residual

FIG. 7. EIT absorption spectrum for different values of cou-pling field and �21=0: �a� �c=0.3�31; �b� �c=2�31.

FIG. 8. First experimental demonstration of EIT in Sr vaporby Boller et al., 1991: top, transmission through cell withoutcoupling field; bottom, with coupling field on. From Boller etal., 1991.

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absorption due to �21 provides a fundamental limit formany of the EIT applications. For the �21��31 case de-picted in Fig. 10�c�, the absorption minimum is absent: inthis limit a constructive interference enhances the ab-sorption coefficient in between the two peaks.

The linear susceptibility given in Eq. �13� dependsonly on the total coherence decay rates, and not on thepopulation decay rates of the atomic states. As a conse-quence, the strong quantum interference effects de-picted in Figs. 3�a�–3�c� are observable even in systemswhere collisional broadening dominates over lifetimebroadening. This implies that EIT effects can be ob-served in dense atomic gases, or even solids, providedthat there is a metastable transition with a relativelylong dephasing rate that satisfies �21��31. EIT in such acollisionally broadened atomic gas was first reported in1991 �Field et al., 1991�.

Amplitude or phase fluctuations of the nonperturba-tive coupling laser can have a detrimental effect on theobservability of EIT. Instead of trying to present a gen-eral analysis of the effect of coupling laser linewidth onlinear susceptibility, we discuss several special cases thatare of practical importance. When the laser linewidth isdue to phase fluctuations that can be modeled using theWiener-Levy phase-diffusion model, it can be shownthat the resulting coupling laser linewidth directly con-tributes to the coherence decay rates �21 and �32 �Ima-moglu, 1991�. In contrast, for a coupling field with largeamplitude fluctuations, the absorption profile could besmeared out. In applications where a nonperturbativeprobe field is used, fluctuations in both laser fields areimportant. If, however, the two lasers are obtained fromthe same fluctuating laser source using electro-optic oracousto-optic modulation, then EIT is preserved to firstorder.

Doppler broadening is ubiquitous in hot atomic gases.If the lambda system is based on two hyperfine-splitmetastable states �1� and �2�, then the Doppler broaden-ing has no adverse effect on EIT, provided that one usescopropagating probe and coupling fields. In other cases,the susceptibility given in Eq. �13� needs to be integratedover a Gaussian density of states corresponding to theGaussian velocity distribution of the atoms. Qualita-tively, the presence of two-photon Doppler broadeningwith width �Dopp��c will wash out the level splittingand the interference. EIT can be recovered by increas-ing the coupling field intensity so as to satisfy �c��Dopp; in this limit we obtain

Im���1��− �p,�p���p=�31��Dopp

2 �31

�c4 . �15�

This strong dependence on reciprocal �c is a direct con-sequence of robust quantum interference. We emphasizethat in this large coupling field limit, we can consider �cas the effective detuning from dressed-state resonances.As a result of quantum interference, this effective detun-ing can be used to suppress absorption much morestrongly than in noninterfering systems, where the ab-sorption coefficient is only proportional to the inversedetuning squared.

FIG. 9. Contour plot of imagi-nary part of susceptibilityIm���1��: left, as a function ofdetunings 1 and 2; right, as afunction of single-photon de-tuning and two-photon de-tuning in units of �31. Whiteareas correspond to low absorp-tion, dark to large absorption.The insensitivity of the inducedtransparency at =0 on thesingle-photon detuning is ap-parent.

FIG. 10. Absorption spectrum for nonvanishing decay of �2�-�1�transition in arbitrary units: �a� �31=1, �21=0; �b� �31=1, �21=0.1; �c� �31=1, �21=10. In all cases �c=0.5.

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B. Effective Hamiltonian and dressed-state picture

It is well known in quantum optics that for a givenmaster equation in the so-called Lindblad form, one canobtain an equivalent stochastic wave-function descrip-tion of the dynamics of the system based on the evolu-tion via a non-Hermitian effective Hamiltonian andquantum-jump processes �Dalibard et al., 1992; Gardineret al., 1992; Carmichael, 1993�. Further simplification ofthe description of the dynamics is obtained when theprobability of a quantum-jump process is negligiblysmall: in this limit, the nonunitary wave-function evolu-tion induced by the effective Hamiltonian captures mostof the essential physics and can be used to calculate lin-ear and nonlinear susceptibilities. This simplification ap-plies for an atomic system in which the ground state�which has near-unity occupancy� is coupled to excitedstates via weak electromagnetic fields. In the case ofEIT, this is exactly the case when atoms in state �1� aredriven by a weak probe field, independent of thestrength of the coupling field.

Our starting point is the Hamiltonian of Eq. �8�. Toincorporate the effect of radiative decay out of state �3�,we introduce an anti-Hermitian term, −i��3�33/2, whichreproduces the correct decay terms in the master equa-tion. Conversely, the physics that this �combined� non-Hermitian Hamiltonian fails to capture is the repopula-tion of atomic states due to spontaneous emission downto states �1� and �2�. Since we cannot describe puredephasing processes using a wave-function model, wecan use the population decay of state �2� �at rate �2� tomimic dephasing. We therefore introduce an additionalanti-Hermitian decay term to obtain an effective Hamil-tonian in the rotating frame

Heff = −�

2��p�31 + �c�32 + H.c.� + �� −

i

2�3��33

+ �� −i

2�2��22. �16�

We reemphasize that for �2=0 and =0, one of theeigenstates of Heff is

�a0� =1

�c2 + �p

2 ��c�1� − �p�2�� , �17�

with an eigenenergy ��0=0 independent of the values ofthe Rabi frequencies and the single-photon detuning.We here have assumed real-valued Rabi frequencies,which are always possible as long as propagation effectsare not considered. In the time-dependent response, weshall see that the emergence of EIT can be understoodas a result of the system’s moving adiabatically frombare state �1� in the absence of fields to �a0� in the pres-ence of the fields.

The structure of this dark eigenstate provides us thesimplest picture with which we can understand coherentpopulation trapping and EIT. The atomic system underthe application of the two laser fields satisfying =0moves into �a0�, which has no contribution from state �3�.

Since the population in state �3� is zero, there is no spon-taneous emission or light scattering and hence no ab-sorption. In the limit of a perturbative probe field where�a0���1�, we have an alternative way of understandingthe vanishing amplitude in �3�: the atom has two ways ofreaching the dissipative state �3�—either directly fromstate �1� or via the path �1�-�3�-�2�-�3�. The latter has com-parable amplitude to the former, since �c is nonpertur-bative. As a result, the amplitudes for these paths canexhibit strong quantum interference.

We can alternatively view the atomic system in a dif-ferent basis—one that would have yielded a diagonalHamiltonian matrix had the dissipation and the probecoupling been vanishingly small. This dressed-state basisis obtained by applying the unitary transformation onthe �bare-basis� state vector ���t��:

��d�t�� = U���t�� , �18�

where

U = �1 0 0

0 cos � sin �

0 − sin � cos �� . �19�

Here � is defined by tan�2��=�c /2. Application of thistransformation to Heff yields

Heff = −�

2�0 �p sin � �p cos �

�p sin � − 2− + i�− −i

2�+−

�p cos � −i

2�+− − 2+ + i�+

� ,

�20�

where − and + denote the detunings of the probe fieldfrom the dressed resonances, �−= ��2 cos2 �+�3 sin2 ��,�+= ��2 sin2 �+�3 cos2 ��, and �+−= ��2−�3�sin 2�. Wecan calculate the absorption rate of the probe photons

by determining the eigenvalues of Heff in the limit ofperturbative �p. The imaginary part of the eigenvalue�Im�−�1�� that corresponds to �1� as �p→0 for �2=0 is

Im��1� =− �p

2�3�+ sin2 � + − cos2 ��2

4−2+

2 + �+�3 sin2 � + −�3 cos2 ��2

� Im���1��− �p,�p�� . �21�

While this expression contains the same information asEq. �13�, appearance of the square of the sum of twoamplitudes in the numerator of Eq. �21� makes the roleof quantum interference clearer.

Heff given in Eq. �20� has exactly the same form as theone describing Fano interference of two autoionizingstates �Fano, 1961; Harris, 1989�. It is the presenceof imaginary off-diagonal terms �23=�32=0.5��3−�2�sin 2� that gives rise to quantum interference in ab-sorption. When �2=0, we find that �23=�2d�3d, where�2d=�2 cos2 �+�3 sin2 � and �3d=�2 sin2 �+�3 cos2 �.In this case, we find that the interference in absorption isdestructive when the probe field is tuned in between the

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two dressed-state resonances, with a perfect cancellationfor =0. When �3=0, we have �23=−�2d�3d; in thiscase, the quantum interference is constructive for probefields tuned in between the two dressed states and de-structive otherwise. Even though absorption is sup-pressed for � ���c, we never obtain complete transpar-ency or a steep dispersion in this ��3=0� case. Finally, for�2=�3, the imaginary off-diagonal terms vanish; in thislimit, there are no interference effects. These resultswere already depicted in Fig. 10.

The complete elimination of absorption for =0 in thelimit �2=0 can therefore be understood by invoking thearguments used by Fano �1961�. Let us assume that thedecay of state �3� is due to spontaneous emission intostate �f�. For an atom initially in state �1�, a probe photonabsorption event results in the generation of a �Raman-�scattered photon and leaves the atom in state �f�. Toreach this final state, the atom can virtually excite eitherof the two dressed states; since excitations are virtual,the probability amplitudes corresponding to each haveto be added, leading to quantum interference. State �f�could either be a fourth atomic state not coherentlycoupled to the others or it could be the state �1� itself. Inthe latter case, the absorption process is Rayleigh scat-tering of probe photons.

Having seen the simplicity of the analysis based oneffective non-Hermitian Hamiltonians, we next addressthe question of nonlinear susceptibilities. Since the latterare by definition calculated in the limit of near-unityground-state occupancy, Heff of Eq. �16� provides a natu-ral starting point. In this section, we shall focus on sum-frequency generation to highlight the different rolequantum interference effects play in nonlinear suscepti-bilities as compared to linear ones. To this end, we in-troduce an additional driving term �eff that couplesstates �1� and �2� coherently. Since we have assumed thistransition to be dipole forbidden, we envision that thiseffective coupling is mediated by two laser fields �at fre-quencies �a and �b� and a set of intermediate �nonreso-nant� states �i�:

�eff = �i

�a�b

2 1

�i1 − �a+

1

�i1 − �b� . �22�

Here �a=�1iEa /� and �b=�i2Eb /�.We can now rewrite the effective Hamiltonian for

sum-frequency generation as

Heffsfg = −

2��eff�21 + �p�31 + �c�32 + H.c.�

+ �� −i

2�3��33 + �� −

i

2�2��22. �23�

To obtain the nonlinear susceptibility, we assume both�eff and �p to be perturbative. The generated polariza-tion at �p is given by

P�t� = ��13a3�t�exp�− i�pt� + c.c., �24�

where a3�t� is the probability amplitude for finding agiven atom in state �3�. This amplitude is calculated us-

ing Heffsfg and has two contributions: the first contribution

proportional to �p simply gives us the same linear sus-ceptibility we calculated in the previous subsection usingthe master-equation approach. The second contributionproportional to �eff�c��a�b�c gives us the nonlinearsusceptibility

��3��− �p,�a,�b,�c�

=2�23�31�

3�0�3

1

�c2 + ��31 + i2���21 + i2 �

��i�1i�i2 1

�i1 − �a+

1

�i1 − �b� . �25�

This third-order nonlinear susceptibility that describessum-frequency generation highlights one of the most im-portant properties of EIT, namely, that the nonlinearsusceptibilities need not vanish when the linear suscep-tibility vanishes due to destructive quantum inter-ference. In fact, for �c

2��32 /2, we notice that

��3��−�p ,�a ,�b ,�c� has a maximum at =0 where��1��−�p ,�p�=0.

It is perhaps illustrative to compare ��3� obtained forthe EIT system with the usual third-order nonlinearsusceptibility for nonresonant sum-frequency genera-tion. As expected, the expressions in the two limitshave the same form, with the exception of detunings:the two- and three-photon detunings in the case ofnonresonant susceptibility �1/�2�3� are replaced by1/ ��c

2+�21�31� in the case of EIT with ==0.

C. Transmission through a medium with EIT

Many of the interesting applications of EIT stem fromthe spectroscopic properties of dark resonances, i.e., nar-row transmission windows, in otherwise opaque media.It is therefore worthwhile to consider the specifics of theEIT transmission profile in more detail. In an opticallythick medium it is not the susceptibility ��1��−�p ,�p� it-self that governs the spectroscopic properties but thecollective response of the entire medium, characterizedby the amplitude transfer function

T��p,z� = exp�ikz��1��− �p,�p�/2� , �26�

with k=2� /� the resonant wave number and z the me-dium length. In contrast to absorbing resonances, thewidth of the spectral window in which an EIT mediumappears transparent decreases with the product of nor-malized density and medium length. Substituting thesusceptibility of an ideal EIT medium with resonantcontrol field into Eq. �26�, one finds that the transmittiv-ity near resonance =0 is a Gaussian with width

�trans =�c

2

�31�31

1��z

���z � 1� . �27�

Here �=3�2 /2� is the absorption cross section of anatom and � is the atom number density. One recognizesa power broadening of the resonance with increasingRabi frequency �c or intensity of the coupling field,

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which can, however, be partially compensated for by in-creasing the density-length product ��z. This importantproperty is illustrated in Fig. 11, where a finite decayrate �21 of the coherence between the two lower levelswas taken into account as well.

The perfect linear dispersion near the EIT resonance,expressed by the linear dependence of the real part of� on , can be used to detect shifts of the EIT resonanceby phase-sensitive detection. Let us define �dis asthe frequency variation associated with an accumulatedphase shift of 2� for the propagation through the me-dium. This is given by

�disp = 2��c

2

�31

1

��z���z � 1� , �28�

much smaller than �trans. This effect, which is also il-lustrated in Fig. 11, plays an important role for applica-tions of EIT in optical detection of magnetic fields�Scully and Fleischhauer, 1992; Fleischhauer and Scully,1994�.

The narrowing of the transparency and dispersivewidth of an EIT medium with increasing density wasexperimentally observed by Lukin et al. �1997� in a for-ward four-wave mixing experiment.

D. Dark-state preparation and Raman adiabatic passage

In the preceding sections we have discussed EIT un-der steady-state conditions. We have seen that atoms ina particular superposition state, the dark state �a0�, canbe decoupled from the interaction. If there is two-photon resonance between atoms and fields, this state isstationary. Properly prepared atoms remain in this stateand render the medium transparent. An important char-

acteristic of EIT, which distinguishes it from coherentpopulation trapping, is the preparation which in EIThappens automatically through Raman adiabatic pas-sage. This process naturally becomes important if an op-tically thick medium is considered.

The formation of a dark state can happen either viaoptical pumping, i.e., by an incoherent process, or via acoherent preparation scheme. The first mechanism is ob-viously required if the atomic ensemble is initially in amixed state. Optical pumping may not, however, be wellsuited to achieve electromagnetically induced transpar-ency in optically thick media. Here radiation trappingcan lead to a dramatic slowdown of the pumping pro-cess. If the dark state has a finite lifetime, the resultingeffective pump rate may become even too slow toachieve transparency at all �Fleischhauer, 1999�. Further-more, the preparation by optical pumping is always as-sociated with a nonrecoverable loss of photons from theprobe field, which could be fatal if the probe field were afew-photon pulse.

The problems associated with radiation trapping andpreparation losses can be avoided if the atoms are ini-tially in a pure state, e.g., state �1�. If �1� is a nondegen-erate ground state with a sufficient energy gap to state�2�, this is in general fulfilled. Under certain conditionsthe dark state is in this case prepared by a coherentmechanism that does not involve spontaneous emissionat all. The process that achieves this is called stimulatedRaman adiabatic passage �STIRAP� and was discoveredand developed by Bergmann et al. �see also Kuklinski etal., 1989 and Gaubatz et al., 1990� after some earlierrelated but independent discoveries of Eberly and co-workers �Oreg et al., 1984�. In the STIRAP process rel-evant here, the atoms are returned to the initial stateafter the interaction with the pulsed probe field, thusconserving the number of photons in the pulse. TheSTIRAP technique is now a widely used method inatomic and molecular physics for the preparation of spe-cific quantum states and for controlling chemical reac-tions. It also has a large range of applications in quan-tum optics and matter-wave interferometry. For recentreviews on this subject see Bergmann et al. �1998� andVitanov et al. �2001�.

In STIRAP, the two coherent fields coupling the statesof the three-level lambda system are considered to betime dependent and in two-photon resonance. In the ba-sis of the bare atomic states ��1�,�2�,�3�� the correspond-ing time-dependent interaction Hamiltonian in therotating-wave approximation and in a proper rotatingframe is

H�t� = −�

2� 0 0 �p�t�0 0 �c�t�

�p�t� �c�t� − 2� . �29�

We assume that the relative phase between the twofields is arbitrary but constant and thus both �p and �ccan be taken real. This also implies that the two fieldsare assumed to be transform-limited pulses, or moreprecisely that the beat note between the two is trans-

FIG. 11. Transmission spectrum through an EIT medium: top,dependence on density length products ��z; bottom, illustra-tion of dispersive width.

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form limited. The essence of STIRAP is that a properadiabatic change of �p�t� and �c�t� allows a completetransfer of population from the initial state �1� to thetarget state �2� or vice versa without populating the in-termediate excited state �3� and thus without spontane-ous emission.

The underlying mechanism can most easily be under-stood if the time-dependent interaction Hamiltonian inthe rotating-wave approximation is written in the basisof its instantaneous eigenstates, given in Eqs. �4�–�6�,with the corresponding eigenvalues given in Eq. �7�. Theinstantaneous eigenstate

�a0�t�� = cos �t��1� − sin �t��2�, tan =�p�t��c�t�

�30�

is the adiabatic dark state, which has no overlap with theexcited state �3� and thus does not lead to spontaneousemission.

An important property of the adiabatic dark state isthat, depending on the value of the mixing angle �t�, itcan coincide with either one of the bare states �1� and �2�as well as any intermediate superposition with fixed rela-tive phase. If =0, which corresponds to a coupling fieldmuch stronger than the pump field, �a0�= �1�. Likewise, if=� /2 we have �a0�=−�2�. Thus if all atoms are initially�i.e., for t=−�� prepared in state �1� and ��c�−���� ��p�−���, i.e., if the coupling pulse is applied first, thesystem is in the dark state at t=−�. If the mixing angle�t� is now rotated from 0 to � /2, the dark state changesfrom �1� to −�2�. If this rotation is sufficiently slow, theadiabatic theorem guarantees that the state vector of thesystem follows this evolution. In this way population canbe transferred with 100% efficiency and withoutspontaneous-emission losses. The only requirements be-sides the correct initial conditions are that the time rateof change of the fields be sufficiently slow, i.e., adiabatic,and that the relative phase between the fields be at leastapproximately constant.

To see how slow the change of the fields has to be tostay within the adiabatic approximation, we transformthe Hamiltonian matrix �29� for the state amplitudes inthe bare basis into the basis of instantaneous eigenstates.Since the transformation matrix R is explicitly time de-pendent for the case of time-dependent fields, we find

H=R−1HR−R−1R, or, explicitly,

H = − ���+�t�

i

2 0

−i

2 0 −

i

2

0i

2 �−�t�

� , �31�

where an overdot means a time derivative. Adiabaticevolution occurs if the off-diagonal coupling, propor-

tional to , is sufficiently small compared to the eigen-values ��±� �7�:

��� d

dta0�t��a±�t��� � ��±�t�� , �32�

which for the case of single-photon resonance =0 sim-plifies to

��t� � �p2�t� + �c

2�t� � ��t�� . �33�

In lowest order of the nonadiabatic coupling and in the

limit ����3, characteristic for long pulses, the excited-

state population is given by ��2 /�2�t�, such that spon-taneous emission leads to an instantaneous loss rate�Fleischhauer and Manka, 1996; Vitanov and Stenholm,1997�

�eff = �3��t��2

�2�t�. �34�

If the pump and coupling pulses are not transformlimited in such a way that the beat-note phase changesin time �Dalton and Knight, 1982�, there will be addi-tional nonadiabatic losses due to this effect.

If all atoms are initially in state �1�, the coupling field��c� is applied before the probe field ��p�, and if thecharacteristic time change is slow on a scale set by therms Rabi frequency, the fields prepare the dark state ofEIT by stimulated Raman adiabatic passage. If the twofields are constant, the dark state is also self-prepared bythe lasers, but this takes a finite amount of preparationenergy from the probe field and transfers it into the me-dium and the coupling field. This preparation energywas first determined by Harris and Luo �1995�. It isgiven by the number of atoms in the probe path N andthe stationary values of the Rabi frequencies,

Eprep = ��pN��p�2

��c�2. �35�

After the preparation energy is transferred to the atomsand the coupling field, the medium remains transparent.On the other hand, if the probe field is also switched offbefore the coupling field, the dark state eventually re-turns to the initial state �1� and the preparation energy istransferred back from the medium and the coupling fieldto the probe field. In this case there is no net loss ofenergy from the probe field but only a temporary trans-fer. As we shall see later in Sec. IV, this temporary trans-fer of energy has some interesting consequences for thepropagation of the fields.

E. EIT inside optical resonators

The large linear dispersion associated with EIT in anoptically thick medium can substantially affect the prop-erties of a resonator system, especially since the mediumdispersion can easily exceed that of an empty cavity. It istherefore natural to investigate the possibility of havingan EIT medium inside a cavity, such that one of thehigh-finesse modes replaces the probe field and couplesthe ground state �1� to state �3�.

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If an empty-cavity resonance �c0 is sufficiently closeto the frequency �0 of electromagnetically inducedtransparency, the presence of the medium will lead to avery strong pulling of the resonance frequency �c of thecombined atom-cavity system towards this value �Lukin,Fleischhauer, et al., 1998�. At the same time the reso-nance width is substantially reduced �Lukin, Fleis-chhauer, et al., 1998�. To see this let us consider the re-sponse function of a resonator containing an EITmedium, i.e., the ratio of circulating to input intensity atfrequency �. For simplicity we assume a unidirectionalring resonator of round-trip length L with a single in-outcoupling mirror. Then

S��� �Icirc���Iin���

=t2

1 + r2�2 − 2r� cos������, �36�

where t and r are the amplitude transmittivity and re-flectivity of the in-out coupling mirror. In the case of alossless mirror, r2+ t2=1. ����=� /c�L+ l Re���1��� is thephase shift and �=exp�−�l Im���1�� /c� the medium ab-sorption per round trip L. l is the length of the EITmedium. Close to two-photon resonance, Re���1����dn /d����−�0�. This yields for the resonance of thecavity+medium system

�c =1

1 + ��c0 +

1 + ��0, where � = �0

dn

d�

l

L. �37�

One recognizes a strong frequency pulling towards theatomic EIT resonance since the dispersion �0dn /d� canbe rather large. Likewise one finds from Eq. �36� thatthe width of the cavity resonance is changed accordingto

�c

�c0=

1 − r���1 − r�

1

1 + �. �38�

The first term accounts for a small enhancement of thecavity width due to the additional losses induced by themedium. Much more important, however, is the secondterm, which describes a substantial reduction of the cav-ity linewidth due to the linear dispersion of the medium.

Since the EIT resonance depends on the atomic sys-tem as well as on the coupling laser, frequency pullingand line narrowing are relative to the coupling laser.These effects have been observed experimentally byMüller et al. �1997�. Figure 12 shows the transmissionspectrum of an empty cavity and a cavity with an EITmedium. Applications for difference-frequency lockingand stabilization have been suggested and the reduc-tion of the beat-note linewidth in a laser systembelow the Shawlow-Townes limit predicted �Lukin,Fleischhauer, et al., 1998�.

F. Enhancement of refractive index, magnetometry, andlasing without inversion

We now undertake a brief survey of some applicationsof EIT and related phenomena which will not be dis-cussed in detail in the following sections. The use of

laser fields to control the refractive index has createdmuch attention. A suggestion for controlling phasematching in nonlinear mixing through the action of anadditional field was made by Tewari and Agarwal �1986�,but it is since the advent of EIT that these ideas havebeen extensively studied. Harris treated the refractiveproperties of an EIT medium theoretically �Harris et al.,1992�. In pulsed-laser experiments the vanishing of thesusceptibility at resonance leads to the elimination ofdistortion for pulses propagating through a very opti-cally thick medium at resonance �Jain et al., 1995�. Re-lated suggestions by Harris for the use of strong off-resonant fields to control the refractive index �Harris,1994b� have been important in the implementation ofmaximal coherence frequency-mixing schemes with off-resonant lasers.

Extensive theoretical work was carried out by Scullyand co-workers �Scully, 1991; Fleischhauer et al., 1992a,1992b; Scully and Zhu, 1992; Rathe et al., 1993� and oth-ers �Wilson-Gordon and Friedman, 1992� on the use ofresonant cw fields for modification of the refractive in-dex. The experiments of several workers have exten-sively studied refractive index modifications in this cwlimit �Moseley et al., 1995; Xiao et al., 1995�.

Working in Doppler-free conditions with the narrowbandwidths attainable through use of cw lasers it is pos-sible to observe very steep EIT-induced dispersion pro-files. This has led to the suggestion that very sensitiveinterferometers and magnetometers be used, since tinyvariations in frequency couple to large changes in refrac-tive index �Scully and Fleischhauer, 1992; Fleischhauerand Scully, 1994; Budker et al., 1998, 1999; Nagel et al.,1998; Sautenkov et al., 2000; Stahler et al., 2001; Af-folderbach et al., 2002�. In the magnetometer schemesthis greatly increases the sensitivity of the measurementof the Zeeman shifts induced by a magnetic field.

FIG. 12. Experimental demonstration of linewidth narrowing:�a� transmission spectrum of empty cavity and with EIT me-dium; �b� magnification of transmission spectrum with EIT.From Müller et al., 1997.

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As mentioned earlier, lasing without inversion isclosely related to the interference important in EIT.There have by now been a number of experimental veri-fications of this concept with the actual realization oflaser action without population inversion in the visiblespectral range including several reports of inversionlessamplification �Gao et al., 1992; Fry et al., 1993; Nottle-man et al., 1993� as well as actual demonstrations of laseroscillation without population inversion �Zibrov et al.,1995; Padmabandu et al., 1996�. An application of lasingwithout inversion to short-wavelength lasing is yet to beshown.

IV. EIT AND PULSE PROPAGATION

So far we have considered EIT only from the point ofview of the atomic system and its linear response to sta-tionary monochromatic fields, but have not paid atten-tion to propagation effects of pulses. Although the sus-ceptibility discussed in the last section already containsall necessary information, it is worthwhile to devote aseparate section to this issue, the more so as the almostperfect transparency at certain frequencies, characteris-tic of EIT, allows pulse propagation in otherwise opti-cally thick media. Here the action of the medium on thelight pulses is—apart from the single frequency forwhich the medium is transparent—quite substantial, anda number of interesting propagation phenomena are en-countered. These effects often have a simple physicalorigin, but their size and their special properties makethem very important for a variety of applications.

A. Linear response: Slow and ultraslow light

Let us first consider the properties associated with thelinear response of an EIT medium to the probe field Ep.We have seen in Sec. III, Eq. �13�, that the most charac-teristic feature of the real part of the susceptibility spec-trum is a linear dependence on the frequency close tothe two-photon resonance =0. For a negligible decay ofthe �1�-�2� coherence one finds

Re���1�� = 2�31

�c2 + O� 2� , �39�

where = �3/4�2���3 is the normalized density and � isthe transition wavelength in vacuum. Since the lineardispersion dn /d�p of the refractive index n=1+Re���is positive, EIT is associated with a reduction of thegroup velocity according to

vgr � �d�p

dkp� =0

=c

n + �p�dn/d�p�, �40�

which was first pointed out by Harris et al. �1992�. At thesame time the index of refraction of an ideal three-levelmedium is unity and thus the phase velocity of the probefield is just the vacuum speed of light:

vph � ��p

kp� =0

=c

n= c . �41�

An important property of the EIT system is that thesecond-order term in Eq. �39� vanishes exactly if there isalso single-photon resonance =0 of the probe field. Asa consequence there is no group velocity dispersion, i.e.,no wave-packet spreading. Using Eq. �39� yields at two-photon resonance

vgr =c

1 + ngrwith ngr = ��c

�31

�c2 , �42�

where k=�� was used. The reduced group velocitygives rise to a group delay in a medium of length L:

!d = L� 1

vgr−

1

c� =

Lngr

c= ��L

�31

�c2 . �43�

It should be noted that under nonideal conditions, i.e., ifthe dark resonance is not perfectly stable, for exampledue to collisional dephasing of the �1�-�2� coherence orfast phase fluctuations in the beat note between couplingand probe fields, the denominator in the expression forngr needs to be replaced by �c

2+�31�21, where �31 and �21are, as defined in Sec. III, the transversal decay rates ofthe probe transition and the �1�-�2� coherence. In thiscase there is a lower limit to vgr for a fixed density �.

Due to the vanishing imaginary part of the suscepti-bility, i.e., perfect transparency, at =0, relatively highatom densities � and low intensities of the coupling fieldIc��c

2 can be used. Thus the group index ngr can berather large compared to unity, and extremely smallgroup velocities are possible. In the first experiments byHarris and co-workers in lead vapor a group velocity ofvgr /c�165 was observed �Kasapi et al., 1995�. In laterexperiments by Meschede et al. a dispersive slope corre-sponding to a value of vgr /c�2000 was found �Schmidtet al., 1996�. Other earlier experiments in which directlyor indirectly slow group velocities were measured wereperformed by Xiao et al. �1995� and by Lukin et al.�1997�. The slowdown of light by EIT attracted a lot ofattention when Hau and collaborators observed thespectacular reduction of the group velocity to 17 m/s ina Bose condensate of Na atoms, corresponding to apulse slowdown of seven orders of magnitude �Hau etal., 1999�. Similarly small values were later obtained in abuffer-gas cell of hot Rb atoms by Kash et al. �1999� andby Budker et al. �1999�. More recently a substantial slow-down of the group velocity was also observed in thesolid state by Turukhin et al. �2001�.

The lossless slowdown of a light pulse in a medium isassociated with a number of important effects. When apulse enters such a medium, it becomes spatially com-pressed in the propagation direction by the ratio ofgroup velocity to the speed of light outside the medium�Harris and Hau, 1999�. This compression emerges be-cause when the pulse enters the sample its front endpropagates much more slowly than its back end. At thesame time, however, the electrical-field strength remainsthe same. The reverse happens when the pulse leaves

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the sample. In the case of the experiment of Hau et al.�1999� the spatial compression was from a kilometer to asubmillimeter scale! Since the refractive index is unity attwo-photon resonance, reflection from the mediumboundary is usually negligible as long as the pulse spec-trum is not too large �Kozlov et al., 2002�.

Although in the absence of losses the time-integratedphoton flux through any plane inside the medium is con-stant, the total number of probe photons inside the me-dium is reduced by a factor vgr /c due to spatial compres-sion. Thus photons or electromagnetic energy must betemporarily stored in the combined system of atoms andcoupling field. It should be noted that the notion of agroup velocity of light is still used even for vgr�c, whereonly a tiny fraction of the original pulse energy remainselectromagnetic.

It is instructive to consider slow-light propagationfrom the point of view of the atoms. From this perspec-tive the physical mechanism for the temporary transferof excitations to and from the medium can be under-stood as stimulated Raman adiabatic return. Before theprobe pulse interacts with three-level atoms, a cw cou-pling field puts all atoms into state �1� by optical pump-ing. In this limit state �1� is identical to the dark state.When the front end of the probe pulse arrives at anatom, the dark state makes a small rotation from state�1� to a superposition between �1� and �2�. In this processenergy is taken out of the probe pulse and transferredinto the atoms and the coupling field. When the probepulse reaches its maximum, the rotation of the darkstate stops and is reversed. Thus energy is returned tothe probe pulse at its back end. The excursion of thedark state away from state �1� and hence the character-istic time of the adiabatic return process depends on thestrength of the coupling field. The weaker the couplingfield, the larger the excursion and thus the larger thepulse delay. To put this picture into a mathematical for-malism one can employ a quasiparticle picture first in-troduced by Mazets and Matisov �1996� and indepen-dently by Fleischhauer and Lukin �2000�, in which thisconcept was first applied to pulse propagation. One de-fines dark ��� and bright ��� polariton fields accordingto

��z,t� = cos "Ep�z,t� − sin "��21�z,t�eikz, �44�

��z,t� = sin "Ep�z,t� + cos "��21�z,t�eikz, �45�

with the mixing angle determined by the group index

tan2 " =��c�31

�c2 = ngr. �46�

Ep is the normalized, slowly varying probe field strength,Ep=Ep�� /2�0, with � being the corresponding carrierfrequency. k=kc

� −kp, where kp is the wave number ofthe probe field and kc

� is the projection of the coupling-field wave vector along the z axis. �21 is the single-atomoff-diagonal density-matrix element between the twolower states.

In the limit of linear response, i.e., in first order ofperturbation in Ep, and under conditions of EIT, i.e., fornegligible absorption, the set of one-dimensionalMaxwell-Bloch equations can be solved �Fleischhauerand Lukin, 2000, 2001�. One finds that only the darkpolariton field � is excited, i.e., ��0, and thus

Ep�z,t� = cos "��z,t� , �47�

�21�z,t� = − sin "��z,t�e−ikz

�. �48�

Furthermore, under conditions of single-photon reso-nance, � obeys the simple shortened wave equation

�t+ c cos2 "

�z���z,t� = 0, �49�

which describes a form-stable propagation with velocity

vgr = c cos2 " . �50�

The slowdown of the group velocity of light in an EITmedium can now be given a very simple interpretation:EIT corresponds to the lossless and form-stable propa-gation of dark-state polaritons. These quasiparticles area coherent mixture of electromagnetic and atomic spinexcitations, the latter referring to an excitation of the�1�-�2� coherence. The admixture of the components de-scribed by the mixing angle " depends on the strength ofthe coupling field as well as the density of atoms anddetermines the propagation velocity. In the limit "→0,corresponding to a strong coupling field, the dark-statepolariton is almost entirely electromagnetic in nature,and the propagation velocity is close to the vacuumspeed of light c. In the opposite limit, "→� /2, the dark-state polariton has the character of a spin excitation andits propagation velocity is close to zero. The concept ofdark- and bright-state polaritons can easily be extendedto a quantized description of the probe field �Fleisch-hauer and Lukin, 2000, 2001� as well as quantized matterfields �Juzeliunas and Carmichael, 2002�, in which casethe polaritons obey approximately Bose commutationrelations. This extension is of relevance for applicationsin quantum information processing and nonlinear quan-tum optics, to be discussed later on.

So far the atoms have been assumed to be at rest.However, EIT in moving media shows a couple of otherinteresting features. First of all, if all atoms move withthe same velocity v ��v��c� relative to the propagationdirection of the light pulse, Galilean transformationrules predict that the group velocity �Eq. �50�� will bemodified according to

vgr = c cos2 " + v sin2 " . �51�

This expression can also be obtained from Eq. �40� ifone takes into account that as a result of the Dopplereffect the susceptibility or the index of refraction be-comes k dependent �spatial dispersion�. Light draggingaccording to Eq. �51� was recently observed by Strekalovet al. �2004�. It is interesting to note that vgr can now bezero or even negative for "�� /2 if the atoms move in

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the opposite direction to the propagation of light. Infact, since extremely large values of the group index canbe achieved in vapor cells, the thermal motion of atomseven at room temperatures can have a significant effecton the effective group velocity. Making use of this effect,Kocharovskaya et al. suggested “freezing” a light pulsein a Doppler-broadened medium, where a specific veloc-ity class is selectively excited �Kocharovskaya et al.,2001�. When the group velocity is decreased, the spectralbandwidth of linear dispersion and transparency de-creases. In the limit of zero group velocity, the disper-sion is infinite, which can clearly be supported only overa vanishingly narrow frequency band. Thus in practiceabsorption and reflection will prevent the full stoppingof a light pulse with this method, unless a time-dependent coupling field is used, which leads to a dy-namical narrowing of the pulse spectrum �see below� oracceleration of the atoms. If the flow of atoms is notparallel to the propagation of light, excitation carried bythe atoms can be transported in a perpendicular direc-tion �Juzeliunas et al., 2003�, as can be observed in vaporcells where the atoms undergo diffusive motion �Zibrovet al., 2002�.

The slowing down of light has a number of importantapplications. A reduction of the group velocity of pho-tons leads to an enhanced interaction time in a nonlinearmedium, which is important in enhancing the efficiencyof nonlinear processes �Harris and Hau, 1999; Lukin,Yelin, and Fleischhauer, 2000; Lukin and Imamoglu,2001�. It should be noted in this context that, althoughthe number of photons in the medium is reduced by theratio of the group velocity to the vacuum speed of light,the electric field is continuous at the boundaries andthus nonlinear polarization is not reduced as a result ofcompression. These applications will be discussed inmore detail in Secs. V and VI. Making use of the sub-stantial pulse deformation at the boundary of an EITmedium is also of interest for the storage of informationcontained in long pulses, which in this way can be com-pressed to a very small spatial volume.

As noted by Harris, pulse compression is associatedwith a substantial enhancement of the spatial field gra-dient in the propagation direction �Harris, 2000�, andthus ponderomotive dipole forces on atoms, resultingfrom off-resonant couplings to the light pulse, are sub-stantially amplified. If I0�z� denotes the probe intensityat the input of the medium, one finds a force enhancedby the ratio c /vgr:

Fdip � −d

dzI�z� = −

c

vgr

d

dzI0�z� . �52�

This has important potential applications in atom andmolecular optics as well as laser cooling.

The light-matter coupling associated with EIT canalso be used for the preparation and detection of coher-ent matter-wave phenomena in ultracold quantum gases.For example, EIT has been suggested as a probe for thediffusion of the relative phase in a two-component Bose-Einstein condensate �Ruostekoski and Walls, 1999a,1999b�. Stopping of a light pulse at a spatial discontinu-

ity of the control field was used by Dutton et al. �2001� tocreate strongly localized shock waves in a Bose-Einsteincondensate and to study their dynamics.

Finally, interesting effects have been predicted byLeonhardt and co-workers when the propagation of ul-traslow light is considered in nonuniformly moving me-dia. Using an effective classical field theory for slowlight, in which losses are neglected, they predicted anAharonov-Bohm-type phase shift in a rotating EIT me-dium with a vortex flow �Leonhardt and Piwnicki, 1999�.Leonhardt and Piwnicki also pointed out an interestinganalogy between slow light in a medium with a vortexflow and general relativistic equations for a black hole�Leonhardt and Piwnicki, 2000a�. This may allow forlaboratory studies of some general relativistic effectswhen adding an inward flow to the vortex �Leonhardtand Piwnicki, 2000b; Visser, 2000�. The practical obser-vation of an optical analogy to the event horizon of ablack hole will, however, most likely be prevented byabsorption and reflection effects.

To assess the potential of slow light it is important toconsider its limitations. A convenient figure of merit forthis is not the achievable group velocity itself, but theratio of achievable delay time !d of a pulse in an EITmedium to its pulse length !p. One upper limit for thedelay time is given by probe absorption due to the finitelifetime of the dark resonance. Furthermore, for apulsed probe field, i.e., for a probe field with a finitespectral width, the absorption of the nonresonant fre-quency components is nonzero even under ideal condi-tions of an infinitely long-lived dark state. Thus, in orderfor the absorption to be negligible, first the decay timeof the �1�-�2� transition must be very small and secondthe spectral width of the pulse �p has to be muchsmaller than the transparency width �trans:

�p ��trans =�c

2

�31�31

1��L

. �53�

It is instructive to express �trans in this condition interms of the group delay !d=ngrL /c:

�trans = ��L1

!d�31

�31. �54�

This discussion shows that it is not possible to bring apulse to a complete stop by using EIT with a stationarycoupling field, since in this case the transparency widthwould vanish, leading to a complete absorption of thepulse. The same argument holds if one tries to stop anultraslow pulse by means of atomic motion antiparallelto the flow of light. Making use of �p!p#1 one finds anupper limit for the ratio of group delay to pulse length�Harris and Hau, 1999�

!d

!p� ��L . �55�

The product ��L is the opacity of the medium in theabsence of EIT, i.e., exp�−��L� is the transmission co-efficient at bare atomic resonance. Thus a noticeable

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time delay of a pulse by EIT requires an optically thickmedium with ��L�1.

B. “Stopping of light” and quantum memories for photons

As discussed in the preceding subsection it is not pos-sible to bring a light pulse to a complete stop with sta-tionary EIT. To achieve this goal the group velocity hasto be changed in time, as was shown by Fleischhauer andLukin �2000�. It should be mentioned that the possibilityof transferring spatial excitation distributions of atomicensembles to light pulses was pointed out before byCsesznegi and Grobe �1997�. In the following the “stop-ping” and “reacceleration” of a light pulse and its poten-tial applications will be discussed in more detail. At thispoint a word of caution is needed, however. The expres-sion “stopping of light” should not be taken literally. Asmentioned before, the reduction of the propagation ve-locity of light in a lossless, passive medium is alwaysassociated with a temporary transfer of its energy to themedium. In the extreme limit of zero velocity relative tothe stationary medium no electromagnetic excitation isleft at all. Nevertheless, the notion of a vanishing groupvelocity of light has here the same justification as thenotion of a group velocity in the case of ultraslow pulsepropagation, in which likewise only a tiny fraction of theoriginal excitation remains in the form of photons.

A key conceptual advance with respect to the stop-ping of a light pulse occurred when it was realized byFleischhauer and Lukin �2000� that the bandwidth limi-tation �Eq. �53�� of EIT can be overcome if the groupvelocity is adiabatically reduced to zero in time. This canbe achieved, for example, by reducing the Rabi fre-quency of the drive field. In this case the spectrum of theprobe pulse narrows proportional to the group velocity,

�p�t� = �p�0�vgr�t�

vgr�0�, �56�

and the spectrum of the probe pulse stays within thetransparency window at all times, provided it fulfills thiscondition initially. The propagation equation for thedark-state polariton is then still given by Eq. �49�, with"→"�t�. Adiabatically rotating "�t� from 0 to � /2 leadsto a deceleration of the polariton to a full stop. At thesame time its character changes from that of an electro-magnetic field to that of a pure spin excitation. Mostimportantly, provided the rotation is adiabatic, all prop-erties of the original light pulse are coherently trans-ferred to the atomic spin system in this process moduloan overall phase determined by that of the couplingfield.

Conditions of adiabaticity have been analyzed byMatsko, Rostovtsev, Kocharovskaya, et al. �2001� and byFleischhauer and Lukin �2001�. The resulting limitationson the rate of change of the coupling field are ratherweak. Furthermore, if the coupling field and thus thegroup velocity is already very small, even an instanta-neous switchoff would lead only to a loss of the verysmall electromagnetic component of the polariton �Liuet al., 2001; Matsko, Rostovtsev, Kocharovskaya, et al.,2001; Fleischhauer and Mewes, 2002�. Reversing theadiabatic rotation of " by increasing the strength of thecoupling field leads to a reacceleration of the dark-statepolariton associated with a change of character fromspinlike to electromagnetic. In Fig. 13 a numerical simu-lation of the stopping of a dark-state polariton and itssuccessive reacceleration is shown. The transfer of exci-tation from the probe pulse to the Raman spin excita-tion and back is apparent. In this way the excitation andall information contained in the original pulse can be

FIG. 13. Numerical simula-tion of dark-state polaritonpropagation with envelopeexp�−�z /10�2�. The mixingangle is rotated from 0 to � /2and back according to cot "�t�= 100 „1 − 0.5 tanh � 0.1 � t −15��+ 0.5 tanh � 0.1 � t −125�� …: top,polariton amplitude; bottomleft, electric field amplitude;bottom right, atomic spin co-herence; all in arbitrary units.Axes are in arbitrary units withc=1.

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reversibly transferred and stored in long-lived spin co-herences. It should be noted that the spin excitationdoes not in general store the photon energy, as most of itis transferred to the coupling field by stimulated Stokesemission.

The classical aspects of stopping and reaccelerating oflight pulses have been experimentally demonstrated byLiu and co-workers in ultracold Na �Liu et al., 2001�, byPhillips et al. in hot Rb vapor �2001�, and in solids byTurukhin and collaborators �Turukhin et al., 2001�. Theexperiment of Liu et al. used a similar setup to that usedfor the demonstration of the group velocity reduction to17 m/s �Hau et al., 1999�. Here a light pulse was slowedand spatially compressed in an ultracold and dense va-por of Na atoms at a temperature just above the point ofBose condensation and then stopped by turning off thecoupling laser. Storage times of up to 1.5 ms for a pulselength of a few �s were achieved. Figure 14 shows theexperimental setup as well as the storage and retrieval ofa light pulse. The use of ultracold gases has advantagessuch as the reduction of two-photon Doppler shifts andhigh densities. Ultracold gases, however, are not neces-sary to achieve light stopping. In the experiment of Phil-lips et al. a Rb vapor gas cell at temperatures of about

70–90 °C was used. To eliminate two-photon Dopplershifts, degenerate Zeeman sublevels were used for states�1� and �2�. Moreover, to reduce the effects of atomicmotion a He buffer gas was employed. Here storagetimes of up to 0.5 ms could be obtained for a pulselength of several �s. The first successful demonstrationof light “storage” in a solid was achieved by Turukhinand co-workers in Pr-doped Y2SiO5 after proper prepa-ration of the inhomogeneously broadened material by aspecial optical hole- and antihole-burning technique �Tu-rukhin et al., 2001�. A direct proof of the coherence ofthe storage mechanism was provided by Mair et al.�2002�, when the phase of the stored light pulse was co-herently modified during the storage time by a magneticfield. Experimental studies of light storage under condi-tions of large single-photon detuning were moreoverperformed by Kozuma et al. �2002�. All of these experi-ments have to be considered as a proof of principle sincethe transfer efficiency or fidelity of the storage is stillrather small due to a variety of limiting effects such asdark-state dephasing, atomic motion, and mode mis-match. An application that does not suffer from theseeffects is the use of the �partially dissipative� stopping of

FIG. 14. �Color in online edition� Storage andretrieval of light in ultracold Na atoms: top,experimental setup for observing light storageand retrieval; bottom, measurements of de-layed �a� and revived �b� probe pulse afterstorage. Dashed line shows intensity of cou-pling fields. For details see Liu et al., 2001.From Liu et al., 2001.

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a light pulse at a spatial discontinuity of the drive field tocreate well-defined shock waves in an atomic Bose-Einstein condensate, as demonstrated by Dutton et al.�2001�.

The most important potential application of lightstopping is certainly in the field of quantum information.The dark-state polariton picture of light stopping alsoholds for quantized radiation. Here quantum states ofphotons are transferred to collective excitations of themedium. For example, a single-photon state of a singleradiation mode �1�ph is mapped to an ensemble of Nthree-level atoms according to

�1�ph � �1,1, . . . ,1� ,

�0�ph �1

N�j=1

N

�1,1, . . . ,2j, . . . ,1� . �57�

In this way the EIT medium can act as a quantummemory for photons. Compared to other proposals forquantum memories, such as single-atom cavity systems�Cirac et al., 1997�, continuous Raman scattering �Schoriet al., 2002�, or storage schemes based on echo tech-niques, the EIT-based system is capable of storing indi-vidual photon wave packets with high fidelity and with-out the need for a strongly coupling resonator. It shouldbe mentioned that a time-symmetric photon-echo tech-nique has been proposed by Moiseev and Kroll �2001�,that is also capable of storing single photons but has notbeen experimentally implemented. Several limitations ofthe EIT storage technique have been studied, includingfinite two-photon detuning �Mewes and Fleischhauer,2002�, decoherence �Fleischhauer and Mewes, 2002;Mewes and Fleischhauer, 2005�, and fluctuations of cou-pling parameters �Sun et al., 2003�. An extension of themodel to three spatial dimensions was given by Duan etal. �2002�.

An interesting extension of the light-stopping schemewas very recently experimentally demonstrated byBajcsy et al. �2003�. After the coherent transfer of a lightpulse to an ensemble of Rb atoms in a vapor cell, twocounterpropagating coupling lasers were applied ratherthan one, forming a standing-wave pattern. At the nodesof this pattern, a periodic structure of spatially narrowabsorption zones emerged. The two counterpropagatingcomponents of the coupling laser regenerated twoStokes pulses with opposite wave vectors. Thus the re-generated light pulses also formed a standing-wave pat-tern, which matched that of the counterpropagatingdrive fields. In this way a stationary pulse of light wascreated and stored for several �s. Although the regen-erated pulse contained only a small fraction of the origi-nal photon number, the electromagnetic field was non-zero during the storage period in contrast to the above-mentioned experiments. This generation of stationarylight pulses may be an important tool for nonlinear op-tical processes with few photons.

Stopping of light can in principle also be used toprepare atomic ensembles in specific nonclassical or

entangled many-particle states �Lukin, Yelin, andFleischhauer, 2000�. An alternative and often experi-mentally simpler way to achieve the same effect is to useabsorptive or dispersive interaction of continuous-wavelight fields with atomic ensembles and detection.Kuzmich et al. proposed the generation of spin-squeezedensembles by absorption of squeezed light �Kuzmich etal., 1997�, which was experimentally demonstrated laterby Hald et al. �1999�. Employing similar ideas Polzik sug-gested the creation of two Einstein-Podolsky-Rosen en-tangled atomic ensembles by absorption of quantum-correlated light fields �Polzik, 1999� and Kozhekin et al.proposed a memory for quantum states of cw light fieldsalong these lines �Kozhekin et al., 2000�. Recently ex-perimental progress toward the implementation of thecw quantum memory has been reported �Schori et al.,2002�. A very intriguing technique for creating nonclas-sical or entangled atomic ensembles is the measurementof the collective spin using off-resonant dispersive inter-actions. Kuzmich et al. performed in this way quantumnondemolition measurements to prepare quantum stateswith sub-shot-noise spin fluctuations �Kuzmich et al.,1999, 2000�. Duan et al. �2000� suggested a detectionscheme in which coherent light simultaneously probestwo atomic ensembles to prepare an Einstein-Podolsky-Rosen correlation between them, with potential applica-tions in continuous teleportation of collective spinstates. Recently, Julsgaard and co-workers have demon-strated this technique in a remarkable experiment �Juls-gaard et al., 2001�.

Finally, Imamoglu �2002� and James and Kwiat �2002�have suggested transferring photons to atomic en-sembles to build single-photon detectors with high effi-ciency, which are able to distinguish between single andmultiple photons. Such a device has particular impor-tance for quantum information processing with linearoptical elements �Knill et al., 2001�.

C. Nonlinear response: Adiabatic pulse propagation andadiabatons

So far we have discussed the interaction of light pulseswith ensembles of three-level atoms only in the linearresponse limit, i.e., assuming a weak probe field. Nowwe consider the nonperturbative situation of probe andcoupling pulses with comparable strength, which showsa number of interesting additional features. New effectsarise, in particular, because in this case the medium hasan effect on the propagation of both pulses.

In almost all theoretical treatments a quasi-one-dimensional situation is considered in which all fieldspropagate parallel to the z direction, a homogeneousmedium is assumed, and atomic motion is disregarded.The properties of the atoms are described by a statevector with slowly varying probability amplitudesCn�z , t�, n� �1,2,3�. For simplicity let us assume that thecarrier frequencies of coupling and probe pulses are onresonance with the corresponding atomic transitions,which allows us to treat the slowly varying field ampli-tudes as real. The evolution of the atomic state is de-

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scribed by a Schrödinger equation for Cn�z , t� with thethree-level Hamiltonian given in Eq. �29�. Because de-cay of the polarization plays an important role for thepropagation of the fields, a decay out of the excited state�3� with rate �3 is included by adding an imaginary termi��3 /2 to the energy of the excited state. The propaga-tion of the field amplitude is most easily written in termsof the corresponding Rabi frequencies. In a slowly vary-ing amplitude and phase approximation one finds theshortened wave equations for the probe ��p� and cou-pling pulse ��c�,

� �

�t+ c

�z��p�z,t� = i

�p

2C2�z,t�C1

*�z,t� , �58�

� �

�t+ c

�z��c�z,t� = i

�c

2C2�z,t�C3

*�z,t� , �59�

where the effect of the atoms on the fields is param-etrized by the resonant absorption coefficients

�p =�p���12�2

2�0�, �c =

�c���32�2

2�0�, �60�

with �ij being the dipole moments of the correspondingtransitions.

If the amplitudes of the fields vary sufficiently slowlythat the adiabatic approximation holds, and if the appro-priate initial conditions are fulfilled, the atoms stay at alltimes in the instantaneous dark state, �a0�t��=cos �t��1�−sin �t��2�, Eq. �17�, with tan �t�=�p�t� /�c�t�. As men-tioned before, an important property of this state is theabsence of a component in state �3�. As a consequence ofthis there is also no dipole moment on either of the twotransitions coupled by the probe and coupling pulses.Atoms in the instantaneous dark state therefore have noeffect at all. Light and matter are exactly decoupled.

As first noted by Harris �1993�, there is another casein which bichromatic pulses are exactly decoupled fromthe atoms even if they do not vary slowly. The dark statecorresponding to a pair of pulses with identical enve-lope, so-called matched pulses, i.e., �p�z , t�=�pf�z , t� and�c�z , t�=�cf�z , t�, is time independent and thus a trueeigenstate of the system. After an appropriate prepara-tion of the medium, matched pulses will thus remainexactly decoupled from the interaction for all times andpropagate with the vacuum speed of light.

As shown by Harris and Luo, a pair of matched pulsesapplied to an atomic ensemble in the ground state,which in this case does not correspond to the dark state,will prepare the atoms by stimulated Raman adiabaticpassage �STIRAP; Harris and Luo, 1995�. During thefirst few single-photon absorption lengths, the front endof the probe pulse experiences a small loss. In this way acounterintuitive pulse ordering is established. This pro-vides asymptotic connectivity of the dark state to theinitial state of the atoms. The leading end of this slightlydeformed pair of pulses will prepare all atoms in thepathway via STIRAP and the pulses can propagate un-affected through the rest of the medium �Eberly et al.,1994; Harris, 1994a�. This is illustrated in Fig. 15 where

numerical calculations are used to show the propagationof two matched pulses in an unprepared medium.

Apart from the preparation at the front end, matchedpulses are stable solutions of the propagation problem.They should therefore be formed whenever pulses ofarbitrary shape are applied to optically thick three-levelmedia. In fact, it has been shown by Harris �1993� thatnonequal pairs of pulses with a strong cw carrier,

�p�z,t� = �1 + f�z,t���pe−i�p�t−z/c�, �61�

�c�z,t� = �1 + g�z,t���ce−i�c�t−z/c�, �62�

tend to adjust their amplitude modulations f�z , t� andg�z , t� in the course of propagation, i.e.,

� f�z,t�g�z,t�

�z→�

= 1. �63�

Fluctuations in the difference of f and g lead to non-adiabatic transitions out of the dark state, defined by thecw components, and will thus be absorbed. Eventuallypulses of identical envelope are formed. The tendency togenerate pulses with identical envelopes is not restrictedto fields with a strong cw carrier but is also found forarbitrary pulses �pulse matching�. The phenomenon ofpulse matching causes a correlation of quantum fluctua-tions in both fields, as discussed, for example, by Agar-wal �1993�, Fleischhauer �1994�, and Jain �1994�.

As noted above, an EIT medium does not couple tothe fields at all in the adiabatic limit. In order to covereffects like the reduction in group velocity, it is neces-sary to include first-order nonadiabatic corrections inthe dynamics of the atoms, which lead to small contribu-tions to the excited-state amplitude. Taking these intoaccount, one finds for the state amplitudes

FIG. 15. Comparison of probe field and coupling field:top, amplitude of probe field as function of time in arbitraryunits; dotted line, at medium entrance z=0; solid line, forz=1000; bottom, the same for the coupling field. Distance ismeasured in units of absorption length and peak values are�c=�p=10�.

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C1 = cos , C2 =2i

�, C3 = − sin , �64�

where �=�p2 +�c

2. It was shown by Grobe et al. �1994�that the nonlinear equations of propagation, Eqs. �58�and �59�, together with Eq. �64� are adiabatically inte-grable even for fields of comparable strength and witharbitrary shape. To see this, we adopt the method ofFleischhauer and Manka �1996� and transform the fieldequations for �p and �c in propagation equations forthe rms Rabi frequency � and the nonadiabatic coupling

. Under nearly adiabatic conditions, ����, a weak-coupling approximation is justified. In this limit, assum-ing furthermore equal coupling strength �p=�c=�, onefinds that the total Rabi frequency fulfills the free-spacepropagation equation

� �

�t+ c

�z���z,t� � 0. �65�

No photons are lost by absorption and there is only acoherent transfer from one field to the other. At the

same time obeys the equation

� �

�t+ c

�z��z,t� = − �

�t�

�2� , �66�

which is exactly integrable. The corresponding solutions,called adiabatons �Grobe et al., 1994�, are particularlysimple if � is approximately constant over the time in-terval of interest. In that case the probe and coupling

pulses have complementary envelopes and propagateswithout changing form with velocity vgr=c / �1+� /�2�.The quasi-form-invariant propagation of an adiabaton isshown in Fig. 16. If the Rabi frequency of the couplingfield is much larger than that of the probe field, i.e., inthe perturbative limit, the adiabaton shows no notice-able dip in the coupling-field strength. This case thusresembles the propagation of a weak pulse in EIT withreduced group velocity.

The first experimental evidence of adiabaton propaga-tion in Pb vapor was reported by Kasapi et al. �1996�.Adiabatons in more complicated configurations, such as

double-lambda systems and double pairs of pulses, werestudied by Cerboneschi and Arimondo �1995� as well asby Hioe and Grobe �1994�. The effect of nonequal cou-pling strength on adiabaton propagation was studied byGrigoryan and Pashayan �2001�. The adiabaton solutionsare approximately stable over many single-photon ab-sorption lengths. However, as shown by Fleischhauerand Manka �1996�, they eventually decay to matchedpulses after sufficiently long propagation distances.

For time scales short enough that decays can be ig-nored, and for equal coupling strength, exact solutionsof the nonlinear propagation problem in V- and �-typesystems exist even beyond the adiabatic approximation.Konopnicki and Eberly �1981� and Konopnicki et al.�1981� found soliton solutions with identical pulse shape,called simultons. An interpretation of simultons in termsof solitons corresponding to an effective two-level prob-lem was given by Fleischhauer and Manka �1996�.Malmistov �1984� and Bol’shov and Likhanskii �1985�derived simulton solutions in three-level systems usingthe inverse-scattering method. Several extensions of si-multons have been discussed. N-soliton solutions weregiven by Steudel �1988�. Inhomogeneously broadenedsystems were considered by Rahman and Eberly �1998�and by Rahman �1999� and extensions to five-level sys-tems were discussed by Hioe and Grobe �1994�.

V. ENHANCED FREQUENCY CONVERSION

A. Overview of atomic coherence-enhanced nonlinearoptics

Harris and co-workers identified the enhancement ofnonlinear optical frequency conversion as a major ben-efit of electromagnetically induced transparency in theirpaper of 1990 �Harris et al., 1990�. Hemmer and co-workers �Hemmer et al., 1995� introduced the double-�scheme as an important tool in EIT-based resonant four-wave mixing. Coherent preparation of a maximal coher-ence was likewise found by a number of researchers,including the groups of Harris �Jain et al., 1996� in theUSA and Hakuta in Japan �Hakuta et al., 1997�, to sig-nificantly improve the conversion efficiency in four-wavemixing. While destructive interference reduces the linearsusceptibility of a laser-dressed system, this is not so forthe nonlinear susceptibility in four-wave mixing, whichin fact undergoes constructive interference. To see howthis leads to improved frequency-mixing efficiency weneed to consider the factors that determine how a gen-erated field can grow effectively in a four-wave mixingprocess.

Laser fields applied close to resonance in an atomicmedium drive various frequency components of the po-larization in that medium, both at the frequency of thedriving fields and at new frequencies given by combina-tions of the applied ones. Figure 17 illustrates fourschemes in which three- or four-level atoms are driven,at single- or two-photon resonance, by laser fields. Thepolarization component at the new frequencies will actas a source of new electromagnetic fields. Thus in the

FIG. 16. Propagation of adiabatons: amplitudes of probe andcoupling fields as a function of time in a comoving frame forngrz /c are �a� 0, �b� 25, �c� 50, and �d� 100 from numericalsolution of propagation equations �Fleischhauer and Manka,1996�. Arbitrary space and time units with c=1.

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schemes illustrated in Fig. 17 the material polarizationP��4� is driven at frequency �4 by fields applied at �1,�2, and �3, and will in its turn drive a new electromag-netic field at �4. The growth of the new field dependsupon the magnitude of the nonlinear source term Pnl��4�and upon the linear polarization Pl��4� of the medium atthis frequency, which will determine the absorption anddispersion of the generated field. The equation that de-scribes the growth of this new field E4 is derived fromMaxwell’s equations within the slowly varying envelopeapproximation �see, for example, Reintjes �1984�� andcan be written in terms of the positive frequency com-ponent of the field as

�E4

�z+

1

c

�E4

�t=

i

2�4

c�0�Pnl + Pl� . �67�

Here the vector character of the field and polarizationwas suppressed for simplicity. The term on the right-hand side contains the effect of the linear polarizationupon the propagation of the generated field and thenonlinear polarization, which is the source of the newfield.

The response of the medium to an electric field is gov-erned by its polarization. In the framework of the den-sity matrices for a three-level atom we can write thepositive frequency component of the polarization as

P = �Tr���� = ���12�21 + �13�31 + �23�32� . �68�

We can express the total polarization P in terms of sus-ceptibilities using the familiar polarization expansion.

Thus we can express the linear response of the atom tothe field at frequency �4 as

P�1���4� = �0��1�E4, �69�

and the nonlinear response as

P�3���4� =32�0�

�3�E1E2E3, �70�

where Ej is the electric-field amplitude associated withthe frequency component �j. The forms of the suscepti-bilities that can be derived by a full quantum treatmentof the atom are extensively discussed in the literature onnonlinear optics �see for, example, Reintjes �1984��.In general, for nonresonant nonlinear mixing the suscep-tibility will be characterized by the appearance of threedetuning terms in the denominator while the linearsusceptibility has only a single detuning term. As wasshown in Sec. III in the case of EIT, it is of coursenecessary to use the dressed susceptibilities that alreadyinclude the effect of the strong coupling field to allorders. It is the difference of these susceptibilities fromthe normal form that give the advantages for nonlinearmixing. The forms of the dressed susceptibilities forEIT are given in Sec. III. These reflect the resonantnature of the fields involved and now contain theimportant interference character due to the dressingfield. Inclusion of collisional dephasing and integrationover the Doppler profile are usually required toyield effective susceptibilities that can be used in acalculation.

FIG. 17. Different schemes for resonantly en-hanced four-wave mixing processes based onEIT and related interference phenomena: �i�sum-frequency generation ��4� out of twopump pulses ��1 and �2� and one strong co-herent drive ��3�; �ii� up-conversion of pump��3� into generated field ��4� due to maxi-mum coherence prepared by two strong drivefields ��1 and �2�; �iii� off-resonant variant of�ii�; �iv� parametric generation of two fields��3 and �4� out of two pump fields ��1 and�2� in forward- or backward-scattering con-figuration. See text for details.

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The form of the equation that describes the �station-ary� growth of the �slowly varying� field E4 can now berecast in terms of these dressed and Doppler integratedsusceptibilities:

dE4

dz= i�4

4c��3�E1E2E3eikz −

�4

2cIm���1��E4

+ i�4

2cRe���1��E4, �71�

where k=k4− �k1+k2+k3� is the wave-vector mismatchdue to the refractive-index effect of all other resonancesin the atom and can also include any other phase-matching effects such as beam geometry and plasma dis-persion. It is easy to see from Eq. �67� that if the non-linear polarization reaches a large value then the sourceof the new field will be strong, and thus the productionof the new field will be enhanced. The absorption andrefraction of the generated field are determined by thelinear polarization of the medium at the generated fre-quency. The refraction has a very important effect uponthe growth of the field by determining whether or notthe generated field remains in phase with the polariza-tion that drives it. This phase-matching condition is es-sential and dictates �at resonance where Re���1��=0� thatwe also require k=0 for efficient growth of the field.For a generated field on resonance both absorption anddispersion can be problematic. In general k will be fi-nite, resulting in a limited length over which the fieldgrows before slipping out of phase with the driving po-larization lcoh=1/k. But in the presence of EIT, andwith the elimination of the linear susceptibility tovacuum values, the result is no absorption and no refrac-tion due to the resonant level, and only nonresonanttransitions to other levels contribute to the wave-vectormismatch.

Let us examine the four-wave mixing schemes shownin Fig. 17 in which EIT or related effects have beenfound to be advantageous. There are several intercon-nected mechanisms through which atomic coherenceprepared by the applied laser fields can increase the con-version efficiency of these wave-mixing processes:

�i� We have here a four-wave mixing scheme in whicha two-photon and a single-photon resonance are usedfor the applied fields, generating a field in resonancewith an atomic transition �Fig. 17�i��. The generated fieldand the single-photon resonant applied field act as probeand coupling fields, respectively, in a lambda EITscheme. EIT then leads to an elimination of absorptionand refraction for the generated field. The nonlinear sus-ceptibility is enhanced by proximity to resonance, sincethis term is subject to constructive interference betweenthe pair of dressed states. This mechanism is relevant toboth cw and pulsed lasers fields �Harris et al., 1990;Zhang et al., 1993�.

�ii� Here a pair of strong fields is applied resonantly ina lambda-configured three-level system close to bothRaman and single-photon resonance �Fig. 17�ii��. Thepresence of EIT eliminates absorption and dispersion inthe propagation of both fields. They can then propagate

without loss or distortion in what would otherwise be anextremely optically dense gas. This enables a dark statewith maximum amplitude and equally phased atomic co-herence to be formed along the entire propagationlength. Four-wave mixing from the maximal coherenceleads to large up-conversion efficiency. This requires theuse of fields strong enough to ensure adiabatic evolutionof the dark state and so is restricted to high-powerpulsed lasers or to Doppler-free cw laser experiments�Harris and Jain, 1997; Merriam et al., 1999, 2000�.

�iii� If two very strong fields are applied close to Ra-man resonance they will still lead to the formation of adark-state maximal coherence even if they are detunedfar from single-photon resonance �detuning ; Fig.17�iii�� provided that they are strong enough so that thecondition �1�2��21 is satisfied �where �21 is thedephasing rate of the �1�-�2� coherence�. In this case EITplays no role, in so far as absorption is concerned, as thefields are far from resonance. Nevertheless, the choice ofthe correct magnitude of Raman detuning can lead tocancellation of the material refraction for the pair ofapplied fields �Harris et al., 1997�. The maximal coher-ence, however, greatly boosts the conversion efficiencyin four-wave mixing �Hakuta et al., 1997� as for �ii�. If themaximal coherence is excited between molecular vibra-tional or rotational states the medium is very suited tohigh-order stimulated Raman sideband generation�Sokolov et al., 2000�. The requirement of high power toachieve adiabatic evolution restricts this to pulsed lasers.

�iv� The third field may also be close to resonant withanother transition. In this case the generated field com-pletes a double-� scheme, which can enhance the mixingfurther. An important variant of the double-� schemewith three applied fields is that explored by Hemmer etal. �1995� and by Zibrov and co-workers, in which onlytwo fields are injected �Zibrov et al., 1999; Fig. 17�iv��.These fields are applied close to single-photon reso-nance with transitions in each of the two � systemsformed in alkali-metal atoms. This scheme with cw laserfields has remarkable properties including, if the appliedfields are copropagating, the growth of both pairs offields and, if the applied fields are counterpropagating,mirrorless oscillation, as experimentally demonstratedby Zibrov et al. �1999�.

The advantages of all of these schemes lie in the re-duction of unwanted absorption and dispersion for thegenerated and the driving fields while the source termfor new fields is boosted. As a consequence of the elimi-nation of drive-field dispersion and absorption, largevalues of coherence can be driven in the case of �ii�, �iii�,and �iv�, in fact to the maximal value �i.e., �12=0.5� in allthe atoms �molecules� within the interaction region. Thefrequency-mixing process then proceeds with the atomiccoherence acting as the local oscillator with which thefinal drive field beats to produce a generated field with ahigh overall conversion efficiency.

In addition to the different four-wave mixing schemesdiscussed in the remainder of this section, other excitingapplications of EIT to nonlinear optics have been pro-posed and experimentally investigated. One recent de-

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velopment is the demonstration of high-efficiency mul-tiorder Raman sideband generation through themodulation of optical fields in a molecular medium pre-pared in a maximally coherent state of the v=0 and v=1 vibrational states �Hakuta et al., 1997; Sokolov et al.,2000�. The possibility of using the broad-bandwidthspectrum of phase-coherent sidebands to synthesize ul-trashort pulses was first predicted by Harris and Sokolov�1997� and then demonstrated by the Stanford group�Sokolov et al., 2001�. Another interesting effect hap-pens when in an EIT medium the light velocity matchesthe speed of sound. As shown by Matsko and collabora-tors, a new type of stimulated Brillouin scattering shouldoccur in this case, which in contrast to the usual situationallows efficient forward scattering �Matsko, Rostovtsev,Fleischhauer, and Scully, 2001�.

B. Nonlinear mixing and frequency up-conversion withelectromagnetically induced transparency

The enhancement of four-wave mixing in a two-photon resonant scheme of the type illustrated in Fig.17�i� was first treated by Harris et al. �1990�. The idea ofimproving four-wave mixing using an additional controlfield primarily to adjust the refractive index of the me-dium was discussed by Tewari and Agarwal �1986�. Intheir scheme, however, the control field is not incorpo-rated directly into the mixing fields and there is no con-structive interference of the nonlinear susceptibility.

Taking the linear and nonlinear susceptibilities de-rived in Sec. III, we can proceed with an analysis of thecase of four-wave mixing in a lambda EIT scheme. Pro-ceeding from Eq. �71� and integrating with appropriateboundary conditions �as is usual in all four-wave mixingprocesses� leads to an expression for the generated in-tensity of the form

I4 =3n�2

8Zoc2 ���3��2�E1�2�E2�2�E3�2F�k,z,R,I� , �72�

where z is the propagation distance, Zo=376.7 � is theimpedance of free space, n is the refractive index at thegenerated frequency �typically close to unity�, R and Iare shorthand for Re���1�� and Im���1��, respectively, andc is the speed of light in vacuum. Here we have intro-duced the phase-matching factor F�k ,z ,R ,I�, whichdepends upon the details of the mixing scheme and thefocal geometry. This general result indicates the impor-tance of phase matching �and hence k� for the gener-ated field intensity.

To obtain clear insight into the effects of EIT we shallfollow the solution provided by Petch et al. �1996�. Forthe case of plane waves propagating in a homogeneousmedium this treatment yields

I4 =3n�2

8Zoc2���3��2�E1�2�E2�2�E3�2

1 + e−�Iz/c − 2e−�Iz/2c cos �k +�R

2c�z�

�I

2c�2

+ k +�R

2c�2

.

�73�

It is implicit that ��3�, Re���1��, and Im���1�� are the laser-dressed expressions introduced in Sec. III with any re-quired inclusion of collisions and Doppler integrationhaving been carried out. In the limit of a large propaga-tion length z, the term in the numerator will go to unity,and if also k=0, the intensity is simply related to thesusceptibilities by

I4 � ���3�

��1��2

, �74�

as was pointed out first by Harris et al. �1990� and sub-sequently �in the form used here� by Petch et al. �1996�.It is very clear that EIT increases this ratio by both in-creasing the value of the numerator and reducing thedenominator. It should be noted that a dramatic changein conversion efficiency as a function of couplingstrength is only apparent in an inhomogeneously broad-ened medium. In a homogeneously broadened medium,although EIT increases the efficiency, the ratio of thesusceptibilities remains independent of the couplingstrength.

In the case of four-wave mixing with EIT we usuallyassume a weak probe field so that excited-state popula-tions and coherences remain small �see Fig. 17�i��. Thetwo-photon transition need not be strongly driven �i.e., asmall two-photon Rabi frequency can be used� but a“strong” coupling laser is required. The coupling laseramplitude must be sufficient that the Rabi frequency iscomparable to or exceeds the inhomogenous spectralwidths in the system �e.g., Doppler width�. For example,a laser intensity of above 1 MW cm−2 is required for atypical Doppler-broadened visible transition. This istrivially achieved even for unfocused pulsed lasers, butdoes present a substantial barrier for the use ofcontinuous-wave lasers unless a specific Doppler-freeconfiguration is employed. The latter is not normallysuitable for a frequency up-conversion scheme if a largefrequency up-conversion factor is to be achieved, e.g.,to the vacuum ultraviolet �VUV�. This is in large partbecause atomic species with high-ionization potentials��10 eV� that are suitable for frequency up-conversionto the VUV are not easily trapped and cooled at highdensity. Recent experiments, by, for instance, Hemmer,report significant progress in cw nonlinear optical pro-cesses using EIT �Hemmer et al., 1995; Babin et al.,1996�. A number of other possibilities, e.g., laser-cooledatoms and Doppler-free geometries, have also been ex-plored.

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In the case of frequency up-conversion to short wave-lengths, Doppler shifts arising from the Maxwellian ve-locity distribution of the atoms or molecules in a gas atfinite temperature lead to a corresponding distributionin the detunings for the atomic ensemble. The calcula-tion of the response of the medium, as characterized bythe susceptibilities, must therefore include the Dopplereffect by performing a weighted sum over possible de-tunings. From this operation the effective values of thesusceptibilities at a given frequency are obtained, andthese quantities can be used to calculate the generatedfield. Interference effects persist in the dressed profilesprovided the coupling laser Rabi frequency is compa-rable to or larger than the inhomogeneous width. This isbecause the Doppler profile follows a Gaussian distribu-tion, which falls off much faster in the wings of the pro-file than the Lorentzian profile arising from lifetimebroadening. Indeed, in the case of Doppler broadening,the effects of increasing �c manifest themselves mostclearly. As discussed by Harris et al. �1990� there is ex-pected to be a steplike increase in the conversion effi-ciency when the magnitude of �c approaches the Dop-pler width of the transition. For coupling strengthsbelow this value the atomic sample remains opaque, as itwould under ordinary resonant conditions; in contrast,for a coupling strength exceeding the Doppler width thefull benefit of the quantum interference is obtained.

In the Doppler-broadened case, therefore, thecoupling-field Rabi frequency must remain greater thanthe Doppler width for a significant fraction of the inter-action time. Pulsed lasers, because of the high peak pow-ers that arise, are required. Moreover, a transform-limited single-mode laser pulse is essential for thecoupling laser fields that drive EIT, since a multimodefield will cause an additional dephasing effect on thecoherence, resulting in a deterioration of the quality ofthe interference, as discussed in Sec. III.

The work of the Stanford group �Sokolov et al., 2001�has highlighted that when pulsed laser fields are usedadditional considerations must be made. Kasapi and co-workers showed �Kasapi et al., 1995� that the group ve-locity of a 20-ns pulse can be modified for pulses propa-gating in the EIT; large reductions, e.g., by factors downto �c /100, in the group velocity have been observed.Another consideration beyond that found in the simplesteady-state case is that the medium can only becometransparent if the pulse contains enough energy to dressall of the atoms in the interaction volume. This prepara-tion energy condition was discussed in Sec. IV. This putsadditional constraints on the laser pulse parameters.

Up-conversion to the UV and vacuum UV has beenenhanced by EIT in a number of experiments. The firstexperiment to show EIT in a resonant scheme, whereboth the EIT effect on opacity and phase matching wereimportant, was reported by Zhang et al. �1993�. Theyemployed a four-wave mixing scheme in hydrogen�equivalent to Fig. 17�i�� in which the EIT was createdon the 3p-1s transition at 103 nm in a lambda scheme bythe application of a field at 656 nm on the 2s-3p transi-tion �Fig. 18�. A field at 103 nm is generated by the four-

wave mixing process that was enhanced by this EIT. Inthese experiments the opening of photoionization chan-nels from both of the involved excited atomic statesleads to loss of the conditions for perfect transparency.Nevertheless, because the photionization cross sectionfor the 2s metastable level is about one order of magni-tude less than that from the 3p state, the effect of EIT isonly partially reduced. It is in general important to keepthe photoionization rate sufficiently small so as not toquench the coherence through electron impact broaden-ing �see Buffa et al. �2003��. In this experiment and sub-sequent work by the same authors, conversion efficien-cies up to 2�10−4 were reported �Zhang et al., 1995�.

A limit to the conversion in an EIT-enhanced four-wave mixing scheme is set by the Doppler width. A largeDoppler width dictates that a comparable Rabi fre-quency is required to create good transparency. This inturn leads to a reduction in the nonlinear susceptibility.This is one of the most important factors limiting theconversion efficiency in the hydrogen scheme �Hakuta,2004�. The requirements on a minimum value of �c�Doppler constrain the conversion efficiency that can beachieved because of a scaling factor by 1/�c

2, which ul-timately leads to diminished values of ��3�. The use ofgases of higher atomic weight at low temperatures istherefore highly desirable in any experiment utilizingEIT for enhancement of four-wave mixing to the VUV.A combination of the low mass of the H atom andthe elevated temperatures required in the dissociationprocess leads to especially large Doppler widths��10 GHz�. To overcome such a severe limit, investiga-tions were undertaken by one of the authors in kryptongas, in which the relevant Doppler width is only 1 GHz.

FIG. 18. Experimental demonstration of sum-frequency gen-eration in H: left, level and coupling scheme; right, calculated�dashed� and measured �solid line� nonlinear susceptibility asfunction of detuning for the 3p-1s transition for differentdensity-length products of the medium �nL�. Only under theleast dense condition is the Autler-Townes split structureapparent in the generated signal. At higher density strongresonant enhancement is seen. From Zhang et al., 1993.

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Conversion efficiencies approaching 10−2 were obtained�Dorman et al., 2000b�. In this case, the conversion effi-ciency appears to be limited by the finite value of kdue to the other atomic levels rather than absorption inthe medium �Dorman et al., 2000a�.

In a Doppler-free medium the effect of constructiveinterference for the nonlinear term is most apparent. Ina system without inhomogeneous broadening perfecttransparency can be induced with �c��31. As �c issmall relative to �31, the nonlinear susceptibility will, be-cause of the constructive interference, have a value es-sentially identical to that of the unmodified resonantatomic system. The transparency dip will be very narrow�see Fig. 7�. The spectral widths of these features aretypically subnatural and are therefore accompanied byvery steep normal dispersion, which corresponds to amuch reduced group velocity. As was pointed out byseveral authors, including Schmidt and Imamoglu �1996�and Harris and Hau �1999�, the nonlinear susceptibilitiesin this case can be extremely large, as there is a con-structive interference. Nonlinear optics at very low lightlevels, i.e., at the few-photon limit, are possible in thisregime. Associated with their first dramatic measure-ments of ultraslow light, Hau et al. �1999� reported anonlinear refractive index in an ultracold Na vapor thatwas 0.18 cm2 W−1. This is already 106 times larger thanthat measured for cold Cs atoms in a non-EIT system,which itself was much greater than the nonlinearity in atypical solid-state system. A discussion of this process inthe few-photon limit is given in Sec. VI.

C. Nonlinear optics with maximal coherence

Harris demonstrated an important extension of theEIT concept that occurs if the atomic medium is stronglydriven by a pair of fields in Raman resonance in a three-level system. Due to the elimination of absorption andrefraction, both fields can propagate into the mediumand large values of the material coherence are createdon the nondriven transition. Considering the system il-lustrated in Fig. 17�ii�, we can imagine that both appliedfields are now strong. Under appropriate adiabatic con-ditions �see Sec. III� the system evolves in the dark stateto produce the maximum possible value for the coher-ence �12=0.5. Adiabatic evolution into the maximallycoherent state is achieved by adjusting either the Ramandetuning or the pulse sequence �i.e., to “counterintui-tive” order�. The pair of fields may also be in single-photon resonance with a third level, in which case theEIT-like elimination of absorption will be important.This situation is equivalent to the formation of a darkstate, since neither of the two strong fields is absorbedby the medium. For sufficiently strong fields thesingle-photon resonance condition need not be satisfiedand a maximum coherence can still be achieved �seeFig. 17�iii�� provided that the condition �1�2� ���21 ismet, where is the detuning of the strong fields fromstate �3�.

Under the conditions of STIRAP, or for matchedpulse propagation, large populations of coherent popu-lation trapped states are created. To achieve this situa-tion the laser electric-field strength must be largeenough to create couplings that will ensure adiabaticatomic evolution and pulses that are sufficiently ener-getic to prepare all of the atoms in the beam path. The�12 coherence thus created will have a magnitude ��12�=0.5 �i.e., all the atoms are in a coherent state� and nega-tive sign �i.e., all the atoms are in the population trappeddark state�. The complex coherence varies in space andtime and can be written

�12 = −�1�2

*

�12 + �2

2exp�i��1 − �2�t − �k1 − k2�z�

= ��12�e−i kzei��1−�2�t. �75�

Under these circumstances mixing of additional fieldswith the atom can become extremely efficient �Harris etal., 1997�.

The preparation energy condition for the creation ofmaximal coherence requires that the pulse energy ex-ceed the following value:

Eprep =f13

f23�AL�� , �76�

where fij are the oscillator strengths of the transitionsand �L is the product of the density and the length�Harris and Luo, 1995�. Essentially the number of pho-tons in the pulse must exceed the number of atoms inthe interaction volume to ensure that all atoms are inthe appropriate dressed state. It should be noted thatthe preparation energy is much smaller for nonlinearconversion processes not using maximal coherence. Fur-thermore, this preparation energy is not lost by dissipa-tion if the pulses evolve adiabatically and can be recov-ered.

An additional field applied to the medium can partici-pate in sum- or difference-frequency mixing with thetwo Raman resonant fields. The importance of the largevalue of coherence is that it is the source polarizationthat drives the new fields generated in the frequency-mixing process. This is described by the following equa-tion:

dE4

dz= − i 4��4n��a4�11 + d4�22�E4 + b4�12E3� , �77�

with coefficients a4, b4, and d4 that depend upon atomicfactors. The second term on the right-hand side�b4�12E3�, the source of the new field, is comparable inmagnitude to the first term that describes the dispersionand loss. Complete conversion can occur over a shortdistance if the factor �12 is large. This significantly re-laxes the constraints usually set by phase matching innonlinear optics. Recently near-unity conversion effi-ciencies to the far UV were reported in an atomic leadsystem where maximum coherence had been created�Jain et al., 1996; Harris et al., 1997; Merriam et al., 1999,2000�. An example of high-conversion-efficiency four-

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wave mixing is shown in Fig. 19 �adapted from Jain et al.,1996�.

An alternative way to look at the origin of the largefour-wave mixing efficiency is to consider the suscepti-bilities. Recalling the density-matrix results of a � sys-tem, Eq. �12�, one recognizes that the polarization at �4proportinal to �31 is a sum of two terms, the first beingproportional to the linear susceptibility and the secondto the nonlinear susceptibility. The drive fields E1 and E2are now implicitly included in �12, which has the maxi-mal value of 0.5. These drive fields are resonant so thestructure of ��3� simplifies to a single nonresonant termin the denominator. The nonlinear susceptibility there-fore shares this single nonresonant term in the denomi-nator with the linear susceptibility and so these twoquantities are of comparable size. This is a very unusualsituation in nonlinear optics, and the length over whichefficient frequency conversion can occur is now reducedto values comparable to the coherence length. A conse-quence of this is that complete nonlinear frequency con-version of the fields can occur in a distance of the orderof the coherence length �determined by the real part ofthe susceptibility�. This is equivalent to having nearvacuum conditions for the refraction �and absorption� ofthe medium while the nonlinearity remains large.

In a molecular medium large coherence between vi-brational or rotational levels has also been achieved us-ing adiabatic pulse pairs in gas-phase hydrogen and deu-terium �Sokolov et al., 2000�. Efficient multiorderRaman sideband generation has been observed to occur.In this case a pair of lasers that are slightly detuned fromRaman resonance are used to adiabatically establish asuperposition of two molecular states. The large energyseparation to the next excited electronic state in thesemolecules �10 eV� means that the situation is similar tothat illustrated in Fig. 17�iii�. This superposition thenmixes with the applied fields to form a broad spectrumof sidebands through multiorder mixing. Molecular co-herence in solid hydrogen has recently also been used toeliminate phase mismatch in a Stokes or anti-Stokesstimulated Raman frequency-conversion scheme�Hakuta et al., 1997�. In this scheme, v=0 and v=1 vi-brational states of the hydrogen molecule electronicground state form the lower states of a Raman scheme.

Since the �1�-�2� dephasing rate is very small in appropri-ately prepared samples of solid hydrogen, interferencethat causes the dispersion to become negligible can oc-cur. Because of the removal of the usual phase mis-match, efficient operation of these frequency-conversionschemes over a broad range of frequencies �infrared tovacuum UV� has been shown.

A recent prediction is of broadband spectral genera-tion associated with strong-field refractive index control�Harris and Sokolov, 1997; Kien et al., 1999�. The obser-vations of very efficient high-order Raman sidebandgeneration point the way to synthesizing very-short-duration light pulses, since the broadband Raman side-band spectrum has been proved to be phase locked�Sokolov et al., 2001�. It is anticipated that high-powersources of sub-femtosecond-duration pulses might beachieved through this technique.

D. Four-wave mixing in double-� systems

Four-wave mixing with all the optical fields close toresonance is achieved in atoms with four levels in theso-called double-lambda configuration �Hemmer et al.,1995�, which is detailed in Fig. 20. The fully resonantcharacter of the light-matter interaction for all of thefields �both those applied and those generated� involvedin four-wave mixing leads potentially to very high con-version efficiencies and to a number of important char-acteristics. Earlier work on double-lambda schemes inwhich all four fields are applied has shown, for instance,that the formation of dark states depends upon the rela-tive phase of the fields �Buckle et al., 1986; Arimondo,1996�. Experiments in the cw limit have been carried outin sodium vapor that show the phase sensitivity of EITin a double-lambda scheme �Korsunsky et al., 1999�.

First we shall consider the case in which three reso-nant fields are applied such that the fourth field is gen-erated in resonance between the highest of the excitedstates and the ground state. Under appropriate condi-tions the presence of the three applied fields eliminatesnot only the resonant absorption or refraction of thegenerated field but also that in the applied fields them-selves. The situation of four-wave mixing in a double-lambda system has been treated theoretically in detail

FIG. 19. Parametric up-conver-sion with maximum coherence:�a� a pair of �near� resonantcoupling ��c� and probe pulses��p� generate maximum coher-ence between states �1� and �2�.From Harris et al., 1997. �b� Thecoherently prepared mediumthen leads to efficient up-conversion of second couplingpulse �c into the �h. Alsoshown is the conversion effi-ciency from 425 to 293 nm as afunction of coupling laser inten-sity. From Jain et al., 1996.

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by Korsunsky and Kosachiov �1999�, who found the rela-tive phases of the four fields to evolve with propagationso as to support lossless propagation in a fashion similarto that predicted by Harris for a pair of fields propagat-ing in resonance in a lambda medium �Harris, 1993�. Thephase and the amplitudes of all four fields, the generatedfield and the applied fields, are predicted to adjust asthey propagate so as to support efficient transfer of en-ergy between the fields, under lossless conditions.

An experimental realization of up-conversion in adouble-lambda scheme driven by three resonantly tunedpulsed fields, with the new field generated in the far UVat 186 nm, was achieved by Merriam et al. �2000� in leadvapor. High conversion efficiencies, more than 0.3 fromthe shortest-wavelength applied field at 233 nm to the186 nm field, were indeed observed. The experimentsconfirm that the double-lambda scheme supports loss-less propagation provided that the pairs of lambda reso-nant fields have matched ratios of Rabi frequency. Forpulses not initially satisfying the condition of matchedRabi frequencies, absorptive loss and nonlinear energytransfer occur until the condition is reached, after whichall fields propagate without further loss and refraction.Limits to the up-converted power density were found,however, to arise from power broadening.

An interesting alternative to the schemes just de-scribed for efficient nonlinear optical conversion in adouble-lambda scheme was identified by Zibrov et al.�1999�. In essence the scheme is very simple. Two fieldsare applied resonantly, one in each of the two lambdasystems. These then efficiently couple to the pair offields that complete the double-lambda configuration.The generated fields correspond to Stokes and anti-Stokes components of the two applied fields. In theoriginal experiment in Rb vapor the applied fields coun-terpropagate, and the resulting nonlinear gain and effi-cient intrinsic feedback in this configuration lead to mir-rorless self-oscillation at extremely low applied fieldstrengths �at the �W level of cw power�. Further analysisof the quantum dynamics in this situation has high-lighted the possibility of using this technique to generatenarrow-band sources of nonclassical radiation �Lukin etal., 1999; Fleischhauer, Lukin, et al., 2000�.

VI. EIT WITH FEW PHOTONS

In the absence of a strongly polarizable medium, pho-tons are essentially noninteracting particles. In commu-nications technology this is a desired property, and op-tics has emerged over the last two decades as thepreferred method for communicating information. Forthe same reason, attempts to use light in informationprocessing tasks such as computation have failed. Forthe latter, one needs strong, dissipation-free light-lightinteractions. In the present section we summarize anddiscuss some of the potentials and limitations of electro-magnetically induced transparency in this respect.

A. Giant Kerr effect

Information processing based on light requires nonlin-ear interactions that would lead, for example, to a Kerreffect or, equivalently, a cross-phase modulation inwhich the phase of a light field is modified by an amountdetermined by the intensity of another optical field. Ifthe nonlinear phase shifts arising from such a Kerr effectwere on the order of � rad, it would be possible toimplement all-optical switching. Such large Kerr nonlin-earities without appreciable absorption can, however,only be obtained for intense laser pulses containingroughly 1010 photons.

One of the principal challenges of nonlinear optics isthe observation of a mutual phase shift exceeding � radusing two light fields, each containing a single photon. Inaddition to new possibilities for addressing fundamentalissues in quantum theory, such strong nondissipative in-teractions could be used to realize a controlled-NOT gatebetween two quantum bits �qubits�, defined, for ex-ample, by the polarization state of a single-photon pulse.However, given the weakness of nonresonant opticalnonlinearities and the dominant role of absorption inresonant processes, the combination of a large third-order Kerr susceptibility and small linear susceptibilityappears to be incompatible.

In this section, we show how EIT can overcome theseseemingly insurmountable obstacles. We have alreadyseen how the nonreciprocity between linear and nonlin-ear susceptibilities in EIT media can be used to enhance

FIG. 20. Generation of fields E1 and E2 byresonant four-wave mixing of two counter-propagating pump fields Ef and Eb: left, leveland coupling scheme; right, calculated thresh-old behavior of output intensity of field E1 asa function of medium length L. The insetshows the spatial distributions of the fieldsabove threshold of parametric oscillation atthe indicated point. Adapted fromFleischhauer et al., 2000.

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generation of radiation at new frequencies. We shall seein this subsection how dissipation-free strong photon-photon interactions can be induced using EIT, due to aconstructive interference in Kerr nonlinearity at two-photon resonance � =0�, where linear susceptibility ex-periences a complete cancellation. This so-called giantKerr effect, proposed by Schmidt and Imamoglu �1996�,can be understood using the four-level system depictedin Fig. 21. The principal element here is once again thelambda subsystem consisting of states �1�, �2�, and �3�.The state �4� has the same parity as �3� and has an elec-tric dipole coupling to �2�; the signal field at frequency �sis applied on this transition, with a Rabi frequency �s.The cross-phase modulation nonlinearity that we are in-terested in is between this signal field and the probe fieldapplied on the �1�-�3� transition. We assume that the ap-plied fields couple only to the respective transitions in-dicated in Fig. 21. Throughout this section, we assumethat the coupling field with Rabi frequency �c is nonper-turbative, remains unchanged, and can be treated classi-cally.

To calculate the third-order susceptibility correspond-ing to the cross-phase modulation Kerr nonlinearity, weassume both �p and �s to be perturbative. As we havediscussed in Sec. III, we can use the effective non-Hermitian Hamiltonian

Heff = −�

2��p�31 + �c�32 + �s�42 + H.c.�

+ �� −i

2�3��33 + �� −

i

2�2��22

+ ���42 −i

2�4��44 �78�

to describe the atomic dynamics in this limit. Here wehave introduced the detuning �42=�42−�s and thespontaneous emission rate �4 out of state �4�. Onceagain, we can solve the Schrödinger equation with theHamiltonian of Eq. �78� to obtain the probability ampli-tude for finding the atom in state �3� �a3�t�� and use it toevaluate the linear and nonlinear components of the po-larization at �p:

P�t� = N�13a3�t�e−i�pt + H.c.

= �0��1��− �p,�p�Epe−i�pt

+ �0��3��− �p,�s,− �s,�p��Es�2Epe−i�pt + H.c.

�79�

Similarly, we can obtain the linear and nonlinear suscep-tibilities at �s. For simplicity, we consider only the reso-nant case in which = =0.

First, we emphasize that the linear susceptibility forboth probe and signal fields vanishes in the limit �2=0.This is trivially true for �s, since it is assumed not tocouple to state �1� atoms, and is the case for �p to theextent that �s remains perturbative. We also note thatself-phase-modulation-type Kerr nonlinearities �i.e.,��3��−�p ,�p ,−�p ,�p� and ��3��−�s ,�s ,−�s ,�s��for both fields are identically zero. The cross-phasemodulation nonlinearity �xpm

�3� =��3��−�p ,�s ,−�s ,�p�=��3��−�s ,�p ,−�p ,�s�, on the other hand, is given by�Schmidt and Imamoglu, 1996�

�xpm�3� =

��13�2��24�2�2�0�

3

1

�c2 1

�42+ i

�4

2�422 � . �80�

To evaluate the significance of this result, it is perhapsillustrative to compare it to the Kerr nonlinear suscepti-bility of a standard three-level atomic system ��3−level

�3� �depicted in Fig. 22�a�. For this system consisting of theground state �a�, an intermediate state of opposite parity�b�, and a final state �c�, we find

Re��3−level�3� � =

��ab�2��bc�2�2�0�

3

1

�ab2 �bc

, �81�

where �ij and �ij denote the dipole matrix element ofand the detuning from the transition �i�-�j�, respectively�i , j=a,b,c�. When we compare the two expressions, weobserve that they have practically identical forms: in theEIT cross-phase modulation scheme, 1/4�ab

2 is re-placed by 1/�c

2. Even though one needs to have �ab2

#�b2 in a three-level scheme to avoid absorption, �c

2

��32 is possible in the EIT cross-phase modulation sys-

tem. This in turn implies that the magnitude of Kerrnonlinearity can be orders of magnitude larger in thelatter case. This is the essence of the giant Kerr effect�Schmidt and Imamoglu, 1996�.

We can understand the origin of this enhancement us-ing the dressed-state picture depicted in Fig. 22�b�.When we apply the dressed-state transformation intro-duced in Sec. III, we find that the initial and final states�which remain unchanged� are coupled via two interme-

FIG. 21. Level scheme for giant Kerr effect.FIG. 22. Cross-phase modulation with and without giant Kerreffect: left, level configuration for cross-phase modulation inan ordinary three-level scheme; right, dressed-state represen-tation of giant Kerr scheme.

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diate states �2d� and �3d�. The two paths contributing tovirtual excitation of state �4� experience a constructiveinterference for �p tuned in between the two dressedstates, partially leading to the above-mentioned en-hancement. Perhaps more importantly, however, the “ef-fective detuning” between the intermediate dressedstates is given by �c and can be much smaller than theirwidth.

Alternatively, we can understand the predicted en-hancement of the Kerr nonlinearity by recalling thesteep dispersion obtained at transparency for �c

2��32. In

this limit, small changes in two-photon detuning thatare caused, for example, by a shift in the energy of state�2� can give rise to a drastic increase in Re���1�� experi-enced by the probe field. We can understand the role ofthe signal field as creating an ac-Stark shift of state �2�,thereby leading to a large change in the index of refrac-tion at �p that is proportional to �s

2.Finally, we note that by optical pumping and by

choosing the polarization of the lasers appropriately, onecan realize the energy-level diagram depicted in Fig. 21practically for all alkali-metal atoms. It has been esti-mated that the enhancement of �xpm

�3� over a standardthree-level atomic medium with identical density and di-pole matrix elements could be as much as six orders ofmagnitude �Schmidt and Imamoglu, 1996�. Naturally, theinterest in this system arises from the possibility of doingnonlinear optics on the single-photon level, without re-quiring a high-finesse cavity; this possibility was dis-cussed in the original proposal �Schmidt and Imamoglu,1996�, but in the limit of continuous-wave fields. Recentexperiments have already demonstrated the enhance-ment of Kerr nonlinearities in the limit of �42=0 �Brajeet al., 2003�; this is the purely dissipative limit of �xpm

�3�

and was discussed by Harris and Yamamoto in the con-text of a two-photon absorber �Harris and Yamamoto,1998�.

As we have already discussed above, the steep disper-sion curve of the probe field is responsible for the en-hancement in the Kerr effect. It is interesting that it isthis steep dispersion, or the slow group velocity thatarises from it, that at the same time gives rise to a strin-gent limit on the available single-photon nonlinearphase shifts for pulsed fields. This limitation arises fromthe fact that the probe and signal pulses travel at vastlydifferent group velocities in the limit �c

2��32, and their

interaction time will be limited by the fact that the slowprobe pulse will separate spatially from the fast signalpulse �Harris and Hau, 1999�.

To make quantitative predictions on the limitation onnonlinear phase shifts arising from disparate group ve-locities, we assume a nearly ideal EIT scheme where�3�2��c

2��32, =0 and concentrate on the limit of

small � �. For this system, we have a probe-field groupvelocity

vp =�c

2

�3

1

��13, �82�

where �13=3�p2 /2� is the peak absorption cross section

of the �1�-�3� transition ��p=2�c /�p�. Given that vp�c,the time it takes for the probe pulse to traverse the me-dium,

!d =�3

�c2��13L , �83�

referred to as the group delay time �Harris and Yama-moto, 1998; Harris and Hau, 1999�, can be much longerthan the traversal time of the signal field Ts=L /c andthe pulse widths of the probe �!p� and signal �!s=!p�fields. As discussed earlier, the pioneering experiment ofHau et al. �1999� already demonstrated !D�1 msec�!p.

The generated nonlinear phase shift is proportional tothe interaction length of the probe and signal fields. Inthe EIT giant Kerr scheme, this is determined bymin�L,Lh�, where Lh is the probe propagation lengththat would give rise to a time delay between the twopulses that is equal to their pulse width !p, i.e.,

Lh = vp!p =�c

2

�3

1

��13!p, �84�

which was originally defined as the Hau length �Harrisand Hau, 1999�. The nonlinear phase shift can be deter-mined from the slowly varying envelope equation �Har-ris and Hau, 1999�.

�Ep

�z+ � 1

vp+

1

c� �Ep

�t= i

�p�xpm

�3� �Es�t −z

c��2

Ep

= � i

4��42 + i�4/2�vp�

���s�t −z

c��2

Ep, �85�

where we have used Eq. �80�.When the group delay is negligible compared to !p

�i.e., L�Lh�, and we assume Gaussian input pulses, wefind for the peak �complex� phase shift for the maximumof the pulse

�xpm → � �4

�42 + i�4/2�ns

A

�24

2 ln 2

!d

!p, �86�

where ns is the number of photons in the pulse and A itscross-sectional area. Equation �86� is, as expected, theresult one would obtain using Eq. �80� without takinginto account the group-velocity mismatch between thetwo pulses. In the opposite limit of L�Lh, we have

�xpm → � �4

�42 + i�4/2�ns

A

�24

8. �87�

This result shows that the maximum nonlinear phaseshift that can be obtained in the EIT system is indepen-dent of the system parameters, such as �c, �, and L�Harris and Hau, 1999�, and is on the order of 0.1 rad for

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single-photon pulses focused into an area A��24. Thisremarkably simple result assumes that �42��4 /2, inwhich case we get a two-photon absorption coefficientthat is comparable to the nonlinear phase shift.

B. Cross-phase modulation using single-photon pulseswith matched group velocities

From the perspective of applications in quantum in-formation processing, the result obtained in the previoussubsection has a somewhat negative message: While EITallows for weak optical pulses containing n�10 photonsto induce large nonlinear phase shifts, it cannot providestrong dissipation-free interactions between two single-photon pulses. The limitation, as the presented analysishas revealed, is due to the disparate group velocities ofthe two interacting pulses. The natural question to ask,then, is whether one can obtain larger nonlinear phaseshifts by assuring that both probe and signal pulsestravel at the same ultraslow group velocity.

Before proceeding with the analysis, we emphasizethat it is possible to consider an atomic medium withtwo coherently driven atomic species, each establishingEIT at different frequencies. If the two atomic speciesare different isotopes of the same element, then the fre-quencies at which the medium becomes transparent candiffer by an amount that is on the order of the hyperfinesplitting. We can then envision a scenario in which twodifferent light pulses, centered at the two transparencyfrequencies, travel with the identical ultraslow group ve-locity through the medium containing an optically densemixture of these two isotopes �A and B� of the sameelement. By adjusting the external magnetic field, it isalso possible to ensure that one of these light pulses thatsees EIT induced by species B will also be �nearly� reso-nant with the �2�-�4� transition of species A. By choosingthis pulse to be the signal pulse, one can realize a giantKerr interaction between two ultraslow light pulses�Lukin and Imamoglu, 2000�. An alternative schemebased on only a single atomic species was recently pro-posed by Petrosyan and Kuritzki �2002�.

Even though the analysis of nonlinear optical pro-cesses using single-photon pulses requires a full quan-tum field-theoretical approach, we first consider the clas-sical limit, following up on the analysis previouslypresented for a fast signal pulse. In the limit of a slowsignal pulse with group velocity vs we obtain

�xpm =�4

�42 + i�4/2ns

A

�24

2 ln 2

!d

!p, �88�

which is identical in form to that given in Eq. �86�. Incontrast to that earlier result, however, the expression inEq. �88� is valid even when !d�!p �Lukin and Imamo-glu, 2000�.

To determine the maximum possible phase shift, werecall that for an optically dense medium the EIT trans-mission peak has a width given by �trans

=�c2 /�3��13L. If we set !p=�trans

−1 , we find

!d

!p$ ��13L . �89�

Since the other factors on the right-hand side of Eq. �88�can be of order unity, we can obtain �xpm#�, providedthat we have ��13L#10. The presence of an opticallythick medium is essential for large photon-photon inter-action. At the same time, we note that this interaction isnecessarily slow and has a �xpm-bandwidth product thatis independent of the optical depth �Lukin and Imamo-glu, 2000�.

Next, we examine the validity of this conclusion usinga full quantum field-theoretic approach �Lukin and Ima-moglu, 2000�. To this end, we replace the classical elec-tric fields describing the signal and probe fields with op-

erators Ei�z , t�= Ei�+��z , t�+ Ei

�−��z , t�, where

Ei�+��z,t� = �

k �kc

2�0Vaki�z,t�eik�z−ct�. �90�

Here i=p,s and V denotes the quantization volume. Wesuppress the polarization index for convenience. In or-der to eliminate dissipation and simplify the Heisenbergequations for the quantized fields, we assume �2=0 and��42���4. To ensure, however, that we can adiabati-cally eliminate atomic degrees of freedom, we also needto impose a finite bandwidth q on the quantized field.With this assumption, the equal space-time commutationrelations explicitly depend on this bandwidth �Lukin andImamoglu, 2000�. Finally, to guarantee negligible pulsespreading, we take !p

−1�q��trans and arrive at a pair

of operator equations for Ep�z , t� and Es�z , t� that havethe identical form to that given in Eq. �85�, provided wereplace c by vs on the right-hand side. These equationscan then be solved to yield

Ep�+��z,t� = Ep

�+��0,t��

� exp i�24

2

�2

!d

4��42Es

�−��0,t��Es�+��0,t��� ,

Es�+��z,t� = Es

�+��0,t��

� exp i�13

2

�2

!d

4��42Ep

�−��0,t��Ep�+��0,t��� ,

�91�

where t�= t−z /vp.Using the expressions in Eq. �91�, we can address the

question of nonlinear phase shift obtained during theinteraction of two single-photon pulses. To this end, weassume an input state of the form

�1i� = �k�kaki

† �0� �92�

for both fields, where we require �k��k�2=1 to ensureproper normalization. The spatiotemporal wave func-tion of each pulse is given by the single-photon ampli-

tude, �i�z , t�= 0�Ei�+��z , t��1i�. To evaluate the correla-

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tions induced by the interaction, we concentrate on thetwo-photon amplitude

�sp�z,t ;z�,t�� = 0�Es�+��z,t�Ep

�+��z�,t���1s,1p� . �93�

One finds that the equal-time �t= t�� local �z=z�� corre-lations can be written as

�sp�z,t ;z,t� = �s�0,t��p�0,t�ei�, �94�

where �=!dq�24�4 /4�A�42. Since the equal spaceand time commutation relations are proportional to q,there is explicit dependence on this quantity. The non-linear phase shift � is the counterpart of the classicalphase �xpm obtained earlier in Eq. �88�; the equivalenceis obtained by replacing !p

−1 with the quantization band-width q �Lukin and Imamoglu, 2000�.

A large single-photon nonlinear phase shift � that ex-ceeds � rad can be used to implement two-qubit quan-tum logic gates �Nielsen and Chuang, 2000�. However,we have seen that creating large phase shifts betweensingle-photon pulses also changes the mode profile ofthe signal �probe� pulse, conditioned on the presence ofthe probe �signal� pulse. Therefore we expect entangle-ment between the internal degrees of freedom of thepulses, such as photon number or polarization, and theexternal degrees of freedom, such as mode profile deter-mined by ��k�, invalidating the simplistic assumption ofideal single-photon switching using large Kerr nonlinear-ity.

C. Few-photon four-wave mixing

Besides the resonantly enhanced Kerr effect discussedin the previous subsections, nonlinear interactions onthe few-photon level have been predicted and analyzedin greater detail in the four-wave mixing schemes shownin Fig. 23. In the first scheme discussed by Harris andHau �1999�, photons from two pulses ��1� and ��3� areconverted into an electromagnetic-field pulse at �4 inthe presence of a strong monochromatic drive field at �2�Fig. 23�a��. In the second scheme �Fig. 23�b��, discussedby Johnsson et al. �2002� for copropagating pump fieldsand by Fleischhauer and Lukin �2000� and by Lukin etal. �1999� for counterpropagating pump fields, the field�2 is initially absent and spontaneously generated alongwith �4.

In the first scheme �see Fig. 23�a��, the input pulse �1experiences a group delay due to EIT on the �1�-�3� tran-sition, while the second one, �3, moves almost with the

vacuum speed of light. Harris and Hau showed thatwithin an undepleted pump approximation the conver-sion efficiency, i.e., the ratio of the number of generatedphotons at �4 to the number of input photons at �1, isgiven by a dimensionless function �� ,�L� multipliedby the number of input photons per atomic cross sec-tion:

n4out

n1in = �� ,�L�

n3in

A3�24. �95�

Here � is the �E-field� absorption coefficient at the �1�-�4�transition and L is the medium length. =L /Lh is theratio of the length Lh introduced in Eq. �84�, i.e., theprobe propagation length corresponding to a time delayequal to the pulse width, to the medium length L. In thelimit of an optically thin medium �L→0 and of a smallgroup delay within the medium →0, the usual result ofperturbative long-pulse nonlinear optics is obtained,which for Gaussian pulses reads

�� ,�L� = ln�2��

�1/2 T1

T12 + T3

2 �L . �96�

Here the T�’s denote the temporal length of the pulses.In the limit of →�, i.e., if the group velocity reductionof �1 is large, so that the �3 pulse sweeps over it veryrapidly, the function is given by

�� ,�L� = ln�2��

�1/2�L

exp�− 2�L� , �97�

i.e., falls off with −1. A maximum of � of about 0.074 isattained when =�L=1, i.e., when the pulse delay is justone pulse length and if the medium has an optical thick-ness of one. In this case about ten photons per atomiccross section are sufficient to obtain perfect conversion.

In the second scheme, where the field �2 is generated�see Fig. 23�b��, a finite detuning from level �3� is neededto suppress absorption. Furthermore, in order to cancelac-Stark shifts the frequencies �1 and �2 can be tunedmidway between two states with appropriate transitionmatrix elements �Johnsson et al., 2002�. Efficient fre-quency conversion �Hemmer et al., 1994; Ham, Shariar,et al., 1997; Babin et al., 1999; Hinze et al., 1999�, genera-tion of squeezing �Lukin et al., 1999�, as well as the pos-sibility of mirrorless oscillations �in counterpropagatinggeometry� have been predicted �Lukin, Hemmer, et al.,1998� and in part experimentally observed �Zibrov et al.,1999�. It has been shown by Fleischhauer and Lukin

FIG. 23. Four-wave mixing schemes for thegeneration of �a� one field of frequency �4,and �b� two fields �2 and �4.

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�2000� that an extremely low input photon flux �thr issufficient to reach the threshold of mirrorless oscilla-tions:

�thr = fN�12, �98�

where N is the number of atoms in the beam path and�12 is the coherence decay rate of the �1�-�2� transition. fis a numerical prefactor of order unity. Johnsson et al.have shown that in the adiabatic limit and for �12=0, aneffective interaction Hamiltonian of the four-wave mix-ing process can be derived, which reads for a one-dimensional model �Johnsson and Fleischhauer, 2002�:

Hint =�gc

2� dz

�1†�3

†�2�4 + H.c.

�3†�3 + �4

†�4

, �99�

where �� are the operators corresponding to the �com-plex� Rabi frequencies of the fields and where a com-mon coupling constant g=��2� /2�%0c of all transitionswas assumed. One recognizes that, in contrast to thecase of ordinary nonresonant nonlinear optics, the sumof the intensities of the resonant fields appears in thedenominator. As a consequence, in a copropagating ge-ometry, the conversion length, i.e., the length after whicha pair of photons in the input fields �1 and �3 is com-pletely converted into a pair of photons in the fields, �2and �4, decreases with decreasing input intensity. Thusweak input fields lead to a faster conversion than stron-ger ones.

Finally, we note that although there had been a num-ber of theoretical proposals on EIT-based quantum non-linear optics, it is only very recently that the first experi-mental demonstrations of the predicted effects havebeen seen. Kuzmich and co-workers �Kuzmich et al.,2003� and independently van der Wal and collaborators�van der Wal et al., 2003� have demonstrated quantumcorrelations between Stokes and anti-Stokes photonsgenerated with a controllable time delay by resonantRaman scattering on atomic ensembles. The intermedi-ate storage and subsequent retrieval of correlated pho-tons in atomic ensembles are essential ingredients of theproposal of Duan and co-workers for a quantum re-peater �Duan et al., 2001�, which is an important tool forlong-distance quantum communication.

D. Few-photon cavity EIT

It is well known in quantum optics that the presenceof high-finesse cavities can be used to enhance photon-photon interactions, for example, for the purpose ofquantum computation �Nielsen and Chuang, 2000�. It istherefore natural to consider few-photon EIT inside acavity, i.e., when the probe or the coupling field is re-placed by a single quantized radiation mode. We shallshow in this subsection that electromagnetically inducedtransparency combined with cavity quantum electrody-namics can lead to a number of interesting linear andnonlinear effects.

Let us first discuss linear phenomena associated withintracavity EIT. Consider a single three-level atomplaced inside a resonator where the quantized resonatormode acts as the coupling field between the internalstates �2� and �3�. As shown by Field �1993�, electromag-netically induced transparency on the probe transition isalready possible with few resonator photons or evenvacuum, provided that the atom and the resonator arestrongly coupled. To see this we note that the fully quan-tum interaction Hamiltonian

Hint = − ��p

2�3�1� − �

g

2a�3�2� + H.c. �100�

separates into effective three-level systems,

�1,n�↔�p

�3,n� ↔gn+1

�2,n + 1� , �101�

where 1,2,3 denote the internal state of the atom and nand n+1 the number of photons in the resonator mode.If g, which characterizes the atom-cavity coupling, is suf-ficiently large and the atom is initially in state �1�, EITcan be achieved for the probe field even for n=0. Equa-tion �101� suggests another interesting application. Theeffective three-level systems have dark states. For ex-ample,

�D� = cos �t��1,0� − sin �t��2,1� , �102�

with tan �t�=�p�t� /g. Thus stimulated Raman adiabaticpassage from state �1,0� to state �2,1� induced by theprobe field offers the possibility of a controlled genera-tion of a single cavity photon. The potential usefulnessof this effect in cavity QED was first pointed out byParkins et al. �1993�. Law and Eberly proposed an appli-cation for the generation of a single photon in a well-defined wave packet �Law and Eberly, 1996�. Recentlysuch a “photon pistol” was experimentally realized�Hennrich et al., 2000; Kuhn et al., 2002�.

The process of transferring excitation from atoms to afield mode is reversible and allows the opposite processof mapping cavity-mode fields onto atomic ground-statecoherences. In this case the resonator mode takes on therole of the probe field, i.e.,

Hint = − �g

2a�3�1� − �

�c

2�3�2� + H.c. �103�

and we have the coupling

�1,n�↔gn

�3,n − 1� ↔�c�t�

�2,n − 1� . �104�

The dark state for n=0 is identical to that given in Eq.�102�, except tan �t�=g /�c�t� in the present case. Themapping provides a possibility for measuring cavityfields �Parkins et al., 1995�. It was shown, furthermore,by Pellizzari et al. �1995� that a quantum-state transferfrom one atom to a second atom via a shared cavitymode can be used to implement a quantum gate �Pelliz-zari et al., 1995�. Finally, Cirac et al. �1997� proposed anapplication to transfer quantum information between at-oms at spatially separated nodes of a network.

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All of the above discussed effects require, however,the strong coupling of the single atom to the resonatormode. This experimentally challenging requirement canbe partly met if an optically thick ensemble of three-level atoms is used. When an ensemble of N three-levelatoms interacts with a single resonator �probe� modeand a classical �coupling� field according to

Hint = −�

2 �j=1

N

�ga�3�jj1� + ��c�3�jj2� + H.c.� �105�

only symmetric collective states are coupled to the initialstate �1,1, . . . ,1�, where all N atoms are in the groundstate �1�:

�1N� � �1, . . . ,1� ,

�1N−12� �1

N�j=1

N

�1, . . . ,2j, . . . ,1� ,

�1N−222� �1

2N�N − 1��

i�j=1

N

�1, . . . ,2i, . . . ,2j, . . . ,1� ,

etc. �106�

If the initial photon number is n=1, again a simplethree-level coupling scheme emerges:

�1N,1�↔gN

�1N−13,0� ↔�c�t�

�1N−12,0� . �107�

Due to the symmetric interaction of all N atomic dipoleswith the resonator mode, the coupling constant is, how-ever, collectively enhanced:

g → gN . �108�

Thus controlled dark-state Raman adiabatic passagefrom a state with a single photon in the resonator to asingle collective excitation and vice versa is possiblewithout the requirement of a strong-coupling limit ofcavity QED, defined here as g2���3, where � is thecavity decay rate. For higher photon numbers, morecomplicated coupling schemes emerge. All of them dopossess, however, dark states which allow a paralleltransfer of arbitrary photon states with n�N to collec-tive excitations, which has important applications forquantum memories for photons and quantum network-ing �Lukin, Yelin, and Fleischhauer, 2000�.

As discussed in previous subsections, EIT can be usedto achieve a resonant enhancement of nonlinear interac-tions leading, for example, to a cross-phase modulation.Large phase shifts produced by the interaction of indi-vidual photons are very appealing for their potential usein quantum information processing. Imamoglu et al.�1997� showed that if a medium with sufficiently largeKerr nonlinearity is put into an optical cavity, it can leadto a photon blockade effect. When a photon enters apreviously empty cavity, the induced refractive index de-tunes the cavity resonance. If this frequency shift islarger than the cavity linewidth, a second photon cannotenter and will be reflected. This could be employed, for

example, to build a controllable single-photon source ora controlled-phase gate �Duan and Kimble, 2004�. Ima-moglu et al. suggested the use of a large ensemble ofatoms in the resonator to enhance the photon blockadeeffect. It was later shown �Gheri et al., l999�, however,that the large dispersion of the EIT medium does notallow for a collective enhancement of the nonlinearatom-cavity coupling. While photon blockade can beachieved using either single-atom or multiatom EIT sys-tems �Werner and Imamoglu, 1999�, the strong-couplinglimit of cavity QED appears to be required in bothcases. It must be emphasized, however, that in contrastto a many-atom cavity-EIT medium, a collection of N�1 two-level atoms in a cavity is a nearly ideal linearsystem, exhibiting a vanishingly small photon blockadeeffect. The survival of nonlinearity in N�1 atom cavityEIT is related to the reduction of the cavity linewidth, orequivalently the width of the atom cavity dark state�Lukin, Fleischhauer, et al., 1998; Werner and Imamoglu,1999�.

VII. SUMMARY AND PERSPECTIVES

EIT has had a profound impact upon optical science.Hopefully we have succeeded in explaining the relation-ship between EIT and the earlier related ideas of coher-ent preparation of atoms by fields and especially linkedit with the concept of the dark state. We stress again thatthe distinct feature of EIT and related phenomena, incontrast to the earlier spectroscopic tools such as coher-ent population trapping, is that they occur in media thatare optically dense and cause not only a modification ofthe excitation state of the matter but also significantchanges to the optical fields themselves. It is this latterpoint that is crucial to understanding the importance ofEIT in optics. EIT provides a new means to change theoptical characteristics of matter, for instance, the degreeof absorption or refraction or the group velocity, and soprovides a way to alter the propagation of optical fieldsand to enhance the generation of new fields.

At the time of writing �2004� we are some 14 years onfrom the first experimental demonstration of the phe-nomenon of EIT. Much research activity has been un-dertaken in the period since this discovery, in an effortto understand the phenomenon and its various implica-tions. More importantly there has been a concerted ef-fort in many laboratories to harness the effect for appli-cations. It is informative to review the extent to whichthis field has grown, and has remained a healthy andactive subject with no sign of declining activity. Therecontinue to be new and interesting proposals and dem-onstrations that extend EIT to fresh applications. It hasbecome a standard tool for the study of optical proper-ties of atomic and molecular gases. For these reasons ofdurability and utility we believe that EIT has earned asignificant place in optical physics that make it a valu-able topic of study for new research students and expe-rienced researchers alike.

We have tried in this review to cover the main ideasbehind the application of EIT and related dark-state

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phenomena to optics. Some omissions were inevitable.Much work has been published in this field, especially inthe theoretical area, and it was simply not possible tocover or cite every single contribution. We humblyapologize to anyone we have failed to cite. Our inten-tion was primarily to provide the reader new to the fieldwith an accessible and comprehensive framework thatwould support further, more detailed, explorations ofthis topic.

Where possible we have tried to connect the theorywith the results of real experiments. Again we have beenselective, choosing illustrative examples rather than at-tempting a comprehensive review of all work. Becauseof space constraints we have not explored in any struc-tured way the experimental techniques used in these ex-periments, either to achieve robust EIT or to investigateEIT phenomena. This omission is regrettable, but thesituations in which EIT can be observed are too diverseto be covered in depth here. The reader is urged to con-sult the cited literature and where necessary other re-views and textbooks to obtain a fuller understanding ofthe experimental methods.

One feature of the work in this field is that the vastmajority of investigations have been carried out in thegas phase. This trend runs somewhat contrary to a gen-eral tendency for nonlinear optics to move towards em-ploying solid-state media. The reason for this is that gas-phase media still offer the physicist unique features;small homogeneous and inhomogeneous linewidthscompared to most solid-state media, transparency overmuch of the IR-VUV range, the possibility of using lasercooling, and high thresholds against optical damage.Moreover, a problem normally identified with gas-phasenonlinear optical media, the low value of the nonlinearsusceptibility compared to many solid-density opticalcrystals, is circumvented by EIT since it permits the useof resonance to enhance the susceptibility.

Having reviewed the subject and examined a numberof applications that have already been thoroughly inves-tigated, we now turn to the future. While predictions arealways risky, we offer some thoughts on what new appli-cations are likely to emerge in this field in the hope ofstimulating further ideas. We shall confine ourselves toareas where there is at least some current activity and sosome evidence upon which to base our extrapolations.

The potential of EIT and maximal coherence to gen-erate high-brightness coherent fields in the short-wavelength range remains only partially explored.EIT has been shown to enhance frequency-conversionefficiencies to the 0.01 level in up-conversion to the100-nm frequency range. By careful choice of the atomicsystem and through judicious optimization of the densityand length of the medium and the use of transform-limited lasers to drive all the fields, it may be possible toimprove the conversion efficiency by a substantial fur-ther factor. This could result in the generation oftransform-limited nanosecond pulses �spectral line-widths of �100 MHz� of VUV radiation with energies oftens of microjoules. These could be of use in a numberof applications in nonlinear spectroscopy, for instance, if

combined with a second longer-wavelength tunable laserto excite two-photon transitions to highly excited states,or perhaps in Raman spectroscopy. The fixed frequencynature of the output would appear to preclude a widerrange of applications.

While EIT-type four-wave mixing enhancement maybe of limited utility, since it results in a fixed frequencyoutput, this is not, in principle, a limit for off-resonantschemes with maximal coherence. The high conversionefficiencies already demonstrated in the far UV may beextended into the VUV. There is a problem withRaman-like excitations: a limit to the degree of up-conversion as the initial excitation is via difference-frequency mixing of the applied fields. The inherentlimit is set at around 170 nm for the shortest wavelengththat can be generated due to the energy-level structureof the atoms and the shortest available wavelengthsfrom lasers. For shorter-wavelength extensions an im-portant problem that must be solved is how to excitemaximal coherence between the ground state and a veryhighly excited upper state. One approach is to use sum-frequency excitation by the applied lasers. The creationof maximal coherence in this case cannot use the con-ventional STIRAP-type adiabatic excitation schemes.Recently a promising alternative has been demon-strated, Stark chirped rapid adiabatic passage �SCRAP�,with the potential for efficient excitation of maximal co-herence between the ground state and a very highly ex-cited state of an atom �Rickes et al., 2000�. A recent firstdemonstration of the use of this technique showed a de-gree of enhancement to extreme-UV generation viathird-harmonic generation �Rickes et al., 2003�, but fur-ther work is needed in this area.

Besides the potential for efficient generation of coher-ent radiation in new frequency domains, resonant non-linear optics based on EIT will be of growing impor-tance for controlled nonlinear interactions at the few-photon level. The above-mentioned limitationsconcerning tunability and accessible frequencies are ofno relevance here. Photons are ideal carriers of quantuminformation, yet processing of this information, for ex-ample, in a quantum computer, is quite difficult. Thereason for this is the weakness of photon-photon inter-actions in the usual nonlinear media. Here EIT may of-fer new directions. Although some theoretical proposalsexist, the full potential of EIT-based nonlinear optics forthese applications has still not been explored, especiallyon the experimental side.

Another feature of EIT, which makes it very attractivefor future applications in quantum information, is thepossibility of transferring coherence and quantum statesfrom light to collective atomic spin excitations. Quan-tum memories for photons �Lukin, Yelin, and Fleisch-hauer, 2000; Fleischhauer and Lukin, 2001�, as discussedin Sec. III, are just one potential avenue; quantum net-works based on them, including quantum repeaters, areanother �Kuzmich et al., 2003; van der Wal et al., 2003�.Combining the state mapping techniques with atom-atom interactions in mesoscopic samples may providenew tools for generating photon wave packets with tai-

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lored quantum states or for manipulating these states. Afirst proposal in this direction using a dipole blockade ofRydberg excitations was given by Lukin et al. �2001�.The controlled coupling of a photon to many atomicspins in EIT could be of interest for the preparation orthe probing of entangled many-particle states. We havejust begun to explore the role of entanglement in many-body physics and quantum information. Laser manipu-lation of spin ensembles via EIT could provide a veryuseful tool in this quest.

EIT also offers new possibilities in linear and nonlin-ear matter-wave optics. As pointed out in Sec. IV, slowlight in atomic gases corresponds to a quasiparticle,which is a mixed electromagnetic-matter excitation. Forvery low group velocities almost all excitation is concen-trated in atomic spins while the propagation propertiesare still mostly determined by the electromagnetic part.Thus slow light in ultracold atomic gases provides a newway to control the propagation of matter waves withpotential applications in matter-wave interferometry�Zimmer and Fleischhauer, 2004� or if atom-atom inter-actions are included, in nonlinear matter-wave optics�Masalas and Fleischhauer, 2004�.

Ultrafast measurements have recently entered the at-tosecond domain �Drescher et al., 2002�. The possibilityof employing the highly efficient multiorder Raman gen-eration that results from an adiabatically prepared maxi-mal vibrational coherence has already been investigated.A key problem to solve before this approach can bewidely employed for ultrafast measurements will be thesynchronization of the train of subfemtosecond pulsesthat are generated with external events. One possibility�suggested by Sokolov, 2002� is to use the pulse train to“measure” processes in the modulated molecules them-selves so that the synchronization is automatically satis-fied.

As we have emphasized throughout this review, coher-ent preparation techniques that are at the heart of EITcan be easily implemented in optically dense atomicsamples in the gas phase. It is generally argued, however,that the range of applications can be significantly en-larged if EIT is implemented in solid-state media. Theobvious roadblock in this quest are the ultrashort coher-ence times of optical transitions in solids. Even for tran-sitions that are not dipole allowed, the coherence timesare typically on subnanosecond time scales. Among themany different physical mechanisms contributing to thisfast dephasing, perhaps the most fundamental one is theinteraction of electrons with lattice vibrations �phonons�.While cooling the samples to liquid-helium tempera-tures reduces the average number of phonons andthereby phonon-induced dephasing, the coherence timesstill remain significantly shorter than their atomic coun-terparts.

Fortunately, there is an exception to the general ruleof ultrashort dephasing times in solids: electron spin de-grees of freedom in a large class of solid-state materialshave relatively long coherence times, ranging from 1 �sin optically active direct-band-gap bulk semiconductorssuch as GaAs �Wolf et al., 2001� to 1 ms in rare-earth

ion-doped �insulating� crystals �Kuznetsova et al., 2002�.In both cases, the ratio of the coherence relaxation ratesof the dipole-allowed and spin-flip transitions can be aslarge as 104, indicating that EIT could be implementedefficiently.

Most of the experimental efforts aimed at demonstrat-ing EIT in solid-state spins have so far focused on rare-earth ion-doped crystals �Ham, Hemmer, and Shahiar,1997; Turukhin et al., 2001� and nitrogen vacancy centersin diamond �Wei and Manson, 1999�. In experiments car-ried out using Pr-doped Y2SiO5 at cryogenic tempera-tures �T=5 K�, Turukhin and co-workers have observedgroup velocities as slow as 45 m/s and a group delay of66 �s for light pulses with a pulse width of 50 �s �Tu-rukhin et al., 2001�. Storage and retrieval of these pulseshave also been demonstrated �Turukhin et al., 2001�.While the ratio of the group delay to the pulse width inthese experiments is on the order of unity, this is animpressive and promising development for solid-stateEIT.

Optoelectronics and photonics technology are largelybased on semiconductor heterostructures formed out ofcolumn-III–V semiconductors. The conduction-bandelectrons of these elements have predominantly s-typewave functions with small spin-orbit coupling. As a re-sult, the spin coherence times are four orders of magni-tude longer than the radiative recombination rate of ex-citons. The possibility of observing efficient EIT andslow light propagation in a doped GaAs quantum well inthe quantum Hall regime has been discussed �Imamoglu,2000�. While this particular realization requires low tem-peratures and high magnetic fields �B�10 T�, long spincoherence times in GaAs have been observed even atroom temperature and with vanishing magnetic fields�Wolf et al., 2001�. In fact, there is substantial activity inthe emerging field of spintronics �Wolf et al., 2001; Zuticet al., 2004�, and it is plausible to argue that EIT couldplay a role by providing long optical delays and/or en-abling “spin-photon” information transfer.

Yet another interesting extension of gas-phase EIT toplasmas was suggested by Harris �1996�. Here classicalinterference effects in parametric wave interactionswhich are analogous to interference effects in three-levelatoms lead to an induced transparency window belowthe plasma frequency. This could have interesting appli-cations for magnetic fusion, plasma heating, and plasmadiagnostics. For some recent work on this see, Litvakand Tokman �2002� and Shvets and Wurtele �2002�.

Even though EIT is by now a well-established tech-nique, new applications continue to appear regularly. Itwas recently proposed, for example, that a “dynamicalphotonic band-gap” structure could be realized in anEIT medium by spatially modifying the refractive indexwith a standing-wave optical field �Andre and Lukin,2002�. This could be achieved by making use of the largeabsorption-free dispersion around the transparency fre-quency. EIT may also allow for the realization of a high-efficiency photon counter. After mapping the quantumstate of a propagating light pulse onto collective hyper-fine excitations of a trapped cold atomic gas, it is pos-

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sible to monitor the resonance fluorescence induced byan additional laser field that only couples to the hyper-fine excited state �Imamoglu, 2002; James and Kwiat,2002�. Even with a fluorescence collection/detection ef-ficiency as low as 10%, it should be possible to achievephoton counting with an efficiency exceeding 99%. Sucha device could be of great use in quantum optical infor-mation processing �Knill et al., 2001�.

ACKNOWLEDGMENTS

We would like to express our gratitude to Steve Har-ris, who has been a source of inspiration through hisinnovative ideas, insights, and encouragement. Our nu-merous discussions and collaborations with Klaas Berg-mann, Peter Knight, Misha Lukin, Marlan Scully, MosheShapiro, and Bruce Shore have played a key role inshaping our understanding of various aspects of EIT.A.I. would also like to thank John Field for truly en-lightening discussions dating from the very early days ofEIT. Finally, we acknowledge K. Boller, K. Hakuta, L.Hau, O. Kocharovskaya, and H. Schmidt.

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