Electromagnetically Induced Transparency: Optics in Coherent Media

Michael Fleischhauer1, Atac Imamoglu2, and Jonathan P. Marangos3

1 Fachbereich Physik, Technische Universitat Kaiserslautern, D-67663 Kaiserslautern, Germany2 Institute of Quantum Electronics, ETH-Honggerberg, HPT G12, CH-8093 Zurich, Switzerland3 Quantum Optics and Laser Science Group, Blackett Laboratory, Imperial College, London SW7 2BW,

UK

(Dated: September 14, 2004)

Coherent preparation by laser light of the 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. These phenomena have placed gas phase systems at the centre of recent advances

in the development of media with radically new optical properties. We will discuss these advances,

and the new possibilities that arise in non-linear optics and quantum information science, when

the optical properties of a medium are modified by coherent preparation. We will develop the

theory of electromagnetically induced transparency by considering the atomic dynamics and the

optical response first for a CW laser. The pulse propagation and the adiabatic evolution of the

field coupled states is then developed. From this we will see how coherently prepared media can

be used to improve frequency conversion in non-linear optical mixing experiments. The extension

of these concepts to very weak optical fields in the few photon limit will then be examined. We

will conclude by discussing future prospects and potential new applications.

Contents

I. Introduction 1

II. Physical concepts underlying Electromagnetically

Induced Transparency 3

A. Interference between excitation pathways 3

B. Dark state of the 3-level Λ-type atom 5

III. Atomic dynamics and optical response 6

A. Master equation and linear susceptibility 6

B. Effective Hamiltonian and dressed-state picture 10

C. Transmission through a medium with EIT 11

D. Dark-state preparation and Raman adiabatic passage 12

E. EIT inside optical resonators 13

F. Enhancement of refractive index, magnetometry and

lasing without inversion 14

IV. EIT and pulse propagation 15

A. Linear response slow and ultra-slow light 15

B. “Stopping of light” and quantum memories for

photons 18

C. Nonlinear response: adiabatic pulse propagation and

adiabatons 20

V. Enhanced Frequency Conversion 22

A. Overview of Atomic Coherence Enhanced Non-Linear

Optics 22

B. Non-linear Mixing and Frequency Up-Conversion with

Electromagnetically Induced transparency 24

C. Non-linear Optics with Maximal Coherence 26

D. Four-Wave Mixing in Double-Λ Systems 28

VI. EIT with few photons 29

A. Giant Kerr effect 29

B. Cross-phase modulation using single-photon pulses

with matched group velocities 31

C. Few-photon four-wave mixing 32

D. Few-photon cavity-EIT 33

VII. Summary and perspectives 35

Acknowledgement 37

References 37

I. INTRODUCTION

Advances in optics have frequently arisen through thedevelopment of new materials with optimised opticalproperties. For instance the introduction of new opti-cal crystals in the 1970’s and 80’s led to substantial in-creases in non-linear optical conversion efficiencies to theultra-violet (UV). Another example has been the devel-opment of periodically-poled crystals that permit quasi-phase matching in otherwise poorly phase-matched non-linear optical processes which greatly enhance non-linearfrequency mixing efficiencies. As we will see coherentpreparation is a new avenue that produces remarkablechanges in the optical properties of a gas phase atomicor molecular medium.

The cause of the modified optical response of an atomicmedium in this case is the laser induced coherence ofthe atomic states which leads to quantum interferencebetween the excitation pathways that control the opti-cal response. We can in this way eliminate the absorp-tion and refraction (linear susceptibility) at the resonantfrequency of a transition. This was termed electromag-netically induced transparency (EIT) by Harris and co-workers (Harris et al., 1990). The importance of EITstems from the fact that it gives rise to greatly enhancednon-linear susceptibility in the spectral region of inducedtransparency of the medium and is associated with steepdispersion. For some readable general accounts of earlierwork in the field the reader might see for example Harris(1997) or Scully (1992). Other more recent reviews onspecific aspects of EIT and its applications can be foundin (Lukin et al., 2000a; Matsko et al., 2001a; Vitanovet al., 2001) as well as in the topical Colloquium (Lukin,

2

2003). It should be emphasized that the modificationof atomic properties due to quantum interference hasbeen studied extensively for 25 years, see e.g. Arimondo(1996). In particular the phenomenon of coherent popu-lation trapping (CPT) observed by Alzetta et al. (1976)is closely related to EIT. In contrast to CPT which is a”spectroscopic” phenomenon that involves only modifica-tions to the material states in an optically thin sample,EIT is a phenomenon specific to optically thick mediawhere 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 χ(1).The imaginary part of this susceptibility Im[χ(1)] de-termines the dissipation of the field by the atomic gas(absorption), whilst the real part Re[χ(1)] determinesthe refractive index. The form of Im[χ(1)] at a dipole-allowed transition as a function of frequency is that ofa Lorentzian function with a width set by the damp-ing. The refractive index Re[χ(1)] follows the familiardispersion profile, with anomalous dispersion (decreasein Re[χ(1)] with field frequency) in the central part ofthe absorption profile within the line-width. Figure 1illustrates both the conventional form of χ(1) and themodified form that results from EIT as will be discussedshortly.

χR

e [

]

χ

Im [

]

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3

-0.4

-0.2

0

0.2

0.4

(ω − ω ) / p 31 γ31

FIG. 1 Susceptibility as a function of the frequency ωp of theapplied field relative to the atomic resonance frequency ω31,for a radiatively broadened two-level system with radiativewidth γ31 (dashed lines) and an EIT system with resonant

coupling field (full line). top: Imaginary part of χ(1) charac-

terizing absorption, bottom: Real part of χ(1) determining therefractive properties of the medium.

In the case of laser excitation where the magnitude of

the electric field can be very large we reach the situa-tion where the interaction energy of the laser couplingdevided 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 inthis sense that we refer to atomic coherence and coher-ent preparation. This is of course in contrast to the rateequation treatment of the state populations often appro-priate when the damping is large or the coupling is weakfor which the coherence of the states can be ignored. Fora full account of the coherent excitation of atoms thereader is recommended to consult Shore (1990).

For a 2-level system, the result of coherent evolutionis characterized by oscillatory population transfer (Rabiflopping). The generalisation of this coherent situationto driven 3-level atoms leads to many new phenomenasome of which, such as Autler-Townes splitting (Autlerand Townes, 1955), dark-states and EIT, will be the sub-ject of this review. These phenomena can be understoodeither within the basis of bare atomic states or new eigen-states which diagonalize the complete atom-field interac-tion Hamiltonian. In both cases we will see that inter-ference between alternative excitation pathways betweenatomic states lead to modified optical response.

The linear and non-linear susceptibilities of a Λ-type3-level system driven by a coherent coupling field willbe derived in section III. Figures 1a and b show theimaginary and real part of the linear susceptibility forthe case of a resonant coupling field as a function of theprobe field detuning from resonance. Figure 2 shows thecorresponding third-order non-linear susceptibility. In-spection of these frequency dependent dressed suscep-tibilities reveal immediately several important features.One recognizes that Im[χ(1)] undergoes destructive in-terference in the region of resonance, i.e. the coherentlydriven medium is transparent to the probe field. The factthat transparency of the sample is attained at resonanceis not in itself of great importance as the same degreeof transparency can be obtained simply by tuning suffi-ciently away from resonance. What is important is thatin the same spectral region as there is a high degree oftransmission the non-linear response χ(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 of theresonance differs markedly from the steep anomalous dis-persion familiar at an undressed resonance (see Fig. 1).Instead there is a normal dispersion in a region of lowabsorption the steepness of which is controlled by thecoupling-laser strength (i.e. very steep for low values ofthe drive laser coupling). Thus despite the transparencythe transmitted laser pulse can still experience strong dis-persive and non-linear effects. It is most significant thatthe refractive index passes through the vacuum value andthe dispersion is steep and linear exactly where absorp-

3

tion is small. This gives rise to effects such as ultra-slowgroup velocities, longitudinal pulse compression and stor-age of light. Furthermore through the destructive (con-structive) interference in Imχ(1) (χ3) and the eliminationof the effect of resonance upon the refractive index, theconditions for efficient non-linear mixing are met.

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

0 2−2 −1 10

0.2

0.4

0.6

0.8

1

χ(3)

|

|

(ω − ω ) / γp31 31

FIG. 2 Absolute value of nonlinear suscpetibility for sum-frequency generation |χ(3)| as a function of ωp, in arbitraryunits. The parameters are identical to those used in Fig. 1.

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 the states will in general beinhibited by dephasing of the complex state amplitudes,but the coherence dephasing rates for gases are relativelysmall when compared to those in solid state media. Nev-ertheless several workers have recently made progress inapplying these techniques to a variety of solid-state sys-tems (Faist et al., 1997; Ham et al., 1997a; Schmidt et al.,1997; Serapaglia et al., 2000; Wei and Manson, 1999;Zhao et al., 1997). Atomic gases remain very impor-tant in optical applications for several key reasons; forexample they are often transparent in the vacuum UVand infra-red, and there exist techniques to cool them toultra-cold temperatures thus eliminating inhomogeneousbroadening. They can also tolerate high optical inten-sities. Gas phase media are for instance necessary forfrequency conversion beyond the high and low frequencycut-off of solid state materials.

The dramatic modifications of the optical propertiesthat are gained through coherent preparation of theatoms or molecules in a medium have ensured a renais-sance in activity in the area of gas phase non-linear op-tics. New opportunities arise also due to the extremelyspectrally narrow features that can result from coher-ent preparation; in particular this gives ultra-low groupvelocities and ultra-large non-linearities. These are ofpotential importance to new techniques in optical infor-mation storage and also of non-linear optics at the fewphoton level both of which may be important to quantuminformation processing.

Throughout this review we will try to point out theimportant applications that arise as a result of EIT anddescribe in outline some of the seminal experiments.

We attempt to cover comprehensively the alteration oflinear and non-linear optical response due to electro-magnetically induced transparency and related phenom-ena. We do not, however, cover the related topic of las-ing without inversion (Gornyi et al., 1989; Harris, 1989;Kocharovskaya and Khanin, 1986; Scully et al., 1989)and the 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 will introduce the underlying phys-ical concepts of EIT through the coupling of near reso-nant laser fields with the states of a 3-level system. Insection III we discuss in detail the dynamics in 3-levelatoms coupled to the applied laser fields and determinethe optical response. In section IV we treat the topic ofpulse propagation in EIT and review the developmentsthat have culminated in the demonstration of ultra-slowgroup velocities down to a few metres per second andeven of “stopping” and “storing” of a pulse of light. Theutility of coherent preparation in enhancing the efficiencyof frequency conversion processes is discussed in sectionV. Then in section VI we will turn to the treatment ofEIT in the few photon limit where it is necessary to ap-ply a fully quantum treatment of the fields. Finally wedraw conclusions and discuss the prospects in quantumoptics and atom optics that may arise as a result of EIT.

II. PHYSICAL CONCEPTS UNDERLYING

ELECTROMAGNETICALLY 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 photoionisation of multi-electron atoms. Dueto the existence of doubly excited states the photoioni-sation spectrum of multi-electron atoms can show a richstructure of resonances. These resonances are broadeneddue to relatively rapid decay (caused by the interactionbetween the excited electrons of these states) to the de-generate continuum states, with lifetimes in the picosec-ond to sub-picosecond range. For the reason that theydecay naturally to the continuum, these states are calledauto-ionising states. Fano introduced to atomic physicsthe concept of the interference between the excitationchannels to the continuum that exist for a single auto-ionising resonance coupled to a flat continuum (Fano,1961) Figure 3(a). In the vicinity of the auto-ionisingresonance the final continuum state can be reached viaeither direct excitation (channel 1) or through the reso-nance with the configuration interaction leading to thedecay that provides a second channel to the final con-tinuum state. The interference between these channelscan be constructive or destructive and leads to frequencydependent suppression or enhancement in the photoion-isation cross-section.

4

Experiments by Madden and Codling (1965) showedthe resulting transparency windows in the auto-ionisingspectrum of helium. Sometime later the auto-ionisinginterference 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). Inthis wave-mixing experiment the absorption was elimi-nated in spectral regions where the non-linear responseremained large, hence an improved efficiency for fre-quency conversion was reported.

continuum

(b)(a)

continuum

1

12

2

FIG. 3 Fano-interferences of excitation channels into a con-tinuum for a single autoionizing resonance (a) and two au-toionizing states (b).

During the 1960’s auto-ionising spectra were muchstudied (see for example (Garton, 1966)). Several au-thors addressed the issue of interaction between two ormore spectral series in the same frequency range wherethe interference between closely spaced resonances needsto be considered (Fano and Cooper, 1965; Shore, 1967).Shapiro provides the first explicit analysis of the case ofinterference between two or more resonances coupled toa single continuum (Shapiro, 1970). In this case the in-terference is mainly between the two transition pathwaysfrom the ground state to the final state via each of the tworesonances (Figure 3(b)). Naturally interference will besignificant only if the spacing between these resonancesis comparable to or less than their widths.

Hahn, King and Harris (Hahn et al., 1990) showed howthis situation could be used to enhance four-wave mixing.In experiments in zinc vapour that showed significantlyincreased non-linear mixing one of the fields was tuned toa transition between an excited bound state and a pair ofclosely spaced auto-ionising states, that had a frequencyseparation much less than their decay widths.

The case of interference between two closely spacedlifetime broadened resonances, decaying to the same con-tinuum, was further analyzed by Harris (1989). Hepointed out that this will lead to lasing without inversionsince the interference between the two decay channelseliminates absorption whilst leaving stimulated emissionfrom the states unchanged. Although we will not discussthe subject of lasing-without inversion further in this re-view, this work was an important step in the story withwhich we are concerned. A breakthrough was then madein the work of Imamoglu and Harris (1989) where it wasrealised that the pair of closely spaced lifetime broadened

resonances were equivalent to dressed states created bycoupling a pair of well separated atomic bound levelswith a resonant laser field (Figure 4). They thus pro-posed that the energy level structure required for quan-tum interference could be engineered by use of an exter-nal laser field. Harris et al. (1990) then showed how thissame situation can be extended to frequency-conversionin a four-wave mixing scheme among atomic bound-stateswith the frequency conversion hugely enhanced. This isachieved through the cancellation of linear-susceptibilityat resonance as shown in Figure 1, whilst the non-linearsusceptibility is enhanced through constructive interfer-ence. The latter paper was the first appearance ofthe term electromagnetically induced transparency (EIT)which was used to describe this cancellation of the lin-ear response by destructive interference in a laser dressedmedium.

dressing

probe

1⟩

2⟩

3⟩

probe

1⟩

a(+)⟩a(-)⟩

Γ Γd

Γd(-)

(+)

FIG. 4 Coherent coupling of a meta-stable state |2〉 to anexcited state |3〉 (left) generates interference of excitationpathways through the doublet of dressed states |a±〉 (Autler-Townes doublet) provided the decay out of state |2〉 is negli-gible as compared to that of state |3〉.

Boller et al. (1991), in discussing the first experimen-tal observation of EIT in Sr vapour point 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 where the dressed statescan be viewed as simply comprising two closely spacedresonances effectively decaying to the same continuum(Boller et al., 1991; Zhang et al., 1995) . If the probe fieldis tuned exactly to the zero field resonance frequency thenthe contributions to the linear susceptibility due to thetwo resonances, which are equally spaced but with op-posite signs of detuning, will be equal and opposite andthus lead to the cancellation of the response at this fre-quency due to a Fano-like interference of the decay chan-nels. An alternative and equivalent picture is to considerthe bare rather than the dressed atomic states. In thisview EIT can be seen as arising through different path-ways between the bare states. The effect of the fields isto transfer a small, but finite amplitude, into state |2〉.The amplitude for |3〉 which is assumed to be the onlydecaying state and thus the only way to absorption isthus driven by two routes; directly via the |1〉− |3〉 path-way, or indirectly via |1〉 − |3〉 − |2〉 − |3〉 pathway (or

5

by higher order variants). Because the coupling field ismuch more intense than the probe, this indirect pathwayhas a probability amplitude that is in fact of equal mag-nitude to the direct pathway, but for resonant fields it isof opposite sign.

B. Dark state of the 3-level Λ-type atom

The use in laser spectroscopy of externally applied elec-tromagnetic fields to change the system Hamiltonian ofcourse pre-dates the idea of using this in non-linear op-tics 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 appliedto an atom began to be extensively studied following theinvention of the laser in the early 1960’s. Hansch andToschek (1970) recognized the existence of these typeof interference processes for three-level atoms coupled tostrong laser fields in computing the system susceptibilityfrom a density matrix treatment of the response. Theyidentified terms in the off-diagonal density matrix ele-ments indicative of the interference, although they didnot explicitly consider the optical and non-linear opticaleffects in a dense medium.

Our interest is in the case of electromagnetic fields, inthe optical frequency range, applied in resonance to thestates of a 3-level atom. We illustrate the three possi-ble coupling schemes in Figures 5 and 6. For consistencystates are labelled so that the |1〉 − |2〉 transition is al-ways the dipole forbidden transition. Of these prototypeschemes we will be most concerned with the Lambda con-figuration Fig. 5, since the Ladder and Vee configurationsillustrated in Fig. 6 are of more limited utility for the ap-plications that will be discussed later.

ω

2

1

3

1

ω

∆ ∆ 2

p

c

Γ 32Γ31

FIG. 5 Generic system for EIT: Lambda-type scheme withprobe field of frequency ωp and coupling field of frequencyωc. ∆1 = ω31 − ωp and ∆2 = ω32 − ωc denote field detuningsfrom atomic resonances and Γik radiative decay rates fromstate |i〉 to state |k〉.

The physics underlaying the cancellation of absorptionin EIT is identical to that involved in the phenomena ofdark-state and coherent population trapping (Lounis and

1⟩ 1⟩

2⟩

2⟩3⟩

3⟩

FIG. 6 Ladder (left) and Vee-type (right) three-level schemeswhich do not show EIT in the strict sense because of theabsence of a (meta) stable dark state.

Cohen-Tannoudji, 1992). We will therefore review brieflythe concept of dark-states. Alzetta et al. (1976) made theearliest observation of the phenomenon of coherent pop-ulation trapping (CPT) shortly followed by Whitley andStroud (1976). Gray et al. (1978) explained these obser-vations using the notion of coherent population trappingin a dark eigen-state of a three level Lambda medium(see Figure 5). In this process a pair of near resonantfields are coupled to the Lambda system and result inthe Hamiltonian H = H0 + Hint, where the Hamiltonianfor the bare atom is H0 and that for the interaction withthe fields is Hint. The Hamiltonian H has a new setof eigenstates when viewed in a proper rotating frame(see below) one of which has the form |a0〉 = α|1〉 − β|2〉which contains no amplitude of the bare state |3〉, andis therefore effectively decoupled from the light fields. Inthe experiments of Alzetta population was pumped intothis state via spontaneous decay from the excited statesand then remains there since the excitation probabilityof this dark-state is cancelled via interference. A veryinformative review of the applications of dark-states andthe coherent population trapping that accompanies themin spectroscopy has been provided by Arimondo (1996).

We would now like to look a bit more closely at thestructure of the laser dressed eigen-states of a 3-levelatom illustrated in Fig. 5. This discussion is intendedto provide a simple physical picture that establishes theconnection between the key ideas of EIT and that of max-imal coherence.

Within the dipole approximation the atom-laser inter-action Hint = µ · E is often expressed in terms of theRabi coupling (or Rabi frequency) Ω = µ ·E0/~, withE0 being the amplitude of the electric field E, and µ thetransition electronic dipole moment. After introducingthe rotating-wave approximation, we can represent theHamiltonian of the 3-level atom interacting with a cou-pling laser with real Rabi-frequency Ωc and a probe laserwith Rabi-frequency Ωp (Fig. 5), in a rotating frame as

Hint = −~

2

0 0 Ωp

0 −2(∆1 −∆2) Ωc

Ωp Ωc −2∆1

. (1)

6

Here ∆1 = ω31−ωp and ∆2 = ω32−ωc are the detuningsof the probe and coupling laser frequencies ωp and ωc

from the corresponding atomic transitions.A succinct way of expressing the eigenstates of the in-

teraction Hamiltonian Eq.(1) is in terms of the “mixingangles” θ and φ that are dependent in a simple way uponthe Rabi couplings as well as the single-photon (∆1 = ∆)and two-photon (δ = ∆1−∆2) detunings (see Figure 5).For two-photon resonance (∆2 = ∆1, or δ = 0) the mix-ing angles are given by:

tanθ =Ωp

Ωc, (2)

tan2φ =

√

Ω2p + Ω2

c

∆. (3)

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

∣

∣a+⟩

= sinθsinφ |1〉+ cosφ |3〉+ cosθsinφ |2〉 , (4)∣

∣a0⟩

= cosθ |1〉 − sinθ |2〉 , (5)∣

∣a−⟩ = sinθcosφ |1〉 − sinφ |3〉+ cosθcosφ |2〉 . (6)

The readers attention is drawn to the following features:Whilst the state

∣

∣a0⟩

remains at zero energy the pairof states |a+〉 and |a−〉 are shifted up and down by anamount ~ω±

~ω± =~

2(∆±

√

∆2 + Ω2p + Ω2

c). (7)

The states |a±〉 retain a component of all of the bareatomic states, but in contrast state

∣

∣a0⟩

has no contri-bution from |3〉 and is therefore the dark-state since ifthe atom 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 pictureabove consider the case of a weak probe, that is Ωp Ωc.In this case sinθ → 0 and cosθ → 1 which results in thedressed eigenstates shown in Figure 4. The ground statebecomes identical to the dark-state

∣

∣a0⟩

= |1〉 from whichexcitation cannot occur. Furthermore, when the probe ison resonance (∆ = 0) then tanφ → 1 (i.e. φ = π/2),

and then |a+〉 = 1/√

2(|2〉+ |3〉) and |a−〉 = 1/√

2(|2〉 −|3〉); these are the usual dressed states relevant to EITin the limit of a strong coupling field and a weak probe.Conversely the picture of dark-states show us that theEIT interference will survive in a Lambda system in the

limit of a pair of strong fields and so is relevant to thesituation of maximal coherence.

To ensure that a dark-state is formed an adiabatic evo-lution of the field is often adopted (Oreg et al., 1984).The physically most interesting example of this involves aso called counter-intuitive pulse sequence. This was firstdemonstrated by Bergmann and his co-workers (Gaubatzet al., 1990; Kuklinski et al., 1989) who termed this Stim-ulated Raman Adiabatic Passage (STIRAP). With thesystem starting with all the amplitude in the ground-state |1〉 the laser field at ωc is first applied. This islike the EIT situation since Ωp Ωc and the dark-stateindeed corresponds exactly to |1〉 with negligible popu-lation either in |2〉 or |3〉. But now Ωp is gradually (i.e.adiabatically) increased and at the same time Ωc is grad-ually decreased so that eventually Ωc Ωp and the dark-state becomes

∣

∣a0⟩

= − |2〉. This population transfer isattained via the ”maximally coherent” dark-state whenΩp = Ωc i.e. θ = π/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 where wewill assume continuous-wave (cw) classical fields interact-ing with a single atom that can be modelled as a lambdasystem (Figure 5). For the derivation of linear and non-linear susceptibilities, we concentrate on the steady-statesolution of the atomic master equation. While a mas-ter equation analysis is general and can be used for thetreatment of non-perturbative field effects, a single-atomwave-function approach is simpler and more illustrative;we will therefore use the latter to discuss the dressed-state interpretation of EIT. As many of the central fea-tures of EIT are related to optically thick media we willdiscuss the light transmission of such a medium. Theprocess of establishing EIT will then be examined andthe connection to Raman adiabatic passage will be es-tablished. Finally, we will discuss EIT inside optical res-onators and briefly review some applications of EIT thatwill not be expanded upon in the following sections ofthe review.

A. Master equation and linear susceptibility

We consider an ensemble of identical atoms whose dy-namics 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 lowestenergy state |1〉 (Fig. 5). State |2〉 has the same parityas |1〉 and is assumed to have a very long coherence time.The highest energy state |3〉 is of opposite parity and has

7

a non-zero electric-dipole coupling to both |1〉 and |2〉. A(near) resonant non-perturbative electromagnetic field offrequency ωc, termed the coupling field, is applied on the|2〉-|3〉 transition. A probe field of frequency ωp is appliedon the |1〉-|3〉 transition: EIT is primarily concerned withthe modification of the linear and nonlinear optical prop-erties of this - typically perturbative - probe field. Weemphasize that most atomic systems have a subspace oftheir state-space that can mimic this simplified picturewhen driven by near resonant polarized electromagneticfields.

The time-dependent interaction Hamiltonian in the in-teraction picture that describes the atom-laser couplingis:

Hint = −~

2

[

Ωp(t)σ31ei∆1t + Ωc(t)σ32e

i∆2t + h.c.]

,(8)

where Ωc(t) and Ωp(t) denote the Rabi frequency as-sociated with the coupling and probe fields. σij =|i〉〈j| is the atomic projection operator (i, j = 1, 2, 3).This semiclassical Hamiltonian can be obtained from thefully quantized interaction-picture interaction Hamilto-nian by replacing annihilation and creation operators byc-numbers.

The dynamics of the laser driven atomic systems isgoverned by the master equation for the atomic densityoperator

dρ

dt=

1

i~

[

Hint, ρ]

+Γ31

2

[

2σ13ρσ31 − σ33ρ− ρσ33

]

+Γ32

2

[

2σ23ρσ32 − σ33ρ− ρσ33

]

+γ2deph

2

[

2σ22ρσ22 − σ22ρ− ρσ22

]

+γ3deph

2[2σ33ρσ33 − σ33ρ− ρσ33

]

, (9)

where the second and third terms on the right hand sidedescribe spontaneous emission from state |3〉 to states|1〉 and |2〉, with rates Γ31 and Γ32, respectively. Weassume here that the thermal occupancy of the relevantradiation field modes is completely negligible: for opti-cal frequencies, this approximation is easily justified. Adetailed derivation of this master equation is given in(Cohen-Tannoudji et al., 1992). We have also introducedenergy conserving dephasing processes with rates γ3deph

and γ2deph; the latter determines one of the fundamen-tal time-scales for practical EIT systems as we will seeshortly. For convenience, we define the total spontaneousemission rate out of state |3〉 as Γ3 = Γ31 + Γ32. The co-herence decay rates are defines as γ31 = Γ3 + γ3deph,γ32 = Γ3 + γ3deph + γ2deph, and γ21 = γ2deph.

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 value ofthe atomic polarization is

~P (t) = −∑

i

〈eri〉/V (10)

=Natom

V

[

µ13ρ31e−iω31t + µ23ρ32e

−iω32t + c.c.]

.

To obtain the expression in the second line, we as-sumed that all Natom atoms contained in the volumeV couple identically to the electromagnetic fields. Thetime-dependent exponentials arise from the conversionto the Schrodinger picture. Assuming µ13 = µ13z andµ23 = µ23z, we let % = Natom/V and obtain Pz(t) = P (t)as

P (t) = %[

µ13ρ31e−iω31t + µ23ρ32e

−iω32t + c.c.]

.(11)

We now focus on the perturbative regime in the probefield and evaluate the off-diagonal density matrix ele-ments ρ31(t), ρ32(t), and ρ12(t) to obtain P (t), or equiv-alently, the linear susceptibility χ(1)(−ωp, ωp). Takingρ11 ' 1 and using a rotating frame to eliminate fast ex-ponential time-dependences, we find

ρ32 =iΩce

i∆1t

(γ32 + i2∆2)ρ12,

ρ12 = − iΩcei∆2t

γ21 + i2(∆2 −∆1)ρ13

ρ31 =iΩpe

i∆1t

(γ31 + i2∆1)+

iΩcei∆2t

(γ31 + i2∆1)ρ21. (12)

As in Sec. I we define the single-photon detuning as ∆ =∆1 = ω31−ωp and two-photon detuning as δ = ∆1−∆2 =ω21−(ωp−ω−c). Keeping track of the terms that oscillatewith exp[−iωpt], we obtain

χ(1)(−ωp, ωp) =|µ13|2%

ε0~×

×[

4δ(|Ωc|2 − 4δ∆)− 4∆γ221

||Ωc|2 + (γ31 + i2∆)(γ21 + i2δ)|2

+i8δ2γ31 + 2γ21(|Ωc|2 + γ21γ31)

||Ωc|2 + (γ31 + i2∆)(γ21 + i2δ)|2

]

. (13)

The linear susceptibility given in Eq. (13) containsmany of the important features of EIT. First, it mustbe mentioned that Eq. (13) predicts Autler-Townes split-ting of an atomic resonance due to the presence of a non-perturbative 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 (δ = 0), both real and imagi-nary parts of the linear susceptibility vanish in the ideallimit of γ21 = 0. As depicted in Figure 7, this result is in-dependent of the strength of the coherent coupling field.It should be noted that changing the Rabi-frequency ofthe coupling laser only changes the spectral profile of ab-sorption, the integral of Im[χ] over ∆ is conserved as Ωc

is varied. In the limit |Ωc| > γ31, the absorption pro-file carries the signatures of an Autler-Townes doublet.Important features in absorption, such as vanishing lossat δ = 0 and enhanced absorption on the low and high

8

energy sides of the doublet, become evident on a closerinspection. For |Ωc| γ31, we obtain a sharp transmis-sion window with a linewidth much narrower than γ31.In this latter case, it becomes apparent that the modifi-cations in the linear susceptibility call for an explanationbased on quantum interference phenomena rather than asimple line splitting. Before proceeding with this discus-sion, we highlight some of the other important propertiesthat emerge in Eq. (13).

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

χ(1)

Im [

]χ

Im

[

]

(1)

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

(a)

(b)

(ω − ω ) / γ31p31

FIG. 7 EIT absorption spectrum for different values of cou-pling field and γ21 = 0: (a) Ωc = 0.3γ31, (b) Ωc = 2γ31.

First and foremost it must be noted that the possi-bility of eliminating absorption in an otherwise opticallythick medium has been demonstrated in several labora-tories in systems ranging from hot atomic vapor cells tomagnetically trapped Bose-condensed atoms. Figure 8shows the absorption profile obtained in the first exper-iment to demonstrate the effect, carried out by Bollerand co-workers (Boller et al., 1991). The application ofa coupling field, opens up a transparency window in anotherwise completely opaque atomic cell. This experi-ment was carried out using an autonionizing state |3〉 ofstrontium with γ31 ∼ 2×1011 s−1. The decay rate of thelower state coherence γ21 ∼ 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 associ-ated dispersion becomes highly asymmetric for ∆ > γ31.Figure 9 shows contour plots of Im[χ(1)] as a function ofthe detunings ∆1 and ∆2 of the two fields as well as thesingle (∆) and two (δ) photon detunings. It is evidentthat for large detuning ∆2 of the coupling field the ab-sorption spectrum is essentially that of a two-level system

probe laser detuning (cm )−1

tran

smis

sion

FIG. 8 First experimental demosntratiopn of EIT in Sr vaporby Boller et al. (1991). top: transmission through cell withoutcoupling field, bottom with coupling field on.

with an additional narrow Raman peak close to the two-photon resonance. Exactly at the two-photon resonancepoint ∆1 = ∆2 the absorption vanishes independent ofthe single-photon detuning.

In the limit ∆2 = 0 and for Ωc γ31, the real partof the susceptibility seen by a probe field varies rapidlyat resonance (∆ ∼ 0). In contrast to the well-known(single) atomic resonances, the enhanced dispersion inthe EIT system is associated with a vanishing absorptioncoefficient, implying that ultra-slow group velocities forlight pulses can be obtained in transparent media whichwill be 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 are carried out using ultra-cold atomic gasesdriven by highly coherent laser fields where Eq. (9) pro-vides a perfect description of the atomic dynamics, it isimportant to analyze the robustness of EIT against thesenon-ideal effects.

In actual atomic systems, the dephasing rate of theforbidden |1〉-|2〉 transition is non-zero due to atomic col-lisions. All the important features of EIT remain observ-able even when γ21 6= 0, provided that the coupling field

9

∆ / γ31

δ / γ

3131∆

/ γ

2

31∆ / γ

1

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

FIG. 9 Contour plot of imaginary part of susceptibilityIm[χ(1)] as function of detunings ∆1 and ∆2 (left) and asfunction of single-photon detuning ∆ and two-photon detun-ing δ in units of γ31. White areas correspond to low ab-sorption, dark to large absorption. The insensitivity of theinduced transparency at δ = 0 on the single-photon detuningis apparent.

Rabi frequency satisfies

|Ωc|2 γ31γ21 . (14)

Figure 10 shows the imaginary part of χ(1)(−ωp, ωp)for γ21 = 0, (Fig. 10a), γ21 = 0.1γ31 (Fig. 10b) andγ21 = 10γ31 (Fig. 10c). In all plots, we take Ωc = 0.5γ31.We first note from Fig. 10b that the absorption profileappears virtually unchanged with respect to the idealcase, provided that the inequality (14) is satisfied. Thetransparency is no longer perfect however, and the resid-ual absorption due to γ21 provides a fundamental limitfor many of the EIT applications. For the γ21 γ31

case depicted in Fig. 10c, the absorption minimum is ab-sent: in this limit a constructive interference enhancesthe absorption coefficient in between the two peaks.

The linear susceptibility given in Eq. 13 depends onlyon the total coherence decay rates, and not on the pop-ulation decay rates of the atomic states. As a conse-quence, the strong quantum interference effects depictedin Figs. 3a-c are observable even in systems where col-lisional broadening dominates over lifetime broadening.This implies that EIT effects can be observed in denseatomic gases, or even solids, provided that there is ametastable transition with a relatively long dephasingrate that satisfies γ21 γ31. EIT in such a collisionallybroadened atomic gas was first reported in 1991 (Fieldet al., 1991).

Amplitude or phase fluctuations of the non-perturbative coupling laser can have a detrimentaleffect on the observability of EIT. Instead of trying topresent a general analysis of the effect of coupling laserlinewidth on linear susceptibility, we discuss severalspecial cases which are of practical importance. Whenthe laser linewidth is due to phase fluctuations thatcan be modelled using the Wiener-Levy phase diffusionmodel, it can be shown that the resulting coupling laserlinewidth directly contributes to the coherence decayrates γ21 and γ32 (Imamoglu, 1991). In contrast, for

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

χ(1)

Im [

]χ(

1)

Im [

]χ(

1)

Im [

]

-1.5 -1 -0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

(a)

(b)

(c)

(ω − ω ) / γ31 31p

FIG. 10 Absorption spectrum for nonvanishing decay of |2〉−|1〉 transition in arbitrary units. (a) γ31 = 1, γ21 = 0, (b)γ31 = 1, γ21 = 0.1, (c) γ31 = 1, γ21 = 10. In all casesΩc = 0.5.

a coupling field with large amplitude fluctuations, theabsorption profile could be smeared out. In applicationswhere a non-perturbative probe field is used, fluctuationsin both laser fields are important. If however, the twolasers are obtained from the same fluctuating lasersource using electro-optic or acousto-optic modulation,then to first order EIT is preserved.

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 broad-ening has no adverse effect on EIT, provided that oneuses co-propagating probe and coupling fields. In othercases, the susceptibility given in Eq. (13) needs to be in-tegrated over a Gaussian density of states correspondingto the Gaussian velocity distribution of the atoms. Qual-itatively, the presence of two-photon Doppler broadeningwith width ∆νdopp > Ωc will wash out the level-splittingand the interference. EIT can be recovered by increasingthe coupling field intensity so as to satisfy Ωc > ∆νdopp;in this limit we obtain

Im[

χ(1)(−ωp, ωp)]

ωp=ω31

∝∆ν2

doppγ31

Ω4c

. (15)

This strong dependence on reciprocal Ωc is a direct con-sequence of robust quantum interference. We emphasizethat in this large coupling field limit, we can considerΩc as the effective detuning from the dressed-state res-onances. Thanks to quantum interference, this effectivedetuning can be used to suppress absorption much more

10

strongly in comparison to non-interfering systems wherethe absorption coefficient is only proportional to the in-verse detuning squared.

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 descriptionof the dynamics of the system based on the evolutionvia a non-Hermitian effective Hamiltonian and quantumjump processes (Carmichael, 1993; Dalibard et al., 1992;Gardiner et al., 1992). Further simplification of the de-scription of the dynamics is obtained when the probabil-ity of a quantum jump process is negligibly small: in thislimit, the non-unitary wave-function evolution inducedby the effective Hamiltonian captures most of the essen-tial physics and can be used to calculate linear and non-linear susceptibilities. This simplification applies for anatomic system where the ground-state (which has near-unity occupancy) is coupled to excited states via weakelectromagnetic fields. In the case of EIT, this is exactlythe case when atoms in state |1〉 are driven by a weakprobe field, independent of the strength of the couplingfield.

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~Γ3σ33/2which reproduces the correct decay terms in the mas-ter equation. Conversely, the physics that this (com-bined) non-Hermitian Hamiltonian fails to capture is there-population of atomic states due to spontaneous emis-sion down to states |1〉 and |2〉. Since we can not describepure dephasing processes using a wave-function model,we can use population decay of state |2〉 (at rate Γ2) tomimic dephasing: we therefore introduce an additionalanti-Hermitian decay term to obtain an effective Hamil-tonian in the rotating frame

Heff = −~

2[Ωpσ31 + Ωcσ32 + h.c.] + ~(∆− i

2Γ3)σ33

+~(δ − i

2Γ2)σ22 . (16)

We reemphasize that for Γ2 = 0 and δ = 0, one of theeigenstates of Heff is

|a0〉 =1

Ω2c + Ω2

p

[

Ωc|1〉 − Ωp|2〉]

, (17)

with an eigenenergy ~ω0 = 0, independent of the values ofthe Rabi frequencies and the single-photon detuning. Wehere have assumed real-valued Rabi-frequencies, which isalways possible as long as propagation effects are notconsidered. In the time-dependent response, we will seethat the emergence of EIT can be understood as a resultof the system moving adiabatically from bare state |1〉 inthe absence of fields to |a0〉 in the presence 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 δ = 0move 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 waysof reaching the dissipative state |3〉; either directly fromstate |1〉 or via the path |1〉 − |3〉 − |2〉 − |3〉. The latterhas comparable amplitude to the former since Ωc is non-perturbative: as a result, the amplitudes for these pathscan exhibit 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 ba-sis is obtained by applying the unitary transformation onthe (bare-basis) state-vector |Ψ(t)〉

|Ψd(t)〉 = U |Ψ(t)〉 , (18)

where

U =

1 0 00 cos φ sin φ0 − sin φ cosφ

. (19)

Here φ is defined by tan(2φ) = Ωc/∆2. Application ofthis transformation to Heff yields

Heff = −~

2

0 Ωp sin φ Ωp cos φΩp sinφ −2∆− + iΓ− − i

2Γ+−Ωp cos φ − i

2Γ+− −2∆+ + iΓ+

.(20)

Here, ∆− and ∆+ denote the detunings of the probe fieldfrom the dressed resonances, Γ− = (Γ2 cos2 φ+Γ3 sin2 φ),Γ+ = (Γ2 sin2 φ+Γ3 cos2 φ), and Γ+− = (Γ2−Γ3) sin 2φ.We can calculate the absorption rate of the probe photonsby determining the eigenvalues of Heff in the limit ofperturbative Ωp. The imaginary part of the eigenvalue(Im(−λ1)) that corresponds to |1〉 as Ωp → 0 for Γ2 = 0is

Im(λ1) =−Ω2

pΓ3(∆+ sin2 φ + ∆− cos2 φ)2

4∆2−∆2

+ + (∆+Γ3 sin2 φ + ∆−Γ3 cos2 φ)2

∝ Im[χ(1)(−ωp, ωp)] . (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 asthe one describing Fano interference of two autoioniz-ing states (Fano, 1961; Harris, 1989). It is the presenceof imaginary off-diagonal terms κ23 = κ32 = 0.5(Γ3 −Γ2) sin 2φ that gives rise to quantum interference in ab-sorption. When Γ2 = 0, we find that κ23 =

√Γ2dΓ3d,

11

where Γ2d = Γ2 cos2 φ + Γ3 sin2 φ and Γ3d = Γ2 sin2 φ +Γ3 cos2 φ. In this case, we find that the interference inabsorption is destructive when the probe field is tuned inbetween the two dressed-state resonances, with a per-fect cancellation for δ = 0. When Γ3 = 0, we haveκ23 = −√Γ2dΓ3d ; in this case, the quantum interfer-ence is constructive for probe fields tuned in betweenthe two dressed-states and destructive otherwise. Eventhough absorption is suppressed for |δ| Ωc, we neverobtain complete transparency or a steep dispersion inthis (Γ3 = 0) case. Finally, for Γ2 = Γ3, the imaginaryoff-diagonal terms vanish; in this limit, there are no in-terference effects. These results were already depicted inFig. 10.

The complete elimination of absorption for δ = 0 inthe limit Γ2 = 0 can therefore be understood by invokingthe arguments used by Fano (Fano, 1961); let’s assumethat the decay of state |3〉 is due to spontaneous emissioninto state |f〉. For an atom initially in state |1〉, a probephoton absorption event results in the generation of a(Raman) scattered photon and leaves the atom in state|f〉. To reach this final state, the atom can virtually ex-cite either of the two dressed-states; since excitations arevirtual, the probability amplitudes corresponding to eachhave to 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 lat-ter are by definition calculated in the limit of near unityground-state occupancy, Heff of Eq. (16) provides a nat-ural starting point. In this section, we will focus on sumfrequency generation to highlight the different role quan-tum interference effects play in nonlinear susceptibilitiesas compared to linear ones. To this end, we introduce anadditional driving term Ωeff that couples states |1〉 and|2〉 coherently. Since we have assumed this transition tobe dipole-forbidden, we envision that this effective cou-pling is mediated by two laser fields (at frequencies ωa

and ωb) and a set of intermediate (non-resonant) states|i〉

Ωeff =∑

i

ΩaΩb

2

[

1

ωi1 − ωa+

1

ωi1 − ωb

]

. (22)

Here Ωa = µ1iEa/~ and Ωb = µi2Eb/~.We can now rewrite the effective Hamiltonian for sum

frequency generation as

Hsfgeff = −~

2

[

Ωeff σ21 + Ωpσ31 + Ωcσ32 + h.c.]

+

+~(∆− i

2Γ3)σ33 + ~(δ − i

2Γ2)σ22 . (23)

To obtain the nonlinear susceptibility, we assume bothΩeff and Ωp to be perturbative. The generated polar-

ization at ωp is given by

P (t) = %µ13a3(t) exp(−iωpt) + c.c., (24)

where a3(t) is the probability amplitude for finding agiven atom in state |3〉: this amplitude is calculated using

Hsfgeff and has two contributions: the first contribution

proportional to Ωp simply gives us the same linear sus-ceptibility we calculated in the previous subsection usingthe master equation approach. The second contributionproportional to ΩeffΩc ∝ ΩaΩbΩc gives us the nonlinearsusceptibility

χ(3)(−ωp, ωa, ωb, ωc) =

=2µ23µ31%

3ε0~3

1

Ω2c + (γ31 + i2∆)(γ21 + i2δ)

×

×∑

i

µ1iµi2

[

1

ωi1 − ωa+

1

ωi1 − ωb

]

. (25)

This third-order nonlinear susceptibility that describessum-frequency generation highlights one of the most im-portant properties of EIT; namely, that the nonlinear sus-ceptibilities need not vanish when the linear susceptibil-ity vanishes due to destructive quantum interference. Infact, for Ω2

c < Γ23/2, we notice that χ(3)(−ωp, ωa, ωb, ωc)

has a maximum at δ = 0 where χ(1)(−ωp, ωp) = 0.

It is perhaps illustrative to compare χ(3) obtained forthe EIT system with the usual third-order nonlinear sus-ceptibility for non-resonant sum-frequency generation.As expected, the expressions in the two limits have thesame form, with the exception of detunings: the two andthree photon detunings in the case of non-resonant sus-ceptibility (1/(∆ω2∆ω3)) is replaced by 1/(Ω2

c + γ21γ31)in the case of EIT with δ = ∆ = 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 χ(1)(−ωp, ωp)itself that governs the spectroscopic properties but thecollective response of the entire medium characterizedby the amplitude transfer function

T (ωp, z) = exp

ikzχ(1)(−ωp, ωp)/2

(26)

with k = 2π/λ the resonant wavenumber and z themedium 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 resonant con-trol field into (26), one finds that the transmittivity near

12

resonance δ = 0 is a Gaussian with width

∆ωtrans =Ω2

c√Γ31γ31

1√%σz

(%σz 1). (27)

Here σ = 3λ2/2π is the absorption cross section of anatom and % the atom number density. One recognizes apower broadening of the resonance with increasing Rabi-frequency Ωc or intensity of the coupling field which canhowever be partially compensated by increasing the den-sity length product %σz. This important property is il-lustrated in Fig. 11, where a finite decay rate γ21 of thecoherence between the two lower levels was taken intoaccount as well.

The perfect linear dispersion near the EIT resonance,expressed by the linear dependence of the real part of χon δ can be used to detect shifts of the EIT resonance byphase sensitive detection. Defining the frequency vari-ation which leads to an accumulated phase shift of 2πafter the medium as dispersive width ∆ωdis, one finds aneven stronger reduction with the density-length product

∆ωdisp = 2πΩ2

c

Γ31

1

%σz(%σz 1) (28)

This effect which is also illustrated in Fig. 11 plays animportant role for applications of EIT in optical detectionof magnetic fields (Fleischhauer and Scully, 1994; Scullyand Fleischhauer, 1992).

T

T

31

31

-0.1 -0.05 0 0.05 0.1

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

∆ω dis

∆ω tr

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

−4 −2 0 2 4

1

=0.1

5

1050

δ / γ

δ / γ

−0.4

0

0.4

−0.1 −0.05 0.050 0.1

= 500 Lρ σ

Lρ σ

FIG. 11 top: Transmission spectrum T (δ) of an EIT mediumfor different density length products %σz. bottom: Illustrationof dispersive width.

The narrowing of the transparency and dispersivewidth of an EIT medium with increasing density was ex-perimentally observed by Lukin et al. (1997) in a forwardfour-wave mixing experiment.

D. Dark-state preparation and Raman adiabatic passage

In the preceeding sections we have discussed EIT un-der steady-state conditions. We have seen that atomsin a particular superposition state, the dark state |a0〉,can be decoupled from the interaction. If there is two-photon resonance between atoms and fields, this stateis stationary. Properly prepared atoms remain in thisstate and render the medium transparent. An impor-tant characteristics of EIT, which distinguishes it fromcoherent population trapping is the preparation whichin EIT happens automatically through Raman adiabaticpassage. This process naturally becomes important if anoptically thick medium is considered.

The formation of a dark state can either happen viaoptical pumping, i.e. by an incoherent process, or viaa coherent preparation scheme. The first mechanism isobviously required if the atomic ensemble is initially ina mixed state. To achieve electromagnetically inducedtransparency in optically thick media, optical pumpingmay however not be well suited. Here radiation trap-ping can lead to a dramatic slow-down of the pumpingprocess. If the dark state has a finite lifetime, the re-sulting effective pump rate may become even too slowto achieve transparency at all (Fleischhauer, 1999). Fur-thermore the preparation by optical pumping is alwaysassociated with a non-recoverable loss of photons fromthe probe field, which could be fatal if the probe field isa few-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 conditions thedark state is in this case prepared by a coherent mech-anism which does not involve spontaneous emission atall: the process which achieves this is called StimulatedRaman Adiabatic Passage (STIRAP) and was discoveredand developed by Bergmann et al. (Gaubatz et al., 1990;Kuklinski et al., 1989) after some earlier related but inde-pendent discoveries of Eberly and coworkers (Oreg et al.,1984). In the STIRAP process relevant here, the atomsare returned to the initial state after the interaction withthe pulsed probe field, thus conserving the number ofphotons in the pulse. The STIRAP technique is by nowa widely used method in atomic and molecular physicsfor the preparation of specific quantum states and con-trolling chemical reactions. It has also a large range ofapplications in quantum optics and matter-wave interfer-ometry. For recent reviews on this subject see Bergmannet al. (1998) and Vitanov 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 rotating

13

wave approximation and in a proper rotating frame is

H(t) = −~

2

0 0 Ωp(t)0 0 Ωc(t)

Ωp(t) Ωc(t) −2∆

. (29)

We assume that the relative phase between the two fieldsis arbitrary but constant and thus both Ωp and Ωc canbe taken real. This also implies that the two fields are as-sumed to be transform limited pulses, or more preciselythat the beat-note between the two is transform limited.The essence of STIRAP is that a proper adiabatic changeof Ωp(t) and Ωc(t) allows a complete transfer of popula-tion from the initial state |1〉 to the target state |2〉 orvice versa without populating the intermediate excitedstate |3〉 and thus without spontaneous 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) withthe corresponding eigenvalues, given in Eq. (7). The in-stantaneous eigenstate

|a0(t)〉 = cos θ(t)|1〉 − sin θ(t)|2〉, tan θ =Ωp(t)

Ωc(t)(30)

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

The important property of the adiabatic dark state isthat depending on the value of the mixing angle θ(t) itcan coincide with either one of the bare states |1〉 and|2〉 as well as any intermediate superposition with fixedrelative phase. If θ = 0, which corresponds to a couplingfield much stronger than the pump field, |a0〉 = |1〉. Like-wise, if θ = π/2 we have |a0〉 = −|2〉. Thus if all atomsare initially (i.e. for t = −∞) prepared in state |1〉 and|Ωc(−∞)| |Ωp(−∞)|, i.e. if the coupling pulse is ap-plied first, the system is in the dark-state at t = −∞. Ifthe mixing angle θ(t) is now rotated from 0 to π/2 thedark-state changes from |1〉 to−|2〉. If this rotation is suf-ficiently slow, the adiabatic theorem guarantees that thestate vector of the system follows this evolution. In thisway population can be transferred with 100% efficiencyand without spontaneous emission losses. The only re-quirements besides the correct initial conditions are thatthe time rate of change of the fields is sufficiently slow,i.e. adiabatic, and that the relative phase between thefields is at least approximately 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 findH = R−1HR−R−1R, or explicitly,

H = −~

ω+(t) i2 θ 0

− i2 θ 0 − i

2 θ

0 i2 θ ω−(t)

, (31)

where an overdot means a time derivative. Adiabaticevolution occurs if the off-diagonal coupling, proportionalto θ is sufficiently small compared to the eigenvalues |ω±|(7)

∣

∣

∣

⟨ d

dta0(t)

∣

∣

∣a±(t)

⟩∣

∣

∣ |ω±(t)| (32)

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

Ω(t) ≡√

Ω2p(t) + Ω2

c(t) |θ(t)|. (33)

In lowest order of the non-adiabatic coupling θ and inthe limit |θ| Γ3, characteristic for long pulses, the

excited state population is given by |θ|2/Ω2(t), such thatspontaneous emission leads to an instantaneous loss rate(Fleischhauer and Manka, 1996; Vitanov and Stenholm,1997)

Γeff = Γ3|θ(t)|2Ω2(t)

. (34)

If the pump and coupling pulses are not transform lim-ited in such a way that the beat-note phase changes intime (Dalton and Knight, 1982), there will be additionalnon-adiabatic losses due to this effect.

If all atoms are initially in state |1〉, the coupling field(Ωc) is applied before the probe field (Ωp), 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 preparation en-ergy from the probe field and transfers it into the mediumand the coupling field. This preparation energy was firstdetermined by Harris and Luo (1995). It is given by thenumber of atoms in the probe path N and the stationaryvalues of the Rabi-frequencies

Eprep = ~ωpN|Ωp|2|Ωc|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 energyis transferred back from the medium and the couplingfield to the probe field. In this case there is no net loss ofenergy from the probe field but only a temporary trans-fer. As we will see later on in Section IV this temporarytransfer of energy has some interesting consequences forthe propagation of the fields.

E. EIT inside optical resonators

The large linear dispersion associated with EIT inan optically thick medium can substantially affect the

14

properties of a resonator system, in particular since themedium dispersion can easily exceed that of an emptycavity. It is therefore natural to investigate the possibil-ity of having an EIT medium inside a cavity, such thatone of the high-finesse modes replaces the probe field andcouples the ground-state |1〉 to state |3〉.

If an empty-cavity resonance ωc0 is sufficiently close tothe frequency ω0 of electromagnetically induced trans-parency, the presence of the medium will lead to avery strong pulling of the resonance frequency ωc of thecombined atom–cavity system towards this value (Lukinet al., 1998a). At the same time the resonance width issubstantially reduced (Lukin et al., 1998a). To see thislet us consider the response function of a resonator con-taing an EIT medium, i.e. the ratio of circulating toinput intensity at frequency ω. For simplicity we assumea uni-directional ring resonator of round-trip length Lwith a single in-out coupling mirror. Then

S(ω) ≡ Icirc(ω)

Iin(ω)=

t2

1 + r2κ2 − 2rκ cos [Φ(ω)], (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. Φ(ω) = ω/c

(

L + lRe[χ(1)])

is the phase shift and κ = exp−ωlIm[χ(1)]/c themedium absorption per round trip L. l is the lengthof the EIT medium. Close to two-photon resonanceRe[χ(1)] ≈ dn

dω (ω − ω0). This yields for the resonanceof the cavity + medium system

ωc =1

1 + ξωc0 +

ξ

1 + ξω0, where ξ = ω0

dn

dω

l

L.(37)

One recognizes a strong frequency pulling towards theatomic EIT resonance since the dispersion ω0dn/dω canbe rather large. Likewise one finds from (36) that thewidth of the cavity resonance is changed according to

∆ωc

∆ωc0=

1− rκ√κ(1− r)

1

1 + ξ. (38)

The first term accounts for a small enhancement of thecavity width owing to the additional losses induced by themedium. Much more important is however 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 systemas well as on the coupling laser, frequency pulling andline-narrowing are relative to the coupling laser. Theseeffects have been observed experimentally in (Mulleret al., 1997). Figure 12 shows the transmission spectrumof an empty cavity and a cavity with an EIT medium.Applications for difference-frequency locking and stabi-lization have been suggested and the reduction of thebeat-note linewidth in a laser system below the Shawlow-Townes limit predicted (Lukin et al., 1998a).

MHz below the two-photon resonance and scanned 10 MHzover the resonance. The transmission of the resonator withand without atomic beam is shown in Fig. 5~a!. The line-width with atomic beam was 140 kHz ~HWHM! @Fig. 5~b!#,which is 50 times smaller than the linewidth without atomicbeam. The transmitted intensity is half of the transmittedintensity of the empty resonator. This was caused by addi-tional losses on resonance due to background gas. Absorp-tion in the background gas was mainly caused by the 6s1/2F53↔6p3/2 F852 transition, which could not be coupledto the 6s1/2 F54 level.

To compare the reduction of the linewidth with the reduc-tion predicted from theory, we had to take into account satu-ration broadening, power buildup in the cavity, and the ratiobetween the interaction length and the length of the resona-tor. The power of the pump field in this scan was 220 mWand the power of the probe field in front of the resonator 450mW. Due to the coupling efficiency of 30% for the emptyresonator and 15% for the resonator with atomic beam andthe power buildup in the resonator, the power of the probefield in the resonator without atomic beam on resonance wasin the range of 4.5 mW and with atomic beam 2.25 mW.Consequently the effective Rabi frequency VR is determinedmainly by the probe field when on resonance and by thepump field when off resonance. This leads to a strong satu-ration of the dispersion and the absorption on resonance.Therefore the edges of the transmission profile @Fig. 5~b!# aresteeper than the edges of a typical Airy function. The right-hand-side curves in Fig. 5 show the calculated absorption ~c!and the calculated additional round-trip phase shift ~d! ~dispersion! for these parameters. In both curves the strongsaturation is easily seen. For these curves we have calculatedthe resonator internal intensity of the probe field and its in-fluence on the absorptive and dispersive properties of CPT.In our calculations we took into account the residual Doppler

broadening of the atomic beam, the power of the pump field,the power of the probe field including the buildup, the pa-rameters of the resonator, and additional absorption due tothe 6s1/2 F53↔6p3/2 F852 transition, which depends onlyon the power of the probe field. We did not take into accountthe possible Zeeman splitting of the lower levels, whichcould be caused by imperfectly compensated magnetic fieldsand the Gaussian mode of the fields. We have assumed aconstant intensity across the beam profile.

Next we increased the intensity of the pump field to 300mW and reduced the intensity of the probe field to 150 mW.The linewidth for these parameters was again 140 kHz, butnow the line profile was more similar to an Airy function~see Fig. 6!. Curve ~a! is a measured transmission profile,while curve ~b! is calculated. The right-hand curves show thecalculated absorption ~c! and the additional round-trip phaseshift ~d! ~ dispersion! induced by the medium for theseparameters. The resonator internal power of the probe fieldon resonance was now about 450 mW, which is similar to thepower of the pump field. Therefore the saturation of the dis-persion and the absorption due to the power buildup is not soimportant. The transmitted intensity is now reduced to 30%of the transmitted intensity of the empty resonator. This caneasily be explained by the higher absorption due to the back-ground gas. This absorption rate only scales with the lowerintensity of the probe field and cannot be saturated by thehigher intensity of the pump field.

The line profile could be varied from a very sharp profilewith steep edges by using more power for the probe field andless for the pump field to a profile which looks more like asmooth Airy function by using more power for the pumpfield and less power for the probe field. This behavior couldeasily be understood by taking into account the powerbuildup in the cavity and the saturation behavior of CPT.

So far we have only discussed how the transmission sig-nal varies with the laser frequency in the case when the reso-

FIG. 5. ~a! Measured transmission of the resonator withoutatomic beam ~upper curve! and with. ~b! Measured transmissionwith atomic beam. The linewidth without atomic beam was 7.5MHz ~HWHM! and with atomic beam 140 kHz ~HWHM!. Theintensities are Ipump5220 mW, Iprobe5450 mW. ~c! Calculatedround-trip absorption a and ~d! additional round-trip phase shift df~ dispersion! induced by the medium. Both curves are stronglybleached out by the power buildup of the probe field. The emptyresonator is resonant at the two-photon resonance.

FIG. 6. ~a! Measured transmission of the resonator with atomicbeam. ~b! Calculated transmission profile. The linewidth was again140 kHz ~HWHM!. ~c! Calculated round-trip absorption a and ~d!

additional round-trip phase shift df ~ dispersion! induced by themedium. With these intensities (Ipump5300 mW, Iprobe5150 mW)both curves are now not so strongly bleached out by the powerbuildup of the probe field. The empty resonator is resonant at thetwo-photon resonance.

2388 56MULLER, MULLER, WICHT, RINKLEFF, AND DANZMANN

Tra

nsm

issi

onT

rans

mis

sion

∆ [MHz]

FIG. 12 Experimental demonstration of linewidth narrow-ing from (Muller et al., 1997): (a) Transmission spectrumof empty cavity and with EIT medium, (b) magnification ofTransmission spectrum with EIT.

F. Enhancement of refractive index, magnetometry and

lasing 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 laserfields to control the refractive index has created muchattention. A suggestion for controlling phase-matchingin non-linear mixing through the action of an additionalfield was made by Tewari and Agarwal (1986), but itis since the advent of EIT that these ideas have beenextensively studied. Harris treated the refractive proper-ties of an EIT medium theoretically (Harris et al., 1992).In pulsed laser experiments the vanishing of the suscep-tibility at resonance leads to the elimination of distor-tion for pulses propagating through a very optically thickmedium at resonance (Jain et al., 1995). Related sugges-tions by Harris for the use of strong off–resonant fieldsto control the refractive index (Harris, 1994b) have beenimportant in the implementation of maximal coherencefrequency mixing schemes with off-resonant lasers.

Extensive theoretical work was carried out by Scullyand co-workers (Fleischhauer et al., 1992a,b; Rathe et al.,1993; Scully, 1991; Scully and Zhu, 1992) and others(Wilson-Gordon and Friedman, 1992) on the use of res-onant cw fields for modification of the refractive index.The experiments of several workers have extensively stud-ied refractive index modifications in this cw limit (Mose-

15

ley et al., 1995; Xiao et al., 1995).Working in Doppler free conditions with the narrow

bandwidths attainable through use of cw lasers it is pos-sible to observe very steep EIT induced dispersion pro-files. This led to the suggestion of implementation ofvery sensitive interferometers and magnetometers sincetiny variations in frequency couple to large changes inrefractive index (Affolderbach et al., 2002; Budker et al.,1999, 1998; Fleischhauer and Scully, 1994; Nagel et al.,1998; Sautenkov et al., 2000; Scully and Fleischhauer,1992; Stahler et al., 2001). In the magnetometer schemesthis greatly increases the sensitivity of the measurementof the Zeeman shifts induced by a magnetic field.

As mentioned earlier, lasing without inversion is closelyrelated to the interference important in EIT. There haveby now been a number of experimental verifications ofthis concept with the actual realization of laser actionwithout population inversion in the visible spectral rangeincluding several reports of inversionless amplification(Fry et al., 1993; Gao et al., 1992) and (Nottleman et al.,1993) as well as actual demonstrations of laser oscilla-tion without population inversion (Zibrov et al., 1995)and (Padmabandu et al., 1996). An application of LWIto short wave-length lasing is yet to be shown.

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 attentionto propagation effects of pulses. Although the suscepti-bility discussed in the last section contains already allnecessary information, it is worthwhile to devote to thisissue a separate section. The more so as the almost per-fect transparency at certain frequencies, characteristic forEIT, allows pulse propagation in otherwise optically thickmedia. Here the action of the medium on the light pulsesis – apart from the single frequency for which the mediumis transparent – quite substantial, and a number of inter-esting propagation phenomena are encountered. Theseeffects have often a simple physical origin, but their sizeand their special properties make them very importantfor a variety of applications.

A. Linear response slow and ultra-slow 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 character-istic feature of the real part of the susceptibility spectrumis a linear dependence on the frequency close to the two-photon resonance δ = 0. For a negligible decay of the|1〉 − |2〉 coherence one finds

Re[χ(1)] = η2Γ31δ

Ω2c

+O(δ2), (39)

where η = 34π2 %λ3 is the normalized density, and λ the

transition wavelength in vacuum. Since the linear disper-sion dn/dωp of the refractive index n =

√

1 + Re[χ] ispositive, EIT is associated with a reduction of the groupvelocity according to

vgr ≡dωp

dkp

∣

∣

∣

∣

δ=0

=c

n + ωpdndωp

, (40)

which has first been pointed out by Harris et al. (1992).At the same time the index of refraction of the idealthree-level medium is unity and thus the phase velocityof the probe field is just the vacuum speed of light

vph ≡ωp

kp

∣

∣

∣

∣

δ=0

=c

n= c. (41)

An important property of the EIT system is that thesecond order term in (39) vanishes exactly if there is alsosingle-photon resonance ∆ = 0 of the probe field. As aconsequence there is no group velocity dispersion, i.e. nowave-packet spreading. Using (39) yields at two-photonresonance

vgr =c

1 + ngrwith ngr = %σc

Γ31

Ω2c

, (42)

where ηk = σ% was used. The reduced group velocitygives rize to a group delay in a medium of length L

τd = L

(

1

vgr− 1

c

)

=Lngr

c= %σL

Γ31

Ω2c

. (43)

It should be noted that under non-ideal conditions, i.e.if the dark-resonance is not perfectly stable, e.g. dueto collisional dephasing of the |1〉 − |2〉 coherence or fastphase fluctuations in the beat-note between coupling andprobe fields, the denominator in the expression for ngr

needs to be replaced by Ω2c + γ31γ21, where γ31 and γ21

are, 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 δ = 0, relatively highatom densities % and low intensities of the coupling fieldIc ∼ Ω2

c can be used. Thus the group index ngr canbe rather large compared to unity and extremely smallgroup velocities are possible. In the first experiments byHarris and coworkers 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 havebeen performed by Xiao et al. (1995) and Lukin et al.

(1997). The slow-down of light by EIT attracted a lotof attention when Hau and collaborators observed thespectacular reduction of the group velocity to 17 m/sin a Bose condensate of Na atoms, corresponding to apulse slow down of seven orders of magnitude (Hau et al.,

16

1999). Similarly small values where later obtained in abuffer-gas cell of hot Rb atoms by Kash et al. (1999)and by Budker et al. (1999). More recently a substantialslow-down of the group velocity was observed also in thesolid-state by Turukhin et al. (2002).

The loss-less slow-down of a light pulse in a mediumis associated with a number of important effects: Whena pulse enters such a medium, it becomes spatially com-pressed in the propagation direction by the ratio of groupvelocity to the speed of light outside the medium (Harrisand Hau, 1999). This compression emerges since whenthe pulse enters the sample its front end propagates muchslower than its back end. At the same time however, theelectrical field strength remains the same. The reversehappens when the pulse leaves the sample. In the caseof the experiment of Hau et al. (1999) the spatial com-pression was from a kilometer to a sub-millimeter scale!Since the refractive index is unity at two-photon reso-nance, reflection from the medium boundary is usuallynegligible as long as the pulse spectrum is not too large(Kozlov et al., 2002).

Although in the absence of losses the time-integratedphoton flux through any plane inside the medium isconstant, the total number of probe photons inside themedium is reduced by the factor vgr/c due to spatial com-pression. Thus photons or electromagnetic energy mustbe temporarily stored in the combined system of atomsand coupling field. It should be noted that the notionof a group velocity of light is still used even for vgr cwhere only a tiny fraction of the original pulse energyremains electromagnetic.

It is instructive to consider slow-light propagation fromthe point of view of the atoms. From this perspective thephysical mechanism for the temporary transfer of excita-tions to and from the medium can be understood as stim-ulated Raman adiabatic return. Before the probe pulseinteracts with the three-level atoms a cw coupling fieldputs all atoms into state |1〉 by optical pumping. In thislimit state |1〉 is identical to the dark-state. When thefront end of the probe pulse arrives at an atom, the dark-state makes a small rotation from state |1〉 to a superposi-tion between |1〉 and |2〉. In this process energy is takenout of the probe pulse and transferred into the atomsand the coupling field. When the probe pulse reachesits maximum, the rotation of the dark-state stops andis reversed. Thus energy is returned to the probe pulseat its back end. The excursion of the dark-state awayfrom state |1〉 and hence the characteristic time of theadiabatic return process depends on the strength of thecoupling field. The weaker the coupling field the largerthe excursion and thus the larger the pulse delay. To putthis picture into a mathematical formalism one can em-ploy a quasi-particle picture first introduced by Mazetsand Matisov (1996) and independently by Fleischhauerand Lukin (2000) which first applied this concept to pulsepropagation. One defines dark (Ψ) and bright polariton(Φ) fields according to

Ψ(z, t) = cos ϑ Ep(z, t)− sin ϑ√

% ρ21(z, t) ei∆kz,(44)

Φ(z, t) = sinϑ Ep(z, t) + cosϑ√

% ρ21(z, t) ei∆kz,(45)

with the mixing angle determined by the group index

tan2 ϑ =%σcΓ31

Ω2c

= ngr. (46)

Ep is the normalized, slowly-varying probe field strength,

Ep = Ep√

(~ω)/(2ε0), with ω being the corresponding

carrier frequency. ∆k = k‖c − kp, where kp is the wave-

number of the probe field and k‖c the projection of the

coupling field wave-vector to the z axis. ρ21 is the single-atom off-diagonal density matrix element between thetwo lower states.

In the limit of linear response, i.e. in first order of per-turbation in Ep, and under conditions of EIT, i.e. for neg-ligible absorption, the set of one-dimensional Maxwell-Bloch equations can be solved (Fleischhauer and Lukin,2000, 2002). One finds that only the dark-polariton fieldΨ is excited, i.e. Φ ≡ 0 and thus

Ep(z, t) = cos ϑΨ(z, t), (47)

ρ21(z, t) = − sin ϑΨ(z, t)e−i∆kz

√%

. (48)

Furthermore under conditions of single-photon resonanceΨ obeys the simple shortened wave equation

[

∂

∂t+ c cos2 ϑ

∂

∂z

]

Ψ(z, t) = 0, (49)

which describes a form-stable propagation with velocity

vgr = c cos2 ϑ. (50)

The slow-down of the group velocity of light in an EITmedium can now be given a very simple interpretation:EIT corresponds to the loss-less and form-stable prop-agation of dark-state polaritons (DSPs). These quasi-particles are a coherent mixture between electromagneticand atomic spin excitations, the latter referring to an ex-citation of the |1〉 − |2〉 coherence. The admixture of thecomponents described by the mixing angle ϑ depends onthe strength of the coupling field as well as the density ofatoms and determines the propagation velocity. In thelimit ϑ → 0, corresponding to a strong coupling field,the DSP is almost entirely of electromagnetic nature andthe propagation velocity is close to the vacuum speed oflight c. In the opposite limit, ϑ→ π/2, the DSP has thecharacter of a spin excitation and its propagation veloc-ity is close to zero. The concept of dark- and bright-statepolaritons can easily be extended to a quantized descrip-tion of the probe field (Fleischhauer and Lukin, 2000,2002) as well as quantized matter fields (Juzeliunas andCarmichael, 2002) in which case the polaritons obey ap-proximately Bose commutation relations. This extensionis of relevance for applications in quantum informationprocessing and nonlinear quantum optics discussed lateron.

17

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 propaga-tion direction of the light pulse, Galilean transformationrules predict that the group velocity (50) will be modifiedaccording to

vgr = c cos2 ϑ + v sin2 ϑ. (51)

This expression can also be obtained from (40) if onetakes into account that due to the Doppler effect thesusceptibility or the index of refraction becomes k-dependent (spatial dispersion). It is interesting to notethat vgr can now be zero or even negative for ϑ 6= π/2if the atoms move against the propagation of light. Infact since extremely large values of the group index canbe achieved in vapor cells, already the thermal motion ofatoms at room temperatures can have a significant effecton the effective group velocity. Making use of this effectKocharovskaya et al. suggested to “freeze” a light pulsein a Doppler-broadened medium, where a specific velocityclass is selectively excited (Kocharovskaya et al., 2001).When the group velocity is decreased the spectral band-width of linear dispersion and transparency decreases. Inthe limit of zero group velocity the dispersion is infinite,which can clearly be supported only over a vanishinglynarrow frequency band. Thus in practice absorption andreflection will prevent the full stop of a light pulse withthis method, unless a time-dependent coupling field isused which leads to a dynamical narrowing of the pulsespectrum (see below) or the atoms are accelerated. If theflow of atoms is not parallel to the propagation of light,excitation carried by the atoms can be transported in aperpendicular direction (Juzeliunas et al., 2003), whichcan be observed also in vapor cells where the atoms un-dergo a diffusive motion (Zibrov et 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 nonlin-ear medium, which is important in enhancing the effi-ciency of such processes (Harris and Hau, 1999; Lukinand Imamoglu, 2001; Lukin et al., 2000b). It should benoted in this context that although the number of pho-tons in the medium is reduced by the ratio of the groupvelocity to the vacuum speed of light, the electric field iscontinuous at the boundaries and thus the nonlinear po-larization is not reduced due to the compression. Theseapplications will be discussed in more detail in Secs. Vand VI. Making use of the substantial pulse deformationat the boundary of an EIT medium is also of interestfor the storage of information contained in long pulses,which in this way can be compressed to a very small spa-tial volume.

As noticed by Harris, the pulse compression is associ-ated with a substantial enhancement of the spatial fieldgradient in the propagation direction (Harris, 2000) andthus pondermotive dipole forces on atoms, resulting fromoff-resonant couplings to the light pulse, are substantially

amplified. If I0(z) denotes the probe intensity at the in-put of the medium one finds an enhanced force by theratio 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 can alsobe used for the preparation and detection of coherentmatter wave phenomena in ultra-cold quantum gases.E.g. EIT has been suggested as a probe for the diffusionof the relative phase in a two-component Bose Einsteincondensate (Ruostekoski and Walls, 1999a,b). Stoppingof a light pulse at a spatial discontinuity of the controlfield was used to create strongly localized shock-waves ina BEC and study their dynamics by Dutton et al. (2001).

Finally interesting effects have been predicted by Leon-hard and coworkers when the propagation of ultra-slowlight is considered in non-uniformly moving media. Us-ing an effective classical field theory for slow light wherelosses are neglected they predicted an Aharonov-Bohmtype phase shift in a rotating EIT medium with a vor-tex flow (Leonhardt and Piwnicki, 1999). Leonhard et

al. also pointed out an interesting analogy between slow-light in a medium with a vortex flow and general rel-ativistic equations for a black hole (Leonhard and Pi-wnicki, 2000). This may allow for laboratory studies ofsome general relativistic effects when adding an inwardflow to the vortex (Leonhardt and Piwnicki, 2000; Visser,2000). The practical observation of an optical analogy ofthe event horizon of a black hole will however most likelybe prevented by absorption and reflection effects.

To assess the potentials of slow light it is importantto consider its limitations. A convenient figure of meritfor this is not the achievable group velocity itself, butthe ratio of achievable delay time τd of a pulse in anEIT medium to its pulse length τp. One upper limit forthe delay time is given by probe absorption due to thefinite lifetime of the dark resonance. Furthermore for apulsed probe field, i.e. for a probe field with a finite spec-tral width the absorption of the non-resonant frequencycomponents is nonzero even under ideal conditions of aninfinitely long-lived dark state. Thus in order for theabsorption to be negligible, firstly the decay time of the|1〉 − |2〉 transition must be very small and secondly thespectral width of the pulse ∆ωp has to be much smallerthan the transparency width ∆ωtrans:

∆ωp ∆ωtrans =Ω2

c√Γ31γ31

1√%σL

. (53)

It is instructive to express ∆ωtrans in this condition interms of the group delay τd = ngrL/c

∆ωtrans =√

%σL1

τd

√

Γ31

γ31. (54)

18

This discussion shows that it is not possible to bring apulse to a complete stop by using EIT with a station-ary coupling field, since in this case the transparencywidth would vanish leading to a complete absorption ofthe pulse. The same argument holds if one tries to stopan ultra-slow pulse by means of atomic motion againstthe flow of light. Making use of ∆ωpτp ≥ 1 one finds anupper limit for the ratio of group delay to pulse length(Harris and Hau, 1999)

τd

τp√

%σL. (55)

The product %σL is the opacity of the medium in theabsence of EIT, i.e. exp−%σL is the transmission co-efficient at bare atomic resonance. Thus a noticeabletime delay of a pulse by EIT requires an optically thickmedium with %σL > 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 possibil-ity to transfer spatial excitation distributions of atomicensembles to light pulses has been pointed out before byCszeznegi and Grobe (1997). In the following the “stop-ping” and “re-acceleration” of a light pulse and its poten-tial applicatons will be discussed in more detail. At thispoint a word of caution is needed however: The notion“stopping of light” should not be taken literally. As men-tioned before, the reduction of the propagation velocityof light in a loss-less, passive medium is always associatedwith a temporary transfer of its energy to the medium.In the extreme limit of zero velocity relative to the sta-tionary medium no electromagnetic excitation is left atall. Nevertheless the notion of a vanishing group velocityof light has here the same justification as the notion of agroup velocity in the case of ultra-slow pulse propagationwhere also only a tiny fraction of the original excitationremains in the form of photons.

A key conceptual advance with respect to the abilityto bring a light pulse to a complete stop occured when itwas realized in (Fleischhauer and Lukin, 2000) that thebandwidth limitation (53) of EIT can be overcome, if thegroup velocity is adiabatically reduced to zero in time.This can be achieved for example by reducing the Rabi-frequency of the drive field. In this case the spectrum ofthe probe pulse narrows proportional to the group veloc-ity

∆ωp(t) = ∆ωp(0)vgr(t)

vgr(0)(56)

and the spectrum of the probe pulse stays within thetransparency window at all times, provided it fulfills this

condition initially. The propagation equation for thedark-state polariton is then still given by eq. (49), withϑ→ ϑ(t). Adiabatically rotating ϑ(t) from 0 to π/2 leadsto a deceleration of the polariton to a full stop. At thesame time its character changes from an electromagneticfield to a pure spin excitation. Most importantly, pro-vided the rotation is adiabatic, all properties of the orig-inal light pulse are coherently transferred to the atomicspin system in this process modulo an overall phase de-termined by that of the coupling field.

−40 −20 0 20 40 60 80 100 1200

50

100

150

0

2

4

6

8

10

x 10−4

−40 −20 0 20 40 60 80 100 1200

50

100

150

0

2

4

6

8

10

x 10−4

−40 −20 0 20 40 60 80 100 1200

50

100

150

0

2

4

6

8

10

x 10−4

t t

E

100

150

50

0−40 0 40 80 120

z120

0

50

150

100

−40 0 40 80

z

ρ

t100

150

120

50

0

Ψ

400−4080

z

21

FIG. 13 Numerical simulation of dark-state polariton prop-agation with envelope exp−(z/10)2. The mixing angle isrotated from 0 to π/2 and back according to cot ϑ(t) =100(1 − 0.5 tanh[0.1(t − 15)] +0.5 tanh[0.1(t − 125)]). top:

polariton amplitude, bottom-left: electric field amplitude ,bottom-right: atomic spin coherence; all in arbitrary units.Axes are in arbitrary units with c = 1.

Conditions of adiabaticity have been analyzed byMatsko et al. (2001c) and Fleischhauer and Lukin (2002).The resulting limitations on the rate of change of the cou-pling field are rather weak. Furthermore if the couplingfield and thus the group velocity is already very smalleven an instantaneous switch-off would only lead to a lossof the very small electromagnetic component of the po-lariton (Fleischhauer and Mewes, 2002; Liu et al., 2001;Matsko et al., 2001c). Reversing the adiabatic rotationof ϑ by increasing the strength of the coupling field leadsto a re-acceleration of the dark-state polariton associatedwith a change of character from spin-like to electromag-netic. In Fig. 13 a numerical simulation of the stopping ofa dark-state polariton and its successive re-accelerationis shown. The transfer of excitation from the probe pulseto the Raman spin excitation and back is apparent. Inthis way the excitation and all information contained inthe original pulse can be reversible transferred and storedin long-lived spin coherences. It should be noted that thespin excitation does in general not store the photon en-ergy as most of it is transferred to the coupling field bystimulated Stokes emission.

19

Close this window to return to the previous window

Figure 1 Experimental set-up and procedure. a, States |1 , |2 and |3 form the three-levelEIT system. The cooled atoms are initially magnetically trapped in state |1 = |3S, F = 1, MF= -1 . Stimulated photon exchanges between the probe and coupling laser fields create a’dark’ superposition of states |1 and |2 , which renders the medium transparent for theresonant probe pulses. b, We apply a 2.2-mm diameter, --polarized coupling laser, resonantwith the |3S,F = 2, MF = +1 |3P,F = 2, MF = 0 transition, and a co-propagating, 1.2-mm

+

BEC

Close this window to return to the previous window

FIG. 14 top: Experimental set-up for observing light storageand retrieval in ultra-cold Na atoms from (Liu et al., 2001).bottom: Measurements of delayed (a) and revived probe pulseafter storage (b). Dashed line shows intensity of couplingfields. For details see (Liu et al., 2001).

The classical aspects of “stopping” and “re-accelerating” of light pulses have been experimentallydemonstrated by Liu and coworkers in ultra-cold Na(Liu et al., 2001), by Phillips et al. in hot Rb vapor(Phillips et al., 2001), and in solids by Turukhin andcollaborators (Turukhin et al., 2002). The experimentof Liu et al. used a similar setup as used for thedemonstration of the group velocity reduction to 17m/s (Hau et al., 1999). Here a light pulse has beenslowed and spatially compressed in an ultra-cold anddense vapor of Na atoms at a temperature just abovethe point of Bose condensation and then “stopped” byturning off the coupling laser. Storage times of up to 1.5ms for pulse length of a few µs were achieved. Fig. 14shows the experimental set-up as well as the storageand retrieval of a light pulse. The use of ultra-coldgases has advantages like the reduction of two-photonDoppler shifts and high densities. They are however notnecessary to achieve light “stopping”. In the experimentof Phillips et al. a Rb vapor gas cell at temperaturesof about 70-900C was used. To eliminate two-photon

Doppler shifts degenerate Zeeman sublevels were usedfor states |1〉 and |2〉. Moreover to reduce effects ofatomic motion a He buffer gas was employed. Herestorage times of up to 0.5 ms could be obtained forpulse length of again several µs. The first successfuldemonstration of light “storage” in a solid was achievedby Turukhin and co-workers in Pr doped Y2SiO5 afterproper preparation of the inhomogeneously broadenedmaterial by a special optical hole- and anti-hole burningtechnique (Turukhin et al., 2002). A direct proof ofthe coherence of the storage mechanism was providedby Mair et al. (2002), where the phase of the storedlight pulse was coherently modified during the storagetime by a magnetic field. Experimental studies of lightstorage under conditions of large single-photon detuningwere performed moreover by Kozuma et al. (2002). Allof these experiments have to be considered as a proof ofprinciple since the transfer efficiency or fidelity of thestorage is still rather small due to a variety of limitingeffects such as dark-state dephasing, atomic motion, andmode mismatch. An application which does not sufferfrom these effects is the use of the (partially dissipative)“stopping” of a light pulse at a spatial discontinuity ofthe drive field to create well defined shock waves in anatomic Bose-Einstein condensate as demonstrated byDutton et al. (2001).

The most important potential application of light“stopping” is certainly in the field of quantum informa-tion. The dark-state polariton picture of light “stopping”also holds for quantized radiation. Here quantum statesof photons 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〉l (57)

|0〉ph ⊗ 1√N

∑Nj=1 |1, 1, ..., 2j , ...1〉.

In this way the EIT medium can act as a quantum mem-ory for photons. As 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),which is also capable of storing single photons but hasnot been experimentally implemented. Several limita-tions of the EIT storage technique have been studiedincluding finite two-photon detuning (Mewes and Fleis-chhauer, 2002), decoherence (Fleischhauer and Mewes,2002; Mewes and Fleischhauer, 2004), and fluctuationsof coupling parameters (Sun et al., 2003). An extensionof the model to three spatial dimensions was given in(Duan et al., 2002).

20

An interesting extension of the light “stopping” schemewas very recently experimentally demonstrated by Ba-jcsy et al. (2003). After the coherent transfer of a lightpulse to an ensemble of Rb atoms in a vapor cell, twocounter-propagating coupling laser were applied ratherthan one, forming a standing-wave pattern. At the nodesof this pattern, a periodic structure of spatially narrowabsorption zones emerged. The two counter-propagatingcomponents of the coupling laser regenerated two Stokespulses with opposite wave-vectors. Thus also the regener-ated light pulses formed a standing-wave pattern, whichmatched that of the counterpropagating drive fields. Inthis way a stationary pulse of light was created and storedfor several µsec. Although the regenerated pulse con-tained only a small fraction of the original photon num-ber, the electromagnetic field was nonzero during thestorage period in contrast to the above mentioned exper-iments. This generation of stationary light pulses maybe an important tool for nonlinear optical processes withfew photons.

“Stopping” of light can in principle also be used to pre-pare atomic ensembles in specific nonclassical or entan-gled many-particle states (Lukin et al., 2000b). An alter-native and experimentally often simpler way to achievethe same is to use absorptive or dispersive interactionof continuous-wave light fields with atomic ensemblesand detection. Kuzmich et al. proposed the generationof spin-squeezed ensembles by absorption of squeezedlight (Kuzmich et al., 1997), which was experimentallydemonstrated later by Hald et al. (1999). Employingsimilar ideas Polzik suggested to create two Einstein-Podolsky-Rosen entangled atomic ensembles by absorp-tion of quantum-correlated light fields (Polzik, 1999) andKozhekin et al. proposed a memory for quantum statesof cw light fields along these lines (Kozhekin et al.,2000). Recently experimental progress toward the im-plementation of the cw quantum memory has been re-ported (Schori et al., 2002). A very intriguing tech-nique to create nonclassical or entangled atomic ensem-bles is the measurement of the collective spin using off-resonant dispersive interactions. Kuzmich et al. per-formed in this way quantum non-demolition measure-ments to prepare quantum states with sub-shot noisespin fluctuations (Kuzmich et al., 2000, 1999). Duanet al. (2000) suggested a detection scheme where coher-ent light probes simultaneously two atomic ensembles toprepare an Einstein-Podolsky-Rosen correlation betweenthem with potential applications in continuous telepor-tation of collective spin states. Recently, Julsgaard andcoworkers have demonstrated this technique in a remark-able experiment (Julsgaard et al., 2001).

Finally Imamoglu (2002) and James and Kwiat (2002)suggested to use the transfer of photons to atomic ensem-bles to build single-photon detectors with high efficiency,which are able to distinguish between single and multi-ple photons. Such a device has particular importancefor quantum information processing with linear opticalelements (Knill et al., 2001).

C. Nonlinear response: adiabatic pulse propagation and

adiabatons

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. Onthe other hand the nonperturbative situation of probeand coupling pulses with comparable strength shows anumber 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 where all fields prop-agate parallel to the z direction, a homogeneous mediumis assumed and atomic motion is disregarded. Theproperties of the atoms are described by a state vec-tor with slowly-varying probability amplitudes Cn(z, t),n ∈ 1, 2, 3. For simplicity let us assume that the car-rier frequencies of coupling and probe pulses are on res-onance with the corresponding atomic transitions, whichallows us to treat the slowly-varying field amplitudes asreal. The evolution of the atomic state is described by aSchrodinger equation for the Cn(z, t) with the three-levelHamiltonian given in Eq. (29). Because decay of the po-larization plays an important role for the propagation ofthe fields, a decay out of the excited state |3〉 with rate Γ3

is included by adding an imaginary term i~Γ3/2 to theenergy of the excited state. The propagation of the fieldamplitude is most easily written in terms of the corre-sponding Rabi-frequencies. In slowly-varying amplitudeand phase approximation one finds the shortened waveequations for the probe (Ωp) and coupling pulse (Ωc)

(

∂

∂t+ c

∂

∂z

)

Ωp(z, t) = iαp

2C2(z, t)C∗

1 (z, t), (58)

(

∂

∂t+ c

∂

∂z

)

Ωc(z, t) = iαc

2C2(z, t)C∗

3 (z, t), (59)

where the effect of the atoms on the fields is parameter-ized by the resonant absorption coefficients

αp =ωp%|µ12|2

2ε0~, αc =

ωc%|µ32|22ε0~

, (60)

with µij being the dipole moments of the correspondingtransitions.

If the amplitudes of the fields vary sufficiently slowlysuch that the adiabatic approximation holds and if theappropriate initial conditions are fulfilled, the atomsstay at all times in the instantaneous dark state,|a0(t)〉 = cos θ(t)|1〉− sin θ(t)|2〉, Eq.(17), with tan θ(t) =Ωp(t)/Ωc(t). As mentioned before an important propertyof this state is the absence of a component in state |3〉.As a consequence of this there is also no dipole momenton either of the two transitions coupled by the probe andcoupling pulses. Atoms in the instantaneous dark statehave therefore no effect at all. Light and matter are ex-actly decoupled.

21

As first noted by Harris (1993) there is another casein which bi-chromatic pulses are exactly decoupled fromthe atoms even if they do not vary slowly. The dark statecorresponding to a pair of pulses with identical envelope,so-called matched pulses, i.e. Ωp(z, t) = Ωpf(z, t) andΩc(z, t) = Ωcf(z, t), is time independent and thus a trueeigenstate of the system. After an appropriate prepa-ration 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, whichin this case does not correspond to the dark state, willprepare the atoms by stimulated Raman adiabatic pas-sage (STIRAP) (Harris and Luo, 1995). During the firstfew single-photon absorption lengths, the front end ofthe probe pulse experiences a small loss. In this waya counter-intuitive pulse ordering is established. Thisprovides asymptotic connectivity of the dark state tothe initial state of the atoms. The leading end of thisslightly deformed pair of pulses will prepare all atoms inthe pathway via STIRAP and the pulses can propagateunaffected through the rest of the medium (Eberly et al.,1994; Harris, 1994a). This is illustrated in Fig. 15 wherenumerical calculations are used to show the propagationof two matched pulses in an unprepared medium.

ampl

itude

[arb

.uni

ts]

pump

z = 0

z = 1000

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

time [arb. units]

ampl

itude

[arb

.uni

ts]

Stokes

z = 0

z = 1000

FIG. 15 Top: Amplitude of probe field as function of time inarb. units at medium entrance z = 0 (dotted line) and forz = 1000 (full line). Bottom: The same for the couplingfield. Distance is measured in units of absorption length andpeak values are Ωc = Ωp = 10Γ.

Apart from the preparation at the front end, matchedpulses are stable solutions of the propagation problem.They should therefore be formed whenever pulses of ar-bitrary shape are applied to optically thick 3-level media.In fact it has been shown by Harris (1993) that nonequalpairs of pulses with a strong cw-carrier

Ωp(z, t) =(

1 + f(z, t))

Ωpe−iωp(t−z/c), (61)

Ωc(z, t) =(

1 + g(z, t))

Ωce−iωc(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 takes place also forarbitrary pulses (pulse-matching). The phenomenon ofpulse matching causes a correlation of quantum fluctua-tions in both fields as discussed e.g. by Agarwal (1993),Fleischhauer (1994), and Jain (1994).

As noted above an EIT medium does not couple to thefields at all in the adiabatic limit. In order to cover ef-fects like the group-velocity reduction, it is necessary toinclude first-order non-adiabatic corrections in the dy-namics of the atoms, which lead to small contributionsto the excited state amplitude. Taking them into accountone finds for the state amplitudes

C1 = cos θ, C2 =2i

Ωθ, C3 = − sin θ, (64)

where Ω =√

Ω2p + Ω2

c . It was shown by Grobe

et al. (1994) that the nonlinear equations of propagation(58,59), together with (64) are adiabatically integrableeven for fields of comparable strength and with arbitraryshape. To see this, we adopt the method of (Fleischhauerand Manka, 1996) and transform the field equations forΩp and Ωc in propagation equations for the rms Rabi-

frequency Ω and the non-adiabatic coupling θ. Undernearly adiabatic conditions |θ| Ω, a weak-coupling ap-proximation is justified. In this limit, assuming further-more equal coupling strength αp = αc = α, one findsthat the total Rabi-frequency fulfills the free-space prop-agation equation

(

∂

∂t+ c

∂

∂z

)

Ω(z, t) ≈ 0. (65)

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

(

∂

∂t+ c

∂

∂z

)

θ(z, t) = −α∂

∂t

(

θ

Ω2

)

, (66)

which is exactly integrable. The corresponding solutions,called adiabatons (Grobe et al., 1994), are particularlysimple if Ω is approximately constant over the time in-terval of interest. In that case the probe and couplingpulses have complementary envelopes and θ propagateswithout changing form with velocity vgr = c/(1+α/Ω2).The quasi form-invariant propagation of an adiabaton is

22

shown 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.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

d

S

P

puls

e am

plitu

de [

arb.

uni

ts]

[arb. units]time

a cb

coupling

probe

time (arb.units)

ampl

itude

(ar

b. u

nits

)

FIG. 16 Propagation of adiabatons: Shown are amplitudes ofprobe and coupling fields as function of time in a co-movingframe for ngrz/c = 0 (a), 25 (b), 50 (c), and 100 (d) from nu-merical solution of propagation equations (Fleischhauer andManka, 1996). Arbitrary space and time units with c = 1.

First experimental evidence of adiabaton propagationin Pb vapor was reported by Kasapi et al. (1996). Adi-abatons in more complicated configurations, such asdouble-Lambda systems and double pairs of pulses, werestudied by Cerboneschi and Arimondo (1995) as well asHioe and Grobe (1994). The effect of non equal cou-pling strength on the adiabaton propagation was studiedby Grigoryan and Pashayan (2001). The adiabaton solu-tions are approximately stable over many single-photonabsorption length. However, as shown by Fleischhauerand Manka (1996), they eventually decay to matchedpulses after sufficiently long propagation distances.

For time scales short enough such that decays canbe ignored, and equal coupling strength, exact solutionsof the nonlinear propagation problem in V and Λ-typesystems exist even beyond the adiabatic approximation.Konopnicki and Eberly (1981) and Konopnicki et al.

(1981) have found soliton solutions with identical pulseshape, called simultons. An interpretation of simultons interms of solitons corresponding to an effective two-levelproblem was given in (Fleischhauer and Manka, 1996).Malmistov (1984) and Bol’shov and Likhanskii (1985) de-rived the simulton solutions in three-level systems usingthe inverse scattering method. Several extension of si-multons have been discussed. N -soliton solutions weregiven by Steudel (1988). Inhomogeneously broadenedsystems were considered by Rahman and Eberly (1998)and Rahman (1999) and extensions to five-level systemshave been discussed by Hioe and Grobe (1994).

V. ENHANCED FREQUENCY CONVERSION

A. Overview of Atomic Coherence Enhanced Non-Linear

Optics

Harris and co-workers identified the enhancement ofnon-linear 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 Λscheme as an important tool to EIT-based resonant four-wave mixing. Coherent preparation of a maximal co-herence 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. Whilst destructive interference reduces the lin-ear susceptibility of the laser dressed system this is notso for the non-linear susceptibility in four-wave mixingwhich in fact undergoes constructive interference. To seehow this leads to improved frequency mixing efficiencywe need to consider the factors which determine how agenerated field can grow effectively in a four-wave mixingprocess.

Laser fields applied close to resonance in an atomicmedium drive various frequency components of the polar-ization in that medium, both at the frequency of the driv-ing fields and at new frequencies given by combinationsof the applied ones. Figure 17 illustrates four schemes inwhich we will be interested where 3- or 4-level atoms aredriven, at single or two-photon resonance, by laser fields.The polarization component at the new frequencies willact as a source of new electromagnetic fields. Thus in theschemes illustrated in Fig. 17 the material polarizationP (ω4) is driven at frequency ω4 by fields applied at ω1, ω2

and ω3, and will in its turn drive a new electromagneticfield at ω4. The growth of the new field depends uponthe magnitude of the non-linear source term P nl(ω4) andupon the linear polarization P l(ω4) 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)) and canbe written in terms of the positive frequency componentof the field as:

∂E4

∂z+

1

c

∂E4

∂t=

i

2

ω4

cε0(Pnl + P l). (67)

Here the vector character of field an polarization wassuppressed for simplicity. The term on the right-handside contains the effect of the linear polarization uponthe propagation of the generated field and the nonlinearpolarization which is the source of the new field.

The response of the medium to an electric field is gov-erned by its polarisation. In the framework of the densitymatrices for a three-level atom we can write the positive

23

1⟩

2⟩

3⟩

(i)

ω4

ω2

ω3

ω1

1⟩

2⟩

4⟩

3⟩

(ii)

ω4

ω2

ω3

ω1

1⟩

2⟩

3⟩

(iii)

ω4

ω2

ω3

ω1

1⟩

2⟩

4⟩

3⟩

(iv)

ω4

ω2

ω3

ω1

FIG. 17 Different schemes for resonantly enhanced four-wavemixing processes based on EIT and related interference phe-nomena: (i) sum-frequency generation (ω4) out of two pumppulses (ω1 and ω2) and one strong coherent drive (ω3); (ii)up-conversion of pump (ω3) into generated field (ω4) due tomaximum coherence prepared by two strong drive fields (ω1

and ω2); (iii) off-resonant variant of (ii); (iv) parametric gen-eration of two fields (ω3 and ω4) out of two pump fields (ω1

and ω2) in forward or backward scattering configuration

frequency component of the polarization as:

P = %Tr[µρ] =

= %(

µ12ρ21 + µ13ρ31 + µ23ρ32

)

. (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 ω4 as:

P (1)(ω4) = ε0χ(1)E4, (69)

and the non-linear response as:

P (3)(ω4) =3

2ε0χ

(3)E1E2E3, (70)

where Ej is the electric field amplitude associated withthe frequency component ωj . The form of the suscepti-bilities that can be derived by a full quantum treatmentof the atom are extensively discussed in the literature onnon-linear optics (see for example Reintjes (1984)). Ingeneral for non-resonant non-linear mixing the suscepti-bility will be characterized by the appearance of threedetuning terms in the denominator whilst the linear sus-ceptibility has only a single detuning term. As was shownin section III in the case of EIT it is of course necessary to

use the dressed susceptibilities that already include theeffect of the strong coupling field to all orders. It is thedifference of these susceptibilities from the normal formthat give the advantages for non-linear mixing. The formof the dressed susceptibilities for EIT are given in sectionIII. These reflect the resonant nature of the fields involvedand now contain the important interference character dueto the dressing field. Inclusion of collisional de-phasingand integration over the Doppler profile is usually re-quired to yield the effective susceptibilities that can beused in a calculation.

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

ω4

4cχ(3)E1E2E3e

i∆kz

−ω4

2cIm[χ(1)]E4 + i

ω4

2cRe[χ(1)]E4, (71)

where ∆k = k4 − (k1 + k2 + k3) is the wave-vector mis-match due to the refractive index effect of all other reso-nances in 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, thus the production of thenew field will be enhanced. The absorption and refractionof the generated field are determined by the linear polar-ization of the medium at the generated frequency. Therefraction has a very important effect upon the growthof the field by determining whether or not the gener-ated field remains in phase with the polarization whichdrives it. This phase-matching condition is essential anddictates (at resonance where Re[χ(1)] = 0) that we alsorequire ∆k=0 for efficient growth of the field. For a gen-erated field on resonance both absorption and dispersioncan be problematic. In general ∆k will be finite result-ing in a limited length over which the field grows be-fore slipping out of phase with the driving polarizationlcoh = 1/∆k. But in the presence of EIT, and with theelimination of the linear susceptibility to vacuum values,the result is no absorption and no refraction due to theresonant level and only non-resonant transitions to otherlevels contribute to the wave-vector mismatch.

We examine various four wave-mixing schemes inwhich EIT or related effects have been found to be ad-vantageous; these are shown in Fig. 17. There are severalinterconnected mechanisms through which atomic coher-ence prepared by the applied laser fields can increase theconversion efficiency of these wave-mixing process es:

(i) We have here a four-wave mixing scheme where atwo-photon and a single photon resonance are used forthe applied fields generating a field in resonance withan atomic transition (figure 17(i)). The generated fieldand the single photon resonant applied field act as probeand coupling fields respectively in a Lambda EIT scheme.EIT then leads to an elimination of absorption and refrac-tion for the generated field. The non-linear susceptibility

24

is enhanced by proximity to resonance since this term issubject to constructive interference between the pair ofdressed states. This mechanism is relevant to both CWand pulsed lasers fields (Harris et al., 1990; Zhang et al.,1993).

(ii) Here a pair of strong fields are applied resonantlyin a Lambda configured three-level system near to bothRaman and single photon resonance (figure 17(ii)). Thepresence of EIT on both fields eliminates absorption anddispersion in their propagation. They can then propagatewithout 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 propagation length.Four-wave mixing from the maximal coherence leads tolarge up-conversion efficiency. This requires the use offields strong enough to ensure adiabatic evolution of thedark state so is restricted to high power pulsed lasers orto Doppler free CW laser experiments (Harris and Jain,1997; Merriam et al., 2000, 1999).

(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 ∆) (figure17(iii)) providing that they are strong enough so thatthe condition Ω1Ω2 > ∆γ21 is satisfied (where γ21 is thedephasing rate of the |1〉 − |2〉 coherence). In this caseEIT plays no role in so far as absorption is concerned asthe fields are far from resonance. Nevertheless the choiceof the correct magnitude of Raman detuning can lead tocancellation of the material refraction for the pair of ap-plied fields (Harris et al., 1997). The maximal coherence,however, greatly boosts the conversion efficiency in four-wave mixing (Hakuta et al., 1997) as for (b). If the max-imal coherence is excited between molecular vibrationalor rotational states the medium is very suited to high-order stimulated Raman side-band generation (Sokolovet al., 2000). The requirement of high power to achieveadiabatic 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-Λ scheme which can enhance the mixingfurther. An important variant of the double-Λ schemewith 3 applied fields is that explored by Hemmer et al.

(1995) and by Zibrov and co-workers where only twofields are injected (Zibrov et al., 1999) (figure 17(iv)).These fields are applied close to single photon resonancewith transitions in each of the two Λ sytems formed inalkali metal atoms. This scheme with CW laser fields hasremarkable properties including, if the applied fields areco-propagating, the growth of both pairs of fields, and ifthe applied fields are counter-propagating mirrorless os-cillation as experimentally demonstrated by Zibrov et al.

(1999).

The advantages of all of these schemes lies in the re-duction of unwanted absorption and dispersion for thegenerated and the driving fields whilst the source termfor new fields is boosted. As a consequence of the elimina-

tion of drive field dispersion and absorption large valuesof coherence can be driven in the case of (ii), (iii) and(iv), in fact to the maximal value ( i.e. ρ12 = 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-velopment is the demonstration of high efficiency multi-order Raman side-band generation through the modula-tion of optical fields in a molecular medium prepared ina maximally coherent state of the v=0 and v=1 vibra-tional states (Hakuta et al., 1997; Sokolov et al., 2000).The possibility of using the broad-bandwidth spectrumof phase coherent side-bands to synthesize ultra-shortpulses was first predicted (Harris and Sokolov, 1997)and then demonstrated by the Stanford group (Sokolovet al., 2001). Another interesting effect happens whenin an EIT medium the light velocity matches the speedof sound. As shown by Matsko and collaborators a newtype of stimulated Brillouin scattering should occur inthis case which in contrast to the usual situation allowsefficient forward scattering (Matsko et al., 2001b).

B. Non-linear Mixing and Frequency Up-Conversion with

Electromagnetically Induced transparency

The enhancement of four-wave mixing in a two-photonresonant scheme of the type illustrated in Fig. 17i wasfirst treated by Harris et al. (1990). The idea of improv-ing four-wave mixing using an additional control field pri-marily to adjust the refractive index of the medium wastreated by Tewari and Agarwal (1986). In their scheme,however, the control field is not incorporated directly intothe mixing fields and there is no constructive interferenceof the non-linear susceptibility.

Taking the linear and non-linear susceptibilities de-rived in section III we can procceed with an analysis ofthe case of four-wave mixing in a Lambda EIT scheme.Proceeding from Eq. (71) and integrating with appro-priate boundary conditions (as is usual in all four-wavemixing processes) leads to an expression for the gener-ated intensity of the form:

I4 =3nω2

8Zoc2

∣

∣

∣χ(3)

∣

∣

∣

2

|E1|2 |E2|2 |E3|2 F (∆k, z,R, I), (72)

where z is the propagation distance, Zo = 376.7Ω is theimpedance of free space, n is the refractive index at thegenerated frequency (typically close to unity), R and I areshorthand for Re[χ(1)] and Im[χ(1)] respectively and c isthe speed of light in vacuum. Here we have introducedthe phase–matching factor F (∆k, z,R, I) that depends

25

upon the details of the mixing scheme and the focal ge-ometry. This general result indicates the importance ofthe phase-matching (and hence ∆k) upon the generatedfield intensity.

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

I4 =3nω2

8Zoc2

∣

∣

∣χ(3)

∣

∣

∣

2

|E1|2 |E2|2 |E3|2 ×

×1 + e−ωIz

c − 2e−ωIz2c cos[(∆k + ωR

2c )z][

ωI2c

]2+[

∆k + ωR2c

]2 . (73)

It is implicit that χ(3), Re[χ(1)] and Im[χ(1)] are the laserdressed expressions introduced in section III with anyrequired 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 unityand if also ∆k = 0 the intensity is simply related to thesusceptibilities by:

I4 ∼∣

∣

∣

∣

χ(3)

χ(1)

∣

∣

∣

∣

2

(74)

as was pointed out first by Harris et al. (1990) andsubsequently (in the form used here) by Petch et al.

(1996). It is very clear that EIT increases this ratioby both increasing the value of the numerator and re-ducing the denominator. It should be noted that adramatic change in conversion efficiency as a functionof coupling strength is only apparent in an inhomoge-neously broadened medium. In a homogeneously broad-ened medium although EIT increases the efficiency, theratio of the susceptibilities remains independent of thecoupling strength.

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 figure 17(i)). Thetwo-photon transition need not be strongly driven (i.e.a small two-photon Rabi frequency can be used) but a”strong” coupling laser is required. The coupling laseramplitude must be sufficient that the Rabi frequencyis comparable to or exceeds the inhomogenous spectralwidths in the system (e.g. Doppler width). For exam-ple a laser intensity of above 1MWcm−2 is required for atypical Doppler broadened visible transition. This is triv-ially achieved even for unfocused pulsed lasers, but doespresent a substantial barrier for the use of continuouswave (CW) lasers unless a specific Doppler free configu-ration is employed. The latter is not normally suitablefor a frequency up-conversion scheme if a large frequencyup-conversion factor is to be achieved e.g. to the vac-uum ultra-violet (VUV). This is in large part becauseatomic species with high ionisation potentials (> 10eV )that are suitable for frequency up-conversion to the VUVare not easily trapped and cooled at high density. Recentexperiments, by for instance Hemmer, report significant

progress in CW non-linear optical processes using EIT(Babin et al., 1996; Hemmer et al., 1995). A number ofother possibilities e.g. laser cooled atoms and Dopplerfree geometries have also been explored.

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 value of thesusceptibilities at a given frequency are obtained, andthese quantities can be used to calculate the generatedfield. The interference effects persist in the dressed pro-files provided 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 profilethan the Lorentzian profile arising from lifetime broaden-ing. Indeed in the case of Doppler broadening the effectsof increasing Ωc manifest themselves most clearly. Asdiscussed by Harris et al. (1990) there is expected to bea step-like increase in the conversion efficiency when themagnitude of Ωc approaches the Doppler width of thetransition. For coupling strengths below this value theatomic sample remains opaque as it would under ordinaryresonant conditions; in contrast for a coupling strengthexceeding the Doppler width the full benefit of the quan-tum interference is obtained.

In the Doppler broadened case therefore the cou-pling field Rabi frequency must remain greater than theDoppler width for a significant fraction of the interactiontime. Pulsed lasers, because of the high peak powersthat arise, are required. Moreover a transform limitedsingle mode laser pulse is essential for the coupling laserfields which drive EIT since a multi-mode field will causean additional dephasing effect on the coherence resultingin a deterioration of the quality of the interference, asdiscussed in Sec. III.

The work of the Stanford group has highlighted thatwhen pulsed laser fields are used additional considera-tions must be made. Kasapi and co-workers showed (Kas-api et al., 1995) that the group velocity of a 20ns pulsecan be modified for pulses propagating in the EIT; largereductions, e.g. by factors down to < c/100, in the groupvelocity have been observed. Another consideration be-yond that found in the simple steady-state case is thatthe medium can only become transparent if the pulsecontains enough energy to dress all of the atoms in theinteraction volume. This preparation energy conditionwas discussed in section IV. This puts additional con-straints 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 were

26

important, was reported by Zhang et al. (1993). Theyemployed a four-wave mixing scheme in Hydrogen (equiv-alent to figure 17(i)), where the EIT was created on the3p-1s transition at 103nm in a Lambda scheme by theapplication of a field at 656nm on the 2s-3p transition(figure 18). A field at 103nm 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 states leadsto loss of the conditions for perfect transparency. Never-theless because the photionization cross-section for the 2smetastable level is about one order of magnitude less thanthat from the 3p state, the effect of EIT is only partiallyreduced. It is in general important to keep the photoion-ization rate sufficiently small so as to not to quench thecoherence through electron impact broadening (see Buffaet al. (2003)). In this experiment, and subsequent workby the same authors conversion efficiencies up to 2×10−4

were reported (Zhang et al., 1995).

Ionization Region243 nm

12.5

×1 2×1014

cm-2

4×1014

cm-2

8×1014

cm-2

×

243 nm

243 nm

103 nm

243 nm

856 nm

856 nm

3p

2s

1s

Γ31

19

-10 0 10

×

∆ω31 (cm-1)

FIG. 18 Experimental demonstration of sum-frequency gen-eration in H from (Zhang et al., 1993). left: Level and cou-pling scheme, right: Calculated (dashed) and measured (fullline) nonlinear susceptibility as function of detuning from 3p- 1s transition for different density length products of themedium (nL), only under the least dense condition is theAutler-Townes split structure apparent in the generated sig-nal, at higher density strong resonant enhancement is seen(Zhang et al., 1993).

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 frequencyis required to create good transparency, this in turn leadsto a reduction in the non-linear susceptibility. This is oneof the most important factors limiting the conversion ef-ficiency in the hydrogen scheme (Hakuta, unpublished).The requirements on a minimum value of Ωc > ∆Doppler

constrains the conversion efficiency that can be achievedbecause of a scaling factor by 1/Ω2

c that ultimately leadsto diminished values of χ(3). The use of gases of higheratomic weight at low temperatures is therefore highly de-sirable in any experiment utilising EIT for enhancementof four-wave mixing to the VUV. A combination of the

low mass of the H atom and the elevated temperaturesrequired in the dissociation process leads to especiallylarge Doppler widths (> 10 GHz). To overcome such asevere limit investigations were undertaken by one of theauthors in krypton gas where the relevant Doppler widthis only 1 GHz. Conversion efficiencies approaching 10−2

were obtained (Dorman et al., 2000b). In this case, theconversion efficiency appears to be limited by the finitevalue of ∆k due to the other atomic levels rather thanabsorption in the medium (Dorman et al., 2000a).

In a Doppler free medium the effect of the construc-tive interference for the non-linear term is most apparent.In a system without inhomogeneous broadening perfecttransparency can be induced with Ωc γ31. As Ωc issmall relative to γ31 the non-linear susceptibility will,because of the constructive interference, have a valueessentially identical to that of the unmodified resonantatomic system. The transparency dip will be very nar-row (see Figure 7). The spectral widths of these featuresare typically sub-natural and are therefore accompaniedby very steep normal dispersion, which corresponds toa much reduced group velocity. As was pointed out byseveral authors including Schmidt and Imamoglu (1996),and Harris and Hau (1999), the non-linear susceptibil-ities in this case can be extremely large as there is aconstructive interference. Non-linear optics at very lowlight levels, i.e. at the few-photon limit, is possible inthis regime. Associated with their first dramatic mea-surements of ultra-slow light Hau et al. (1999) reporteda non-linear refractive index in an ultra-cold Na vapourthat was 0.18cm2W−1, this is already 106 times largerthan that measured for cold Cs atoms in a non-EIT sys-tem, which itself was much greater than the non-linearityin a typical solid state system. The discussion of this pro-cess in the few photon limit is given in section VI.

C. Non-linear 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 3-levelsystem. Due to the elimination of absorption and re-fraction, both fields can propagate into the medium andlarge values of the material coherence are created on thenon-driven transition. Considering the system illustratedin figure 17 (ii) we can imagine that both applied fieldsare now strong. Under appropriate adiabatic conditions(see section III) the system evolves in the dark-state toproduce the maximum possible value for the coherenceρ12 = 0.5. Adiabatic evolution into the maximally coher-ent state is achieved by adjusting either the Raman de-tuning or the pulse sequence (i.e. to ”counter–intuitive”order). The pair of fields may also be in single photon res-onance with a third level, in which case the EIT-like elim-ination of absorption will be important. This situation isequivalent to the formation of a dark-state, since neitherof the two strong fields is absorbed by the medium. For

27

sufficiently strong fields the single-photon resonance con-dition need not be satisfied and a maximum coherencecan still be achieved (see figure 17(iii)) provided that thecondition Ω1Ω2 > |∆|γ21 is met, where ∆ is the detuningof the strong fields from state |3〉.

Under the conditions of STIRAP, or for matched pulsepropagation, large populations of coherent populationtrapped states are created. To achieve this situation thelaser electric field strength must be large enough to cre-ate couplings that will ensure adiabatic atomic evolution,and laser pulses that are sufficiently energetic to prepareall of the atoms in the beam path. The ρ12 coherencethus created will have a magnitude |ρ12| = 0.5 (i.e. allthe atoms are in a coherent state) and negative sign (i.e.all the atoms are in the population trapped dark-state).The complex coherence varies in space and time and canbe written:

ρ12 = − Ω1Ω∗2

√

Ω21 + Ω2

2

exp

i(ω1 − ω2)t− (k1 − k2)z

= |ρ12|e−iδkzei(ω1−ω2)t. (75)

Under these circumstances mixing of additional fieldswith the atom can become extremely efficient (Harriset al., 1997).

The preparation energy condition for the creation of amaximal coherence requires that the pulse energy mustexceed the following value:

Eprep =f13

f23%AL~ω, (76)

where fij are the oscillators strengths of the transitionsand %L the product of the density and the length (Harrisand Luo, 1995). Essentially the number of photons inthe pulse must exceed the number of atoms in the inter-action volume to ensure all atoms are in the appropriatedressed state. It should be noted that the preparation en-ergy is much smaller for non-linear conversion processesnot using maximal coherence. Furthermore this prepara-tion energy is not lost by dissipation if the pulses evolveadiabatically and can be recovered.

An additional field applied to the medium can par-ticipate in sum- or difference frequency mixing with thetwo Raman resonant fields. The importance of the largevalue of coherence is that it is the source polarization thatdrives the new fields generated in the frequency mixingprocess. This is described by the following equation,

dE4

dz= −iη4~ω4n

[

(a4ρ11 + d4ρ22)E4 + b4ρ12E3

]

, (77)

with some coefficients a4, b4 and d4 that depend uponatomic factors. The second term on the right hand side(b4ρ12E3), 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 ρ12 is large, this significantly re-laxes the constraints usually set by phase-matching in

non-linear optics. Recently near unity conversion efficien-cies to the far-UV were reported in an atomic lead sys-tem where maximum coherence had been created (Harriset al., 1997; Jain et al., 1996; Merriam et al., 2000, 1999).An example of high conversion efficiency four-wave mix-ing is shown in figure 19 (after Jain et al. (1996)).

CouplingLaser

406 nm

425 nm

1112cm-1

10660cm-1

0cm-1

(a) (b)

00

10

20

30

40

50

10 20

Coupling Laser Intensity (MW/cm2)

Con

vers

ion

Effi

cien

cy (

%)

30 40 50

Ωc

ProbeLaser

283 nm

293 nm

Ωp

ΩhΩc

3⟩

2⟩

1⟩

3⟩

2⟩

1⟩

FIG. 19 Parametric up-conversion with maximum coherencefrom (Harris et al., 1997): A pair of (near) resonant cou-pling (Ωc) and probe pulses (Ωp) as shown in (a) generatemaximum coherence between states |1〉 and 2〉. The coher-ently prepared medium then leads to efficient up-conversionof second coupling pulse Ωc into Ωh (b). Also shown is theconversion efficieny from 425 nm to 293 nm as function ofcoupling laser intensity (after Jain et al. (1996)).

An alternative way to look at the origin of the largefour-wave mixing efficiency is to consider the susceptibil-ities. Recalling the density matrix results of a Λ system,eq.(12), one recognizes that the polarization at ω4 pro-portinal to ρ31 is a sum of two terms, the first beingproportional to the linear susceptibility and the secondto the nonlinear susceptibility. The drive fields E1 andE2 are now implicitly included in ρ12 that has taken themaximal value of 0.5. These drive fields are resonant sothe structure of χ(3) simplifies to a single non-resonantterm in the denominator. The non-linear susceptibilitytherefore shares this single non-resonant term in the de-nominator with the linear susceptibility and so these twoquantities are of comparable size. This is a very unusualsituation in non-linear optics and the length over whichefficient frequency conversion can occur is now reducedto values comparable to the coherence length. A con-sequence of this is that complete non-linear frequencyconversion of the fields can occur in a distance of the or-der of the coherence length (determined by the real partof the susceptibility). This is equivalent to having nearvacuum conditions for the refraction (and absorption) ofthe medium whilst the non-linearity remains large.

In a molecular medium large coherence between vibra-tional or rotational levels have also been achieved us-ing adiabatic pulse pairs in gas phase hydrogen and deu-terium (Sokolov et al., 2000). Efficient multi-order Ra-man side-band generation has been observed to occur. Inthis 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 ( 10eV) means that the situation is similar to

28

that illustrated in figure 17(iii). This superposition thenmixes with the applied fields to form a broad spectrumof sidebands through multi-order mixing. Molecular co-herence in solid hydrogen has recently also been usedto eliminate phase-mismatch in a Stokes or anti-Stokesstimulated Raman frequency-conversion scheme (Hakutaet al., 1997). In this scheme, v=0 and v=1 vibrationalstates of the hydrogen molecule electronic ground stateform the lower states of a Raman scheme. Since the|1〉−|2〉 dephasing rate is very small, in appropriately pre-pared samples of solid hydrogen, interference that causesthe dispersion to become negligible can occur. Because ofthe removal of the usual phase-mismatch, efficient oper-ation of these frequency conversion schemes over a broadrange of frequencies (infra-red to vacuum UV) has beenshown.

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 side-band gen-eration lead the way to synthesizing very short durationlight pulses since the broad-band Raman side-band spec-trum has been proved to be phase-locked (Sokolov et al.,2001). It is anticipated that high power sources of sub-femtosecond duration pulses might be achieved throughthis technique.

D. Four-Wave Mixing in Double-Λ Systems

Four-wave mixing with all the optical fields close to res-onance is achieved in atoms with 4-levels in the so calleddouble-Lambda configuration (Hemmer et al., 1995),which is detailed in Fig. 20. The fully resonant char-acter of the light-matter interaction for all of the fields(both those applied and those generated) involved infour-wave mixing leads potentially to very high conver-sion efficiencies and to a number of important character-istics. Earlier work on double-Lambda schemes in whichall four fields are applied has shown for instance thatthe formation of dark-states depends upon the relativephase of the fields (Arimondo, 1996; Buckle et al., 1986).Experiments in the CW limit have been carried out insodium vapour that show the phase-sensitivity of EIT ina double-Lambda scheme (Korsunsky et al., 1999).

First we will consider the case where three resonantfields are applied such that the fourth field is generatedin resonance between the highest of the excited statesand the ground state. Under appropriate conditions thepresence of the three applied fields eliminates not onlythe resonant absorption/refraction of the generated fieldbut also for the applied fields themselves. The situa-tion of four-wave mixing in a double-lambda has beentreated theoretically in detail by Korsunsky and Kosa-chiov (1999), where the relative phases of the four fieldswere found to evolve with propagation so as to supportthe lossless propagation in a fashion similar to that pre-dicted by Harris for a pair of fields propagating in reso-

nance in a Lambda medium (Harris, 1993). The phaseand the amplitudes of all four fields, the generated fieldand the applied fields, are predicted to adjust as theypropagate so as to support efficient transfer of energybetween the fields, under loss-less conditions.

An experimental realization of the up-conversion in adouble-Lambda scheme driven by three resonantly tunedpulsed fields with the new field generated in the far-UVat 186nm was achieved by Merriam et al. (2000) in leadvapour. High conversion efficiencies, more than 0.3 fromthe shortest wavelength applied field at 233nm to the186nm field, were indeed observed. The experiments con-firm that the double-Lambda scheme supports loss-lesspropagation provided that the pairs of Lambda resonantfields have matched ratios of Rabi frequency. For pulsesnot initially satisfying the condition of matched Rabi fre-quencies absorptive loss and non-linear energy transferoccur until the condition is reached after which all fieldspropagate without further loss and refraction. Limits tothe up-converted power density were found, however, toarise from power broadening.

a1

a2

El

Ef

E2

b1 1.20

00 10.2 0.4

z/L0.6 0.8

0.2

1.0

0.4

0.6

0.8

0.05

0.1

0.15

0.2

0.25

1.3 1.4 1.5kL/∆

1.6 1.7 1.8

b2Eb

∆EI(L)2

Ef(0)2

E2

Eb Ef

E1

FIG. 20 Generation of fields E1 and E2 by resonant four-wave mixing of two counter-propagating pump fields Ef andEb: left: level and coupling scheme; right: Calculated thresh-old behaviour of output intensity of field E1 as function ofmedium length L. The insert shows the spatial distributionsof the fields above threshold of parametric oscillation at theindicated point (after Fleischhauer et al. (2000)).

An interesting alternative to the schemes just de-scribed for efficient non-linear 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-Stokescomponents of the two applied fields. In the origi-nal experiment in Rb vapour the applied fields counter-propagate and the resulting non–linear gain and efficientintrinsic feedback in this configuration lead to mirror-lessself oscillation at extremely low applied field strengths (atthe µW level of CW power). Further analysis of the quan-tum dynamics in this situation has highlighted the pos-sibility of using this technique to generate narrow-bandsources of non-classical radiation (Lukin et al., 1999).

29

VI. EIT WITH FEW PHOTONS

In the absence of a strongly polarizable medium, pho-tons are essentially non-interacting particles. In commu-nication technology this is a desired property and opticshas emerged over the last two decades as the preferredmethod for communicating information. For the samereason, attempts to use light in information processingtasks such as computation has failed. For the latter, oneneeds strong, dissipation free light-light interactions. Inthe present section we summarize and discuss some of thepotentials and limitations of electromagnetically inducedtransparency in this respect.

A. Giant Kerr effect

Information processing based on light requires nonlin-ear interactions that would lead e.g. to a Kerr effect, orequivalently, a cross-phase modulation where the phaseof a light field is modified by an amount determined bythe intensity of another optical field. If the nonlinearphase shifts arising from such a Kerr effect were on theorder of π-radians, it would be possible to implement all-optical switching. Such large Kerr nonlinearities withoutappreciable absorption can however, only be obtained forintense laser pulses containing roughly 1010 photons.

One of the principal challenges of nonlinear optics isthe observation of a mutual phase shift exceeding π-radians using two light fields each containing a singlephoton. In addition to new possibilities for addressingfundamental issues in quantum theory, such strong non-dissipative interactions can be used to realize a controllednot gate between two quantum bits (qubits), definedfor example by the polarization state of a single-photonpulse. However, given the weakness of non-resonant op-tical nonlinearities and the dominant role of absorptionin resonant processes, the combination of a large thirdorder Kerr susceptibility and small linear susceptibilityappears to be incompatible.

In this section, we show how EIT can overcome theseseemingly unsurmountable obstacles. We have alreadyseen how the non-reciprocity between linear and non-linear susceptibilities in EIT media can be used to en-hance generation of radiation at new frequencies. Wewill see in this subsection how dissipation-free strongphoton-photon interactions can be induced using EITdue to a constructive interference in Kerr nonlinearityat two-photon resonance (δ = 0) where linear suscepti-bility experiences a complete cancellation. This so-called“giant Kerr” effect, proposed by Schmidt and Imamoglu(1996) can be understood using the four-level systemdepicted in Fig. 21: the principal element here is onceagain the Lambda subsystem consisting of states |1〉, |2〉,and |3〉. The state |4〉 has the same parity as |3〉 andhas an electric-dipole coupling to |2〉; the signal field atfrequency ωs is applied on this transition, with a Rabifrequency Ωs. The cross-phase modulation nonlinearity

that we are interested in is between this signal field andthe probe field applied on the |1〉− |3〉 transition. We as-sume that the applied fields only couple to the respectivetransitions indicated in fig. 21. Throughout this section,we assume that the coupling field with Rabi frequencyΩc is non-perturbative, remains unchanged, and can betreated classically.

4

12

3

Ωp

Ωc

Ωs

FIG. 21 Level scheme for giant Kerr effect.

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

Heff = −~

2

[

Ωpσ31 + Ωcσ32 + Ωsσ42 + h.c.]

(78)

+~(∆− i

2Γ3)σ33 + ~(δ − i

2Γ2)σ22 + ~(∆ω42 −

i

2Γ4)σ44

to describe the atomic dynamics in this limit. Here, wehave introduced the detuning ∆ω42 = ω42 − ωs and thespontaneous emission rate Γ4 out of state |4〉. Onceagain, we can solve the Schrodinger equation with theHamiltonian of Eq. (78) to obtain the probability ampli-tude for finding the atom in state |3〉 (a3(t)) and use itto evaluate the linear and nonlinear components of thepolarization at ωp:

P (t) = Nµ13a3(t)e−iωpt + h.c.

= ε0χ(1)(−ωp, ωp)Epe

−iωpt (79)

+ ε0χ(3)(−ωp, ωs,−ωs, ωp)|Es|2 Epe

−iωpt + h.c. .

Similarly, we can obtain the linear and nonlinear sus-ceptibilities at ωs. For simplicity, we consider only theresonant case where ∆ = δ = 0.

First we emphasize that the linear susceptibility forboth probe and signal fields vanish in the limit Γ2 = 0.This is trivially true for ωs since it is assumed notto couple to state |1〉 atoms, and is the case for ωp

to the extent that ωs remains perturbative. We alsonote that self-phase modulation type Kerr nonlinearities(i.e. χ(3)(−ωp, ωp,−ωp, ωp) and χ(3)(−ωs, ωs,−ωs, ωs))for both fields are identically zero. The cross-phase mod-

ulation nonlinearity χ(3)xpm = χ(3)(−ωp, ωs,−ωs, ωp) =

30

χ(3)(−ωs, ωp,−ωp, ωs) on the other hand is given by(Schmidt and Imamoglu, 1996)

χ(3)xpm =

|µ13|2|µ24|2%2ε0~3

1

Ω2c

[

1

∆ω42+ i

Γ4

2∆ω242

]

. (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 (χ(3)3−level)

depicted in Figure 22a: for this system consisting of theground state |a〉 and an intermediate state of oppositeparity |b〉 and a final state |c〉, we find

Re[χ(3)3−level] =

|µab|2|µbc|2%2ε0~3

1

∆ω2ab∆ωbc

, (81)

where µij and ∆ωij 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-xpm scheme 1/(4∆ω2

ab) is replaced by 1/Ω2c . Even

though one needs to have ∆ω2ab ≥ Γ2

b in a three-levelscheme to avoid absorption, Ω2

c Γ23 is possible in the

EIT-xpm system. This in turn implies that the mag-nitude of Kerr nonlinearity can be orders of magnitudelarger in the latter case: this is the essence of the giantKerr effect (Schmidt and Imamoglu, 1996).

∆ ω

Ω

Ω

Ω c

s

p

Ω

Ω

ab

s

p

a

b

1

2d3d

4c

FIG. 22 Level configuration for cross-phase modulation inordinary three-levels cheme (left) and dressed-state represen-tation of giant-Kerr scheme (right).

We can understand the origin of this enhancementusing the dressed-state picture depicted in Figure 22b:when we apply the dressed-state transformation intro-duced in Sec. III, we find that the initial and finalstates (which remain unchanged) are coupled via twointermediate states |2d〉 and |3d〉. The two paths con-tributing to virtual excitation of state |4〉 experience aconstructive interference for ωp tuned in between the twodressed-states, partially leading to the above mentionedenhancement. Perhaps more importantly however, the“effective detuning” between the intermediate dressed-states is given by Ωc and can be much smaller than theirwidth.

Alternatively, we can understand the predicted en-hancement of the Kerr nonlinearity by recalling the steepdispersion obtained at transparency for Ω2

c Γ23: in this

limit, small changes in two-photon detuning δ that arecaused for example by a shift in the energy of state |2〉can give rise to drastic increase in Re[χ(1)] experiencedby the probe field. We can understand the role of the sig-nal field as creating an ac-Stark shift of state |2〉, therebyleading to a large change in the index of refraction at ωp

that is proportional to Ω2s.

Finally, we note that by optical pumping and by choos-ing the polarization of the lasers appropriately, the en-ergy level diagram depicted in Fig. 21 can be realizedpractically for all alkali atoms. It has been estimated

that the enhancement of χ(3)xpm over a standard three-level

atomic medium with identical density and dipole matrixelements could be as much as six orders of magnitude(Schmidt and Imamoglu, 1996). Naturally, the interestin this system arises from the possibility of doing nonlin-ear optics on the single photon level, without requiring ahigh-finesse cavity; this possibility was already discussedin the original proposal (Schmidt and Imamoglu, 1996),but in the limit of continuous-wave fields. Recent exper-iments have already demonstrated the enhancement ofKerr nonlinearities in the limit of ∆ω42 = 0 (Braje et al.,

2003); this is the purely dissipative limit of χ(3)xpm and was

discussed by Harris and Yamamoto in the context of atwo-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 itis this 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 nonlinear phaseshifts for pulsed fields. This limitation arises from thefact that the probe and signal pulses travel at vastly dif-ferent group velocities in the limit Ω2

c Γ23 and their

interaction time will be limited by the fact that the slow

probe 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 whereΓ3Γ2 Ω2

c Γ23, ∆ = 0, and concentrate on the limit

of small |δ|. For this system, we have a probe field groupvelocity

vp =Ω2

c

Γ3

1

%σ13, (82)

where σ13 = 3λ2p/2π is the peak absorption cross-section

of the |1〉 − |3〉 transition (λp = 2πc/ωp). Given thatvp c, the time it takes for the probe pulse to traversethe medium

τd =Γ3

Ω2c

%σ13L , (83)

31

referred to as the group delay time (Harris and Hau, 1999;Harris and Yamamoto, 1998), can be much longer thanthe traversal time of the signal field Ts = L/c and thepulse-widths of the probe (τp) and signal (τs = τp) fields.As discussed earlier, the pioneering experiment of Hauet al. (1999) already demonstrated τD > 1 msec τp .

The generated nonlinear phase shift is proportionalto the interaction length of the probe and signal fields.In the 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 τp; i.e.

Lh = vpτp =Ω2

c

Γ3

1

%σ13τp , (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 (Harrisand Hau, 1999)

∂Ep

∂z+

(

1

vp+

1

c

)

∂Ep

∂t= i

π

λpχ(3)

xpm

∣

∣

∣Es

(

t− z

c

)∣

∣

∣

2

Ep

=

(

i

4(∆ω42 + iΓ4/2)vp

)

∣

∣

∣Ωs

(

t− z

c

)∣

∣

∣

2

Ep , (85)

where we have used Eq. (80).When the group delay is negligible compared to τp (i.e.

L Lh), and we assume Gaussian input pulses, we findfor the peak (complex) phase shift for the maximum ofthe pulse

φxpm →(

Γ4

∆ω42 + iΓ4/2

)

ns

A

σ24

2

√

ln 2

π

τd

τp, (86)

where ns is the number of photons in the pulse and A itscross section area. Eq.(86) is, as expected, the result onewould obtain using Eq. (80) without taking into accountthe group velocity mismatch between the two pulses. Inthe opposite limit of L Lh, we have

φxpm →(

Γ4

∆ω42 + iΓ4/2

)

ns

A

σ24

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 Ωc, %, L) (Har-ris and Hau, 1999), and is on the order of 0.1 radiansfor single photon-pulses focused into an area A ∼ σ24.This remarkably simple result assumes that we take∆ω42 ∼ Γ4/2, in which case we get a two-photon ab-sorption coefficient that is comparable to the nonlinearphase shift.

B. Cross-phase modulation using single-photon pulses with

matched group velocities

From the perspective of applications in quantum in-formation processing, the result obtained in the previous

subsection 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 askthen is whether one can obtain larger nonlinear phaseshifts by assuring that both probe and signal pulses travelat the same ultra-slow group velocity.

Before proceeding with the analysis, we emphasize thatit is possible to consider an atomic medium with two co-herently driven atomic species, each establishing EIT atdifferent frequencies. If the two atomic species are dif-ferent isotopes of the same element, then the frequenciesat which the medium becomes transparent can differ byan amount that is on the order of the hyperfine split-ting. We can then envision a scenario where two differentlight pulses, centered at the two transparency frequen-cies, travel with the identical ultra-slow group velocitythrough the medium containing an optically dense mix-ture of these two isotopes (A and B) of the same element.By adjusting the external magnetic field, it is also pos-sible to ensure that one of these light pulses that seesEIT induced by species B will also be (nearly) resonantwith the |2〉−|4〉 transition of species A. By choosing thispulse to be the signal pulse, one can realize giant Kerr in-teraction between two ultra-slow light pulses (Lukin andImamoglu, 2000). An alternative scheme based on only asingle atomic species was recently proposed by Petrosyanand Kuritzki (Petrosyan and Kurizki, 2002).

Even though the analysis of nonlinear optical processesusing single-photon pulses requires a full quantum fieldtheoretical approach, we first consider the classical limitfollowing up on the analysis previously presented for afast signal pulse. In the limit of a slow signal pulse withgroup velocity vs we obtain

φxpm =Γ4

∆ω42 + iΓ4/2

ns

A

σ24

2

√

ln 2

π

τd

τp, (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 τd τp (Lukin and Imamoglu,2000).

To determine the maximum possible phase shift, werecall that for an optically dense medium the EITtransmission peak has a width given by ∆ωtrans =Ω2

c/(Γ3

√%σ13L). If we set τp = ∆ω−1

trans, we find

τd

τp≤

√

%σ13L . (89)

Since the other factors on the right hand side of Eq. 88can be of order unity, we can obtain φxpm ≥ π, providedthat we have %σ13L ≥ 10: presence of an optically thickmedia is essential for large photon-photon interaction.At the same time, we note that this interaction is nec-essarily slow and has a φxpm-bandwidth product that is

32

independent of the optical depth (Lukin and Imamoglu,2000).

Next, we examine the validity of this conclusion us-ing a full quantum field theoretic approach (Lukin andImamoglu, 2000). To this end, we replace the classicalelectric fields describing the signal and probe fields with

operators Ei(z, t) = E(+)i (z, t) + E

(−)i (z, t), where

E(+)i (z, t) =

∑

k

√

~kc

2ε0Vaki(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 Γ2 = 0 and|∆ω42| Γ4. To ensure however that we can adiabati-cally eliminate atomic degrees of freedom, we also needto impose a finite bandwidth ∆q to 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 τ−1

p < ∆q ∆ωtrans and arrive at a

pair of operator equations for Ep(z, t) and Es(z, t) thathave the identical form as the one given in Eq. (85), pro-vided we replace c by vs on the right-hand-side. Theseequations can then be solved to yield

E(+)p (z, t) = E(+)

p (0, t′)

× exp

[

iµ2

24

~2

τd

4π∆ω42E(−)

s (0, t′)E(+)s (0, t′)

]

,

E(+)s (z, t) = E(+)

s (0, t′)

× exp

[

iµ2

13

~2

τd

4π∆ω42E(−)

p (0, t′)E(+)p (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 the in-teraction of two single-photon pulses. To this end, weassume an input state of the form

|1i〉 =∑

k

ξka†ki |0〉 , (92)

for both fields, where we require∑

k |ξk|2 = 1 to en-sure proper normalization. The spatio-temporal wave-function of each pulse is given by the single-photon am-

plitude Ψi(z, t) = 〈0|E(+)i (z, t)|1i〉. To evaluate the cor-

relations induced by the interaction, we concentrate onthe two-photon amplitude

Ψsp(z, t; z′, t′) = 〈0|E(+)s (z, t)E(+)

p (z′, t′)|1s, 1p〉. (93)

One finds that the equal-time (t = t′) local (z = z′)correlations can be written as

Ψsp(z, t; z, t) = Ψs(0, t)Ψp(0, t)eiφ, (94)

where φ = τd∆q σ24Γ4/(4πA∆ω42). Since the equalspace and time commutation relations are proportionalto ∆q, there is explicit dependence on this quantity. Thenonlinear phase shift φ is the counterpart of the classicalphase φxpm obtained earlier in Eq. (88); the equivalenceis obtained by replacing τ−1

p with the quantization band-width ∆q (Lukin and Imamoglu, 2000).

A large single-photon nonlinear phase shift φ that ex-ceeds π radians can be used to implement two-qubitquantum logic gates (Nielsen and Chuang, 2000). How-ever, we have seen that creating large phase shifts be-tween single photon pulses also changes the mode profileof the signal (probe) pulse, conditioned on the presenceof the probe (signal) pulse. Therefore, we expect entan-glement 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 ξk, 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 on thefew-photon level have been predicted and analyzed inmore detail in the four-wave mixing schemes shown infigure 23. In the first scheme discussed by Harris andHau (1999) photons from two pulses (ω1) and (ω3) areconverted into an electromagnetic field pulse at ω4 in thepresence of a strong monochromatic drive field at ω2 (fig-ure 23(a)). In the second scheme (figure 23(b)) discussedin (Johnsson et al., 2002) for co-propagating pump fieldsand in (Fleischhauer and Lukin, 2000; Lukin et al., 1999)for counter-propagating pump fields, the field ω2 is ini-tially absent and spontaneously generated along with ω4.

ω1ω2

1

3

4

3ωω

(a)

4

21

ω ω1

2

(b)

2

3

4

5

∆ 3ω4ω

FIG. 23 Four-wave mixing schemes for the generation of onefield of frequeny ω4 (a), and for the generation of two field ω2

and ω4 (b).

In the first scheme, see Fig. 23(a), the input pulse ω1

experiences a group delay due to EIT on the |1〉 − |3〉transition, while the second one, ω3, moves almost withthe vacuum speed of light. Harris and Hau showed withinan undepleted pump approximation, that the conversionefficiency, i.e. the ratio of the number of generated pho-tons at ω4 to the number of input photons at ω1 is givenby a dimensionless function Φ(η, κL) multiplied by the

33

number of input photons per atomic cross section.

nout4

nin1

= Φ(η, κL)nin

3

A3σ24 (95)

Here κ is the (E-field) absorption coefficient at the |1〉 −|4〉 transition and L the medium length. η = L/Lh isthe ratio 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 κL→ 0 and of a smallgroup delay within the medium η → 0, the usual resultof perturbative long-pulse nonlinear optics is obtained,which for Gaussian pulses reads

Φ(η, κL) =

[

ln(2)

π

]1/2T1

√

T 21 + T 2

3

ηκL. (96)

Here the Tµ’s denote the temporal length of the pulses.In the limit of η →∞, i.e. if the group velocity reductionof ω1 is large, such that the ω3 pulse sweeps over it veryrapidly, the function is given by

Φ(η, κL) =

[

ln(2)

π

]1/2κL

ηexp(

−2κL)

(97)

i.e. falls off with η−1. A maximum of Φ of about 0.074is attained when η = κL = 1, i.e. when the pulse de-lay is just one pulse length and if the medium has anoptical thickness of one. In this case about 10 photonsper atomic cross section are sufficient to obtain perfectconversion.

In the second scheme, where the field ω2 is generated,see Figure 23(b), a finite detuning from level |3〉 is neededto suppress absorption. Furthermore in order to cancelac-Stark shifts the frequencies ω1 and ω2 can be tunedmidway between two states with appropriate transitionmatrix elements (Johnsson et al., 2002). Efficient fre-quency conversion (Babin et al., 1999; Ham et al., 1997b;Hemmer et al., 1994; Hinze et al., 1999), generation ofsqueezing (Lukin et al., 1999), as well as the possibilityof mirrorless oscillations (in counter-propagating geom-etry) have been predicted (Lukin et al., 1998b) and inpart experimentally observed (Zibrov et al., 1999). Ithas been shown in (Fleischhauer and Lukin, 2000) thatan extremely low input photon flux φthr is sufficient toreach the threshold of mirrorless oscillations

φthr = fNγ12 (98)

where N is the number of atoms in the beam path andγ12 is the coherence decay rate of the |1〉− |2〉 transition.f is a numerical pre-factor of order unity. Johnsson et al.have shown that in the adiabatic limit and for γ12 = 0 aneffective interaction hamiltonian of the four-wave mixingprocess can be derived which reads for a one-dimensionalmodel (Johnsson and Fleischhauer, 2002)

Hint =~gc

2∆

∫

dzن

1Ω†3Ω2Ω4 + h.c.

Ω†3Ω3 + Ω†

4Ω4

, (99)

where Ωµ are the operators corresponding to the (com-plex) Rabi-frequencies of the fields and a common cou-pling constant g = %µ2ω/2~ε0c of all transitions was as-sumed. One recognizes that in contrast to the case ofusual non-resonant nonlinear optics, the sum of the in-tensities of the resonant fields appears in the denomi-nator. As a consequence in a co-propagating geometry,the conversion length, i.e. the length after which a pairof photons in the input fields ω1 and ω3 is completelyconverted into a pair of photons in the fields ω2 and ω4

decreases with decreasing input intensity. Thus weak in-put fields lead to a faster conversion than stronger 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 since very recently that therehave been first experimental demonstrations of the pre-dicted effects. Kuzmich and coworkers (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 photons gen-erated with a controllable time delay by resonant Ra-man scattering on atomic ensembles. The intermediatestorage and subsequent retrieval of correlated photons inatomic ensembles are essential ingredients of the proposalof Duan and co-workers for a quantum repeater (Duanet al., 2001), which is an important tool for long-distancequantum communication.

D. Few-photon cavity-EIT

It is well known in quantum optics that presence ofhigh-finesse cavities can be used to enhance photon-photon interactions for example for the purpose of quan-tum 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 willshow in the present subsection that electromagneticallyinduced transparency combined with cavity quantumelectrodynamics can lead to a number of interesting lin-ear and nonlinear effects.

Let us first discuss linear phenomena associated withintracavity EIT. Consider a single three-level atom placedinside a resonator where the quantized resonator modeacts as the coupling field between the internal states |2〉and |3〉. As shown by Field (1993) electromagneticallyinduced transparency on the probe transition is possi-ble already with few resonator photons or even vacuum,provided that the atom and the resonator are stronglycoupled. To see this we note that the fully quantum in-teraction Hamiltonian

Hint = −~Ωp

2|3〉〈1| − ~

g

2a |3〉〈2|+ h.c. (100)

separates into effective three-level systems

|1, n〉 Ωp←→ |3, n〉 g√

n+1←→ |2, n + 1〉 (101)

34

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, issufficiently large and the atom is initially in state |1〉,EIT can be achieved for the probe field even for n = 0.Eq.(101) suggests another interesting application: Theeffective three-level systems have dark states. For exam-ple

|D〉 = cos θ(t)|1, 0〉 − sin θ(t)|2, 1〉 (102)

with tan θ(t) = Ωp(t)/g. Thus stimulated Raman adia-batic passage from state |1, 0〉 to state |2, 1〉 induced bythe probe field offers the possibility of a controlled gener-ation of a single cavity photon. The potential usefulnessof this effect in cavity QED has first been pointed outby Parkins et al. (Parkins et al., 1993). Law and Eberlyproposed an application for the generation of a singlephoton in a well-defined wavepacket (Law and Eberly,1996). Recently such a “photon pistol” was experimen-tally 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| − ~

Ωc

2|3〉〈2|+ h.c. (103)

and we have the coupling

|1, n〉 g√

n←→ |3, n− 1〉 Ωc(t)←→ |2, n− 1〉 (104)

The dark state for n = 0 is identical to that given inEq. (102), except tan θ(t) = g/Ωc(t) in the present case.The mapping provides a possibility for measuring cavityfields (Parkins et al., 1995). It was shown furthermore byPellizzari et al (1995) that a quantum-state transfer fromone atom to a second atom via a shared cavity mode canbe used to implement a quantum gate (Pellizzari et al.,1995). Finally Cirac et al. (Cirac et al., 1997) proposedan application to transfer quantum information betweenatoms at spatially separated nodes of a network.

All of the above discussed effects require however thestrong coupling of the single atom to the resonator mode.This experimentally challenging requirement can partlybe alleviated if an optically thick ensemble of three levelatoms is used. When an ensemble of N three-level atomsinteracts with a single resonator (probe) mode and a clas-sical (coupling) field according to

Hint = −~

2

N∑

j=1

[ga |3〉jj〈1|+ ~Ωc |3〉jj〈2|+ h.c.] (105)

only symmetric collective states are coupled to the initialstate |1, 1, ...1〉, where all N atoms are in the ground state|1〉:

|1N 〉 ≡∣

∣

∣1 . . . 1

⟩

,

|1N−12〉 ≡ 1√

N

N∑

j=1

∣

∣

∣1 . . . 2j . . . 1

⟩

, (106)

|1N−22

2〉 ≡ 1√

2N(N − 1)

N∑

i6=j=1

∣

∣

∣1 . . . 2i . . . 2j . . . 1

⟩

,

etc.

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

|1N , 1〉 g√

N←→ |1N−13, 0〉 Ωc(t)←→ |1N−1

2, 0〉. (107)

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

g −→ g√

N. (108)

Thus controlled dark-state Raman adiabatic passagefrom a state with a single photon in the resonator toa single collective excitation and vice versa is possiblewithout the requirement of a strong coupling limit ofcavity-QED, defined here as g2 κΓ3, where κ is thecavity decay rate. For higher photon numbers more com-plicated coupling schemes emerge. All of them do pos-sess however dark states which allow a parallel transferof arbitrary photon states with n N to collective ex-citations, which has important applications for quantummemories for photons and quantum networking (Lukinet al., 2000b).

As discussed in previous subsections, EIT can be usedto achieve a resonant enhancement of nonlinear interac-tions leading e.g. to a cross-phase modulation. Largephase shifts produced by the interaction of individualphotons are very appealing for their potential use inquantum information processing. Imamoglu et al. (1997)showed that if a medium with sufficiently large Kerr non-linearity is put into an optical cavity, it can lead to a pho-ton blockade effect. When a photon enters a previouslyempty cavity, the induced refractive index detunes thecavity resonance. If this frequency shift is larger than thecavity linewidth a second photon cannot enter and willbe reflected. This can be employed e.g. to build a con-trollable single-photon source or a controlled-phase gate(Duan and Kimble, 2003). Imamoglu et al. suggestedthe use of a large ensemble of atoms in the resonator toenhance the photon blockade effect. It was later shown(Gheri et al., 1999) however, that the large dispersion ofthe EIT medium does not allow for a collective enhance-ment of the nonlinear atom-cavity coupling. While pho-ton blockade can be achieved using either single atom ormulti-atom EIT systems (Werner and Imamoglu, 1999),the strong coupling limit of cavity-QED appears to be re-quired in both cases. It must be emphasized however thatin contrast to a many-atom cavity-EIT medium, a collec-tion of N 1 two-level atoms in a cavity is a nearly ideallinear system, exhibiting vanishingly small photon block-ade effect. The survival of the nonlinearity in the N 1

35

atom cavity-EIT is related to the reduction of the cavity-linewidth, or equivalently the width of the atom-cavitydark state (Lukin et al., 1998a; Werner and Imamoglu,1999).

VII. SUMMARY AND PERSPECTIVES

EIT has undoubtedly made a mark upon optical sci-ence. Hopefully we have succeeded in explaining the re-lationship between EIT and the earlier related ideas ofcoherent preparation of atoms by fields and especiallylinked it with the concept of the dark-state. We stressagain the distinct feature of EIT and related phenomena,in contrast to the earlier spectroscopic tools such as co-herent population trapping, is that they occur in mediathat are optically dense and cause not only a modificationof the 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 fourteenyears on from the first experimental demonstration of thephenomenon of EIT. Much research activity has been un-dertaken in the period since this discovery to understandthe phenomenon and its various implications. More im-portantly there has been a concerted effort in many labo-ratories to harness the effect for applications. It is infor-mative to review the extent to which this field has grownrapidly in these years, and has so remained a healthy andactive subject for the last decade with no sign of declin-ing activity. There are constantly new and interestingproposals and demonstrations that extend EIT to freshapplications. It has become a standard tool in the kit forresearchers studying the optical properties of atomic andmolecular gases. For these reasons of durability and util-ity we believe that EIT has earned a significant place inoptical physics that make it a valuable topic of study fornew research students and experienced researchers alike.

We have tried in this review to cover the main ideasbehind the application of EIT and related dark-state phe-nomena to optics. It is inevitable in a work such as thisthat there have been omissions. Much work has indeedbeen published in this field, especially in the theoreticalarea, and it was simply not possible to cover or cite everysingle contribution. We humbly apologize to anyone wehave failed to cite. Our intention was primarily to providethe reader new to the field with an accessible and com-prehensive framework that will support them when theymake further more detailed explorations of this topic. Wehope that in this we have succeeded and that we have pro-vided a useful introduction to the theoretical ideas thatunderpin the subject.

Where possible we have tried to examine the results

of real experiments. Again we have been selective choos-ing the most convenient illustrative examples rather thanattempting a comprehensive review of all work. The de-tails of the experiments discussed here are unfortunatelyrather sparse due to the constraints of space. Indeed wehave not explored in any structured way the experimen-tal techniques that are required to achieve robust EITand thus to investigate these phenomena. This omissionis regrettable, but comes about since there are in factmany types of situation where EIT can be observed, e.g.in dense gases with transform limited high power pulsedlasers, with CW lasers interacting with vapours usingDoppler free geometries or indeed with CW lasers in coldgases. It is thus not practicable to cover so many diverseexperimental situations and to understand the types ofexperiments the reader must consult the cited literatureand where necessary other reviews and textbooks to ob-tain a fuller understanding of the required experimentalmethods.

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 non-linear optics to move towards em-ploying solid-state media. The reason for this is thatgas phase media still offer the physicist unique features;small homogeneous and inhomogeneous linewidths com-pared to most solid state media, transparency over muchof the IR-VUV range, possibilities to laser cool and highthresholds against optical damage. Moreover a problemnormally identified for gas phase non-linear optical me-dia, the low value of the non-linear susceptibility com-pared to many solid-density optical crystals, is circum-vented by EIT since it permits the physicists to use res-onance to hugely enhance the susceptibility.

Having reviewed the subject and examined a numberof the applications that have already been thoroughlyinvestigated we now turn to examine the future. It is ofcourse a rather risky undertaking to make any predictionsabout what new applications are likely to emerge andlikely as not we will be wrong about much of what follows.Nevertheless we feel it is of use to other scientists in thisfield at least to offer some thoughts on this subject inthe hope of stimulating further ideas. For safety we willconfine ourselves to areas where there is at least somecurrent activity and so some evidence upon which to baseour extrapolations.

The potential of EIT and maximal coherence to gen-erate high brightness coherent fields in the short wave-length range remains only partially explored. EIT hasbeen shown to enhance frequency conversion efficienciesto the 0.01 level in up-conversion to the 100nm fre-quency range. By careful choice of the atomic systemand through judicious optimisation of the density andlength of the medium and the use of transform-limitedlasers to drive all the fields it may be possible to improvethe conversion efficiency by a substantial further factor.This could result in the generation of transform limitednano-second pulses (spectral linewidths of ∼ 100MHz)

36

of VUV radiation with energies of 10’s of microjoules.These could be of utility in a number of applicationsin non-linear spectroscopy, for instance if combined witha second longer wavelength tunable laser to excite two-photon transitions to highly excited states, or perhapsin Raman spectroscopy. The fixed frequency nature ofthe output would appear to preclude a wider range ofapplications.

Whilst 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 in the Ra-man like excitations since there is a limit to the degreeof up-conversion as the initial excitation is via differencefrequency mixing of the applied fields. There is thereforean inherent limit set at around 170nm for the shortestwavelength that can be generated due to the energy levelstructure of the atoms and the shortest available wave-lengths from lasers. For shorter wavelength extensionsan important problem that must be solved will be howto excite maximal coherence between the ground-stateand a very highly excited upper state. This should usesum-frequency excitation by the applied lasers. The cre-ation of maximal coherence in this case cannot use theconventional STIRAP type adiabatic excitation schemes.Recently a promising alternative has been demonstrated,Stark Chirped Rapid Adiabatic Passage (SCRAP), withthe potential for efficient excitation of maximal coherencebetween the ground-state and a very highly excited stateof an atom (Rickes et al., 2000). A recent first demon-stration of the use of this technique showed a degree ofenhancement to XUV generation via third-harmonic gen-eration (Rickes et al., 2003), but further work is neededin this area.

Besides the potential for efficient generation of coher-ent radiation in new frequency domains, resonant nonlin-ear optics based on EIT will be of growing importance forcontrolled nonlinear interactions at the few photon level.The above mentioned limitations concerning tunabilityand accessible frequencies are of no relevance here. Pho-tons are ideal carriers of quantum information, yet pro-cessing of this information e.g. in a quantum computeris quite difficult. The reason for this is the weaknessof photon-photon interactions in usual nonlinear media.Here EIT may offer new directions. Although some the-oretical proposals exist, the full potential of EIT basednonlinear optics for these applications is still not ex-plored. In particular on the experimental side little hasbeen done in this direction.

Another feature of EIT, which makes it very attrac-tive for future applications in quantum information isthe possibility to transfer coherence and quantum statesfrom light to collective atomic spin excitations. Quan-tum memories for photons (Fleischhauer and Lukin,2002; Lukin et al., 2000b) as discussed in Sec.III arejust one potential avenue, quantum networks based on

them including quantum repeaters are another (Kuzmichet al., 2003; van der Wal et al., 2003). Combining thestate mapping techniques with atom-atom interactionsin mesoscopic samples may provide new tools to gener-ate photon wave-packets with tailored quantum states orto manipulate these states. A first proposal in this direc-tion using a dipole-blockade of Rydberg excitations wasgiven in (Lukin et al., 2001). The controlled coupling ofa photon to many atomic spins in EIT could be of inter-est for the preparation or the probing of entangled manyparticle states. We have just begun to explore the role ofentanglement in many-body physics and quantum infor-mation. Laser manipulation of spin ensembles via EITcould provide a very useful tool in this quest.

EIT also offers new possibilities in linear and nonlinearmatter-wave optics. As pointed out in Sec.IV slow lightin atomic gases corresponds to a quasi-particle which isa mixed electromagnetic-matter excitation. For very lowgroup velocities almost all excitation is concentrated inatomic spins while the propagation properties are stillmostly determined by the electromagnetic part. Thusslow light in ultra-cold atomic gases provides a new wayto control the propagation of matter waves with poten-tial applications in matter-wave interferometry (Zimmerand Fleischhauer, 2004) or if atom-atom interactions areincluded in nonlinear matter-wave optics (Masalas andFleischhauer, 2004).

Ultra-fast measurements have recently entered the at-tosecond domain (Drescher et al., 2002). The possibil-ity of employing the highly efficient multi-order Ramangeneration that results from an adiabatically preparedmaximal vibrational coherence has already been investi-gated. A key problem to solve before this can be widelyemployed for ultra-fast measurements will be the syn-chronisation of the train of sub-femtosecond pulses thatare generated with external events. One possibility (sug-gested by A. Sokolov) is to use the pulse train to ”mea-sure” processes in the modulated molecules themselvesso that the synchronisation is automatically satisfied.

As we have emphasized throughout this article, coher-ent preparation techniques that are at the heart of EITcan be easily implemented in optically dense atomic sam-ples in the gas phase. It is generally argued however, thatthe range of its applications can be significantly enlargedif EIT is implemented in solid-state media. The obvi-ous roadblock in this quest are the ultrashort coherencetimes of optical transitions in solids: even for transitionsthat are not dipole-allowed, the coherence times are typi-cally in the sub-nanosecond timescales. Among the manydifferent physical mechanisms contributing to this fastdephasing, perhaps the most fundamental one is the in-teraction of electrons with lattice vibrations (phonons).While cooling the samples to liquid helium tempera-tures reduces the average number of phonons and therebyphonon-induced-dephasing, the coherence times still re-main significantly shorter than their atomic counterparts.

Fortunately, there is an exception to the general ruleof ultrashort dephasing times in solids: electron spin

37

degrees-of-freedom in a large class of solid-state materialshave relatively long coherence times, ranging from 1µsecin optically active direct-band-gap bulk semiconductorssuch as GaAs (Wolf et al., 2001), to 1msec in rare-earthion 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 can 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 et al., 1997a; Turukhinet al., 2002) and nitrogen-vacancy centers in diamond(Wei and Manson, 1999). In the experiments carriedout using Pr doped Y2SiO5 at cryogenic temperatures(T= 5K), Turukhin and co-workers have observed groupvelocities as slow as 45 m/s and a group delay of 66µsec,for light pulses with a pulsewidth of 50µsec (Turukhinet al., 2002). Storage and retrieval of these pulses havealso been demonstrated (Turukhin et al., 2002). Whilethe ratio of the group delay to the pulsewidth in theseexperiments is on the order of unity, this is an impressiveand promising development for solid-state EIT.

Optoelectronics and photonics technology is largelybased on semiconductor heterostructures formed out ofcolumn III-V semiconductors. The conduction band elec-trons of these elements have predominantly s-type wave-functions with small spin-orbit coupling. As a result, thespin coherence times are 4 orders of magnitude longerthan the radiative recombination rate of excitons. Thepossibility of observing efficient EIT and slow light prop-agation in a doped GaAs quantum well in the quantumHall regime has been discussed (Imamoglu, 2000). Whilethis particular realization requires low temperatures andhigh magnetic fields (B ∼ 10 Tesla), long spin coher-ence times in GaAs have been observed even at roomtemperature and with vanishing magnetic fields (Wolfet al., 2001). In fact, there is substantial activity in theemerging field of spintronics (Wolf et al., 2001) and it isplausible to argue that EIT can play a role by providinglong optical delays and/or enabling “spin-photon” infor-mation transfer.

Yet another interesting extension of gas-phase EIT toplasmas was suggested by Harris (1996). Here classicalinterference effects in parametric wave interactions whichare analogous to interference effects in 3-level atoms leadto an induced transparency window below the plasmafrequency. This could have interesting applications formagnetic fusion, plasma heating and plasma diagnostics.For some recent work on this see e.g. (Litvak and Tok-man, 2002; Shvets and Wurtele, 2002).

Even though EIT is by now a well established tech-nique, new applications continue to appear almost regu-larly. It was recently proposed for example that a “dy-namical photonic band-gap” structure can be realizedin an EIT medium by spatially modifying the refrac-tive index with a standing wave-optical field (Andre andLukin, 2002). This is achieved by making use of the large

absorption-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-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 can then be possible to achievephoton counting with efficiency exceeding 99%. Such adevice could be of great use in quantum optical informa-tion processing (Knill et al., 2001).

Acknowledgement

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

References

Affolderbach, C., M. Stahler, S. Knappe, and R. Wynands,2002, Applied Physics B 75, 605.

Agarwal, G. S., 1993, Phys. Rev. Lett. 71, 1351.Alzetta, G., A. Gozzini, L. Moi, and G. Orriols, 1976, Nuovo

Cimento B. 36, 5.Andre, A., and M. Lukin, 2002, Phys. Rev. Lett. 89, 143602.Arimondo, E., 1996, Progress in Optics 35.Armstrong, J. A., and J. J. Wynne, 1974, Phys. Rev. Lett.

33(20), 1183, see the improved χ(3) fitting: Armstrong,Lloyd and Brian Lee Beers, Phys. Rev. Lett. Vol 34(20):1290–1291(1975).

Autler, S. H., and C. H. Townes, 1955, Physical Review 100,703.

Babin, S., U. Hinze, E. Tiemann, and B. Wellegehausen, 1996,Opt. Letts. 21, 1186.

Babin, S. A., E. V. Podivilov, D. A. Shapiro, U. Hinze, E. Tie-mann, and B. Wellegehausen, 1999, Phys. Rev. A 59, 1355.

Bajcsy, M., A. S. Zibrov, and M. D. Lukin, 2003, Nature 426,638.

Bergmann, K., H. Theuer, and B. W. Shore, 1998, Rev. Mod.Phys. 70, 1003.

Boller, K. J., A. Imamoglu, and S. E. Harris, 1991, Phys. Rev.Lett. 66(20), 2593.

Bol’shov, L. A., and V. V. Likhanskii, 1985, Sov. J. Quant.Electron. 15, 889.

Braje, D. A., V. Balic, G. Y. Yin, and S. E. Harris, 2003,Phys. Rev. A 68, 041801(R).

Buckle, S. J., S. M. Barnett, P. L. Knight, M. A. Lauder, andD. T. Pegg, 1986, Opt. Acta 33, 2473.

38

Budker, D., D. F. Kimball, S. M. Rochester, and V. V.Yashchuk, 1999, Phjys. Rev. Lett. 83, 1767.

Budker, D., V. Yashchuk, and M. Zolotorev, 1998, Phys. Rev.Lett. 81, 5788.

Buffa, R., M. P. Anscombe, and J. P. Marangos, 2003, Phys-ical Review A 67, 033801.

Carmichael, H. J., 1993.Cerboneschi, E., and E. Arimondo, 1995, Phys. Rev. A 52,

R1823.Cirac, J. I., P. Zoller, H. J. Kimble, and H. Mabuchi, 1997,

Phys. Rev. Lett. 78, 3221.Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, 1992,

Atom-Photon Interactions (Wiley, New York).Cszeznegi, J. R., and R. Grobe, 1997, Phys. Rev. Lett. 79,

3162.Dalibard, J., Y. Castin, and K. Molmer, 1992, Phys. Rev.

Lett. 68, 580.Dalton, B. J., and P. L. Knight, 1982, Opt. Comm. 41, 411.Dorman, C., I. Kucukkara, and J. P. Marangos, 2000a, Opt.

Commun. 180, 263.Dorman, C., I. Kucukkara, and J. P. Marangos, 2000b, Phys.

Rev. A 61, 013802.Drescher, M., M. Hentschel, R. Kienberger, M. Uiber-

acker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh,U. Kleineberg, U. Heinzmann, and F. Krausz, 2002, Na-ture 419, 803.

Duan, L. M., J. I. Cirac, and P. Zoller, 2002, Phys. Rev. A66, 023818.

Duan, L. M., J. I. Cirac, P. Zoller, and E. S. Polzik, 2000,Phys. Rev. Lett. 85, 5643.

Duan, L. M., and H. J. Kimble, 2003, quant-ph/0309187.Duan, L. M., M. D. Lukin, J. I. Cirac, and P. Zoller, 2001,

Nature 414, 413.Dutton, Z., M. Budde, C. Slowe, and L. V. Hau, 2001, Science

293, 5530.Eberly, J. H., M. L. Pons, and H. R. Haq, 1994, Phys. Rev.

Lett. 72, 56.Faist, J., F. Capasso, C. Sirtori, K. W. West, and L. Pfeiffer,

1997, Nature 390, 589.Fano, U., 1961, Phys. Rev. 124, 1866.Fano, U., and J. Cooper, 1965, Physical Review A 137, 1364.Field, J. E., 1993, Phys. Rev. A 47, 5064.Field, J. E., K. Hahn, and S. E. Harris, 1991, Phys. Rev. Lett.

67, 3733.Fleischhauer, M., 1994, Phys. Rev. Lett. 72, 989.Fleischhauer, M., 1999, Optics Express 4, 107.Fleischhauer, M., C. H. Keitel, M. O. Scully, and C. Su, 1992a,

Opt. Commun. 87, 238.Fleischhauer, M., C. H. Keitel, M. O. Scully, C. Su, B. T.

Ulrich, and S. Y. Zhu, 1992b, Phys. Rev. A 46, 1468.Fleischhauer, M., and M. D. Lukin, 2000, Phys. Rev. Lett.

84, 5094.Fleischhauer, M., and M. D. Lukin, 2002, Phys. Rev. A 64,

022314.Fleischhauer, M., M. D. Lukin, A. B. Matsko, and M. O.

Scully, 2000, Phys. Rev. Lett. 84, 3558.Fleischhauer, M., and A. S. Manka, 1996, Phys. Rev. A 54,

794.Fleischhauer, M., and C. Mewes, 2002, in Proceedings of the

Enrico Fermi School on Experimental Quantum Computa-

tion and Information (IOS Press), pp. 511–529.Fleischhauer, M., and M. O. Scully, 1994, Phys. Rev. A 49,

1973.Fry, E. S., X. Li, D. Nikonov, G. G. Padmabandu, M. O.

Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson,and S.-Y. Zhu, 1993, Phys. Rev. Lett. 70, 3235.

Gao, J. C., C. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang,Y. Jiang, D. Wang, and D. Jiang, 1992, Optics. Commun.93, 323.

Gardiner, C. W., A. S. Parkins, and P. Zoller, 1992, Phys.Rev. A 46, 4363.

Garton, W. R. S., 1966, Advances in Atomic and MolecularPhysics 2, 93.

Gaubatz, U., P. Rudecki, S. Schiemman, and K. Bergmann,1990, J. Chem. Phys. 92, 5363.

Gheri, K. M., W. Alge, and P. Grangier, 1999, Phys. Rev. A60, R2673.

Gornyi, M. B., B. G. Matisov, and Y. V. Rozhdestvenskii,1989, Sov. Phys. JETP 68, 728.

Gray, H. R., R. M. Whitley, and C. R. Stroud, 1978, OpticsLetters 3(6), 218.

Grigoryan, G. G., and Y. T. Pashayan, 2001, Phys. Rev. A64, 013816.

Grobe, R., F. T. Hioe, and J. H. Eberly, 1994, Phys. Rev.Lett. 73, 3183.

Hahn, K. H., D. A. King, and S. E. Harris, 1990, Phys. Rev.Lett. 65(22), 2777.

Hakuta, K., unpublished, private communication.Hakuta, K., M. Suzuki, M. Katsuragawa, and J. Z. Li, 1997,

Phys. Rev. Lett. 79, 209.Hald, J., J. L. Sorensen, C. Schori, and E. S. Polzik, 1999,

Phys. Rev. Lett. 83, 1319.Ham, B. S., P. R. Hemmer, and M. S. Shahiar, 1997a, Opt.

Commun 144, 227.Ham, B. S., M. S. Shariar, M. K. Kim, and P. R. Hemmer,

1997b, Opt. Lett. 22, 1849.Hansch, T. W., and P. Toschek, 1970, Z. Phys. 236, 213.Harris, S. E., 1989, Phys. Rev. Lett. 62(9), 1033.Harris, S. E., 1993, Phys. Rev. Lett. 70, 552.Harris, S. E., 1994a, Phys. Rev. Lett. 72, 52.Harris, S. E., 1994b, Opt. Lett 19, 2018.Harris, S. E., 1996, Phys. Rev. Lett. 77, 5357.Harris, S. E., 1997, Phys. Today 36, 1.Harris, S. E., 2000, Phys. Rev. Lett. 85, 4032.Harris, S. E., J. E. Field, and A. Imamoglu, 1990, Phys Rev

Lett 64(10), 1107.Harris, S. E., J. E. Field, and A. Kasapi, 1992, Phys. Rev. A

46, R29.Harris, S. E., and L. V. Hau, 1999, Phys. Rev. Lett. 82, 4611.Harris, S. E., and M. Jain, 1997, Opt. Letts. 22, 636.Harris, S. E., and Z.-F. Luo, 1995, Phys. Rev. A 52, R928.Harris, S. E., and A. V. Sokolov, 1997, Phys. Rev. A. 55,

R4019.Harris, S. E., and Y. Yamamoto, 1998, Phys. Rev. Lett. 81,

3611.Harris, S. E., G. Y. Yin, M. Jain, H. Xia, and A. J. B. Mer-

riam, 1997, Philos. Trans. R. Soc. London 355, 2291.Hau, L. V., S. E. Harris, Z. Dutton, and C. H. Behroozi, 1999,

Nature 397, 594.Hemmer, P. R., K. Z. Cheng, J. Kierstead, M. S. Shariar, and

M. K. Kim, 1994, Opt. Lett. 19, 296.Hemmer, P. R., D. P. Katz, J. Donoghue, M. Cronin-Golomb,

M. S. Shariar, and P. Kumar, 1995, Opt. Letts. 20, 769.Hennrich, M., T. Legero, A. Kuhn, and G. Rempe, 2000,

Phys. Rev. Lett. 85, 4872.Hinze, U., L. Meyer, B. N. Chichkov, E. Tiemann, and

B. Wellegehausen, 1999, Opt. Comm. 166, 127.Hioe, F. T., and R. Grobe, 1994, Phys. Rev. Lett. 73, 2559.

39

Imamoglu, A., 1991, Ph.D. thesis, Stanford University.Imamoglu, A., 2000, Opt. Comm. 179, 179.Imamoglu, A., 2002, Phys. Rev. Lett. 89, 163602.Imamoglu, A., and S. E. Harris, 1989, Optics Letters 14(24),

1344.Imamoglu, A., H. Schmidt, G. Woods, and M. Deutsch, 1997,

Phys. Rev. Lett. 79, 1467.Jain, M., 1994, Phys. Rev. A 50, 1899.Jain, M., A. J. Merriam, A. Kasapi, G. Y. Yin, and S. E.

Harris, 1995, Phys. Rev. Lett. 75, 4385.Jain, M., H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris,

1996, Phys. Rev. Lett. 77, 4326.James, D. F. V., and P. Kwiat, 2002, Phys. Rev. Lett. 89,

183601.Johnsson, M., E. Korsunsky, and M. Fleischhauer, 2002, Opt.

Comm. 212, 335.Johnsson, M. T., and M. Fleischhauer, 2002, Phys. Rev. A

66, 043808.Julsgaard, B., A. Kozhekin, and E. S. Polzik, 2001, Nature

413, 400.Juzeliunas, G., and H. J. Carmichael, 2002, Phys. Rev. A 65,

021601.Juzeliunas, G., M. Masalas, and M. Fleischhauer, 2003, Phys.

Rev. A 67, 023809.Kasapi, A., M. Jain, G. Y. Yin, and S. E. Harris, 1995, Phys.

Rev. Lett. 74, 2447.Kasapi, A., G. Y. Yin, M. Jain, and S. E. Harris, 1996, Phys.

Rev. A 53, 4547.Kash, M. M., V. A. Sautenkov, A. S. Zibrov, L. Hollberg,

G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, andM. O. Scully, 1999, Phys. Rev. Lett. 82, 5229.

Kien, F. E., J. Q. Liang, M. Katsuragawa, K. Ohtsuki,K. Hakuta, and A. V. Sokolov, 1999, Phys. Rev. A. 60,1562.

Knight, P. L., 1990, Phys. Rep. 190, 1.Knill, E., R. Laflamme, and G. J. Milburn, 2001, Nature 409,

46.Kocharovskaya, O., 1992, Physics Reports 219, 175.Kocharovskaya, O., Y. Rostovtsev, and M. O. Scully, 2001,

Phys. Rev. Lett. 86, 628.Kocharovskaya, O. A., and Y. I. Khanin, 1986, Sov. Phys.

JETP 63, 945.Konopnicki, M. J., P. D. Drummond, and J. H. Eberly, 1981,

Opt. Comm. 36, 313.Konopnicki, M. J., and J. H. Eberly, 1981, Phys. Rev. A 24,

2567.Korsunsky, E. A., and D. V. Kosachiov, 1999, Phys. Rev. A

60, 4996.Korsunsky, E. A., N. Leinfellner, A. Huss, S. Baluschev, and

L. Windholz, 1999, Phys. Rev. A 59, 2302.Kozhekin, A. E., K. Molmer, and E. Polzik, 2000, Phys. Rev.

A 62, 033809.Kozlov, V. V., S. Wallentowitz, and S. Raghavan, 2002, Phys.

Lett. A 296, 210.Kozuma, M., D. Akasmatsu, L. Denf, E. W. Hagley, and

M. G. Payne, 2002, Phys. Rev. A 66, 031801.Kuhn, A., M. Hennrich, and G. Rempe, 2002, Phys. Rev.

Lett. .Kuklinski, J. R., U. Gaubatz, T. F. Hioe, and K. Bergmann,

1989, Phys. Rev. A 40(11), 6741.Kuzmich, A., W. P. Bowen, A. D. Boozer, A. Bocha, C. W.

Chou, L. M. Duan, and H. J. Kimble, 2003, Nature 423,731.

Kuzmich, A., L. Mandel, and N. P. Bigelow, 2000, Phys. Rev.

Lett. 85, 1594.Kuzmich, A., L. Mandel, J. Janis, Y. E. Young, R. Ejnisman,

and N. P. Bigelow, 1999, Phys. Rev. A 60, 2346.Kuzmich, A., K. Molmer, and E. S. Polzik, 1997, Phys. Rev.

Lett. 79, 4782.Kuznetsova, E., O. Kocharovskaya, P. Hemmer, and M. O.

Scully, 2002, Phys. Rev. A 66, 063802.Law, C. K., and J. H. Eberly, 1996, Phys. Rev. Lett. 76, 1055.Leonhard, U., and P. Piwnicki, 2000, Phys. Rev. Lett. 84,

822.Leonhardt, U., and P. Piwnicki, 1999, Phys. Rev. A 60, 4301.Leonhardt, U., and P. Piwnicki, 2000, Phys. Rev. Lett. 85,

5253.Litvak, A., and M. D. Tokman, 2002, Phys. Rev. Lett. 88,

095003.Liu, C., Z. Dutton, C. H. Behroozi, and L. V. Hau, 2001,

Nature 409, 490.Lounis, B., and C. Cohen-Tannoudji, 1992, J. Phys. II 2,

1992.Lukin, M. D., 2003, Reviews in Modern Physics 75, 457.Lukin, M. D., M. Fleischhauer, H. G. R. A. S. Zibrov, V. L.

Velichansky, L. Hollberg, and M. O. Scully, 1997, Phys.Rev. Lett. 79, 2959.

Lukin, M. D., M. Fleischhauer, R. Cote, L. M. Duan,D. Jaksch, J. I. Cirac, and P. Zoller, 2001, Phys. Rev. Lett.87, 037901.

Lukin, M. D., M. Fleischhauer, M. O. Scully, and V. L.Velichansky, 1998a, Opt. Lett. 23, 295.

Lukin, M. D., P. Hemmer, and M. O. Scully, 2000a, Adv.Atom. Mol. Opt. Physics 42, 347.

Lukin, M. D., P. R. Hemmer, M. Loffler, and M. O. Scully,1998b, Phys. Rev. Lett. 81, 2675.

Lukin, M. D., and A. Imamoglu, 2000, Phys. Rev. Lett. 84,1419.

Lukin, M. D., and A. Imamoglu, 2001, Nature 273, 273.Lukin, M. D., A. B. Matsko, M. Fleischhauer, and M. O.

Scully, 1999, Phys. Rev. Lett. 82, 1847.Lukin, M. D., S. F. Yelin, and M. Fleischhauer, 2000b, Phys.

Rev. Lett. 84, 4232.Madden, R. B., and K. Codling, 1965, Phys. Rep. 141, 364.Mair, A., J. Hager, D. F. Phillips, R. L. Walsworth, and M. D.

Lukin, 2002, Phys. Rev. A 65, 031802.Malmistov, A. I., 1984, Sov. J. Quant. Electron. 14, 385.Mandel, P., 1993, Contemp. Physics 34, 235.Masalas, M., and M. Fleischhauer, 2004, Phys. Rev. A 69,

061801.Matsko, A. B., O. Kocharovskaya, Y. Rostovtsev, A. S. Z.

G. R. Welch, and M. O. Scully, 2001a, Adv. Atom. Mol.Opt. Physics 46, 191.

Matsko, A. B., Y. V. Rostovtsev, M. Fleischhauer, and M. O.Scully, 2001b, Phys. Rev. Lett. 86, 2006.

Matsko, A. B., Y. V. Rostovtsev, O. Kocharovskaya, A. S.Zibrov, and M. O. Scully, 2001c, Phys. Rev. A 64, 043809.

Mazets, I. E., and B. G. Matisov, 1996, JETP Lett. 64, 515.Merriam, A. J., S. J. Sharpe, M. Shverdin, D. Manuszak,

G. Y. Yin, and S. E. Harris, 2000, Phys. Rev. Lett. 84,5308.

Merriam, A. J., S. J. Sharpe, H. Xia, D. Manuszak, G. Y.Yin, and S. E. Harris, 1999, Opt. Lett. 24, 625.

Mewes, C., and M. Fleischhauer, 2002, Phys. Rev. A 66,033820.

Mewes, C., and M. Fleischhauer, 2004, Decoherence in collec-tive quantum memories for photons, quant-ph/0408018.

Moiseev, S. A., and N. S. Kroll, 2001, Phys. Rev. Lett. 87,

40

173601.Moseley, R. R., S. Shepherd, D. J. Fulton, B. D. Sinclair, and

M. H. Dunn, 1995, Phys. Rev. Lett 74, 670.Muller, G., M. Muller, A. Wicht, R.-H. Rinkleff, and K. Danz-

mann, 1997, Phys. Rev. A 56, 2385.Nagel, A., L. Graf, A. Naumov, E. Mariotti, V. Biancalana,

D. Meschede, and R. Wynands, 1998, Europhys. Lett. 44,31.

Nielsen, M., and I. Chuang, 2000, Quantum Computation and

Quantum Information (Cambridge University Press, Cam-bridge).

Nottleman, A., C. Peters, and W. Lange, 1993, Phys. Rev.Lett. 70, 1783.

Oreg, J., F. T. Hioe, and J. H. Eberly, 1984, Phys. Rev. A29, 690.

Padmabandu, G. G., G. R. Welch, I. N. Shubin, E. S. Fry,D. E. Nikonov, M. D. Lukin, and M. O. Scully, 1996, Phys.Rev. Lett. 76, 2053.

Parkins, A. S., P. Marte, P. Zoller, O. Carnal, and H. J. Kim-ble, 1995, Phys. Rev. A 51, 1578.

Parkins, A. S., P. Marte, P. Zoller, and H. J. Kimble, 1993,Phys. Rev. Lett. 71, 3095.

Pellizzari, T., S. A. Gardiner, J. I. Cirac, and P. Zoller, 1995,Phys. Rev. Lett. 75, 3788.

Petch, J. C., C. H. Keitel, P. L. Knight, and J. P. Marangos,1996, Phys. Rev. A 53(1), 543.

Petrosyan, D., and G. Kurizki, 2002, Phys. Rev. A 65, 033833.Phillips, D., A. Fleischhauer, A. Mair, R. Walsworth, and

M. D. Lukin, 2001, Phys. Rev. Lett. 86, 783.Polzik, E. S., 1999, Phys. Rev. A 59, 4202.Rahman, A., 1999, Phys. Rev. A 60, 4187.Rahman, A., and J. H. Eberly, 1998, Phys. Rev. A .Rathe, U., M. Fleischhauer, S. Y. Zhu, T. W. Hansch, and

M. O. Scully, 1993, Phys. Rev. A 47, 4994.Reintjes, J. F., 1984, Academic Press Boston.Rickes, T., J. P. Marangos, and T. Halfmann, 2003, Optics

Communicatio 277, 133.Rickes, T., L. P. Yatsenko, S. Steuerwald, T. Halfmann,

B. W. Shore, N. V. Vitanov, and K. Bergmann, 2000,J.Chem.Phys .

Ruostekoski, J., and D. F. Walls, 1999a, Eur. Phys. J. D 5,335.

Ruostekoski, J., and D. F. Walls, 1999b, Phys. Rev. A 59,R2571.

Sautenkov, V. A., M. D. Lukin, C. J. Bednar, I. Novikova,E. Mikhailov, M. Fleischhauer, V. L. Velichansky, G. R.Welch, and M. O. Scully, 2000, Phys. Rev. A 62, 023810.

Schmidt, H., K. L. Chapman, A. C. Gossard, andA. Imamoglu, 1997, Appl. Phys. Lett. 70, 3455.

Schmidt, H., and A. Imamoglu, 1996, Opt. Letts. 21, 1936.Schmidt, O., R. Wynands, Z. Hussein, and D. Meschede,

1996, Phys. Rev. A 53, R27.Schori, C., B. Julsgaard, J. L. Sorensen, and E. S. Polzik,

2002, Phys. Rev. Lett. 89, 057903.Scully, M. O., 1991, Phys. Rev. Lett. 67, 1855.Scully, M. O., 1992, Phys. Rep. 219, 191.Scully, M. O., and M. Fleischhauer, 1992, Phys. Rev. Lett.

69, 1360.Scully, M. O., S. Zhu, and A. Gavrielides, 1989, Phys. Rev.

Lett 62, 2813.Scully, M. O., and S. Y. Zhu, 1992, Opt. Commun. 87, 134.

Scully, M. O., and M. S. Zubairy, 1997, Cambridge UniversityPress .

Serapaglia, G. B., E. Paspalakis, C. Sirtori, K. L. Vodopy-

anov, and C. C. Phillips, 2000, Phys. Rev. Lett 84, 1019.Shapiro, M., 1970, The Journal of Chemical Physics 56, 2582.Shore, B. W., 1967, Reviews of Modern Physics 39, 439.Shore, B. W., 1990, Theory of Coherent Atomic Excitation

(John Wiley & Sons, New York).Shvets, G., and J. S. Wurtele, 2002, Phys. Rev. Lett. 89,

115003.Sokolov, A. V., D. R. Walker, D. D. Yavuz, G. Y. Yin, and

S. E. Harris, 2000, Phys. Rev. Lett. 85, 562.Sokolov, A. V., D. R. Walker, D. D. Yavuz, G. Y. Yin, and

S. E. Harris, 2001, Phys. Rev. Lett. 87, 033402.Stahler, M., S. Knappe, C. Affolderbach, W. Kemp, and

R. Wynands, 2001, Europhys. Lett. 54, 323.Steudel, H., 1988, J. Mod. Optics 35, 193.Sun, C. P., S. Yi, and L. You, 2003, Phys. Rev. A 67, 063815.Tewari, S. P., and G. S. Agarwal, 1986, Phys. Rev. Lett. 56,

1811.Turukhin, A. V., V. S. Sudarhanam, M. S. Shahriar, J. A.

Musser, B. S. Ham, and P. R. Hemmer, 2002, Phys. Rev.Lett. 88, 023602.

Visser, M., 2000, Phys. Rev. Lett. 85, 5252.Vitanov, N. V., M. Fleischhauer, B. W. Shore, and

K. Bergmann, 2001, Adv. Atom. Mol. Opt. Phys. 46, 55.Vitanov, N. V., and S. Stenholm, 1997, Phys. Rev. A 56,

1463.van der Wal, C., M. D. Eisaman, A. Andre, R. L. Walsworth,

D. F. Phillips, A. S. Zibrov, and M. D. Lukin, 2003, Science301, 196.

Wei, C., and N. B. Manson, 1999, Physical Review A 60,2540.

Werner, M., and A. Imamoglu, 1999, Phys. Rev. A 61,011801.

Whitley, R. M., and C. R. Stroud, 1976, Phys. Rev. A. 14,1498.

Wilson-Gordon, A. D., and H. Friedman, 1992, Opt. Com-mun. 94, 238.

Wolf, S. A., D. D. Awschalom, R. A. Buhrman, J. M.Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-nova, and D. M. Treger, 2001, Science 294, 1488.

Xiao, M., Y. Li, S. Jin, and J. Gea-Banacloche, 1995, Phys.Rev. Lett. 74, 666.

Zhang, G. Z., K. Hakuta, and B. P. Stoicheff, 1993, Phys.Rev. Lett. 71, 3009.

Zhang, G. Z., M. Katsuragawa, K.Hakuta, R. I. Thompson,and B. P. Stoicheff, 1995, Phys. Rev. A 52, 1584.

Zhao, Y., C. Wu, B. S. Ham, M. K. Kim, and E. Awad, 1997,Phys. Rev. Lett. 79, 64.

Zibrov, A. S., M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O.Scully, V. L. Velichansky, and H. G. Robinson, 1995, Phys.Rev. Lett. 75, 1499.

Zibrov, A. S., M. D. Lukin, and M. O. Scully, 1999, Phys.Rev. Lett. 83, 4049.

Zibrov, A. S., A. B. Matsko, O. Kocharovskaya, Y. V. Ros-tovtsev, G. R. Welch, and M. O. Scully, 2002, Phys. Rev.Lett. 88, 103601.

Zimmer, F., and M. Fleischhauer, 2004, Phys. Rev. Lett. 92,253201.