JQI summer school · 2016. 4. 21. · three-level system driven by a coherent coupling ﬁeld ......

JQI summer schoolAug 12, 2013

Mohammad Hafezi

Electromagnetically induced transparency (EIT)(classical and quantum picture)

Optomechanics: Optomechanically induced transparency (OMIT)

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The speed and level will be dynamically tuned

1. Introduction to EIT, a simple picture2. Semi-classical description3. Quantum description 4. Optomechanics (?)

Plan:

medium

What is EIT?

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

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

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

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

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

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

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

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

634 Fleischhauer, Imamoglu, and Marangos: Electromagnetically induced transparency

Rev. Mod. Phys., Vol. 77, No. 2, April 2005

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

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

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

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

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

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

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

Hint = −%

2% 0 0 $p

0 − 2!&1 − &2" $c

$p $c − 2&1& . !1"

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

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

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

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

637Fleischhauer, Imamoglu, and Marangos: Electromagnetically induced transparency


• Transparency• Slow light

Simple picture: Dark state

|a

|b

E

(t)Control field Probe

(Quantum field)

|c

H = |ab| + E|cb|

|B |a+ E|c

|D E|a |c

H = |Bb|

Light-matter interaction can be described by a Hamiltonian:

dividual atomic density matrices,

ρensemble =N∏

j=1

ρ(1) ⊗ . . . ⊗ ρ(N). (4.1)

For an ensemble of N atoms in a volume V, we can define a macroscopicpolarization,

P (ri) =1

Vi

∑

j

〈dj〉, (4.2)

which depends on the total dipole moment of the set of atoms j within asmall volume Vi centered at position ri. Using the optical Bloch equations,we can calculate the polarization for a given field; self-consistency requiresthat this polarization then act as a source term in the Maxwell equationsgoverning light propagation through the atomic cloud:

∇× E = −∂B

∂t(4.3)

∇× B = −1

c2

∂

∂t

(

E +1

ε0P

)

. (4.4)

Similarly, the wave equation for E acquires a source term corresponding tothe backaction of the atoms on the field:

∇2E−1

c2

∂2E

∂t2=

1

ε0c2

∂2P

∂t2. (4.5)

The wave equation is second order in time and space, but applicationof the slowly varying envelope approximation can reduce it to first order.An electric field which is nearly monochromatic can be parameterized by anenvelope function and its central frequency ν,

E(r, t) = E(r, t)eikz−iνt + c.c., (4.6)

where E varies slowly in z and t compared to the optical frequency ν andwavevector k = ν/c. We also introduce a slowly varying polarization ampli-tude,

P(r, t) = P(r, t)eikz−iνt + c.c. (4.7)

Substituting these expressions into Eq. (4.5) and keeping only the lowestorder derivatives of the envelopes, we find the evolution equation for theslowly varying amplitudes,

1

2ik∇2

⊥E +∂E∂z

+1

c

∂E∂t

=ik

2ε0P (4.8)

8

Maxwell equation in the medium:


ρensemble =N∏

j=1

ρ(1) ⊗ . . . ⊗ ρ(N). (4.1)


P (ri) =1

Vi

∑

j

〈dj〉, (4.2)


∇× E = −∂B

∂t(4.3)

∇× B = −1

c2

∂

∂t

(

E +1

ε0P

)

. (4.4)


∇2E−1

c2

∂2E

∂t2=

1

ε0c2

∂2P

∂t2. (4.5)






1

2ik∇2

⊥E +∂E∂z

+1

c

∂E∂t

=ik

2ε0P (4.8)

8


ρensemble =N∏

j=1

ρ(1) ⊗ . . . ⊗ ρ(N). (4.1)


P (ri) =1

Vi

∑

j

〈dj〉, (4.2)


∇× E = −∂B

∂t(4.3)

∇× B = −1

c2

∂

∂t

(

E +1

ε0P

)

. (4.4)


∇2E−1

c2

∂2E

∂t2=

1

ε0c2

∂2P

∂t2. (4.5)






1

2ik∇2

⊥E +∂E∂z

+1

c

∂E∂t

=ik

2ε0P (4.8)

8

Slowly varying envelope approximation:


ρensemble =N∏

j=1

ρ(1) ⊗ . . . ⊗ ρ(N). (4.1)


P (ri) =1

Vi

∑

j

〈dj〉, (4.2)


∇× E = −∂B

∂t(4.3)

∇× B = −1

c2

∂

∂t

(

E +1

ε0P

)

. (4.4)


∇2E−1

c2

∂2E

∂t2=

1

ε0c2

∂2P

∂t2. (4.5)






1

2ik∇2

⊥E +∂E∂z

+1

c

∂E∂t

=ik

2ε0P (4.8)

8


ρensemble =N∏

j=1

ρ(1) ⊗ . . . ⊗ ρ(N). (4.1)


P (ri) =1

Vi

∑

j

〈dj〉, (4.2)


∇× E = −∂B

∂t(4.3)

∇× B = −1

c2

∂

∂t

(

E +1

ε0P

)

. (4.4)


∇2E−1

c2

∂2E

∂t2=

1

ε0c2

∂2P

∂t2. (4.5)






1

2ik∇2

⊥E +∂E∂z

+1

c

∂E∂t

=ik

2ε0P (4.8)

8relation to microscopic properties:

Two properties of this equation should be noted:(1) In the first term, ∇2

⊥ = ∂2

∂x2 + ∂2

∂y2 leads to transverse effects such as

focussing or diffraction. For a plane wave, the first term vanishes. (2)The slowly varying polarization P corresponds exactly to the dipole operatorin the rotating frame. For example, a two level atom leads to a polarizationP = (N/V )µ∗ρ12e−ikz.

4.1 Linear optical propagation

The simplest possible case of light propagation in a resonant field consistsof an applied field sufficiently weak that the atoms only respond to it lin-early. We found earlier that when the applied field is weak, perturbationtheory may be used to solve exactly for the atomic density matrix compo-nents in the Fourier domain. We shall take a similar approach here, wherethe Fourier components of the polarization may be written in terms of theFourier components of the off-diagonal elements of the (two-level) atomicdensity matrix,

P =N

Vµ∗ρ12(δν)e−ikz. (4.9)

If the atoms respond linearly, the polarization must be proportional to theapplied field. The proportionality constant defines the susceptibility,

χ(δν) =P(δν)

ε0E(δν). (4.10)

This quantity χ entirely characterizes the atom-photon interaction in thelinear regime, and no further information is needed to find an exact solution.

For plane wave propagation (where transverse effects may be disre-garded), the Eq. (4.8) reduces to

∂E∂z

+1

c

∂E∂t

=ik

2ε0P. (4.11)

This first order equation may be trivially solved by moving to the Fourierdomain,

∂E∂z

=iδν

cE +

ik

2χ(δν)E︸︷︷︸

∝P

, (4.12)

where the frequency components of the electric field obey

E(δν, z) = E(δν, 0)eiz(δν/c+kχ(δν)/2. (4.13)

9


⊥ = ∂2

∂x2 + ∂2





P =N



χ(δν) =P(δν)

ε0E(δν). (4.10)



∂E∂z

+1

c

∂E∂t

=ik

2ε0P. (4.11)


∂E∂z

=iδν

cE +

ik

2χ(δν)E︸︷︷︸

∝P

, (4.12)



9


ρensemble =N∏

j=1

ρ(1) ⊗ . . . ⊗ ρ(N). (4.1)


P (ri) =1

Vi

∑

j

〈dj〉, (4.2)


∇× E = −∂B

∂t(4.3)

∇× B = −1

c2

∂

∂t

(

E +1

ε0P

)

. (4.4)


∇2E−1

c2

∂2E

∂t2=

1

ε0c2

∂2P

∂t2. (4.5)






1

2ik∇2

⊥E +∂E∂z

+1

c

∂E∂t

=ik

2ε0P (4.8)

8

Assuming the response of the system is linear and the solution is a plane wave:


⊥ = ∂2

∂x2 + ∂2





P =N



χ(δν) =P(δν)

ε0E(δν). (4.10)



∂E∂z

+1

c

∂E∂t

=ik

2ε0P. (4.11)


∂E∂z

=iδν

cE +

ik

2χ(δν)E︸︷︷︸

∝P

, (4.12)



9

By transforming back to the time domain, we obtain a general solution forlinear propagation,

E(t, z) =

∫

d(δν)e−iδνtE(δν, 0)eiz(δν/c+kχ(δν)/2. (4.14)

Note that we only need χ(δν) to solve the problem exactly. Although thisexpression is not particularly transparent, the relevant physics may be ex-tracted from the general solution by examining a few special cases.

4.1.1 Monochromatic fields

A resonant, continuous-wave field has a delta function frequency distribu-tion, E(δν) ∝ δ(δν), so that

E(t, z) = Eeikχ(0)z/2. (4.15)

In general χ may be a complex quantity, and its real and imaginary partshave qualitatively different effects on the propagating field. In particular,Im[χ] leads to exponential attenuation or amplification of the beam intensitywith distance at a rate

α = k Im[χ(0)]. (4.16)

The real component Re[χ] shifts the phase of the field linearly with distance,which can be understood as a modification of the wavevector k → k +kRe[χ]/2, or a change in the index of refraction

n = 1 +Re[χ(0)]

2. (4.17)

Note that these formulae are only valid when the slowly varying envelopeapproximation holds, i.e. |Im[χ(0)]|, |Re[χ(0)]| # 1.

4.1.2 Slowly varying fields

If the applied field has a finite bandwidth, the frequency dependence ofthe susceptibility begins to matter. For nearly monochromatic fields, thesusceptibility may be expanded around its resonant value,

χ(δν) ≈ χ(0) +dχ

dνδν + . . . (4.18)

Each term leads to different effects: the first corresponds to refraction, thesecond to group velocity, the third to group velocity dispersion, and so on.

10

Solving the equation in the Fourier domain:

By transforming back to the time domain, we obtain a general solution forlinear propagation,

E(t, z) =

∫

d(δν)e−iδνtE(δν, 0)eiz(δν/c+kχ(δν)/2. (4.14)

Note that we only need χ(δν) to solve the problem exactly. Although thisexpression is not particularly transparent, the relevant physics may be ex-tracted from the general solution by examining a few special cases.

4.1.1 Monochromatic fields

A resonant, continuous-wave field has a delta function frequency distribu-tion, E(δν) ∝ δ(δν), so that

E(t, z) = Eeikχ(0)z/2. (4.15)

In general χ may be a complex quantity, and its real and imaginary partshave qualitatively different effects on the propagating field. In particular,Im[χ] leads to exponential attenuation or amplification of the beam intensitywith distance at a rate

α = k Im[χ(0)]. (4.16)

The real component Re[χ] shifts the phase of the field linearly with distance,which can be understood as a modification of the wavevector k → k +kRe[χ]/2, or a change in the index of refraction

n = 1 +Re[χ(0)]

2. (4.17)

Note that these formulae are only valid when the slowly varying envelopeapproximation holds, i.e. |Im[χ(0)]|, |Re[χ(0)]| # 1.

4.1.2 Slowly varying fields

If the applied field has a finite bandwidth, the frequency dependence ofthe susceptibility begins to matter. For nearly monochromatic fields, thesusceptibility may be expanded around its resonant value,

χ(δν) ≈ χ(0) +dχ

dνδν + . . . (4.18)

Each term leads to different effects: the first corresponds to refraction, thesecond to group velocity, the third to group velocity dispersion, and so on.

10

For example, if we keep only the second order terms, we find

E(t, z) =

∫

d(δν)E(δν, 0)e−iδνteiz(δν/c+k(χ(0)+ dχdν

δν)/2 (4.19)

= eizk(χ(0)/2∫

d(δν)E(δν, 0)e−iδν(t−z/vg ) (4.20)

= eizk(χ(0)/2E(t − z/vg, z = 0), (4.21)

so the envelope propagates at the so-called group velocity

vg =c

1 + ν2

dχdν

. (4.22)

4.1.3 Two level systems

Consider light propagation through a dilute gas of identical, noninteractingtwo-level atoms. In the steady state, we can solve the optical Bloch equationsfor the off-diagonal density matrix elements to obtain the polarization andthus the susceptibility,

χ(δν) = iN

V

µ2

!ε0

(

ρ011 − ρ0

22

) 1

γ12 − iδν. (4.23)

The susceptibility exhibits a Lorentzian lineshape with linewidth γ12. Farfrom resonance, δν " γ12, the imaginary (absorptive) component Im[χ] ∝1/δν2 is much smaller than the real (reactive) component Re[χ] ∝ 1/δν, sothe medium is almost entirely refractive and causes little absorption.

The magnitude of the susceptibility on resonance is

χ(0) = iµ2

!ε0

N

V

1

γ12(4.24)

= i3

16π2

N

Vλ3 γ

γ12. (4.25)

In the latter equation we have used the definition of γ to express χ in termsof physically intuitive quantities. The electric field amplitude propagates as

E(z) = E(0)e(−Im[χ(0)]+iRe[χ(0)])kz/2 (4.26)

and we define the corresponding intensity loss

I(z) = I(0)e−N σz/V (4.27)

11

Group velocity:


E(t, z) =

∫


δν)/2 (4.19)

= eizk(χ(0)/2∫


= eizk(χ(0)/2E(t − z/vg, z = 0), (4.21)


vg =c

1 + ν2

dχdν

. (4.22)



χ(δν) = iN

V

µ2

!ε0

(

ρ011 − ρ0

22

) 1

γ12 − iδν. (4.23)



χ(0) = iµ2

!ε0

N

V

1

γ12(4.24)

= i3

16π2

N

Vλ3 γ

γ12. (4.25)


E(z) = E(0)e(−Im[χ(0)]+iRe[χ(0)])kz/2 (4.26)


I(z) = I(0)e−N σz/V (4.27)

11


E(t, z) =

∫


δν)/2 (4.19)

= eizk(χ(0)/2∫


= eizk(χ(0)/2E(t − z/vg, z = 0), (4.21)


vg =c

1 + ν2

dχdν

. (4.22)



χ(δν) = iN

V

µ2

!ε0

(

ρ011 − ρ0

22

) 1

γ12 − iδν. (4.23)



χ(0) = iµ2

!ε0

N

V

1

γ12(4.24)

= i3

16π2

N

Vλ3 γ

γ12. (4.25)


E(z) = E(0)e(−Im[χ(0)]+iRe[χ(0)])kz/2 (4.26)


I(z) = I(0)e−N σz/V (4.27)

11

Two-level atom susceptibility:

Three-level atom:|1⟩

|2⟩

|3⟩

Ω1

Ω2 ν1

ν2

Λ System

γ2

γ3

∆ = ν − ω2 2 23 ∆ = ν − ω1 1 13

δ = ∆ −∆ 1 2

Figure 6.4: The Λ system under consideration, with the single- and two-photon detunings shown.

system in the appropriate rotating frame,

H = −∆1|3〉〈3| − δ|2〉〈2| − (Ω1|1〉〈3| + Ω2|2〉〈3| + h.c.) , (6.10)

where we have set the energy of |1〉 to zero. Neglecting decay terms for themoment, a general state of the form

|ψ〉 = c1(t)|1〉 + c2(t)|2〉 + c3(t)|3〉 (6.11)

obeys the Schrodinger equation for the Hamiltonian, yielding equations ofmotion for the population amplitudes,

c1 = iΩ∗1c3 (6.12)

c2 = iδc2 + iΩ∗2c3 (6.13)

c3 = i∆1c3 + iΩ1c1 + iΩ2c2. (6.14)

The Λ system has a particularly simple solution for large but similarsingle photon detunings ∆1 ∼ ∆2 = ∆ % Ω1,Ω2, γ3, δ. Since ∆ is muchlarger than all other time scales in the problem, to lowest order in Ω1,2/∆the excited state amplitude will be constant, with

c3 = −Ω1c1 + Ω2c2

∆. (6.15)

7

rotating frame:

Δ1

Ω

|1|2

|3

2

Ω1

Δ2

a weak probe:

The non-Hermitian Schrodinger equation yields the following equationsof motion for the stochastic wavefunction amplitudes:

c1 = iΩ∗1c3 (6.26)

c2 = − (γ2

2− iδ)

︸︷︷︸

Γ12

c2 + iΩ∗2c3 (6.27)

c3 = − (γ3

2− i∆1)

︸︷︷︸

Γ13

c3 + iΩ1c1 + iΩ2c2. (6.28)

These equations may be solved in different regimes using different methods.If Ω1 is weak and Ω2 is continuous wave, they may be solved exactly usingFourier transforms. If the applied fields have envelopes which vary slowly intime, adiabatic elimination provides a method for finding an approximatesolution. Regardless of the precise mathematical manipulations employed,it is important to always check at the end that the probability for quantumjumps is indeed small.

We will now consider the behavior of the Λ system in a few special cases.For a weak probe field Ω1 and a constant field Ω2 of arbitrary strength, tozeroth order in Ω1 a system which starts in |1〉 will remain in |1〉, so c1(t) ≈ 1.The remaining probability amplitudes can be found by Fourier transformingthe equations of motion, so that

(γ2

2− i(δ − ω))

︸︷︷︸

Γ12

c2 = iΩ∗2c3 (6.29)

(γ3

2− i(∆1 − ω))

︸︷︷︸

Γ13

c3 = iΩ1 + iΩ2c2. (6.30)

Note that the Fourier frequency ω enters in the same place as the detuning,so we will continue to make reference to the complex decay rates Γ12 andΓ13, bearing in mind that the dependence on frequency is implied. Theresults for the density matrix components follow easily:

ρ11 = |c1|2 = 1 (6.31)

ρ12 = c∗1c2 = −Ω1Ω∗

2

Γ12Γ13 + |Ω2|2(6.32)

ρ13 = c∗1c3 = iΩ1Γ12

Γ12Γ13 + |Ω2|2. (6.33)

10


c1 = iΩ∗1c3 (6.26)

c2 = − (γ2

2− iδ)

︸︷︷︸

Γ12

c2 + iΩ∗2c3 (6.27)

c3 = − (γ3

2− i∆1)

︸︷︷︸

Γ13

c3 + iΩ1c1 + iΩ2c2. (6.28)



(γ2

2− i(δ − ω))

︸︷︷︸

Γ12

c2 = iΩ∗2c3 (6.29)

(γ3

2− i(∆1 − ω))

︸︷︷︸

Γ13

c3 = iΩ1 + iΩ2c2. (6.30)


ρ11 = |c1|2 = 1 (6.31)

ρ12 = c∗1c2 = −Ω1Ω∗

2

Γ12Γ13 + |Ω2|2(6.32)

ρ13 = c∗1c3 = iΩ1Γ12

Γ12Γ13 + |Ω2|2. (6.33)

10

atomic equations of motion

|1⟩

|2⟩

|3⟩

Ω1

Ω2 ν1

ν2

Λ System

γ2

γ3

∆ = ν − ω2 2 23 ∆ = ν − ω1 1 13

δ = ∆ −∆ 1 2

Figure 6.4: The Λ system under consideration, with the single- and two-photon detunings shown.

system in the appropriate rotating frame,

H = −∆1|3〉〈3| − δ|2〉〈2| − (Ω1|1〉〈3| + Ω2|2〉〈3| + h.c.) , (6.10)

where we have set the energy of |1〉 to zero. Neglecting decay terms for themoment, a general state of the form

|ψ〉 = c1(t)|1〉 + c2(t)|2〉 + c3(t)|3〉 (6.11)

obeys the Schrodinger equation for the Hamiltonian, yielding equations ofmotion for the population amplitudes,

c1 = iΩ∗1c3 (6.12)

c2 = iδc2 + iΩ∗2c3 (6.13)

c3 = i∆1c3 + iΩ1c1 + iΩ2c2. (6.14)

The Λ system has a particularly simple solution for large but similarsingle photon detunings ∆1 ∼ ∆2 = ∆ % Ω1,Ω2, γ3, δ. Since ∆ is muchlarger than all other time scales in the problem, to lowest order in Ω1,2/∆the excited state amplitude will be constant, with

c3 = −Ω1c1 + Ω2c2

∆. (6.15)

7

Using Stochastic wavefunction approach, instead of master equationignoring jump form the excited state:


c1 = iΩ∗1c3 (6.26)

c2 = − (γ2

2− iδ)

︸︷︷︸

Γ12

c2 + iΩ∗2c3 (6.27)

c3 = − (γ3

2− i∆1)

︸︷︷︸

Γ13

c3 + iΩ1c1 + iΩ2c2. (6.28)



(γ2

2− i(δ − ω))

︸︷︷︸

Γ12

c2 = iΩ∗2c3 (6.29)

(γ3

2− i(∆1 − ω))

︸︷︷︸

Γ13

c3 = iΩ1 + iΩ2c2. (6.30)


ρ11 = |c1|2 = 1 (6.31)

ρ12 = c∗1c2 = −Ω1Ω∗

2

Γ12Γ13 + |Ω2|2(6.32)

ρ13 = c∗1c3 = iΩ1Γ12

Γ12Γ13 + |Ω2|2. (6.33)

10

mentum kicks in opposite directions to the atom.

6.2.2 Dark States

The Λ system examined above in the far-detuned regime exhibits a verydifferent response to resonant excitation. As we bring the single photondetunings to zero ∆1,∆2 → 0, the two-photon resonance approaches thesingle-photon resonance and begins to broaden. Simultaneously, the ACStark shifts increase. The overall effects, however, remain the same providedthat ∆ " γ13,Ω1,2. When this condition is no longer satisfied, the systemresponse changes qualitatively (see Figure 6.6).

Probe frequency ν1

Ab

sorp

tio

n ~

Im

[χ]

ν = ω

ν = ν + ω − ω1 2 13 23

1 13

Single photon

resonance Two-photon resonance

Probe field susceptibility as ∆ → 0

∆ large

∆ = 0

Figure 6.6: As the detuning ∆ is reduced, the two-photon resonance movestowards the single-photon resonance and broadens. When the the two res-onances begin to overlap, the system no longer behaves like a far-detunedsystem, and qualitatively new effects appear.

Consider an experiment where we illuminate the three-level atom withan applied field Ω2 which is precisely on resonance, ∆2 = 0. We then sweepthe frequency of the probe beam ∆1 = δ, and observe how the susceptibilitychanges as a function of ∆1. Using the expression we previously derived forthe optical coherence on the ν1 transition,

ρ13

Ω1=

i

Γ13

(

1 −|Ω2|2/Γ13

Γ12 + |Ω2|2/Γ13

)

, (6.48)

15

All other density matrix elements are of order |Ω1|2 or higher, and conse-quently may be disregarded. The same results may be obtained directlyfrom the master equation, where in general the complex decay rates wouldinclude effects of dephasing as well as decay.

Large detuning ∆: Raman Physics

The three-level system can be considerably simplified in the case of a largesingle-photon detuning ∆. The susceptibility on the probe transition, χ1 =(|µ|2/!ε0)ρ13/Ω1, is proportional to

ρ13

Ω1=

i

Γ13

(Γ12

Γ12 + |Ω2|2/Γ13

)

, (6.34)

which we can rewrite as

ρ13

Ω1=

i

Γ13

(

1 −|Ω2|2/Γ13

Γ12 + |Ω2|2/Γ13

)

. (6.35)

The first term i/Γ13 contains the single photon physics for a two-level systemof |1〉 and |3〉 coupled by Ω1, and appears as a broad resonance at ν1 = ω13

with linewidth γ13. The second term deserves greater care since it describesthe effects of two-photon transitions. In the limit that ∆ # γ13, the complexdecay rate Γ13 ≈ −i∆, so the second term becomes

ρ13

Ω1−

i

Γ13≈

i|Ω2|2/∆2

Γ12 + (|Ω2|2/∆2)(γ13 + i∆)(6.36)

≈i|Ω2|2/∆2

γ12 + γ13(|Ω2|2/∆2)

︸︷︷︸

γeff

−i (δ − |Ω2|2/∆)

︸︷︷︸

δ′

. (6.37)

The real and complex parts of the denominator correspond to the effectivewidth and detuning of the two-photon resonance. Because we have includeddecay, the susceptibility remains finite even on resonance δ → |Ω2|2/∆.The width of the two-photon resonance is set by both the metastable statedecoherence γ12 and a term resulting from excited state decay, γ13|Ω2|2/∆2,which strongly suppressed by the large single-photon detuning.

Another important physical effect can be derived from our susceptibil-ity calculation. For a system exactly on resonance, δ = |Ω2|2/∆, with nodephasing γ12 = 0, γ13 = γ3/2, the susceptibility

ρ13

Ω1∼

2i

γ3(6.38)

11

Limit of large detuning:




ρ13

Ω1=

i

Γ13

(Γ12

Γ12 + |Ω2|2/Γ13

)

, (6.34)


ρ13

Ω1=

i

Γ13

(

1 −|Ω2|2/Γ13

Γ12 + |Ω2|2/Γ13

)

. (6.35)



ρ13

Ω1−

i

Γ13≈

i|Ω2|2/∆2

Γ12 + (|Ω2|2/∆2)(γ13 + i∆)(6.36)

≈i|Ω2|2/∆2

γ12 + γ13(|Ω2|2/∆2)

︸︷︷︸

γeff

−i (δ − |Ω2|2/∆)

︸︷︷︸

δ′

. (6.37)



ρ13

Ω1∼

2i

γ3(6.38)

11




ρ13

Ω1=

i

Γ13

(Γ12

Γ12 + |Ω2|2/Γ13

)

, (6.34)


ρ13

Ω1=

i

Γ13

(

1 −|Ω2|2/Γ13

Γ12 + |Ω2|2/Γ13

)

. (6.35)



ρ13

Ω1−

i

Γ13≈

i|Ω2|2/∆2

Γ12 + (|Ω2|2/∆2)(γ13 + i∆)(6.36)

≈i|Ω2|2/∆2

γ12 + γ13(|Ω2|2/∆2)

︸︷︷︸

γeff

−i (δ − |Ω2|2/∆)

︸︷︷︸

δ′

. (6.37)



ρ13

Ω1∼

2i

γ3(6.38)

11

we find that the absorption vanishes precisely on resonance δ = 0:

Im[ρ13

Ω1] =

γ12

γ13γ12 + |Ω2|2→ 0 as γ12 → 0. (6.49)

For Ω2 " γ13, the width of this absorption dip is γ12 + |Ω2|2/γ13, whereasfor Ω2 # γ13, the width is approximately γ12 + |Ω2|. Note that the widthof the dip can be very narrow " γ13; this is an indication that two-photonphysics is essential to the dark resonance. Although the optical coherenceρ13 vanishes at zero detuning, the coherence on the forbidden transition,ρ12 → −Ω1/Ω2, remains finite, indicating that atomic coherence also playsa crucial role in this phenomenon. Far from the resonance condition, theusual far-detuned behavior applies, so we expect the probe to reveal a broadabsorption peak with a narrow dark resonance in the center (see Figure 6.7).

Im[χ

]

Re[χ

]

Probe frequency ν1ν = ω1 13

Probe susceptibility for ∆ =02

Figure 6.7: The real and imaginary part of the probe beam susceptibil-ity are shown under resonant illumination by Ω2. The absorption dip atzero detuning is known as a dark resonance or electromagnetically inducedtransparency.

Physical picture of the dark resonance

To gain some intuition for the physics behind the dark resonance, we againresort to the dressed state interpretation with a non-Hermitian effectiveHamiltonian. If the metastable decoherence rate vanishes, γ12 = 0, and theapplied fields are precisely on resonance, ∆ = δ = 0, the effective Hamilto-

16

On resonance:

mentum kicks in opposite directions to the atom.

6.2.2 Dark States

The Λ system examined above in the far-detuned regime exhibits a verydifferent response to resonant excitation. As we bring the single photondetunings to zero ∆1,∆2 → 0, the two-photon resonance approaches thesingle-photon resonance and begins to broaden. Simultaneously, the ACStark shifts increase. The overall effects, however, remain the same providedthat ∆ " γ13,Ω1,2. When this condition is no longer satisfied, the systemresponse changes qualitatively (see Figure 6.6).

Probe frequency ν1

Abso

rpti

on ~

Im

[χ]

ν = ω

ν = ν + ω − ω1 2 13 23

1 13

Single photon

resonance Two-photon resonance

Probe field susceptibility as ∆ → 0

∆ large

∆ = 0

Figure 6.6: As the detuning ∆ is reduced, the two-photon resonance movestowards the single-photon resonance and broadens. When the the two res-onances begin to overlap, the system no longer behaves like a far-detunedsystem, and qualitatively new effects appear.

Consider an experiment where we illuminate the three-level atom withan applied field Ω2 which is precisely on resonance, ∆2 = 0. We then sweepthe frequency of the probe beam ∆1 = δ, and observe how the susceptibilitychanges as a function of ∆1. Using the expression we previously derived forthe optical coherence on the ν1 transition,

ρ13

Ω1=

i

Γ13

(

1 −|Ω2|2/Γ13

Γ12 + |Ω2|2/Γ13

)

, (6.48)

15

Transparency

Dressed state picture and splitting with (use board)

nian becomes

Heff = −i!γ3

2|3〉〈3| − (Ω1|1〉〈3| + Ω2|2〉〈3| + h.c.) (6.50)

= −i!γ3

2|3〉〈3| − (Ω1|1〉 + Ω2|2〉)

︸︷︷︸

|B〉

〈3| − |3〉 (Ω∗1〈1| + Ω∗

2〈2|)︸︷︷︸

〈B|

.(6.51)

Note that the excited state |3〉 only couples to a particular superpositionof the lower states, |B〉. Furthermore, if we can construct a state |D〉 suchthat 〈D|B〉 = 0, this new orthogonal superposition will be decoupled fromthe excited state and thus insensitive to its decay. This so-called dark state,

|D〉 =Ω2|1〉 − Ω1|2〉√

|Ω1|2 + |Ω2|2(6.52)

is an eigenstate of the effective Hamiltonian with zero energy:

Heff |D〉 = 0. (6.53)

A system in state |D〉 has vanishing absorption and polarization on the |1〉to |3〉 transition, but finite polarization on the forbidden transition. Quan-titatively, the density matrix for the dark state,

ρ = |D〉〈D|, (6.54)

has off-diagonal elements

ρ12 = −Ω∗

2Ω1

|Ω1|2 + |Ω2|2. (6.55)

In the limit that Ω1 $ Ω2, we reproduce our previous result. Consequently,we see that the vanishing absorption on resonance is associated with theproduction of dark states which are decoupled from both light beams. Thiscan also be viewed as a quantum interference phenomenon, where simulta-neous excitations of |D〉 by Ω1 and Ω2 destructively interfere, leaving thedark state unaffected by the applied fields.

Dispersive properties of the dark resonance: ”slow light”

Thus far we have considered only the imaginary part of the probe suscepti-bility in the vicinity of the dark resonance. The real part leads to dispersiveeffects, since for ∆2 = 0 and δ small

Re[χ] ∝ Re[ρ13

Ω1] ≈

δ

|Ω2|2 + γ12γ13. (6.56)

17

Slow light:

In the situation relevant to the dark resonance, the slope of the refractiveindex n ∝ Re[χ] can be controlled by changing the intensity of Ω2. In par-ticular, this slope determines the group velocity since vg = dω

dk = cn(ω)+ω dn

dω

(where n(ω) is the index of refraction) and n(ω) ∝ Re[χ]. The slope dndω

diverges as γ12 → 0 and Ω2 → 0:

vg ≈c

1 + γ13 ν3π(N/V )(λ/2π)3

|Ω2|2+γ13γ12

(6.57)

so that the group velocity is reduced to zero as Ω2 → 0 (see problem set 3for a derivation of this formula). Simultaneously, the width ∆w of the theresonance also vanishes, since

∆w ≈|Ω2|2

γ13→ 0. (6.58)

Note that when the linewidth is dynamically reduced while a pulse is trav-elling through the medium, careful considerations show that the spectrumof the pulse itself is also dynamically reduced so that the pulse spectrumfits inside the transparency window at all times and no dissipation is takingplace (see problem set 3 and references therein for further details).

Since the two-photon resonance can be much narrower than the single-photon linewidth, the group velocity of the probe pulse Ω1 can be reducedsubstantially below the speed of light. As resonant probe light enters themedium it compresses by a factor vg/c, then slowly propagates through theatomic cloud, dragging along a spin coherence associated with the dark state.Since absorption vanishes, virtually all of the probe light is transmitted, witha significant delay due to the slow group velocity. This phenomenon is knownas “electro-magnetically induced transparency” or “slow light”.

Our discussion has not explained how the system is prepared in the darkstate. In fact, if the probe light turns on slowly enough, the system willadiabatically follow from the initial state |1〉 to |D〉, in essence preparing it-self. This phenomenon is an example of stimulated Raman adiabatic passage(STIRAP), whereby slowly varying applied fields can be used to efficientlyand robustly manipulate the populations and atomic coherences of a system.Unlike π pulses based on Rabi oscillations, which are exquisitely sensitiveto pulse amplitude and timing, STIRAP techniques do not depend on thedetails of pulse shape.

To understand the basic mechanism for dark state formation, considerthe asymptotic behavior of the dark state under different applied powers:

Ω2 → 0,Ω1 fixed ⇒ |D〉 = −|2〉 (6.59)

18


dk = cn(ω)+ω dn

dω



vg ≈c

1 + γ13 ν3π(N/V )(λ/2π)3

|Ω2|2+γ13γ12

(6.57)


∆w ≈|Ω2|2

γ13→ 0. (6.58)





Ω2 → 0,Ω1 fixed ⇒ |D〉 = −|2〉 (6.59)

18

divergence when


dk = cn(ω)+ω dn

dω



vg ≈c

1 + γ13 ν3π(N/V )(λ/2π)3

|Ω2|2+γ13γ12

(6.57)


∆w ≈|Ω2|2

γ13→ 0. (6.58)





Ω2 → 0,Ω1 fixed ⇒ |D〉 = −|2〉 (6.59)

18

the bandwidth shrinks:

what are the conditions under which an entire pulse can be trapped?

Quantum description:

what happens to the excitations?

can the stored pulse be retrieved?

what happens to the quantum properties of light?

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

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

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

"!z,t" = cos $Ep!z,t" − sin $%!%21!z,t"ei&kz, !44"

#!z,t" = sin $Ep!z,t" + cos $%!%21!z,t"ei&kz, !45"

with the mixing angle determined by the group index

tan2 $ =!'c(31

)c2 = ngr. !46"

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

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

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

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

Ep!z,t" = cos $"!z,t" , !47"

%21!z,t" = − sin $"!z,t"e−i&kz

%!. !48"

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

( !

!t+ c cos2 $

!

!z)"!z,t" = 0, !49"

which describes a form-stable propagation with velocity

vgr = c cos2 $ . !50"

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

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

vgr = c cos2 $ + v sin2 $ . !51"

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



keeping both photonic and atomic operatorsintroducing bright and dark polaritons: (explain Fock state of excitations)

controllable mixing angle (dynamical):

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

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

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

"!z,t" = cos $Ep!z,t" − sin $%!%21!z,t"ei&kz, !44"

#!z,t" = sin $Ep!z,t" + cos $%!%21!z,t"ei&kz, !45"

with the mixing angle determined by the group index

tan2 $ =!'c(31

)c2 = ngr. !46"

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

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

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

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

Ep!z,t" = cos $"!z,t" , !47"

%21!z,t" = − sin $"!z,t"e−i&kz

%!. !48"

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

( !

!t+ c cos2 $

!

!z)"!z,t" = 0, !49"

which describes a form-stable propagation with velocity

vgr = c cos2 $ . !50"

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

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

vgr = c cos2 $ + v sin2 $ . !51"

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



under EIT conditions:

We have already argued that for EIT to be effective ineliminating dissipation, the light pulse spectrum shouldbe contained within a relatively narrow transparencywindow [Fig. 1(b)]. A vanishing control beam intensityimplies that the transparency window would become in-finitely narrow and eventually disappear. How can oneavoid loss in such a case? The essence of adiabatic fol-lowing in polaritons is that a dynamic reduction in groupvelocity is accompanied by narrowing of the polaritonfrequency spectrum, such that it is not destroyed even ifvg!0. To see why it happens, we note that during theprocess of adiabatic slowing the spatial profile and, inparticular, the spatial width of the wave packet remainsunaffected (see Fig. 3), as long as the group velocityvg(t) is only a function of time (Fleischhauer and Lukin,2002). At the same time, the amplitude of the electricfield gets reduced and its temporal profile is stretcheddue to the reduction of the group velocity.

The spectrum of the signal field is reduced in propor-tion to vg /c!!"!2, i.e., by exactly the same factor as thetransparency bandwidth #$. Therefore, the conditionsfor adiabatic following are very simple: the entire pulseshould be within the medium at the beginning of thetrapping procedure, and its spectrum should be con-tained within the original transparency window. Onceagain, these conditions are satisfied only if an opticallydense medium (12) is used. It is also worth noting thatthe rate at which the group velocity is turned to zero canbe quite fast, especially if the initial group velocity of thelight pulse is much smaller than c . The adiabaticity con-ditions have been analyzed in detail by Matsko et al.(2001a), and by Fleischhauer and Lukin (2002).

The concept of adiabatic passage in multilevel systemswas first introduced by Oreg et al. (1984) and was ex-perimentally rediscovered by Gaubatz et al. (1990). Itsapplication for quantum state transfer was first pointedout by Parkins et al. (1993). Extensions and detailed

analysis of such techniques were considered by Parkinsand Kimble (1999). Recent experimental progress to-ward implementation of these ideas (Kuhn et al., 2002)should be especially noted. Csesznegi and Grobe (1997)pointed out that the spatial profile of an atomic Ramancoherence can be mirrored into the electromagnetic fieldby coherent scattering, whereas time-varying fields canbe used to create spatially nonhomogeneous matter ex-citations. These techniques were reviewed by Bergmannet al. (1998). There is by now a considerable literatureinvestigating various aspects of storage in atomic en-sembles (Juzelinas and Carmichael, 2002; Mewes andFleischauer, 2002) as well as nonclassical light genera-tion (Poulsen and Molmer, 2001) using these techniques.See also the review by Fleischhauer and Mewes (2001).

Finally, it should be remarked here that the essentialpoint of this technique is not to store the energy or mo-mentum carried by photons but their quantum states. Infact, in practice almost no energy or momentum is actu-ally stored in the EIT medium. Instead, both are beingtransferred into (or borrowed from) the control beam insuch a way that an entire optical pulse is coherently con-verted into a low-energy spin wave. This is the key fea-ture that distinguishes the present approach from earlierstudies in optics [involving, e.g., traditional photon echotechniques (Boyd, 1992) or nuclear physics (Shvydkoet al., 1996)], and that enables potential applications inquantum information science. A different proposal to‘‘freeze’’ light pulses in a moving medium was suggestedby Kocharovskaya et al. (2001) and the possibility to ob-serve phenomena resembling black holes was consid-ered by Leonhardt (2001).

E. Collective enhancement and stored states

The above considerations indicate that, in principle,complete storage and retrieval of the input state is pos-

FIG. 3. A dark-state polariton can be stoppedand reaccelerated by ramping the control fieldintensity, as shown in (a). The coherent am-plitudes of the polariton %, the electric fieldE , and the spin components s are plotted in(b)–(d).

462 M. D. Lukin: Colloquium: Trapping and manipulating photon states in atomic ensembles


Tempting: Increase the propagation (interaction) time!

Can we use this to improve nonlinearity and reach few photon nonlinearity, single photon switch, quantum gates, etc.?

input photons output photons

control photons

photon blockade, single-photon transistor, quantum phase gate …

“classical” nonlinear fiber, self-focusing,

solitons, parametric down conversion in crystal

Quantum Many-body effects:

Tonks gas, Mott insulator …

Photon Number

Weak Strong (Quantum)(classical)

λ2/Α

ƒ λ2/Α

ƒʹλ2/Α222

Quantum Dot+ PC nanocavityVuckovic, Waks, Englund,....

Cavity QEDKimble, Haroche,Rempe,...

Free Space: Rb, dye molecule, QDWrigge et al Nat. Phys.4 60 (2008)Tey et al. Nat. Phys. 4, 924 (2008)

1D waveguide: optical, plasmonsGhosh et al. PRL 94, 093902 (2005)LeKien et al. PRA 77, 033826 (2005)

Akimov, Mukherjee, et al. Nat Phys (2008)Evanescent coupling Rauschenbautel, JQI

Vahala, Painter

Few photon nonlinearity

photonic excitation are converted to polaronic excitations, one-to-one map

interaction time will increase by the slowing factor

while the effective Kerr nonlinearity decreases by the same amount

Therefore, the net effect is zero.

Estimation of nonlinear effect:

EIT, slow light ...is due to interference between three modes:optical field - atomic polarization - ground state excitation which can be dynamically controlled via an external field.

Most of the physics can be explain by wave mechanics.Δ1

Ω

|1|2

|3

2

Ω1

Δ2

Can we generalize such idea to other systems? yes, we can!

Optomechanics

(on board)

4

b

a2a1

ex

h

ex

inin

a

R

(z) optical waveguide

mechanical cavity

optical cavities

a

L

(z)

a

R

(z+d)

a

L

(z+d)

a

ex

in

m(t)

bmechanical

cavity

optical waveguide

a

R

(z)

a

L

(z)

a

R

(z+d)

a

L

(z+d)

(a) (b)

|n , n >m 1

|n , n + 1>m 1

|n + 1, n >m 1

m(t)a

R

(z,t)

(d)(c)

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|t| , |r| , |t| , |r| ,

2

2

2

2

Ω = 0Ω = 0Ω = κ/10Ω = κ/10

δ/κ

-0.1 0 0.10

0.5

1

“active”optical cavity

m

m

Figure 1. (a) Illustration of a double optical cavity system forming the unitcell of the optomechanical array. A two-way optical waveguide is coupled toa pair of optical cavity modes a1 and a2, whose resonance frequencies differby the frequency of the mechanical mode b. Both optical modes leak energyinto the waveguide at a rate ex and have an inherent decay rate in. Themechanical resonator optomechanically couples the two optical resonances witha cross-coupling rate of h. (b) A simplified system diagram where the classicallydriven cavity mode a2 is effectively eliminated to yield an optomechanicaldriving amplitude m between the mechanical mode and the cavity mode a1.(c) Frequency-dependent reflectance (black curve) and transmittance (red) ofa single array element, in the case of no optomechanical driving amplitudem = 0 (dotted line) and an amplitude of m = ex/10 (solid line). The inherentcavity decay is chosen to be in = 0.1ex. (Inset) The optomechanical couplingcreates a transparency window of width 42

m/ex for a single element andenables perfect transmission on resonance, k = 0. (d) Energy level structureof the simplified system. The number of photons and phonons are denoted byn1 and nm, respectively. The optomechanical driving amplitude m couplesstates |nm + 1, n1i $ |nm, n1 + 1i, while the light in the waveguide couples states|nm, n1i $ |nm, n1 + 1i. The two couplings create a set of 3-type transitionsanalogous to that in EIT.

New Journal of Physics 13 (2011) 023003 (http://www.njp.org/)

7

0 0.5 1fraction of excitations

in

aj

bj

aj+1

ex

m(t)

bj+1

aj-1

bj-1

0 0.5 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 0 0.5 1 0 0.5 1

K (π/d)

ω (π

c /

a)

Ωm = 0 Ω

m = κ/4 Ω

m = κ/2 Ω

m = κ

ω (π

c /

a)

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K (π/d)

κ = ω/4Ω = κ/10

d

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10.40

0.42

0.44

0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60

0

40

5

fraction of excitationsK (π/d)

κ

4Ωm

κ

2

~

~

(a) (b)

(c) (d) (e) (f )

m

Figure 2. (a) Illustration of an OMC array. A two-way optical waveguide iscoupled to a periodic array of optomechanical elements spaced by a distance d.The optical cavity modes a j of each element leak energy into the waveguideat a rate ex and have an inherent decay rate in. The mechanical resonator ofeach element has frequency !m and is optomechanically coupled to the cavitymode through a tuning cavity (shown in figure 1) with strength m. (b) The bandstructure of the system, for a range of driving strengths between m = 0 andm = . The blue shaded regions indicate band gaps, while the color of the bandselucidates the fractional occupation (red for energy in the optical waveguide,green for the optical cavity and blue for mechanical excitations). The dynamiccompression of the bandwidth is clearly visible as m ! 0. (c) Band structurefor the case m = /10 is shown in greater detail. (d) The fractional occupationfor each band in (c) is plotted separately. It can be seen that the polaritonicslow-light band is mostly mechanical in nature, with a small mixing with thewaveguide modes and negligible mixing with the optical cavity mode. Zoom-insof panels (c) and (d) are shown in (e) and (f).

New Journal of Physics 13 (2011) 023003 (http://www.njp.org/)

• Lukin RMP (2003)

• Fleischhauer et al RMP (2005)

• Scully quantum optics book and Lukin lecture notes

• Chang et al. NJP (2011)

• Weis et al, Science (2010), Safavi-Naeini et al, Nature (2011)

References:

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