Standing Wave Electromagnetically Induced Transparency

Standing WaveElectromagnetically

Induced Transparency

Kristian Rymann Hansen

QUANTOPDepartment of Physics and Astronomy

University of Aarhus, Denmark

M. Sc. ThesisSeptember 2006

ii

Acknowledgements

Many people deserve my gratitude for helping me in one way or another duringthe time I spent writing this thesis.

First of all, my supervisor Prof. Klaus Mølmer for his patience and unwa-vering optimism. Even though he is an extremely busy man, he always had5 minutes (which occasionally turned into 2 hours) to discuss my latest, crazyidea and generously share his great insights with a poor, bewildered student.His sense of humor and optimistic outlook on life have made it a pleasure towork under his guidance.

I would also like to thank my parents for supporting me, and my sister forbeing the best sister one could want. Your love and support has been invaluableto me, and will continue to be so in the future.

I have had the privilege of being a member of Quantop while working onthis thesis. Every member of this group has been very friendly and forthcoming,making it a real pleasure to have made your acquaintance. I would like to thankall of you for all the stimulating and fun discussions (whether physics relatedor not), pleasant social gatherings and exhausting sumo-wrestling tournaments.Although I cannot mention you all by name here, I would like to single out LineHjortshøj Pedersen for her assistance in proofreading this thesis.

Finally, all of my friends, both on and off campus, deserve my gratitude forbeing a source of joy and fresh perspectives on things. Your friendship is verymuch appreciated.

Kristian Rymann HansenSeptember, 2006

Contents

1 Introduction 11.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Interaction of light and atoms 32.1 Quantization of the electromagnetic field . . . . . . . . . . . . . . 3

2.1.1 The potential formulation of electrodynamics . . . . . . . 32.1.2 Vacuum solutions . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Canonical quantization . . . . . . . . . . . . . . . . . . . . 7

2.2 Interaction with atoms . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Minimal-coupling Hamiltonian . . . . . . . . . . . . . . . . 102.2.2 The dipole approximation . . . . . . . . . . . . . . . . . . . 112.2.3 The atomic Hamiltonian . . . . . . . . . . . . . . . . . . . . 122.2.4 The Heisenberg-Langevin equations . . . . . . . . . . . . . 13

3 Dark states in 3-level atoms 213.1 The 3-level lambda system . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 The interaction Hamiltonian . . . . . . . . . . . . . . . . . 213.1.2 Dark states in the lambda system . . . . . . . . . . . . . . . 233.1.3 Dynamics of the lambda system . . . . . . . . . . . . . . . 243.1.4 Heisenberg-Langevin equations . . . . . . . . . . . . . . . 26

3.2 Electromagnetically induced transparency . . . . . . . . . . . . . . 283.2.1 Maxwell’s equations in matter . . . . . . . . . . . . . . . . 293.2.2 Group velocity and dispersion . . . . . . . . . . . . . . . . 313.2.3 Susceptibility of a gas of lambda atoms . . . . . . . . . . . 32

4 Electromagnetically induced transparency 374.1 Solving the propagation problem . . . . . . . . . . . . . . . . . . . 38

4.1.1 Heisenberg-Langevin equations for continuum variables . 384.1.2 Wave equations for the fields . . . . . . . . . . . . . . . . . 394.1.3 The weak probe approximation . . . . . . . . . . . . . . . . 414.1.4 The adiabatic approximation . . . . . . . . . . . . . . . . . 42

4.2 Corrections to the analytic solution . . . . . . . . . . . . . . . . . . 444.2.1 Non-adiabatic corrections . . . . . . . . . . . . . . . . . . . 454.2.2 Adiabatons . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.3 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . 54

iii

iv CONTENTS

4.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.1 Weak probe . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.2 Strong probe . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Dark state polaritons 675.1 Using dark states as a quantum memory . . . . . . . . . . . . . . . 67

5.1.1 Many-atom Hamiltonian . . . . . . . . . . . . . . . . . . . 685.1.2 Number-state representation . . . . . . . . . . . . . . . . . 695.1.3 Generalized dark states . . . . . . . . . . . . . . . . . . . . 695.1.4 Quantum memory . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Polariton fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2.1 Field equations for the polariton fields . . . . . . . . . . . . 725.2.2 The adiabatic approximation . . . . . . . . . . . . . . . . . 73

5.3 Corrections to the adiabatic solution . . . . . . . . . . . . . . . . . 755.3.1 Validity of the weak probe approximation . . . . . . . . . . 755.3.2 Effect of a small dephasing rate . . . . . . . . . . . . . . . . 765.3.3 Retardation of the coupling laser . . . . . . . . . . . . . . . 775.3.4 Non-adiabatic corrections . . . . . . . . . . . . . . . . . . . 78

5.4 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4.1 Initial conditions and medium parameters . . . . . . . . . 815.4.2 Adiabatic switching . . . . . . . . . . . . . . . . . . . . . . 825.4.3 Ultra fast switching . . . . . . . . . . . . . . . . . . . . . . . 84

6 Standing wave polaritons in thermal gasses 916.1 Heisenberg-Langevin equations . . . . . . . . . . . . . . . . . . . . 92

6.1.1 Fourier decomposition of the atomic operators . . . . . . . 936.1.2 The weak probe approximation . . . . . . . . . . . . . . . . 946.1.3 Effect of atomic motion . . . . . . . . . . . . . . . . . . . . 946.1.4 Adiabatic elimination of the optical coherence . . . . . . . 95

6.2 Polariton fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.1 Wave equation for the polariton field . . . . . . . . . . . . 97

6.3 Solving the coupled wave equations . . . . . . . . . . . . . . . . . 996.3.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 1006.3.2 Solution with negligible diffusion . . . . . . . . . . . . . . 1006.3.3 Solution including diffusion . . . . . . . . . . . . . . . . . . 103

6.4 The physical origin of the diffusion . . . . . . . . . . . . . . . . . . 107

7 Standing wave polaritons in ultra cold gasses 1097.1 Heisenberg-Langevin equations . . . . . . . . . . . . . . . . . . . . 109

7.1.1 The weak probe approximation . . . . . . . . . . . . . . . . 1117.1.2 The adiabatic approximation . . . . . . . . . . . . . . . . . 111

7.2 Polariton fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.2.1 Wave equations for the polariton field . . . . . . . . . . . . 1137.2.2 Low group velocity limit . . . . . . . . . . . . . . . . . . . 113

7.3 Solving the coupled wave equations . . . . . . . . . . . . . . . . . 114

CONTENTS v

7.3.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 1167.3.2 Probe retrieval by a standing wave coupling field . . . . . 1177.3.3 Probe retrieval by a quasi-standing wave coupling field . . 1187.3.4 Calculation of the Raman coherence . . . . . . . . . . . . . 118

7.4 Corrections to the adiabatic solution . . . . . . . . . . . . . . . . . 1217.4.1 Non-adiabatic corrections . . . . . . . . . . . . . . . . . . . 121

7.5 Comparison with the thermal gas case . . . . . . . . . . . . . . . . 125

8 Summary and outlook 1278.1 Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 1278.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

A Operator relations in the Heisenberg picture 131

B Derivation of the non-adiabatic corrections 133

C Computer programs 135C.1 Dimensionless quantities . . . . . . . . . . . . . . . . . . . . . . . . 135

C.1.1 Calculation of parameters . . . . . . . . . . . . . . . . . . . 136C.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C H A P T E R 1Introduction

The invention of the laser in the 1960s heralded a renaissance in atomic physics.Lasers provided atomic physicists with a tool of unprecedented precision andcontrol, opening new avenues of research ranging from the basic research ori-ented, such as high precision spectroscopy and laser cooling, to the immediatelyapplicable, such as the atomic clock.

One such new area of research began in 1976 with the discovery by Alzettaet al. [1] that the fluorescence from a gas of sodium atoms vanished when thesplitting of the hyperfine levels of the atoms matched the mode spacing of theapplied multimode laser. This discovery, known as Coherent Population Trap-ping (CPT), lead to the realization that an otherwise opaque medium could berendered transparent, an effect known as Electromagnetically Induced Trans-parency (EIT). It is this effect which is the topic of this thesis.

The first experimental demonstration of EIT was reported in 1991 by Boller etal. [2] using a gas of strontium atoms. Subsequent experiments [3, 4, 5] focusednot only on demonstrating transparency of the medium, but also on measur-ing the associated reduction of the speed of light, culminating in the impressiveexperiment by Hau et al. [6] in which light speeds as low as 17 m/s were mea-sured.

Shortly after, it became apparent that EIT had potential applications in thefield of quantum information processing, a very hot topic of physics indeed, asit was demonstrated that the laser pulses could not only be slowed down, butin fact brought to a complete stop in the medium. It thus became possible to usethe medium as a quantum memory, transferring the quantum state of the lightto the atoms and back again. Early experimental demonstrations [7, 8] usedatomic gasses as media, but more recent experiments [9, 10] have demonstratedlight storage in a solid state medium.

All the experiments mentioned so far have used traveling wave lasers. Re-cently, however, an experiment has been performed using standing wave laserfields [11]. By using standing wave fields, a stationary laser pulse was createdin the medium. This interesting effect is the focal point of the work presented inthis thesis.

1

2 Chapter 1. Introduction

1.1 Outline of the thesis

The thesis can be divided into three main parts. The first part (chapters 2 and 3)serves as an introduction to the quantum theory of light and matter interactionsas well as to the basics of EIT. In chapter 2, the quantum theory of the interactionbetween light and atoms is outlined with the aim of introducing the so-calledHeisenberg-Langevin equations, which will form the basis for the rest of thethesis. Chapter 3 introduces the 3-level lambda atom, and the susceptibility of amedium comprised of these atoms is calculated, thereby showing the presenceof EIT in such a medium.

The second part (chapters 4 and 5) deals with the detailed theory of EITand light storage using traveling wave laser fields. In chapter 4 the theory ofEIT given in [22] is reviewed, followed by calculations of various non-adiabaticcorrections to the theory. A novel approach to the phenomenon of adiabatonsis also presented, as well as a detailed discussion of the effect of atomic motion.Finally, the analytical results are compared to numerical solutions. Chapter 5reviews the so-called dark state polariton theory of light storage given by [22]and discusses the approximations made as well as various corrections to thetheory, clearing up a few mistakes in [22] underway. Again, the analytical resultsare compared to a numerical simulation of the light storage process.

The third part (chapters 6 and 7) of the thesis deals with EIT and light storageusing standing wave laser fields, and contains the main results of the thesis.Chapter 6 presents and improves the theoretical treatment given in [33] of EITusing standing wave fields in the case of a thermal gas medium, while chapter7 presents a novel theory for EIT using standing wave fields in ultra cold gassesor solid state media.

C H A P T E R 2Interaction of light

and atoms

In this chapter we shall review the quantum theory of the interaction of lightand atoms. First the canonical quantization of the free electromagnetic fieldwill be described, followed by the interaction with the atoms. Working in theHeisenberg picture, we shall derive the so-called Heisenberg-Langevin equa-tions which describe the evolution of an atom interacting with a quasi-resonantmode of the quantized electromagnetic field, including the interaction with thevacuum modes responsible for spontaneous emission. The treatment presentedhere is based on the monographs [12], [13] and [14].

2.1 Quantization of the electromagnetic field

In quantum theory all observables are represented by operators on a Hilbertspace. Therefore the electric field E and the magnetic field B of classical elec-trodynamics must also be operators in a quantum theory of electrodynamics.By rewriting the basic equations of classical electrodynamics, it becomes appar-ent that we can describe the electromagnetic field as a collection of harmonicoscillators, and the transition to a quantum theory becomes clear.

2.1.1 The potential formulation of electrodynamicsThe Maxwell equations of classical electrodynamics are [15]

∇ ·E =ρ

ε0(2.1a)

∇ ·B = 0 (2.1b)

∇×E = −∂B∂t

(2.1c)

∇×B =1c2

∂E∂t

+ µ0J. (2.1d)

3

4 Chapter 2. Interaction of light and atoms

where E and B are the electric and magnetic fields, and ρ and J are the chargeand current densities.

By combining (2.1a) and (2.1d) we obtain the continuity equation expressingcharge conservation

∂ρ

∂t+∇ · J = 0. (2.2)

It is well known from classical electrodynamics [15] that we can express thefields in terms of a scalar potential φ and a vector potential A. These potentialsare defined by

B = ∇×A (2.3)

andE = −∇φ− ∂A

∂t. (2.4)

To obtain differential equations for the potentials, we insert (2.3) and (2.4) intothe Maxwell equations (2.1). This yields the following equations:

∇ (∇ ·A)−∇2A +1c2

∂

∂t∇φ +

1c2

∂2A∂t2

= µ0J (2.5)

∇2φ +∂

∂t(∇ ·A) = − ρ

ε0. (2.6)

These equations allow us to calculate the potentials A and φ given the chargeand current densities ρ and J. They are rather complicated, but fortunately wecan impose additional conditions on A and φ which will simplify the equations.

Since the electric and magnetic fields are derivatives of the potentials, thelatter are not unique. It is easy to verify that any set of potentials A, φ and A′,φ′, related by a gauge transformation defined by

A = A′ −∇Λ (2.7)

andφ = φ′ +

∂Λ∂t

, (2.8)

where Λ(r, t) is any scalar function of r and t, will yield the same fields E and B.This gauge freedom allows us to fix the divergence of A. In the Coulomb gaugewe choose

∇ ·A = 0. (2.9)

In this case the gauge function Λ must satisfy

∇2Λ = ∇ ·A′ (2.10)

where A′ is the vector potential in some other gauge. In the Coulomb gauge theequations for A and φ take the simpler form

−∇2A +1c2

∂

∂t∇φ +

1c2

∂2A∂t2

= µ0J (2.11)

2.1 Quantization of the electromagnetic field 5

∇2φ = − ρ

ε0. (2.12)

The differential equation for the scalar potential φ is now Poisson’s equation, fa-miliar from electrostatics, whose solutions are well known [15]. The equation forthe vector potential A is still complicated by the term involving φ. Fortunatelythis term can be removed by the use of Helmholtz’ theorem [16] which states thatany vector field can be written as the sum of two parts, one of which has zerodivergence and the other zero curl. We can therefore write the current density Jas

J = JT + JL (2.13)

where∇ · JT = 0 (2.14)

and∇× JL = 0. (2.15)

JT is called the transverse component, and JL the longitudinal component, ofthe current density. The same can be done for the vector potential A and thefields E and B. From equation (2.3) it follows that B is determined solely bythe transverse part of the vector potential AT , and in the Coulomb gauge (2.9)the vector potential is purely transverse (AL = 0). For this reason we will leaveout the subscript T for the vector potential in the following. By using the vectoridentity [15]

∇2a = ∇ (∇ · a)−∇× (∇× a) (2.16)

we see that the terms involving A are the transverse part of the lhs. of equation(2.11), and the term containing φ is the longitudinal part. We can therefore splitequation (2.11) into two equations

−∇2A +1c2

∂2A∂t2

= µ0JT (2.17)

1c2

∂

∂t∇φ = µ0JL. (2.18)

The vector potential thus satisfies an inhomogeneous wave equation with thetransverse current density as source term. The scalar potential satisfies bothequations (2.12) and (2.18), but by taking the divergence on both sides of (2.18)and using (2.2) we see that these two equations are equivalent.

2.1.2 Vacuum solutionsTo facilitate the transition to the quantum theory of electrodynamics we considerthe vacuum solutions of the potentials. In the absence of sources (ρ = 0,J = 0)we have φ = 0. The vector potential obeys the homogeneous wave equation

−∇2A +1c2

∂2A∂t2

= 0. (2.19)


We consider a cubic region in space of volume V and impose periodic boundaryconditions. Proceeding by the method of separation of the variables r and t, wecan expand the solution of (2.19) in the series

A (r, t) =∑m

√1

2ε0ωm(αm(t)um(r) + α∗m(t)u∗m(r)) (2.20)

where the sum is over all allowed modes m. The mode functions um(r) satisfythe Helmholtz equation

∇2um +ω2

m

c2um = 0 (2.21)

which has plane wave solutions

um =em√V

exp (ikm · r) . (2.22)

Here km is the wave vector and em the polarization vector of mode m. Theangular frequency ωm and the magnitude of the wave vector are related by thevacuum dispersion relation ω2

m = k2c2. The mode functions satisfy the orthogo-nality condition ∫

V

um(r) · un(r)dV = δmn. (2.23)

The time dependent expansion coefficients αm obey the differential equation

∂2αm

∂t2+ ω2

mαm = 0 (2.24)

which has the solutions

αm(t) = αm(0)e−iωmt. (2.25)

The Coulomb gauge condition (2.9) implies em · km = 0, giving two indepen-dent directions of polarization. Therefore the subscript m signifies the variousmodes, including the polarization. Inserting the solutions (2.22) and (2.25) into(2.20) the vector potential becomes

A(r, t) =∑m

√1

2ε0V ωmem

(αm(0)ei(km·r−ωmt) + α∗m(0)e−i(km·r−ωmt)

)(2.26)

The electric and magnetic fields are found from (2.3) and (2.4)

E = −∂A∂t

(2.27)

B = ∇×A. (2.28)


It is clear from these expressions that the fields are purely transverse in this case,and we therefore drop the subscript T . The expansions for the fields are then

E(r, t) = i∑m

√ωm

2ε0(αm(t)um(r)− α∗m(t)u∗m(r)) (2.29)

B(r, t) =∑m

√1

2ε0ωm∇× (αm(t)um(r) + α∗m(t)u∗m(r)) . (2.30)

2.1.3 Canonical quantizationRecall that the Hamiltonian of the electromagnetic field is [12]

H =12

∫

V

(ε0E2 +

1µ0

B2

)dV. (2.31)

Inserting the expansions (2.29) and (2.30) and using the orthogonality of themode functions (2.23), the Hamiltonian can be written as

H =∑m

ωm (αmα∗m + α∗mαm) (2.32)

where we have retained the order of α∗m, αm for later purposes.We now introduce the real canonically conjugate dynamical variables for

each mode qm, pm and write the expansion coefficients in terms of these

αm =1√2ωm

(ωmqm + ipm) . (2.33)

Writing the Hamiltonian in terms of qm, pm we can cast it into a more recogniz-able form

H =12

∑m

(p2

m + ω2mq2

m

)(2.34)

This is the Hamiltonian of a collection of harmonic oscillators. It is now clearhow to quantize the theory, since the theory of the quantum mechanical har-monic oscillator is well known from basic quantum mechanics [17]. The canon-ically conjugate dynamical variables qm, pm become operators which obey thecanonical commutator

[qm, pn] = ihδmn. (2.35)

We now define the annihilation and creation operators

am =1√

2hωm

(ωmqm + ipm) , a†m =1√

2hωm

(ωmqm − ipm) (2.36)

and observe that these operators are obtained from the expansion coefficientsαm by the substitution

αm →√

ham α∗m →√

ha†m. (2.37)


These operators obey the commutation relations

[am, a†n] = δmn, (2.38a)[am, an] = 0, (2.38b)

[a†m, a†n] = 0. (2.38c)

The Hamilton operator for the quantized electromagnetic field is then

H =∑m

hωm

(a†mam +

12

)=

∑m

hωm

(Nm +

12

). (2.39)

where the number operator for mode m is defined by

Nm = a†mam (2.40)

and represents the number of photons in mode m.The eigenstates of this Hamiltonian are called number states or Fock states.

The properties of these states and of the creation and annihilation operators arewell known from quantum mechanics [17]. The number states are characterizedby a set of integers nm, which gives the number of photons in each mode, andare written as

|nm〉 =∏m

|nm〉. (2.41)

The action of the annihilation and creation operators on the number states aregiven by

aj |nm〉 =√

nj |nj − 1〉∏

m 6=j

|nm〉 (2.42)

a†j |nm〉 =√

nj + 1|nj + 1〉∏

m 6=j

|nm〉. (2.43)

We see that the effect of the annihilation operator aj is to remove a photon frommode j and leave all other modes intact. Similarly, the effect of the creationoperator a†j is to add a photon to mode j. The number states are eigenstates ofthe number operator Nj with eigenvalue nj

Nj |nm〉 = nj |nm〉. (2.44)

The vacuum state of the electromagnetic field is denoted by |0〉 and is thelowest energy state of the field. However it is easily seen from (2.39) that itsenergy is

E0 =12

∑m

hωm (2.45)

which is called the vacuum or zero-point energy. Since there is an infinite numberof modes, each having a non-zero frequency ωm, the vacuum energy is also in-finite, a rather disturbing feature of the theory. Fortunately, it turns out that the


energy of the electromagnetic field, as determined through intensity measure-ments, depends only on the difference between the energy of the vacuum stateand the state in question [13]. This fact allows us to renormalize the Hamiltonian(2.39) by setting the vacuum energy to zero. The renormalized Hamiltonian isthus

H =∑m

hωma†mam =∑m

hωmNm. (2.46)

This does not mean that the vacuum energy is entirely inconsequential. Thereare measurable quantities that depend on the vacuum energy1, but since theseeffects do not influence the problems we are considering in this thesis, we willnot discuss these matters any further.

The vector potential and the fields, now represented by operators A, E andB, are easily obtained by using the substitution (2.37) in the classical expressionsfor the fields (2.20), (2.29) and (2.30)

A(r, t) =∑m

√h

2ε0ωm

(am(t)um(r) + a†m(t)u∗m(r)

)(2.47)

E(r, t) = i∑m

√hωm

2ε0

(am(t)um(r)− a†m(t)u∗m(r)

)(2.48)

B(r, t) =∑m

√h

2ε0ωm∇× (

am(t)um(r) + a†m(t)u∗m(r)). (2.49)

Note that we are working in the Heisenberg picture, in which all operators aretime dependent and the state vectors are time independent, as opposed to theSchrödinger picture where the operators are time independent and the state vec-tors time dependent. In the Heisenberg picture, the operators evolve accordingto the Heisenberg equation of motion [18]

˙A =

1ih

[A, H

]+

∂A

∂t(2.50)

where A is an operator representing any dynamical variable of the system. Mostoperators do not depend explicitly on time, and so the last term almost alwaysvanishes. In the Schrödinger picture, the state vectors evolve in time accordingto the well known Schrödinger equation [18]

ih∂

∂t|ψ〉 = H|ψ〉. (2.51)

These two pictures are entirely equivalent, and we shall mainly be using theHeisenberg picture in this thesis.

1One example is the Casimir force [19]


2.2 Interaction with atoms

So far we have only considered free electromagnetic fields. In the following wewill study the interaction between the electromagnetic field and atoms.

2.2.1 Minimal-coupling Hamiltonian

Consider an atom consisting of Z electrons of charge −e surrounding a nucleuscontaining Z protons of charge +e, along with a number of neutrons. The nu-cleus has a much greater mass than the electrons, and is therefore considered tobe fixed at a position given by the vector R. The classical charge and currentdensities are given by

ρ (r) = −e

Z∑

j=1

δ (r− rj) + Zeδ (r−R) , (2.52)

J (r) = −e

Z∑

j=1

rjδ (r− rj) (2.53)

where the sum is over all the electrons, and rj denotes the position of electron j.Since we have a charge distribution, we can no longer assume φ = 0. The scalarpotential is the solution to Poisson’s equation (2.12) and has the familiar form

φ (r) =1

4πε0

−

Z∑

j=1

e

|r− rj | +Ze

|r−R|

. (2.54)

As a consequence, the electric field E is no longer purely transverse.In the Coulomb gauge (2.9) the Hamiltonian of the atom-field system takes

the form [13]

H =1

2me

Z∑

j=1

(pj + eA(rj)

)2

+12

∫ρ(r)φ(r)dV

+12

∫ (ε0ET (r)2 +

1µ0

B(r)2)

dV

(2.55)

which is known as the minimal-coupling Hamiltonian. The pj ’s are the momen-tum operators of the electrons, e is the elementary charge and me is the electronmass. This Hamiltonian can be split into three parts

H = HA + HF + HI (2.56)

2.2 Interaction with atoms 11

where

HA =1

2me

Z∑

j=1

p2j +

12

∫ρ (r)φ (r) dV, (2.57a)

HF =12

∫ (ε0E2

T +1µ0

B2

)dV, (2.57b)

HI =e

me

Z∑

j=1

A · pj +e2

2me

Z∑

j

A2. (2.57c)

The first term HA is the atomic Hamiltonian describing an atom where onlythe Coulomb interactions between the constituent particles are included. Thesecond term HF is the free field Hamiltonian treated in the previous section, andfinally the third term HI is the part of the Hamiltonian describing the interactionbetween the atom and the electromagnetic field.

2.2.2 The dipole approximationThe minimal-coupling Hamiltonian can be transformed into a form more suit-able for calculations. This is achieved by a unitary operator U such that

H = U†H ′U , (2.58)

and the state vectors are transformed according to

|ψ〉 = U†|ψ′〉. (2.59)

It is easy to see that the results obtained from the transformed theory are exactlythe same as those obtained from the original theory.

The advantage of the transformed Hamiltonian is that it contains the fieldsE and B instead of the vector potential A. Performing the transformation is arather lengthy procedure, the details of which can be found in [13]. Discardingthe small interactions with the magnetic field, the Hamiltonian takes the form

H = HA + HF + HI (2.60)

where the parts HA, HF and HI are given by

HA =1

2me

Z∑

j=1

p2j +

12

∫ρ(r)φ(r)dV (2.61a)

HF =12

∫ (ε0E2

T +1µ0

B2

)dV (2.61b)

HI = e

Z∑

j=1

rj · ET (rj). (2.61c)


In the interaction part of the Hamiltonian HI we can exploit the fact that thewavelength of the electromagnetic field is much longer than the typical size ofan atom, which is of the order of the Bohr radius a0. The electromagnetic fieldis thus essentially constant over the spatial extent of the atom, and it is thereforea good approximation to substitute the electric field in HI with the field at theposition of the nucleus, R. With this approximation, known as the electric dipoleapproximation, the interaction part of the Hamiltonian becomes

HI = e

Z∑

j=1

rj · ET (R) = −d · ET (R) (2.62)

where d is the electric dipole moment operator of the atom, defined by

d = −e

Z∑

j=1

rj . (2.63)

Changing the notation so that R → r and suppressing the subscript T , sinceonly the transverse part of the electric field appears in the Hamiltonian, we writethe interaction part as

HI = −d · E (r) (2.64)

This form, known as the electric dipole Hamiltonian, will form the basis for thecalculations of this thesis.

2.2.3 The atomic Hamiltonian

The atomic part HA of the Hamiltonian (2.60) has a set of eigenstates |µ〉, describ-ing the various internal states of the atom, with corresponding energy eigenval-ues hωµ

HA|µ〉 = hωµ|µ〉. (2.65)

The Hamiltonian HA is Hermitian which implies that the eigenstates form acomplete set of orthonormal states. As such, they satisfy the closure relation

∑µ

|µ〉〈µ| = 1. (2.66)

We can use this fact to express HA in terms of the eigenstates and their corre-sponding eigenvalues. By applying (2.66) to HA we get

HA =∑µ,ν

|µ〉〈µ|HA|ν〉〈ν|. (2.67)

From the orthonormality of the eigenstates and the eigenvalue equation (2.65)we know that

〈µ|HA|ν〉 = hωµδµν , (2.68)


and the Hamiltonian therefore takes the form

HA =∑

µ

hωµ|µ〉〈µ|. (2.69)

In a similar fashion, we can use the closure relation to obtain an expression forthe electric dipole moment operator

d =∑µ,ν

dµν |µ〉〈ν| (2.70)

wheredµν = 〈µ|d|ν〉 (2.71)

is a matrix element of the electric dipole moment operator. The eigenstates ofHA are states of definite parity and therefore all terms in (2.70) with ν = µ mustvanish, since the dipole operator as defined in (2.63) has odd parity.

Another operator which will prove to be convenient is defined by

σµν = |µ〉〈ν|, (2.72)

which we shall refer to as the atomic operator. From the orthonormality of theeigenstates it is easily seen that the atomic operators obey the commutation re-lation

[σαβ , σµν ] = δβµσαν − δαν σµβ . (2.73)

With the above definitions, we can write the Hamiltonian (2.60) in terms of cre-ation/annihilation operators and atomic operators as

H = HA + HF + HI (2.74)

where the atomic, field and interaction parts are

HA =∑

µ

hωµσµµ (2.75)

HF =∑m

hωma†mam (2.76)

HI = −E(r) ·∑µ,ν

dµν σµν (2.77)

2.2.4 The Heisenberg-Langevin equationsWe now turn our attention to deriving the Heisenberg equations of motion forthe atomic operators σµν . In particular, it will be shown how the interactionwith the vacuum field modes leads to the phenomenon of spontaneous emis-sion. The situation we will consider is that of a single field mode quasi-resonantwith a particular transition in the atom. In this case the only states that can be


populated are the ground state, labeled |1〉, and the excited state |2〉. All otherstates of the atom are considered to be far enough away from resonance thatthe transition probability becomes negligible, allowing us to treat the atom as atwo-level system.

Recall the Hamiltonian (2.74) of the atom-field system. The electric field E(r)is split into two parts

E(r) = EL(r) + EV (r) (2.78)

where EL is the single mode external field (presumably a laser mode) and EV

denotes the vacuum field. This in turn leads to a splitting of the interaction partof the Hamiltonian HI

HI = HL + HV (2.79)

where

HL = −EL(r) · (d12σ12 + h.a.) (2.80)

describes the interaction between the atom and the laser field and

HV = −EV (r) · (d12σ12 + h.a.) (2.81)

is the interaction with the vacuum field. As before, the fields are given by anexpansion in cavity modes

EL(r) = i

√hωL

2ε0

(buL(r)− b†u∗L(r)

), (2.82)

EV (r) = i∑

m6=L

√hωm

2ε0

(amum(r)− a†mu∗m(r)

)(2.83)

where the subscript L denotes the laser mode and b, b† are the correspondingannihilation and creation operators, respectively.

The atomic part of the Hamiltonian becomes

HA = hω21σ22 (2.84)

where hω21 is the energy difference between the excited state and the groundstate, which has been chosen as the zero-point of the atomic energy. The freefield part of the Hamiltonian is

HF =∑m

hωma†mam (2.85)

where the sum is over all field modes, including the laser mode.


We can now write down the Heisenberg equations of motion for the atomicoperators and the field operators:

˙σµν =1ih

[σµν , H

]=

1ih

([σµν , HA

]+

[σµν , HL

]+

[σµν , HV

])(2.86a)

˙b =

1ih

[b, H

]=

1ih

([b, HF

]+

[b, HL

])(2.86b)

˙am =1ih

[am, H

]=

1ih

([am, HF

]+

[am, HV

])(2.86c)

Using the commutation relations for the atomic operators (2.73) and the fieldoperators (2.38) we obtain the following equations of motion:

˙σ11 =i

h

(EL + EV

) · (d12σ12 − h.a.)

(2.87a)

˙σ22 = − i

h

(EL + EV

) · (d12σ12 − h.a.)

(2.87b)

˙σ12 = −iω21σ12 +i

h

(EL + EV

) · d21

(σ11 − σ22

)(2.87c)

˙am = −iωmam +√

ωm

2ε0hu∗m · (d12σ12 + h.a.

). (2.87d)

In the absence of the electric field, the time evolution of the operators is simply

σfµν(t) = σµν(0)eiωµνt (2.88a)

afm(t) = am(0)e−iωmt, (2.88b)

where the superscript f signifies evolution in the absence of the electric field.This motivates the introduction of slowly varying operators defined by

σµµ(t) = σµµ(t) (2.89a)

σ12(t) = σ12(t)e−iωLt (2.89b)

am(t) = am(t)e−iωmt, (2.89c)

where the operators σµν and am on the rhs. of the equations are the slowly vary-ing operators. Their time evolution is expected to be slow compared to the timescale ω−1 in the presence of interactions. We also define a slowly varying fieldoperator for the laser mode

EL = EaLe−iωLt + Ec

LeiωLt (2.90)

where the slowly varying field operators Ea,cL are defined as

EaL = i

√hωL

2ε0uL(r)b Ec

L = (EaL)† , (2.91)

and the superscripts a, c denotes the part of the field operator EL containing theannihilation and creation operator, respectively.


Inserting the expressions (2.90) and (2.83) for the fields, the equations of mo-tion for the slowly varying operators become

σ11 =i

h

(Ea

Le−iωLt + h.a.) · (d12σ12e

−iωLt − h.a.)

−∑

m 6=L

√ωm

2hε0

(ame−iωmtum(r)− h.a.

) · (d12σ12e−iωLt − h.a.

) (2.92a)

σ22 = − i

h

(Ea

Le−iωLt + h.a.) · (d12σ12e

−iωLt − h.a.)

+∑

m 6=L

√ωm

2hε0


) · (d12σ12e−iωLt − h.a.

) (2.92b)

σ12 = i(ωL − ω21)σ12 +i

heiωLt

(Ea

Le−iωLt + h.a.) · d21

(σ11 − σ22

)

− eiωLt∑

m 6=L

√ωm

2hε0


) · d21

(σ11 − σ22

) (2.92c)

am = eiωmt

√ωm

2hε0u∗m(r) · (d12σ12e

−iωLt + h.a.). (2.92d)

The rhs. of the above equations contain terms with time dependent phase fac-tors that vary either as ei(ω+ωL)t or as ei(ω−ωL)t, where ω is either ωL or ωm.Since the operators on the lhs. vary slowly on the time scale ωL, the terms in-volving ei(ω+ωL)t make a negligible contribution compared to the terms involv-ing ei(ω−ωL)t and are therefore discarded. This approximation is known as therotating-wave approximation (RWA). The equations of motion in the RWA take theform

σ11 =i

h

(Ec

L(r) · d12σ12 − h.a.)

+∑

m 6=L

√ωm

2hε0

(um · d21amσ21e

−i(ωm−ωL)t + h.a.) (2.93a)

σ22 = − i

h

(Ec

L(r) · d12σ12 − h.a.)

−∑

m 6=L

√ωm

2hε0

(um · d21amσ21e

−i(ωm−ωL)t + h.a.) (2.93b)

σ12 = iδLσ12 +i

hEa

L · d21(σ11 − σ22)

−∑

m 6=L

√ωm

2hε0um(r) · d21am(σ11 − σ22)e−i(ωm−ωL)t

(2.93c)

am =√

ωm

2hε0u∗m(r) · d12σ12e

i(ωm−ωL)t (2.93d)

where δL = ωL − ω21 is the detuning of the laser from resonance.


To see the physical significance of this approximation, consider the interac-tion part of the Hamiltonian (2.77) written in terms of the slowly varying oper-ators (2.89) and invoking the RWA

HI,RWA = − (EaL(r) · d21σ21 + Ec

L(r) · d12σ12) (2.94)

From this expression it is easily seen that the interaction Hamiltonian couplesthe states |1〉|n〉, |2〉|n − 1〉. This corresponds to processes that conserve energy.The two terms discarded in the RWA correspond to processes that do not con-serve energy.

Looking at the equations of motion (2.93), the solution of them appears to bean impossible task. The atomic operators σµν are coupled to an infinite num-ber of vacuum field modes. Fortunately we can apply an approximation thatallows us to disregard the precise evolution of the vacuum field modes whilestill modeling their influence on the atom.

Integrating equation (2.93d) from t0 to t, we get

am(t) = am(t0) +√

ωm

2hε0u∗m · d12

∫ ∆t

0

σ12(t− τ)ei(ωm−ωL)(t−τ)dτ (2.95)

where ∆t = t − t0. Inserting this solution into the mode expansion for thevacuum field (2.83) we can write the vacuum field EV (r) as a sum of two con-tributions

EV (r) = EV,f (r) + EV,s(r) (2.96)

where the free field part EV,f is given by

EV,f (r, t) = i∑

m6=L

√hωm

2ε0

(am(t0)e−iωm∆tum(r)− h.a.

)(2.97)

and the source field part EV,s is given by

EV,s(r, t) =i∑

m 6=L

ωm

2ε0

[(u∗m(r) · d12)um(r)e−iωLt×

∫ ∆t

0

σ12(t− τ)e−i(ωm−ωL)τdτ − h.a.].

(2.98)

We see that EV,f is the vacuum field propagating in the absence of the atom,while EV,s is the field generated by the atom from t0 to t, hence the names of thetwo parts of the vacuum field.

Inserting (2.95), the equations of motion for σµν (2.93a)-(2.93c) become

σ11 =i

h

(Ec

L(r) · d12σ12 − h.a.)

+ F11 + G11 (2.99a)

σ22 = − i

h

(Ec

L(r) · d12σ12 − h.a.)

+ F22 + G22 (2.99b)

σ12 = iδLσ12 +i

hEa

L · d21(σ11 − σ22) + F12 + G12 (2.99c)


where F11 = −F22 is given by

F11 =∑

m 6=L

√ωm

2hε0

(um · d21σ21am(t0)e−iδmt + h.a.

), (2.100)

and G11 = −G22 is

G11 =∑

m6=L

ωm

2hε0|um · d21|2

(σ21

∫ ∆t

0

σ12(t− τ)e−iδmτdτ + h.a.

). (2.101)

Finally F12 and G12 are given by

F12 =∑

m6=L

√ωm

2hε0um · d21 (σ22 − σ11) am(t0), (2.102)

G12 =∑

m 6=L

ωm

2hε0|um · d21|2 (σ22 − σ11)

∫ ∆t

0

σ12(t− τ)eiδmτdτ, (2.103)

where δm = ωm − ωL. Note that we have made use of the fact that σµν and am

commute at equal times2 to write all am to the right and all a†m to the left of theatomic operators σµν before inserting (2.95). This is called normal ordering andwill prove to be convenient later.

To proceed further we consider the time integral that appears in the terms(2.101) and (2.103) ∫ ∆t

0

σ12(t− τ)e±iδmτdτ. (2.104)

Since the atomic operator σµν is slowly varying, we can pick a time interval ∆tsmall enough so that we can make the replacement

σ12(t− τ) ' σ12(t) (2.105)

in the integral (2.104), allowing us to take the atomic operator outside the inte-gral such that

∫ ∆t

0

σ12(t− τ)e±iδmτdτ ' σ12(t)∫ ∆t

0

e±iδmτdτ. (2.106)

Note that in order for the operator σ12 to be slowly varying, we must have ωL 'ω21 and we therefore take ωL = ω21 in the following.

This approximation, known as the Markov approximation, makes the evolu-tion of σµν depend only on the value of σµν at the present time t and not on itsvalue in the past, and the terms (2.101) and (2.103) become

G11 =∑

m 6=L

ωm

2hε0|um · d21|2σ22

(∫ ∆t

0

eiδmτdτ + c.c

), (2.107)

2This is not necessarily true for operators taken at different times. In particular, am and σµν atdifferent times do not commute.


G12 = −∑

m 6=L

ωm

2hε0|um · d21|2σ12

∫ ∆t

0

eiδmτdτ (2.108)

The time integral on the rhs. of (2.106) is elementary and gives∫ ∆

0

te±iδmτdτ =sin(δm∆t)

δm+ i

cos(δm∆t)− 1δm

. (2.109)

Recall that the expressions (2.101) and (2.103) involves summing over the in-finitely many vacuum modes. If the quantization volume V is large, we canapproximate the discrete sum with an integral over a continuous distribution ofmodes, with the discrete variable ωm replaced by a continuous variable ω. Wenow assume that ∆t À ω−1

21 . In this case the first term on the rhs. of (2.109)approaches a delta function. The second term vanishes if ω − ω21 = 0, but isotherwise effectively equal to (ω−ω21)−1 when integrated over a large range ofmodes due to the rapidly oscillating cosine function. Therefore when the timeintegral (2.109) appears in an integral over all the vacuum modes, we make thereplacement

∫ ∆t

0

e±i(ωm−ω21)τdτ → πδ(ω − ω21)∓ iP(

1ω − ω21

)(2.110)

where P denotes the Cauchy principal value (see [16] for a definition).Converting the sums in (2.101) and (2.103) into integrals, using (2.110) and

inserting the expression for the mode functions (2.22), we obtain the followingexpressions for G11 and G12

G11 = −G22 = γσ22 (2.111)

G12 =(−γ

2+ i∆

)σ12 (2.112)

where

γ =|d21|2ω3

21

3πε0hc3(2.113)

is the spontaneous decay rate and h∆ is the radiative shift of the excited state. Theradiative shift is quite small and can be thought of as a modification to the reso-nance frequency ω21. For this reason, we shall ignore it in the remainder of thisthesis.3

We can now write down the equations of motion for the atomic operators:

σ11 = γσ22 +i

h(Ec

L(r) · d12σ12 − h.a.) + F11 (2.114a)

σ22 = −γσ22 − i

h(Ec

L(r) · d12σ12 − h.a.) + F22 (2.114b)

σ12 = −(γ

2− iδL

)σ12 +

i

hEa

L · d21 (σ11 − σ22) + F12 (2.114c)

3The calculation of the radiative shift given here is inadequate. See [20] for further details.


These equations are known as the Heisenberg-Langevin equations for the two-level atom coupled to a quasi-resonant electric field mode. Along with theequation of motion for the laser mode, they form a closed set of equations forthe system. The effect of the vacuum field modes on the atom are given by thespontaneous decay rate γ and the noise operators Fµν given by (2.100) and (2.102).The expectation values of the noise operators can be calculated by using

am(t0)|0〉 = 0; 〈0|a†m(t0) = 0. (2.115)

From this, it follows easily that4

〈Fµν〉 = 0. (2.116)

Taking the expectation value of the Heisenberg-Langevin equations, we obtainthe master equation for the density matrix of the system.5 For this reason, onemight be tempted to omit the noise operators altogether, but although their ex-pectation value is zero, the noise operators are essential when dealing with op-erator equations, because without them the commutation relation (2.73) wouldbe violated.

4Note that this calculation was made easy by the fact that we chose the normal ordering of theoperators.

5Also known as the optical Bloch equations.

C H A P T E R 3Dark states in 3-level atoms

Having established the basic theory of the interaction between atoms and theelectromagnetic field, we now apply this theory to the particular case of a 3-levellambda atom coupled to two nearly resonant field modes. It will be shown thatthis system has a stationary state, known as a dark state, which does not coupleto the electromagnetic field. Consequently, no excitation of the atom can occurwhen it is in this state. We shall also extend our formalism of the previous chap-ter to this system, deriving the corresponding Heisenberg-Langevin equations.Finally, we shall see how the presence of the dark state leads to the strange phe-nomenon of EIT, in which resonant light can travel through a normally opticallythick medium with a substantially reduced velocity.

3.1 The 3-level lambda system

The three states of the lambda system (see figure 3.1) consist of the ground state|b〉, the excited state |a〉 and an intermediate state |c〉. The excited state can de-cay to each of the two lower states through spontaneous emission, but the in-termediate state is considered to be metastable due to a vanishing dipole matrixelement (dbc = 0).

The atom is interacting with two laser fields, one of which is nearly resonantwith the |b〉 ↔ |a〉 transition, the other with the |c〉 ↔ |a〉 transition. The formeris called the probe laser, while the latter is called the coupling laser.

3.1.1 The interaction HamiltonianAs a first step, we shall consider the case where the fields can be treated asclassical fields obeying Maxwell’s equations. The atom, however, is still to betreated quantum mechanically. This approximation, known as the semiclassicalapproximation, is valid if the fields are sufficiently strong. In this case the electricfield operator in the interaction Hamiltonian (2.77) is replaced by the classicalfield E(r, t), which is given as a sum of two laser fields

E(r, t) = Ep(r, t) + Ec(r, t), (3.1)

21

22 Chapter 3. Dark states in 3-level atoms

|a〉

|b〉

|c〉

Ωp

Ωc

Figure 3.1: The 3-level lambda system. The probe laser couples the ground state |b〉to the excited state |a〉, and the coupling laser couples the intermediate state |c〉 tothe excited state. The Rabi frequencies of the probe and coupling lasers are denotedΩp and Ωc, respectively.

where Ep is the probe field given by

Ep(r, t) = Ep,0ei(kp·r−ωpt) + c.c. (3.2)

and Ec is the coupling field given by

Ec(r, t) = Ec,0ei(kc·r−ωct) + c.c. (3.3)

Here Ep,0 and Ec,0 determine the amplitudes of the two fields, ωp, ωc are theirrespective angular frequencies and kp, kc the corresponding wave vectors.

The semiclassical Hamiltonian for the lambda system in the rotating-waveapproximation is thus given by

H = hωcbσcc + hωabσaa − h(Ωpe

−iωptσab + Ωce−iωctσac + h.a.

), (3.4)

where the Rabi frequencies Ωp, Ωc of the probe and coupling fields are defined by

Ωp = Ep,0 · dabeikp·r/h Ωc = Ec,0 · dace

ikc·r/h, (3.5)

while hωcb is the energy of the intermediate state |c〉 and hωab is the energy ofthe excited state |a〉. The energy of the ground state |b〉 is chosen to be zero.

The Hamiltonian (3.4) is time dependent as is evident from the rapidly oscil-lating phase factors in the interaction part. As a consequence we cannot separatethe solution to the Schrödinger equation (2.51) into space and time dependentparts. However, by using a unitary transformation we can transform the Hamil-tonian (3.4) into a time independent form. This unitary transformation is knownas a rotating frame transformation, and in the case of the lambda system it has theform

U = e−i((ωp−ωc)σcc+ωpσaa)t. (3.6)

The states of the system in the rotating frame |ψ′〉 are related to the states in thestationary frame |ψ〉 by

|ψ〉 = U |ψ′〉 (3.7)

3.1 The 3-level lambda system 23

Inserting this definition into the Schrödinger equation and rearranging the termswe get

ih∂

∂t|ψ′〉 =

(U†HU − ihU† ∂U

∂t

)|ψ′〉 = H ′|ψ′〉. (3.8)

We see that the states in the rotating frame obey a Schrödinger equation with atransformed Hamiltonian given by

H ′ = −h∆σcc − hδpσaa − h (Ωpσab + Ωcσac + h.a.) , (3.9)

where δp = ωp − ωab is the detuning of the probe laser and ∆ = ωp − ωc − ωcb isthe two-photon detuning.

3.1.2 Dark states in the lambda systemUsing the eigenstates |b〉, |c〉 and |a〉 of the atomic part of the Hamiltonian inthe stationary frame (3.4) as a basis, we can write a matrix representation of therotating frame Hamiltonian (3.9) as

H ′ = −h

0 0 Ω∗p0 ∆ Ω∗c

Ωp Ωc δp

(3.10)

Assuming two-photon resonance (∆ = 0) we can diagonalize the Hamiltonian.The eigenstates are

|D′〉 = cos θ|b〉 − sin θ|c〉 (3.11a)|B′

+〉 = sin θ sin φ|b〉+ cos θ sin φ|c〉+ cosφ|a〉 (3.11b)|B′−〉 = sin θ cos φ|b〉+ cos θ cosφ|c〉 − sin φ|a〉 (3.11c)

where the angles θ and φ are given by

tan θ =Ωp

Ωc, tan 2φ =

2√|Ωp|2 + |Ωc|2

δp. (3.12)

The corresponding energy eigenvalues are given by

εD = 0, (3.13a)

ε+ =h

2

(−δp +

√δ2p + 4 (|Ωp|2 + |Ωc|2)

), (3.13b)

ε− =h

2

(−δp −

√δ2p + 4 (|Ωp|2 + |Ωc|2)

). (3.13c)

The time evolution of the eigenstates in the rotating frame is

|α′, t〉 = |α′, t = 0〉e−iεαt/h (3.14)


where |α′〉 is one of the eigenstates (3.11) of the rotating frame Hamiltonian.Transforming back to the stationary frame, the state |D〉 is

|D〉 = cos θ|b〉 − sin θ|c〉e−i(ωp−ωc)t (3.15)

Notice that this state does not contain the excited state |a〉. Consequently, if theatom is initially in this state, it will never be excited. For this reason, the state|D〉 is referred to as a dark state. Although the dark state in the rotating frame|D′〉 is an eigenstate of the Hamiltonian (3.9), the dark state in the stationaryframe is not an eigenstate of the Hamiltonian (3.4). It is, however, a stationarystate in the sense that the atom will remain in the dark state for all time, as isevident from the expression (3.15).

3.1.3 Dynamics of the lambda systemHaving established the existence of the dark state, the next problem we need toaddress is the time evolution of the lambda system. In particular, the questionarises whether a lambda system initially in the ground state |b〉 eventually endsup in the dark state |D〉. To properly answer this question, we need to solve theHeisenberg-Langevin equations for the lambda system. This is a rather difficulttask, so before undertaking it we shall first employ an easier, but less accurate,approach.

Instead of trying to solve the full Heisenberg-Langevin equations, we shallinstead solve the Schrödinger equation for the lambda atom. The Hamiltonian(3.9) does not take spontaneous emission into account, but we can add a decayof the excited state phenomenologically. This method has the advantage that theproblem involves solving three coupled differential equations for the probabilityamplitudes of the three states, instead of the nine coupled differential equationsfor the atomic operators comprising the Heisenberg-Langevin equations. Thedisadvantage is that the method cannot account for decay into the two lowerstates |b〉 and |c〉. The system is thus considered an open system in which theexcited state |a〉 decays to states outside the lambda system.

With the spontaneous decay of the excited state added and assuming zerotwo-photon detuning (∆ = 0), the Hamiltonian for the system takes the form

H ′ = −h (δp + iγ) σaa − h (Ωpσab + Ωcσac + h.a.) (3.16)

where γ is the spontaneous decay rate of the excited state.Instead of using the basis states |a〉, |b〉 and |c〉, we shall find it more illumi-

nating to define a different basis using the dark state |D′〉 as one of the basisstates. This basis, which we shall call the dark/bright basis, is given by

|D′〉 =Ωc

Ωt|b〉 − Ωp

Ωt|c〉, (3.17a)

|B′〉 =Ω∗pΩt|b〉+

Ω∗cΩt|c〉, (3.17b)

|E′〉 = |a〉, (3.17c)


where Ωt =√|Ωp|2 + |Ωc|2 is the total Rabi frequency.

It is easy to verify that these three states are orthogonal and therefore spanthe Hilbert space. In the dark/bright basis the Hamiltonian (3.16) takes the form

H ′ = −h (δp + iγ) |E′〉〈E′| − hΩt (|E′〉〈B′|+ h.a.) . (3.18)

Notice that the dark state does not appear in the Hamiltonian. Instead we havea coupling between the bright state |B′〉 and the excited state |E′〉. The matrixrepresentation of the Hamiltonian (3.18) in the dark/bright basis is

H ′ = −h

0 0 00 0 Ωt

0 Ωt δp + iγ

(3.19)

We expand the quantum state in the rotating frame in the dark/bright basis

|ψ′(t)〉 = cD(t)|D′〉+ cB(t)|B′〉+ cE(t)|E′〉 (3.20)

with time dependent expansion coefficients cD, cB and cE . Inserting this expan-sion into the Schrödinger equation gives three coupled differential equations forthe amplitudes

cD

cB

cE

=

0 0 00 0 iΩt

0 iΩt iδ − γ

×

cD

cB

cE

(3.21)

where a dot denotes differentiation with respect to time. If the atom is initiallyin the ground state |b〉, the initial conditions on the expansion coefficients are

cD(0) = 〈D′|b〉 =Ω∗cΩt

, (3.22a)

cB(0) = 〈B′|b〉 =Ωp

Ωt, (3.22b)

cE(0) = 〈E′|b〉 = 0. (3.22c)

With these initial conditions the solution is

cD(t) =Ω∗cΩt

(3.23a)

cB(t) =Ωp

ΩtΩ′te−

12 (γ−iδp)t

(12

(γ − iδp) sin(Ω′tt) + Ω′t cos(Ω′tt))

(3.23b)

cE(t) = iΩp

Ω′te−

12 (γ−iδp)t sin(Ω′tt) (3.23c)

where

Ω′t =

√Ω2

t −14(γ − iδp)2. (3.24)

It is clear from this solution that the population in the excited state decays withrate γ, and that a portion of the population is ”trapped“ in the dark state. Thisphenomenon is therefore known as coherent population trapping (CPT).


3.1.4 Heisenberg-Langevin equations

As mentioned earlier, the disadvantage of the method we have employed hereis that spontaneous emission into the states |b〉 and |c〉 cannot be taken into ac-count. The question still remains whether all the population will eventually endup in the dark state. To answer this question we write down the Heisenberg-Langevin equations for the lambda system.

As in chapter 2 the Hamiltonian is split into several parts

H = HA + HF + HL + HV (3.25)

where HA and HF are the parts describing the free atom and the free electro-magnetic field, respectively. The term HL describes the interaction between theatom and the two laser fields, while the last term HV is the interaction with thevacuum field. In the standard basis the individual parts are given by

HA = hωcbσcc + hωabσaa (3.26a)

HF =∑m

hωma†mam (3.26b)

HL = −(Ep + Ec

) · (dbaσba + dcaσca + h.a.) (3.26c)

HV = −EV · (dbaσba + dcaσca + h.a.) . (3.26d)

As in equation (2.90) we introduce slowly varying field operators for the probeand coupling fields

Ep,c = Eap,ce

−iωp,ct + Ecp,ce

iωp,ct (3.27)

with

Eap,c = i

√hωp,c

2ε0up,c(r)bp,c Ec

p,c =(Ea

p,c

)† (3.28)

The slowly varying atomic operators σµν are defined by

σµµ(t) = σµµ(t) (3.29a)

σba(t) = σba(t)e−iωpt (3.29b)

σca(t) = σca(t)e−iωct (3.29c)

σbc(t) = σbc(t)e−i(ωp−ωc)t (3.29d)

In the RWA, the Heisenberg-Langevin equations for the slowly varying atomic


operators are

σaa =−i

h

(Ec

p · dbaσba + Ecc · dcaσca − h.a.

)− γσaa + Faa (3.30a)

σbb =i

h

(Ec

p · dbaσba − h.a.)

+ γbσaa + Fbb (3.30b)

σcc =i

h

(Ec

c · dcaσca − h.a.)

+ γcσaa + Fcc (3.30c)

σba =i

h

(Ea

p · dab(σbb − σaa) + Eac · dacσbc

)− Γbaσba + Fba (3.30d)

σca =i

h

(Ea

c · dac(σcc − σaa) + Eap · dabσcb

)− Γcaσca + Fca (3.30e)

σbc =i

h

(Ec

c · dcaσba −Eap · dabσac

)− Γbcσbc + Fbc (3.30f)

where γ = γb + γc is the total spontaneous emission rate of the excited state, andwhere we have introduced the complex decay rates

Γba = γba − iδp, (3.31a)Γca = γca − iδc, (3.31b)Γbc = γbc − i∆, (3.31c)

which include the detuning of the probe laser δp = ωp−ωab, the detuning of thecoupling laser δc = ωc − ωac and the two-photon detuning ∆ = ωp − ωc − ωcb =δp − δc. Note that we have also introduced a dephasing rate γbc of the operatorσbc. Since we consider state |c〉 to be metastable, this dephasing rate is not dueto spontaneous decay of state |c〉, but rather due to external influences such ascollisions with other atoms and stray electromagnetic fields.

Since the frequencies ωab and ωac are typically optical frequencies, the oper-ators σba and σca are called optical coherences, while the operator associated withthe two-photon transition between |b〉 and |c〉 is called the Raman coherence.

Taking the expectation value of the atomic operators, and assuming that thefields can be treated classically, we obtain the master equations for the densitymatrix elements [18]

ρaa = i (Ωpρba + Ωcρca − c.c.)− γρaa (3.32a)ρbb = −i (Ωpρba − c.c.) + γbρaa (3.32b)ρcc = −i (Ωcρca − c.c.) + γcρaa (3.32c)ρab = iΩp(ρbb − ρaa) + iΩcρcb − Γbaρab (3.32d)ρac = iΩc(ρcc − ρaa) + iΩpρbc − Γcaρac (3.32e)ρcb = i(Ω∗cρab − Ωpρca)− Γbcρcb (3.32f)

It will prove convenient to write the master equation in the dark/bright basis.Assuming zero two photon detuning (∆ = 0), the Hamiltonian in the rotatingframe is

H ′ = −hδpσEE − hΩ(σEB + σBE). (3.33)


For convenience, we assume equal partial decay rates (γb = γc) and negligibledephasing rate (γbc = 0). In this case the master equation in the dark/brightbasis is

ρEE = −iΩ (ρEB − c.c.)− γρEE (3.34a)

ρBB = iΩ(ρEB − c.c.) +γ

2ρEE (3.34b)

ρDD =γ

2ρEE (3.34c)

ρEB = iδpρEB + iΩ(ρBB − ρEE)− γ

2ρEB (3.34d)

ρED = iδpρED + iΩρBD − γ

2ρED (3.34e)

ρBD = iΩρED (3.34f)

We wish to find the steady state solution ρstµν for the density matrix elements by

setting ρµν = 0 on the lhs. of equations (3.34). From (3.34c) we see immediatelythat ρst

EE = 0. Inserting this in (3.34a) yields ρstEB = ρst

BE . Equation (3.34d) andits complex conjugate gives two equations

0 = iδpρEB + iΩρBB − γ

2ρEB (3.35)

0 = −iδpρEB − iΩρBB − γ

2ρEB . (3.36)

Adding these two equations gives us ρstEB = 0 and, by inserting this into one of

the equations, ρstBB = 0. From the normalization condition

ρDD + ρBB + ρEE = 1 (3.37)

we get the result ρstDD = 1. We have thus shown that the lambda system evolves

into the dark state, regardless of its initial state, if the two-photon detuning anddephasing rate are zero.

3.2 Electromagnetically induced transparency

In the previous section the effect of the electromagnetic field on a 3-level lambdaatom was described. We now turn our attention to the effect an ensemble ofatoms has on the fields. In the case of two-photon resonance, the atoms aredriven into the dark state and consequently no absorption of the probe fieldoccurs. This effect is called Electromagnetically Induced Transparency (EIT). Sincean atom in a dark state does not interact with the electromagnetic field, onemight naively expect that the medium will have no effect on the probe field atall. As we shall see in this section, this is not the case. Although the absorptionof the probe field is suppressed under ideal conditions, the velocity with whicha probe pulse travels through the medium can be substantially reduced. In thissection we shall describe this effect by calculating the electric susceptibility of a

3.2 Electromagnetically induced transparency 29

medium consisting of an ensemble of 3-level lambda atoms. We shall also findunder what conditions EIT occurs.

3.2.1 Maxwell’s equations in matterThe Maxwell equations for the electromagnetic field in the presence of a mediumare [15]

∇ ·D = ρf (3.38a)∇ ·B = 0 (3.38b)

∇×E = −∂B∂t

(3.38c)

∇×H = Jf +∂D∂t

(3.38d)

where ρf is the free charge density and Jf is the free current density. The auxil-iary fields D and H are defined by

D = ε0E + P H =1µ0

B−M. (3.39)

The polarization P is the electric dipole moment density, and the magnetizationM is the magnetic dipole moment density, of the medium.

We shall consider source free regions in space (ρf = 0 and Jf = 0) and alsoassume that the medium is nonmagnetic (M = 0).1 In this case we can combineMaxwell’s equations to obtain a wave equation for the electric field2

∇2E−∇(∇ ·E) = µ0ε0∂2E∂t2

+ µ0∂2P∂t2

. (3.40)

Taking the Fourier transform with respect to time of the wave equation yields

∇2E−∇(∇ · E) = −µ0ε0ω2E− µ0ω

2P (3.41)

where the tilde signifies the Fourier transform defined by

f(ω) =12π

∫ ∞

−∞f(t)eiωtdt (3.42)

and ω is the angular frequency.It turns out that the Fourier amplitudes of the polarization and the field are

related by the electric susceptibility χe defined by3

P(ω) = ε0χe(ω)E(ω) (3.43)1This is really an approximation, but a very good one since the dominant interaction in the cases

of interest is due to the electric field.2A similar wave equation can be derived for the magnetic field, but since only the electric field

enters the Hamiltonian (3.25), we shall not bother with the magnetic field here.3The electric susceptibility is a scalar only in the case of isotropic materials which we assume

here. For non-isotropic materials it is a tensor.


It should be noted that the susceptibility depends not only on the frequency, butalso on the intensity of the field. However for weak fields the susceptibility is toa good approximation independent of the field strength, and is therefore knownas the linear susceptibility.

Using the definition of the susceptibility (3.43), the Fourier transform of theauxiliary field D can be written as

D = ε0(1 + χe)E. (3.44)

If we assume that the medium is not only isotropic but also homogeneous, wefind that ∇ ·E = 0 such that the wave equation becomes

∇2E + µ0ε0

(1 + χe(ω)

)ω2E = 0. (3.45)

This equation has plane wave solutions of the form

E(r, ω) = a(ω)eik·r (3.46)

where the magnitude k of the wave vector k is related to the angular frequencyω by the dispersion relation

k(ω) = ω√

µ0ε0(1 + χe (ω)) =ωn (ω)

c. (3.47)

The index of refraction n for a non-magnetic medium is thus defined as

n(ω) =√

1 + χe (ω). (3.48)

It turns out that the electric susceptibility is generally a complex function, andwe shall write it as

χe(ω) = χ′e(ω) + iχ′′e (ω) (3.49)

where χ′e and χ′′e are the real and imaginary parts of the complex susceptibility,respectively. This in turn makes the wave vector k complex, and with a similarnotation we write the wave vector as

k(ω) = k′(ω) + ik′′(ω), (3.50)

and the complex index of refraction as

n(ω) = n′(ω) + in′′(ω). (3.51)

Inserting (3.50) into the plane wave solution (3.46) we see that

E(r, ω) = a(ω)e−k′′·reik′·r, (3.52)

from which it is clear that the complex part of the susceptibility is responsiblefor attenuation of the wave as it propagates.

The solution in the space-time domain is obtained by taking the inverseFourier transform of (3.46)

E(r, t) =∫ ∞

−∞a(ω)ei(k·r−ωt)dω, (3.53)

which is the most general plane wave solution to the wave equation (3.40).


3.2.2 Group velocity and dispersionWe shall now consider a plane wave solution where the Fourier amplitude a(ω)is a sharply peaked function of ω around a maximum value at ω0. Furthermorewe assume that the wave is traveling in the z direction. For this purpose wewrite the electric field as

E(z, t) = F(z, t)ei(k0z−ω0t). (3.54)

By comparison with (3.53) we see that the function F(z, t) is defined by

F(z, t) =∫ ∞

−∞a(ω)ei[(k−k0)z−(ω−ω0)t]dω. (3.55)

Since it is assumed that the Fourier amplitude a(ω) is narrow, the functionF(z, t) is seen to vary much slower in space and time than the exponential ap-pearing in (3.54). For this reason we can think of F as a slowly varying envelopeof a monochromatic field of frequency ω0. The speed with which this envelopemoves is called the group velocity. To see how the group velocity is related to thesusceptibility, we assume that the imaginary part of the latter is very small andexpand the wave vector, now real, to first order in ω around ω0

k′(ω) ' k′0 +dk′

dω

∣∣∣ω0

(ω − ω0) . (3.56)

Inserting this expansion into (3.55) we obtain an approximate expression for theenvelope function

F(z, t) '∫ ∞

−∞a(ω)e−i(ω−ω0)(t−z/vg)dω (3.57)

If we pick a particular value F0 at a particular point in space and time (z1, t1),such that

F0 =∫ ∞

−∞ae−i(ω−ω0)(t1−z1/vg)dω, (3.58)

we will find the same point at a later time at the space-time coordinates (z2, t2).It is clear that the two sets of coordinates must be related by

z2 − z1 = vg(t2 − t1) ⇒ ∆z = vg∆t. (3.59)

From this it is clear that the envelope as a whole travels without changing shapewith the group velocity vg defined by

1vg

=dk′

dω

∣∣∣ω0

. (3.60)

Written in terms of the real part of the index of refraction, the group velocity is

vg(ω0) =c

n′(ω0) + ω0dn′dω |ω0

. (3.61)


It should be stressed that as the pulse length is shortened, the approxima-tion (3.56) loses its validity and the pulse no longer retains its original shapeas it propagates through the medium. This is because a shorter pulse containsa greater range of frequencies. Each frequency component of the pulse travelswith its own velocity, called the phase velocity, which is

vphase =ω

k′. (3.62)

This causes the pulse to spread out, a phenomenon known as dispersion.Both the group and the phase velocity are defined in terms of the real part

of the wave vector k′. If the medium absorbs strongly, that is if the imaginarypart of the wave vector is large, the concept of group velocity loses its meaningbecause the pulse will become heavily distorted in this case.

3.2.3 Susceptibility of a gas of lambda atomsIn order to calculate the electric susceptibility of an ensemble of lambda atoms,we need to calculate the polarization of the medium quantum mechanically.Since the polarization is the dipole moment density of the medium, it is givenby the expectation value of the dipole moment operator

P(r, t) = %〈d〉 (3.63)

where % is the atomic density of the gas. Given the field, the susceptibility canthen be calculated from (3.43).

The complex polarization is given in terms of the slowly varying densitymatrix elements as

P(r, t) = %(dbaρabe

−iωpt + dcaρace−iωct

)(3.64)

with the complex probe field given by

Ep(r, t) = Ep,0ei(k·r−ωpt) (3.65)

As we discussed in the previous sections, the slowly varying density ma-trix elements ρµν evolve into a steady state on the characteristic timescale γ−1.Since we are interested in the medium’s response to fields with a pulse durationmuch longer than this, we can ignore the transient behavior. Using the steadystate solution of the density matrix, we can take the Fourier transform of thepolarization and the probe field and then use the definition of the electric sus-ceptibility (3.43) to obtain

%dbaρstab = ε0χe(ωp)Ep,0e

ik·r. (3.66)

We shall also limit the calculation to the case where the probe laser is muchweaker than the coupling laser (Ωp ¿ Ωc). This allows us to solve the densitymatrix equations perturbatively, yielding the linear susceptibility. Introducing


the small dimensionless parameter ε = Ωp/Ωc, we expand the density matrixelements in powers of ε

ρµν = ρ(0)µν + ρ(1)

µν ε + ρ(2)µν ε2 + . . . (3.67)

The zeroth order term ρ(0)µν is just the initial value of the density matrix element.

Taking the initial state of the atoms to be the ground state |b〉, the initial valuesof the density matrix elements are zero, except for ρbb = 1.

To first order in ε we obtain two coupled equations for the steady state solu-tion for the density matrix elements ρst

ab and ρstcb

0 = iΩp + iΩcρstcb − Γbaρst

ab (3.68a)

0 = iΩ∗cρstab − Γbcρ

stcb. (3.68b)

The solution for ρstab is

ρstab = iΩp

Γbc

ΓbaΓbc + |Ωc|2 . (3.69)

Upon insertion into (3.66), we obtain an expression for the complex linear sus-ceptibility

χe(ωp) =%|dab|2

hε0

iΓbc

ΓbaΓbc + |Ωc|2 . (3.70)

The real and imaginary parts of the susceptibility are then

χ′e =%|dab|2

hε0

∆|Ωc|2 − δp

(∆2 + γ2

bc

)

(|Ωc|2 + γbaγbc − δp∆)2 + (δpγbc + γba∆)2(3.71)

χ′′e =%|dab|2

hε0

γba∆2 + γbc

(|Ωc|2 + γbaγbc

)

(|Ωc|2 + γbaγbc − δp∆)2 + (δpγbc + γba∆)2(3.72)

Figure 3.2 shows a plot of the real and imaginary parts of the susceptibility. Itis clear that the imaginary part of the susceptibility, and hence the absorption ofthe probe pulse, becomes very small when the probe field is close to resonance.We also note that the graph of the real part of the susceptibility has a very steepslope around resonance, which leads to a reduction of the group velocity of theprobe pulse.

If we assume that the coupling laser intensity is high enough to satisfy

|Ωc|2 À γbaγbc, (3.73)

and that the laser fields are close to resonance, we can expand the expressionsfor the susceptibility to lowest non-vanishing order in the detunings. We find

χ′e =%|dab|2

hε0

∆|Ωc|2 , (3.74)

χ′′e =%|dab|2

hε0

(γba∆2

|Ωc|4 +γbc

|Ωc|2)

. (3.75)


−4 −2 0 2 4−0.5

0

0.5

1

δp / γ

ba

χ

Figure 3.2: Plot of the real (dashed curve) and imaginary (solid curve) parts of thecomplex susceptibility in units of %|dab|2

hε0as a function of probe laser detuning. Pa-

rameters are: Ωc = γba, δc = 0, γbc = 0.

We see that near two-photon resonance, the index of refraction is close to unityand we can make the approximation

n(ωp) ' 1 +12χe(ωp). (3.76)

This leads to a group velocity near two-photon resonance given by

vg(ωp) =c

1 + ωp%|dab|2

2hε0|Ωc|2=

c

1 + ng(ωp), (3.77)

where the group index of refraction is defined by

ng(ωp) = ωp%|dab|2

2hε0|Ωc|2 . (3.78)

It is easy to see that for low coupling laser intensities, the group velocity of theprobe pulse is substantially reduced, whereas for high coupling laser intensities,the group velocity approaches the speed of light in vacuum. There is, however,a limit on the achievable group velocity reduction. At exact resonance (δp,c = 0)the imaginary part of the susceptibility is

χ′′e =%|dab|2

hε0

γbc

|Ωc|2 + γbaγbc. (3.79)

To have vanishing absorption, the coupling laser Rabi frequency must thereforesatisfy (3.73) and this sets a lower limit on the group velocity.


Another requirement for the presence of EIT is that the two-photon detuningmust be small. To determine how small, we calculate the intensity transmissionT of the medium. This quantity is defined by

T = e−2k′′L ' e−ωpc χ′′e L, (3.80)

where L is the length of the medium. Assuming that the condition (3.73) issatisfied, the transmission coefficient is

T = e− ∆2

2δω2tr , (3.81)

where the transparency width δωtr is defined by

δωtr =

√c|Ωc|2γbangL

. (3.82)

This leads to the requirement that

|∆| ¿ δωtr. (3.83)

It is thus only when the two conditions (3.73) and (3.83) are satisfied that themedium exhibits EIT.

In the absence of the coupling laser, the susceptibility is

χe(ωp) =i%|dab|2hε0Γba

, (3.84)

which is the familiar result for a two-level atom [13]. Near resonance, the sus-ceptibility becomes purely imaginary, which leads to absorption of the electro-magnetic field. The depth to which the resonant probe field can propagate inthe medium in the absence of EIT is given by the absorption length la defined by

la =2ε0hcγba

%|dab|2ωp. (3.85)

From the expression for the group velocity under EIT conditions (3.77), we cansee that to achieve very low group velocities, a medium characterized by a verysmall absorption length is desirable since, in that case, the coupling laser Rabifrequency can be high enough to satisfy condition (3.73), but small enough toobtain a very low group velocity.

The opacity α of the medium in the absence of EIT is defined by

α = L/la =%|dab|2ωpL

2ε0hcγba. (3.86)

This quantity will turn out to be a useful figure of merit for a given medium, aswe shall see later.

C H A P T E R 4

Electromagneticallyinduced transparency

In chapter 3 we investigated the EIT phenomenon by calculating the susceptibil-ity of an ensemble of lambda atoms. The calculation rested on two approxima-tions, the first being the steady state assumption and the second the assumptionof a weak probe field.

In this chapter we shall examine the EIT phenomenon more thoroughly bya different method invented by M. D. Lukin and M. Fleischhauer [21, 22]. Thismethod involves solving the coupled Maxwell/Heisenberg-Langevin equationsperturbatively, and has some important advantages over a treatment based onthe linear susceptibility. First of all, the perturbative solution of the Heisenberg-Langevin equations goes beyond the steady state solution used in chapter 3.This makes the method more easily adaptable to cases involving time dependentcoupling laser Rabi frequencies, which will be important when we consider lightstorage in the next chapter. Secondly, the theory of Lukin and Fleischhauer is apure quantum theory, unlike the semiclassical susceptibility model, allowing usto take the effect of quantum noise into account. Again, this will be important ifthe light storage discussed in the next chapter is to be used for quantum memorypurposes. Thirdly, the method can be extended to go beyond the weak probeapproximation, which we shall do in this chapter.

In the first part of the chapter, we shall review the results of Lukin and Fleis-chhauer. In the second part, we shall go beyond their results and present calcu-lations of various corrections to their solution. These corrections include non-adiabatic corrections in the case of short probe pulses, the effect of a strong probefield on the coupling field and the effect of atomic motion.

Finally we shall compare the analytical solution to a full numerical solutionof the problem, and examine the regimes in which the approximations made inthe analytical solution begin to break down.

37

38 Chapter 4. Electromagnetically induced transparency

4.1 Solving the propagation problem

4.1.1 Heisenberg-Langevin equations for continuum variablesWe consider an ensemble of N lambda atoms interacting with the probe andcoupling lasers. The two lasers are considered to be copropagating. In section3.1.4 we found the Hamiltonian (3.25) for a single lambda atom interacting withthe two fields . Assuming that the atoms do not interact among each other, theHamiltonian for the N atom problem is

H = HF +N∑

j=1

(Hj

A + HjL + Hj

V

). (4.1)

As before, HF and HA describe the free electromagnetic field and atom, HL

describes the interaction of an atom with the probe and coupling lasers, and HV

describes the interaction with the vacuum field modes. The individual termsare given by

HF =∑m

hωma†mam (4.2a)

HjA = hωcbσ

jcc + hωabσ

jaa (4.2b)

HjL = −

(Ep(rj) + Ec(rj)

)·(dbaσj

ba + dcaσjca + h.a.

)(4.2c)

HjV = −EV (rj) ·

(dbaσj


)(4.2d)

We introduce operators that are slowly varying in both space and time for thefields and for the atomic operators. The field operators for the probe and cou-pling fields are written as

Ep,c(r, t) =√

hωp,c

2ε0V

(ep,cEp,c(r, t)ei(kp,c·r−ωp,ct) + h.a.

)(4.3)

where ep,c is the polarization vector, and Ep,c is the slowly varying field opera-tor.

The slowly varying atomic operators are defined as

σjµµ(t) = σj

µµ(t) (4.4a)

σjba(t) = σj

ba(t)ei(kp·rj−ωpt) (4.4b)

σjca(t) = σj

ca(t)ei(kc·rj−ωct) (4.4c)

σjbc(t) = σj

bc(t)ei((kp−kc)·rj−(ωp−ωc)t) (4.4d)

where σjµν is the slowly varying atomic operator for the j’th atom.

We choose the size of the quantization volume V small enough so that thevariation of the slowly varying operators is negligible over the entire volume,

4.1 Solving the propagation problem 39

yet large enough to contain a large number of atoms. The number of atomscontained within a volume of size V centered around r is denoted by Nr, andmay depend on position. If the number of atoms Nr À 1, we can introducecontinuum variables by the definition [21, 22]

σµν(r, t) =1

Nr

∑

j∈V

σjµν(t), (4.5)

where the sum is over all atoms in the small volume V centered around r.We can now write down the Heisenberg-Langevin equations for the continuumvariables

σaa = −igp

(E†

pσba − h.a.)− igc

(E†

cσca − h.a.)− γσaa + Faa (4.6a)

σbb = igp

(E†

pσba − h.a.)


σcc = igc

(E†

cσca − h.a.)

+ γcσaa + Fcc (4.6c)σba = igpEp (σbb − σaa) + igcEcσbc − Γbaσba + Fba (4.6d)

σca = igcEc (σcc − σaa) + igpEpσ†bc − Γcaσca + Fca (4.6e)

σbc = i(gcE

†cσba − gpEpσ

†ca


Here we have introduced the atom-field coupling constants gp,c for the probeand coupling lasers. They are defined by

gp =√

ωp

2hε0Vep · dba gc =

√ωc

2hε0Vec · dca, (4.7)

and are chosen to be real by an appropriate choice of phase for the states |b〉 and|c〉.

4.1.2 Wave equations for the fields

To solve the propagation problem self-consistently, we also need the equationsof motion for the field operators. These turn out to be nothing more than thefamiliar Maxwell equations [23]. As we did for the classical fields in section3.2.1, we can derive a wave equation for the electric field operator. Recallingthat only the transverse part of the electric field is quantized, the wave equationis

∇2E− 1c2

∂2E∂t2

=1

ε0c2

∂2P∂t2

. (4.8)

The polarization operator P is given by the sum over the dipole moment op-erators dj of all the atoms contained in the small volume V centered around r

P(r, t) =1V

∑

j∈V

dj . (4.9)


The dipole moment operator can be written in terms of the slowly varyingatomic operators

dj = dbaσjba(t)ei(kp·rj−ωpt) + dcaσj

ca(t)ei(kc·rj−ωct) + h.a. (4.10)

Since the volume V is very small, we can replace the position vector rj of eachatom within the volume with r. This allows us to write the polarization operatorin terms of the continuum atomic operators

P(r, t) =Nr

V

(dbaσbaei(kp·r−ωpt) + dcaσcaei(kc·r−ωct) + h.a.

)(4.11)

It is evident from this expression that the polarization can be split into two parts.One that oscillates in time with frequency ωp and one with frequency ωc. Thisimplies that only the part of the polarization that oscillates with ωp contributesto the evolution of the probe field, while only the part that oscillates with ωc

contributes to the evolution of the coupling field.We shall assume that the two fields propagate along the z axis. Inserting the

slowly varying field operators, the wave equation for the probe field becomes

√hωp

2ε0Vep

(∂2Ep

∂z2+ 2ikp

∂Ep

∂z− 1

c2

∂2Ep

∂t2+

2iωp

c2

∂Ep

∂t

)

=1

ε0c2

Nr

Vdba

(∂2σba

∂t2− 2iωp

∂σba

∂t− ω2

pσba

). (4.12)

We now introduce characteristic time and length scales Tp and Lp for the slowlyvarying field operator. Introducing the normalized time and space variables t′

and z′ given by

t′ =t

Tpz′ =

z

Lp, (4.13)

and writing the wave equation in terms of the normalized coordinates, we seethat we can neglect the terms involving the second derivative on the lhs. of(4.12), and also the two terms on the rhs. involving the time derivatives, pro-vided that the characteristic time and length scales satisfy the conditions

Tp,c À ω−1p,c Lp,c À k−1

p,c . (4.14)

This approximation is known as the slowly varying amplitude approximation. Withthis approximation the wave equations for the probe and coupling fields become

(∂

∂t+ c

∂

∂z

)Ep(z, t) = igpNr(z)σba(z, t) (4.15a)

(∂

∂t+ c

∂

∂z

)Ec(z, t) = igcNr(z)σca(z, t) (4.15b)


4.1.3 The weak probe approximationObtaining an exact analytical solution of the coupled Heisenberg-Langevin equa-tions (4.6) and wave equations (4.15) is impossible. Instead we limit ourselvesto cases where the probe laser is much weaker than the coupling laser, and thephoton density is much lower than the atomic density. As shown in [21, 22] thisallows us to use a perturbation method to obtain an approximate solution. Weshall call this approximation the weak probe approximation.

To obtain a perturbative solution of the propagation problem, we introducethe small parameter ε defined as

ε =Ep,max

Ec,max, (4.16)

where Ep,max and Ec,max are the maximum “values” of the probe and couplingfield operators, respectively. We then write the fields and atomic operators as apower series expansion in ε

σµν = σ(0)µν + εσ(1)

µν + ε2σ(2)µν + . . . (4.17)

Ep = E(0)p + εE(1)

p + ε2E(2)p + . . . (4.18)

Ec = E(0)c + εE(1)

c + ε2E(2)c + . . . (4.19)

The zeroth order terms for the atomic operators correspond to the initial values,and the zeroth order terms for the field operators correspond to the case of van-ishing probe field. We assume that all atoms are initially in the ground state |b〉.We can therefore write the state of the system as the product state

|Ψ〉 = |b1, b2, . . . , bN 〉|Φ〉 (4.20)

where |b1, b2, . . . , bN 〉 is the total ground state of the atoms, and |Φ〉 is the stateof the electromagnetic field. The initial conditions for the atomic operators are

σjµν(t = 0) = |µj〉〈νj |. (4.21)

Since the state of the system (4.20) contains only the ground state |b〉 of theatoms, we can make the replacements

σ(0)bb → 1 σ(0)

µν → 0 for µν 6= bb (4.22)

in the Heisenberg-Langevin equations (4.6), where 1 and 0 are the identity andthe null operators, respectively. It should be stressed that the replacements (4.22)are not to be understood as operator identities. It is only because the state of thesystem is given by (4.20) that it is permissible to use these replacements.

It it shown in [24] that the effect of the noise operators is negligible, providedthat the dephasing rate γbc is small. We shall therefore disregard the noise oper-ators in the following calculations. If one is merely interested in the expectation


values of the observables, the noise operators can be discarded since their ex-pectation values are zero. We shall also assume that the strong coupling fieldcan be treated as a classical field. In this case the field operator Ec is simply thecomplex slowly varying amplitude of the classical coupling field.

To first order in ε, the Heisenberg-Langevin equations (4.6d) and (4.6f) canbe written

σba =1

igcE∗c

[∂

∂t+ Γbc

]σbc, (4.23a)

σbc = −gpEp

gcEc+

1igcEc

[∂

∂t+ Γba

]σba. (4.23b)

Inserting (4.23a) into (4.23b) we obtain a differential equation for σbc

σbc = −gpEp

gcEc+

1igcEc

[∂

∂t+ Γba

](1

igcE∗c

[∂

∂t+ Γbc

]σbc

). (4.24)

4.1.4 The adiabatic approximation

To solve equation (4.24), we assume that the slowly varying field operators varyon a characteristic timescale Tp which is sufficiently long to satisfy the conditionT−1

p ¿ γba. By using the normalized time t′ and inserting equation (4.24) iter-atively into itself, we can write it as a power series expansion in (γbaTp)−1. Inthe adiabatic approximation we assume that the characteristic timescale Tp is longenough to neglect all but the zeroth order term in the expansion, and in this casewe find

σbc ' −gpEp

gcEc. (4.25)

Note that in the above expression we have also made the assumption that thedephasing rate Γbc is small enough to satisfy the condition

|ΓbaΓbc||Ωc|2 ¿ 1. (4.26)

To calculate the propagation of the coupling field we need to know σca to firstorder in ε. We find this by exploiting an operator relation. At t = 0 we have theoperator relation

σca(t = 0) = σ†bc(t = 0)σba(t = 0), (4.27)

which is easily seen from (4.21). As shown in appendix A this relation is true forall times, that is

σca(t) = σ†bc(t)σba(t). (4.28)

Using the power series expansion of the atomic operators (4.17) in the operatorrelation (4.28) we see that the first order term in the expansion of σca vanishes.


This implies that to first order in ε the coupling field obeys the vacuum waveequation (

∂

∂t+ c

∂

∂z

)Ec(z, t) = 0. (4.29)

To find a wave equation for the probe field, we insert the adiabatic approxima-tion (4.25) for σbc into (4.23a) and assume that the coupling field is time inde-pendent. This yields

σba =igp

|Ωc|2[

∂

∂t+ Γbc

]Ep. (4.30)

Inserting this expression into the wave equation (4.15a) for the probe field, wearrive at a wave equation for the probe field accurate to first order in ε and tozeroth order in the adiabatic expansion:

(∂

∂t+ vg

∂

∂z

)Ep(z, t) = − Γbcng

1 + ngEp(z, t). (4.31)

Here we have introduced the group velocity vg , given by

vg =c

1 + g2pNr

|Ωc|2=

c

1 + ng, (4.32)

and the group index of refraction

ng =g2

pNr

|Ωc|2 . (4.33)

By inserting the definition (4.7) of the coupling constant gp it is easy to see thatthe group velocity and group index of refraction defined above agree with theexpressions we found in chapter 3.

To solve the partial differential equation (4.31) we take the Fourier transformwith respect to time. This yields an ordinary differential equation which canreadily be solved. Inverting the Fourier transform finally gives the solution

Ep(z, t) = e−Γbc

c

∫ z0 ng(z′)dz′Ep

(0, t−

∫ z

0

dz′

vg(z′)

), (4.34)

where Ep(0, t) is the boundary condition for the probe field. The group velocitymay depend on z, since the atomic density is not necessarily constant.

We see immediately that the temporal profile of the probe pulse is unchangedby the propagation through the medium. The spatial profile, however, is com-pressed as the probe pulse enters the medium due to the reduced group velocity.This can be seen by considering a probe pulse with a temporal width Tp. If weassume that the group velocity of the pulse inside the medium is constant overthe entire width of the pulse, it is easy to see that the spatial width of the pulse


inside the medium Lm is related to the spatial width of the pulse in vacuum Lv

byLm =

vg

cLv. (4.35)

The probe pulse is also delayed by the medium relative to a pulse that travelsthe same distance in vacuum. If the length of the medium is L, this delay Td isgiven by

Td =∫ L

0

1vg(z)

dz − L

c=

1c

∫ L

0

ng(z)dz. (4.36)

If the group velocity is constant inside the medium, the delay is simply

Td =ngL

c. (4.37)

Finally, we see that a small dephasing rate γbc causes attenuation of the probepulse, while a small two-photon detuning causes a phase shift. The magnitudeof these effects is related to the probe delay, which can easily be seen by insertingthe expression (4.36) for the probe delay into the solution (4.34)

Ep(L, t) = e−ΓbcTdEp

(0, t−

∫ L

0

dz′

vg(z′)

). (4.38)

4.2 Corrections to the analytic solution

The solution (4.34) of the coupled Maxwell/Heisenberg-Langevin equations isthe one given by Lukin and Fleischhauer [21, 22]. In deriving it a number ofapproximations were made. These approximations are

• The slowly-varying amplitude approximation.

• The weak-probe approximation.

• The adiabatic approximation.

• Small dephasing rate and detuning.

In the following we shall discuss the validity of these approximations, and alsocalculate corrections to the solution (4.34), giving a more accurate solution thanthe one presented in [21, 22].

The slowly-varying amplitude approximation requires that the laser pulsessatisfy the conditions (4.14). For typical optical transitions, the conditions forthe temporal and spatial widths are

Tp À 10−15 s Lp À 10−7 m. (4.39)

In experiments the typical temporal width of the pulses involved are on theorder of 1 µs, so the condition for the temporal width is always well satisfied.

4.2 Corrections to the analytic solution 45

The spatial width of the pulse is related to the temporal width by Lp = vgTp.Group velocities as low as 10 m/s are experimentally realizable [5, 6], and in thiscase the spatial width of the pulse inside the medium is approximately 10 µm.Even in this extreme case, the condition for the spatial width of the pulse issatisfied by a factor of 100. We shall therefore assume throughout this thesisthat the slowly-varying approximation is always valid.1

The assumption of a small dephasing rate and small detunings is necessaryto insure the existence of EIT, as we saw in section 3.2.3. For this reason, theregime in which this assumption is invalid is not very interesting, since theprobe pulse is rapidly absorbed in this case. The value of the dephasing rateγbc depends on the particular atom and transitions comprising the lambda sys-tem, ranging from several hundred MHz in some of the early experiments [2, 3],to a few kHz in more recent experiments [5].

The weak probe approximation and the adiabatic approximation are, unlikethe slowly-varying amplitude approximation, not always well satisfied in ex-periments on EIT. In the following we shall therefore calculate corrections tothe solution (4.34), and thereby also find more rigorous criteria that determineunder what conditions these corrections become significant.

4.2.1 Non-adiabatic corrections

In this section we shall explore what happens when the temporal width of theprobe pulse becomes small enough that the adiabatic approximation begins tobreak down. Our starting point is the equation (4.24) for the Raman coher-ence σbc. In order to simplify the calculations and to isolate the effects of non-adiabaticity, we assume zero detunings (δp = δc = 0) and negligible dephasingrate (γbc = 0). Using the normalized time t′, we can write (4.24) as

σbc = −gpEp

gcEc− 1|Ωc|2

[T−1

p

∂

∂t′+ γba

]T−1

p

∂

∂t′σbc. (4.40)

By inserting the equation iteratively into itself and keeping terms to second or-der in (γbaTp)−1 we obtain an approximate expression for the Raman coherence

σbc ' gp

gcEc

(−Ep +

γba

|Ωc|2Tp

∂Ep

∂t′+

γ2ba

|Ωc|4T 2p

( |Ωc|2γ2

ba

− 1)

∂2Ep

∂t′2

). (4.41)

Inserting this expression into (4.23a) and (4.15a) we obtain the wave equationfor the probe field including the first two non-adiabatic corrections:

(∂

∂t+ vg

∂

∂z

)Ep =

ng

1 + ng

(γba

|Ωc|2∂2Ep

∂t2+

γ2ba

|Ωc|4( |Ωc|2

γ2ba

− 1)

∂3Ep

∂t3

). (4.42)

1It is not inconceivable that experiments can be performed with pulses short enough to violatethe slowly varying approximation, but such considerations are beyond the scope of this thesis.


To solve this partial differential equation, we Fourier transform it with respectto time. This yields an ordinary differential equation which is easily solvable.The solution is

Ep(z, ν) = Ep(0, ν) exp(−iν

∫ z

0

dz′

vg(z′)

)×

exp(−

[γba

|Ωc|2cν2 + iγ2

ba

|Ωc|4c( |Ωc|2

γ2ba

− 1)

ν3

] ∫ z

0

ng(z′)dz′)

(4.43)

In principle all we have to do now is to invert the Fourier transform to obtainthe solution in the time domain. Unfortunately, doing this analytically is verydifficult. To proceed further we shall exploit the fact that the corrections to theadiabatic solution (4.34) are assumed to be small. The condition for the non-adiabatic corrections to be small is that the first term in the second exponential of(4.43) is small for all relevant values of ν. Since the boundary condition Ep(0, t)has a typical timescale of Tp, the width of the Fourier transform Ep(0, ν) has awidth of approximately T−1

p , so the largest relevant value of ν is T−1p . Therefore

the condition for small non-adiabatic corrections is

γba

|Ωc|2T 2p c

∫ L

0

ng(z′)dz′ ¿ 1. (4.44)

Recalling the definition (4.36) of the probe delay Td, this condition can be writtenas

Td

Tp¿ √

α, (4.45)

where α is the opacity of the medium in the absence of EIT and is given by

α =g2

p

γbac

∫ L

0

Nr(z′)dz′. (4.46)

The condition (4.45) can also be written in terms of the transparency width (3.73)as

δωp ¿ δωtr, (4.47)

where δωp = T−1p is the width of the probe pulse spectrum. It is thus clear

that the condition for adiabaticity is that the probe pulse spectrum is completelycontained within the transparency window of the medium.

When condition (4.45) is satisfied, we can use a Taylor expansion of the sec-ond exponential in (4.43). This yields the approximate expression

Ep(z, ν) = Ep(0, ν) exp(−iν

∫ z

0

dz′

vg(z′)

)×

(1−

[γba

|Ωc|2cν2 + iγ2

ba

|Ωc|4c( |Ωc|2

γ2ba

− 1)

ν3

] ∫ z

0

ng(z′)dz′) (4.48)


This expression can be easily inverted to yield the time domain solution

Ep(z, t) = Ep

(0, t−

∫ z

0

dz′

vg(z′)

)+

γba

|Ωc|2c∫ z

0

ng(z′)dz′∂2

∂t2Ep

(0, t−

∫ z

0

dz′

vg(z′)

)+

γ2ba

|Ωc|4c( |Ωc|2

γ2ba

− 1) ∫ z

0

ng(z′)dz′∂3

∂t3Ep

(0, t−

∫ z

0

dz′

vg(z′)

). (4.49)

Taking the medium to have length L, we can write the solution for the probefield at the exit of the medium as

Ep(L, t′) = Ep(0, Tpt′ − τ) +

1α

T 2d

T 2p

∂2

∂t′2Ep(0, Tpt

′ − τ)+

1α2

( |Ωc|2γ2

ba

− 1)

T 3d

T 3p

∂3

∂t′3Ep(0, Tpt

′ − τ), (4.50)

where we have reintroduced the normalized time t′, and τ is the time it takesthe pulse to travel through the medium

τ =∫ L

0

dz

vg(z). (4.51)

As an example let us assume that the boundary condition for the probe pulse isgiven by

Ep(0, t) = Ep,0 exp(− (t/Tp)

2)

. (4.52)

The opacity and vacuum Rabi frequency of the medium are chosen to be

α = 2500 g2pNr = 106γ2

ba, (4.53)

which roughly correspond to the parameters in experiments such as [5].The effect of the lowest order non-adiabatic correction is shown in figure 4.1.

It shows the temporal profile of the probe pulse at the exit of the medium whenonly the first two terms in the solution (4.50) are calculated. It is evident thatthe effect of the lowest order non-adiabatic correction is to cause a broadeningof the probe pulse.

Figure 4.2 shows the temporal profile of the probe pulse at the exit of themedium when the first two non-adiabatic corrections are taken into account. Wecan see that the effect of the second order term in (4.50) is to cause attenuationof the front end of the pulse as it travels through the medium.

4.2.2 AdiabatonsSo far we have assumed that the probe field is weak enough compared to thecoupling field that it is a good approximation to keep terms in the Heisenberg-Langevin equations (4.6) to first order in ε only. However, to obtain very low


−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

(t−τ) / Tp

Ep /

Ep,

0

Figure 4.1: Plot of the temporal profile of the probe pulse at the exit of the medium.The dotted curve shows the adiabatic solution, and the solid curve shows the solu-tion with only the lowest order non-adiabatic correction. Parameters are: Td/Tp =0.2×√α, α = 2500, vg = c/104, g2

pNr/γ2ba = 106.

group velocities, the coupling laser intensity must be quite low. Since the probefield must be much weaker than the coupling field in order for the weak probeapproximation to be valid, this poses a problem in experiments with ultra slowlight propagation because the probe field must at the same time be strong enoughto be detected. Consequently, in many experiments the weak probe approxima-tion is not very well satisfied. For this reason it is interesting to investigatetheoretically the effect a stronger probe field has on EIT.

To do this we shall solve the Heisenberg-Langevin equations (4.6) and thewave equations for the fields (4.15) to second order in ε. In section 4.1.3 wehave already solved the Heisenberg-Langevin equations to first order under theassumption of a small dephasing rate Γbc. We can use these results to calculatethe atomic operators and the fields to second order. We begin by calculating σca

to second order, using the operator relation

σca = σ†bcσba. (4.54)

Inserting the power series expansion (4.17) into this relation yields

(σ(0)

ca + εσ(1)ca + ε2σ(2)

ca

)=

(σ

(0)bc + εσ

(1)bc + ε2σ

(2)bc

)† (σ0

ba + εσ(1)ba + ε2σ

(2)ba

).

(4.55)Equating terms with like powers of ε we see that the first order term in the powerseries expansion of σca is

σ(1)ca = σ

(0)†bc σ

(1)ba + σ

(1)†bc σ

(0)ba , (4.56)


−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

(t−τ) / Tp

Ep /

Ep,

0

Figure 4.2: Plot of the temporal profile of the probe pulse at the exit of the medium.The dotted curve shows the adiabatic solution, and the solid curve shows the solu-tion with the first two non-adiabatic corrections included. Parameters are the sameas in figure 4.1

and the second order term is

σ(2)ca = σ

(0)†bc σ

(2)ba + σ

(2)†bc σ

(0)ba + σ

(1)†bc σ

(1)ba . (4.57)

Remembering that the state of the system is given by (4.20), we can make use ofthe operator substitutions (4.22) to write the first and second order terms as

σ(1)ca = 0, (4.58a)

σ(2)ca = σ

(1)†bc σ

(1)ba . (4.58b)

In a similar fashion we can calculate σaa and σcc to second order. To calculateσcc we make use of the operator relation

σcc = σ†bcσbc (4.59)

and after using the operator substitutions (4.22) we find

σ(1)cc = 0, (4.60a)

σ(2)cc = σ

(1)†bc σ

(1)bc . (4.60b)

To calculate σaa we use the operator relation

σaa = σ†baσba, (4.61)

and we find the first and second order terms

σ(1)aa = 0, (4.62a)

σ(2)aa = σ

(1)†ba σ

(1)ba . (4.62b)


To calculate σbb to second order we use the operator relation2

σaa + σbb + σcc = 1. (4.63)

From this we immediately find the first and second order terms

σ(1)bb = 0, (4.64a)

σ(2)bb = −σ

(1)†bc σ

(1)bc − σ

(1)†ba σ

(1)ba . (4.64b)

To calculate the higher order corrections to the probe and coupling fields, weinsert the power series expansions for the atomic operators and the fields intothe wave equations (4.15). We thereby find the wave equations to second orderin ε

[∂

∂t+ c

∂

∂z

] (εE(1)

p + ε2E(2)p

)= igpNr

(εσ

(1)ba + ε2σ

(2)ba

), (4.65a)

[∂

∂t+ c

∂

∂z

] (E(0)

c + εE(1)c + ε2E(2)

c

)= igcNrε

2σ(2)ca . (4.65b)

We see immediately that the first order correction to the coupling field E(1)c van-

ishes. In order to calculate the second order term in the expansion of the probefield, we need to know σ

(2)ba . This can be calculated by inserting the power series

expansion of the atomic operators into the Heisenberg-Langevin equations (4.6),which gives the following coupled differential equations for σ

(2)ba and σ

(2)bc :

σ(2)ba =

1

igcE(0)†c

[∂

∂t+ Γbc

]σ

(2)bc , (4.66a)

σ(2)bc = −gpE

(2)p

gcE(0)c

+1

igcE(0)c

[∂

∂t+ Γba

]σ

(2)ba . (4.66b)

These equations, along with the wave equation for the probe field, are identicalin form to the equations (4.23) for the first order term in the expansion of theprobe field. They can thus be solved in exactly the same way, and we find thatthe second order correction to the probe field vanishes.

The wave equation for the coupling field to second order in ε is[

∂

∂t+ c

∂

∂z

]Ec = igcNrσ

(1)†bc σ

(1)ba . (4.67)

Inserting the wave equation for the probe field yields[

∂

∂t+ c

∂

∂z

]Ec =

gc

gpσ

(1)†bc

[∂

∂t+ c

∂

∂z

]E(1)

p . (4.68)

2One might be tempted to use the operator relation σbcσ†bc. This will not work, however, becausewe will not be able to use the operator substitution (4.22) in this case.


We could in principle attempt to solve this equation under general non-adiabaticconditions. This is quite difficult, however, and we shall restrict ourselves tostudy the solution under adiabatic conditions. In this case the Raman coherenceσ

(1)bc is given by

σ(1)bc ' −gpE

(1)p

gcE(0)c

. (4.69)

Inserting this expression into (4.68) yields

[∂

∂t+ c

∂

∂z

]Ec = −E

(1)†p

E(0)†c

[∂

∂t+ c

∂

∂z

]E(1)

p . (4.70)

It is permissible to replace E(0)c with Ec on the rhs. of (4.70), since the two quan-

tities are identical to zeroth order in ε. Adding the Hermitian conjugate of theequation finally yields

[∂

∂t+ c

∂

∂z

]|Ec|2 = −

[∂

∂t+ c

∂

∂z

]|E(1)

p |2. (4.71)

This differential equation is easy to solve by Fourier transforming with respectto time. The solution is

|Ec(z, t)|2 + |E(1)p (z, t)|2 = |Ec(0, t− z/c)|2 + |E(1)

p (0, t− z/c)|2. (4.72)

We can also calculate the expectation value of the populations 〈σµµ〉 in the adi-abatic limit. Using equations (4.60), (4.62) and (4.64) we find to zeroth order in(γbaTp)−1

〈σcc〉 =|Ωp|2|Ωc|2 , (4.73a)

〈σaa〉 =|Γ2

bc||Ωc|2

|Ωp|2|Ωc|2 , (4.73b)

〈σbb〉 = 1−(

1 +|Γbc|2|Ωc|2

) |Ωp|2|Ωc|2 . (4.73c)

As an example, let us assume a vanishing dephasing rate (Γbc = 0). The bound-ary conditions for the probe and coupling fields are taken to be

Ep(0, t) = Ep,0e−(t/Tp)2 , (4.74a)

Ec(0, t) = Ec,0. (4.74b)

We also assume that the medium is homogenous. In this case the solution is

|Ep(z, t)|2 = |Ep,0|2 exp

(−2

(t− z/vg

Tp

)2)

(4.75)


|Ec(z, t)|2|Ec,0|2 = 1− ε2

[exp

(−2

(t− z/vg

Tp

)2)− exp

(−2

(t− z/c

Tp

)2)]

(4.76)

Assuming that the coupling constants gp and gc are identical, the populations〈σbb〉 and 〈σcc〉 are given by

〈σbb〉 = 1− ε2 exp

(−2

(t− z/vg

Tp

)2)

(4.77)

〈σcc〉 = ε2 exp

(−2

(t− z/vg

Tp

)2)

(4.78)

Figure 4.3 shows a plot of the coupling field intensity as a function of time ata fixed position z0 = 4vgTp. We see that as the probe pulse enters the medium(around t = 0), the coupling field intensity increases momentarily. Later, as theprobe field approaches z0 = 4vgTp, the coupling field decreases as the probepulse passes through the point.

−2 0 2 4 60.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

t / Tp

Ec2 /

Ec,

02

Figure 4.3: Temporal profile of the coupling field intensity at z0 = 4vgTp. Theadiabaton is the dip in intensity occurring as the probe pulse passes z0. The initialincrease in intensity is caused by the probe pulse entering the medium. Parametersare: vg = c/105, ε = 0.2

The spatial profile of the coupling field intensity at time t = 4Tp is shown infigure 4.4. The dip in the intensity of the coupling field which accompanies theprobe pulse inside the medium is known as an adiabaton [25]. Such adiabatonpulse pairs have also been observed experimentally, e.g. in the experiment byKasapi et al. [3].


0 2 4 6 80.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

z / vgT

p

Ec2 /

Ec,

02

Figure 4.4: Spatial profile of the coupling field intensity at time t = 4Tp. The dip inintensity is the adiabaton. Parameters are the same as in figure 4.3

The behavior of the coupling field can be understood qualitatively as fol-lows: As the probe pulse travels through the medium, the atoms in front of itare in the ground state |b〉. When the probe pulse arrives, population is trans-ferred by two-photon Raman transitions from state |b〉 to state |c〉. In this pro-cess, probe field photons are absorbed and coupling field photons are emitted.At the back end of the probe pulse, this process is reversed. Population is trans-ferred back from state |c〉 to state |b〉, coupling field photons are absorbed andprobe field photons are emitted.

In the regions before and after the probe pulse, the atoms are in state |b〉 andthe coupling field consequently propagates unhindered with the vacuum speedof light c. As a portion of the coupling field approaches the back end of the probepulse, coupling field photons are absorbed leading to a decrease in the couplingfield intensity. At the front end of the probe pulse, coupling field photons areemitted, restoring the intensity of the coupling field to the value it had beforeencountering the probe pulse. This explains the adiabaton, the dip in couplingfield intensity accompanying the probe pulse.

To understand the initial increase in coupling field intensity, which is evidentin figure 4.3, we need to consider what happens as the probe pulse enters themedium at the boundary. Initially only the front end of the probe pulse hasentered the medium and the coupling field entering the medium encounters aregion in which population is transferred from state |b〉 to state |c〉, leading to anincrease in coupling field intensity. This increase continues until the peak of theprobe pulse has entered the medium.

Once the back end of the probe pulse begins to penetrate into the medium,the coupling field entering the medium will first encounter a smaller region inwhich atoms are returned from state |c〉 to state |b〉, and then the larger region


containing the front end of the probe pulse where atoms are transferred fromstate |b〉 to state |c〉. This still leads to a net increase in coupling field intensity,but it is now smaller than when the peak of the probe pulse has just entered themedium. Once the entire probe pulse is emerged in the medium, the adiabaton“pulse pair” has formed. The derivation of (4.71) required the adiabatic approx-imation to be valid. This is of course the reason for the name “adiabaton” givento this phenomenon.

Another way to think of the atoms response to the probe and coupling fieldsunder adiabatic conditions is in terms of the dark state introduced in chapter3. Recalling the expression for the dark state (3.15), we see that when only thecoupling field is present, the ground state |b〉 of the atoms coincides with thedark state |D〉. According to the adiabatic theorem [17] the atoms remain in theinstantaneous dark state at all times, provided the probe field changes slowlyenough, and the excited state |a〉 is never populated.

4.2.3 The Doppler effect

So far we have assumed that the atoms constituting the medium are stationary.However, most experiments on EIT are done with atomic gasses, and hence weneed to consider the effect of atomic motion. To do this we consider an atom atthe position rj with velocity vj . The position vector rj can then be written

rj(t) = rj(t0) + vj(t− t0), (4.79)

where rj(t0) is the position of the atom at the earlier time t0. We then insert thisexpression into the definitions of the slowly varying field and atomic operators(4.3) and (4.4). The continuum variables are now defined by summing over allatoms with a velocity close to v and contained within a small volume V aroundr. The number of atoms meeting these requirements are denoted by Nr,v, andthe definition of continuum variables analogous to (4.5) is

σµν(v, r, t) =1

Nr,v

∑

j

σjµν(t). (4.80)

It is easy to see that the Heisenberg-Langevin equations for each “velocity class”is found by making the substitutions

ωp,c → ωp,c − kp,c · v (4.81)


in the Heisenberg-Langevin equations (4.6) for stationary atoms. Thus the Heisen-berg-Langevin equations for each velocity class become

σaa = −igp

(E†

pσba − h.a.)− igc

(E†

cσca − h.a.)− γaσaa + Faa (4.82a)

σbb = igp

(E†

pσba − h.a.)


σcc = igc

(E†

cσca − h.a.)

+ γcσaa + Fcc (4.82c)σba = igpEp (σbb − σaa) + igcEcσbc − Γbaσba + Fba (4.82d)σca = igcEc (σcc − σaa) + igpEpσcb − Γcaσca + Fca (4.82e)

σbc = i(gcE

†cσba − gpEpσac


where we have introduced the velocity dependent complex decay rates

Γba = γba − i(δp − kp · v) (4.83a)Γca = γca − i(δc − kc · v) (4.83b)Γbc = γbc − i(∆− (kp − kc) · v). (4.83c)

After solving the Heisenberg-Langevin equations for each velocity class, we canthen find the continuum operator σµν(r, t) for all the atoms within the volumeV . This is done simply by integrating over the distribution of velocities

σµν(r, t) =∫

f(v)σµν(v, r, t)dv, (4.84)

where f(v) is the distribution of velocities in the gas.As before we take the probe and coupling fields to be copropagating along

the z axis. The wave equations for the fields are then(

∂

∂t+ c

∂

∂z

)Ep(z, t) = igpNr(z)

∫ ∞

−∞f(v)σba(v, z, t)dv (4.85)

(∂

∂t+ c

∂

∂z

)Ec(z, t) = igcNr(z)

∫ ∞

−∞f(v)σca(v, z, t)dv (4.86)

where v is the z component of the velocity v.We can now solve the equations by invoking the weak probe approxima-

tion and the adiabatic approximation as we did in section 4.1. The relevantHeisenberg-Langevin equations in the weak probe approximation are

σba(v) =1

iΩ∗c

[∂

∂t+ Γbc(v)

]σbc(v) (4.87a)

σbc(v) = −gpEp

Ωc+

1iΩc

[∂

∂t+ Γba(v)

]σba(v). (4.87b)

Using the adiabatic approximation we can obtain an approximate expression forσbc(v)

σbc(v) = −(

1 +ΓbaΓbc

|Ωc|2)−1

gpEp

Ωc. (4.88)


Note that we cannot disregard the second term in the parenthesis, as we did insection 4.1.4, because this would make σbc independent of v which is too crudean approximation and would remove the effect of atomic motion altogether.

Inserting (4.88) into the wave equation for the probe field (4.85) yields(

∂

∂t+ c

∂

∂z

)Ep = −ang

∂Ep

∂t− bngEp, (4.89)

where

a =∫ ∞

−∞

f(v)dv

1 + ΓbaΓbc

|Ωc|2, b =

∫ ∞

−∞

f(v)Γbcdv

1 + ΓbaΓbc

|Ωc|2. (4.90)

Introducing the effective group index n′g , the effective group velocity v′g and the effec-tive dephasing rate Γ defined by

n′g = ang, v′g =c

1 + n′g, Γ =

b

a, (4.91)

the wave equation (4.89) can be written(

∂

∂t+ v′g

∂

∂z

)Ep = − n′g

1 + n′gΓEp. (4.92)

The solution is easily found to be

Ep(z, t) = e−Γc

∫ z0 n′g(z′)dz′Ep

(0, t−

∫ z

0

dz′

v′g(z′)

). (4.93)

All that remains is to evaluate the two integrals a and b. We assume zero de-tuning (δp = ∆ = 0) and take the distribution function f(v) to be the Maxwelldistribution [26]

f(v) =1√πu

e−v2/u2. (4.94)

The width u of the distribution is determined by the temperature T of the gasand the mass M of the atoms and is given by

u =

√2kbT

M, (4.95)

where kb is Boltzmann’s constant.Unfortunately, the two integrals (4.90) are impossible to evaluate analytically

in this case, but it is easy to verify that they are both purely real. We can, how-ever, obtain an approximate analytical expression for the integrals.

Since the distribution function f(v) falls off exponentially for large v, weassume that the coupling laser Rabi frequency is large enough to allow us toexpand the denominator in the two integrals for all relevant v. Specifically weuse the expansion

11 + x

' 1− x. (4.96)

4.3 Numerical solution 57

With this approximation the integral a is

a ' 1− γbaγbc

|Ωc|2 +kp(kp − kc)|Ωc|2

∫ ∞

−∞

v2

√πu

e−v2/u2dv (4.97)

which is easily evaluated to yield

a ' 1− γbaγbc

|Ωc|2 +kp(kp − kc)u2

2|Ωc|2 . (4.98)

The second integral b is approximated in a similar fashion and the result is

b ' γbc

(1− γbaγbc

|Ωc|2 +kp(kp − kc)u2

|Ωc|2)

+γba(kp − kc)2u2

2|Ωc|2 . (4.99)

From the solution (4.93) it is evident that the effect of atomic motion is to causea small alteration of the group velocity as well as attenuation of the probe pulse.We can also see the importance of using copropagating lasers when performingexperiments with thermal gasses, since the magnitude of the effect depends onthe difference between the wave vectors of the probe and coupling laser fields.If the two lower states |b〉 and |c〉 are nearly degenerate, this difference is verysmall for copropagating lasers, and the experiments are almost insensitive toatomic motion. This configuration is known as a Doppler free configuration.

If the two lower states are not nearly degenerate, or if the lasers are notcopropagating, a larger coupling laser intensity is required to overcome theDoppler effect, and it becomes impossible to reach very low group velocities.This was the case in some of the early experiments on EIT [2, 3]. In later exper-iments, such as the one by Kash et al. [5], the lambda system was composed ofhyperfine levels in alkali atoms. The very small hyperfine splitting between theground state |b〉 and the intermediate state |c〉 drastically reduces the effect ofatomic motion in the thermal gas when copropagating lasers are employed.

The dependence of the Doppler effect on the relative propagation directionof the probe and coupling lasers will be important later, when we considerstanding wave coupling fields.

4.3 Numerical solution

To check the validity of the analytical solutions presented in this chapter, thepropagation problem has been solved numerically. In particular we shall checkthe validity of the weak probe approximation and the adiabatic approximation.To do this we need to convert the operator equations of motion for the atomicand field operators into scalar equations by taking the expectation value of theoperators in question.


The wave equations (4.15) become(

∂

∂t+ c

∂

∂z

)Ωp(z, t) = ig2

pNr(z)σba(z, t) (4.100a)(

∂

∂t+ c

∂

∂z

)Ωc(z, t) = ig2

cNr(z)σca(z, t) (4.100b)

and the Heisenberg-Langevin equations (4.6) become the scalar equations3

σaa = i(Ωpσ∗ba − c.c.) + i(Ωcσ

∗ca − c.c.)− γσaa (4.101a)

σbb = −i(Ωpσ∗ba − c.c.) + γbσaa (4.101b)

σcc = −i(Ωcσ∗ca − c.c.) + γcσaa (4.101c)

σba = iΩp(σbb − σaa) + iΩcσbc − Γbaσba (4.101d)σca = iΩc(σcc − σaa) + iΩpσ

∗bc − Γcaσca (4.101e)

σbc = i(Ω∗cσba − Ωpσ∗ca)− Γbcσbc (4.101f)

We also introduce dimensionless quantities by introducing the normalized fieldsE and F defined by

Ωp(z, t) = Ωp,0E(z, t) Ωc(z, t) = Ωc,0F (z, t), (4.102)

and the scaled space and time coordinates t′ and z′ defined by

t′ =t

Taz′ =

z

vgTp. (4.103)

The timescale Ta is determined by the characteristic timescale on which theatomic variables σµν evolve, since this timescale is faster than the characteris-tic temporal length of the probe pulse Tp. Consequently, the timescale is chosento be

Ta = γ−1ba . (4.104)

Since we are interested in examining the regime in which the adiabatic approx-imation breaks down, we introduce the adiabaticity parameter η as a measure ofnon-adiabaticity. It is defined by the relation

Td

Tp= η

√α. (4.105)

Recalling the condition for adiabaticity (4.45) we see that if η ¿ 1, the adiabaticapproximation is valid.

We also introduce the Rabi frequency ratio ε defined by

ε =Ωp,0

Ωc,0. (4.106)

3In this section, σµν denotes the expectation value of the corresponding operator.


By specifying values for η and ε, as well as the length L of the medium and theexpected group velocity vg , the pulse length Tp and the Rabi frequencies Ωc,0,Ωp,0 can be uniquely determined.

Finally, we must specify values for the various decay rates as well as for thevacuum Rabi frequencies gp,c. The decay rates are chosen to be real, implyingzero detuning, and to be related by

γca = γb = γc = γba γ = 2γba γbc = 0. (4.107)

The vacuum Rabi frequencies are taken to be

g2pNr = g2

cNr = 5 · 106γ2ba. (4.108)

The boundary condition for the probe field at z = 0 is chosen to be a Gaussian

E(0, t) = exp(− (t− 3Tp)

2/T 2

p

), (4.109)

and the boundary condition for the coupling field is

F (0, t) = 1. (4.110)

The initial conditions for the atomic variables are

σbb(z, 0) = 1, σµν(z, 0) = 0 for µν 6= bb. (4.111)

The numerical solution presented in this section was obtained by solving thecoupled Maxwell/Heisenberg-Langevin equations subject to the given bound-ary conditions (4.109) and (4.110) for the fields and initial conditions (4.111) forthe atoms.

The Heisenberg-Langevin equations were propagated in time using a fourthorder Runge-Kutta algorithm, while the wave equations were solved using aLax-Wendroff finite difference scheme [27]. Additional details regarding thealgorithms used can be found in appendix C.

Numerical solutions have been obtained for a weak (ε = 0.1) and strong(ε = 0.5) probe field, and in both cases for both adiabatic (η = 0.1) and non-adiabatic (η = 0.2) pulses.

In all four cases the medium is characterized by the decay rates given by(4.107) and the vacuum Rabi frequencies given by (4.108). The length of themedium L = 4vgTp and the expected group velocity vg = c/105.

The results are presented by plotting the normalized probe and couplingfields at the exit (z = L) of the medium as a function of time. The numericalsolution for the probe field is compared to both the adiabatic solution (4.34) andthe solution including non-adiabatic corrections (4.50). The numerical solutionfor the coupling field is compared to the analytical adiabaton solution (4.72)using the adiabatic solution for the probe field, and also to a semi-analytical so-lution in which the numerical solution for the probe field is inserted into (4.72).


4.3.1 Weak probeFigure 4.5 shows the numerical solution for a weak (ε = 0.1), adiabatic (η =0.1) probe pulse compared to the analytical solutions. In this case the temporallength of the probe pulse Tp ' 8γ−1

ba . We see that the adiabatic solution is in goodagreement with the numerical solution, but a small broadening of the pulse isvisible. This broadening is well described by the lowest order non-adiabaticcorrection to the adiabatic solution.

The numerical solution for the coupling field in this case is shown in figure4.6. We find good agreement between the numerical and analytical solutions,and excellent agreement between the numerical and semi-analytical solutions.Note that the shape of the adiabaton mirrors the probe pulse shape.

Figure 4.7 shows the numerical solution for a weak (ε = 0.1), non-adiabatic(η = 0.2) probe pulse. Here the temporal length of the probe pulse Tp ' 2γ−1

ba

is close to the characteristic timescale of the evolution of the atomic variables.The numerical solution deviates from the adiabatic solution in this case but isin fairly good agreement with the non-adiabatic analytical solution. We alsosee that the probe pulse is not only broadened but also becomes slightly asym-metric, as predicted by the non-adiabatic analytical solution. This asymmetryis again mirrored in the coupling field adiabaton shown in figure 4.8, but theagreement between the numerical and analytical solutions is still quite good.


−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−τ−3Tp) / T

p

Ωp /

Ωp,

0

Figure 4.5: Numerical solution (solid curve) for a weak (ε = 0.1), adiabatic (η = 0.1)probe field, compared to the adiabatic (dotted curve) and non-adiabatic (dashedcurve) analytical solutions. The numerical solution is in good agreement with theadiabatic solution.

0 10 20 30 40 50 60 70 800.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01

t / γba−1

|Ωc|2 /

|Ωc,

0|2

Figure 4.6: Numerical solution (solid curve) for the coupling field, compared to theanalytical solution (dotted curve) and the semi-analytical solution (dashed curve).The parameters are the same as in figure 4.5. The numerical solution is in goodagreement with the analytical solution.


−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−τ−3Tp) / T

p

Ωp /

Ωp,

0

Figure 4.7: Numerical solution (solid curve) for a weak (ε = 0.1), non-adiabatic(η = 0.2) probe field, compared to the adiabatic (dotted curve) and non-adiabatic(dashed curve) solutions. The non-adiabatic corrections are significant in this case.

0 5 10 15 200.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01

t / γba−1

|Ωc|2 /

|Ωc,

0|2

Figure 4.8: Numerical solution (solid curve) for the coupling field, compared to theanalytical solution (dotted curve) and the semi-analytical solution (dashed curve).The parameters are the same as in figure 4.7. The non-adiabatic corrections affectthe shape of the adiabaton.


4.3.2 Strong probeThe results for a strong (ε = 0.5), adiabatic (η = 0.1) probe pulse are shown infigure 4.9. The agreement between both adiabatic and non-adiabatic analyticalsolutions and the numerical solution is not as good in this case due to the break-down of the weak probe approximation. Additional broadening and dampingof the probe pulse is clearly evident, and we also note that the probe delay isslightly shorter than expected from the analytical solution, indicating that theprobe group velocity depends on the total electromagnetic field intensity, notjust the coupling field intensity.

The numerical solution for the coupling field shown in figure 4.10 also dif-fers from the analytical solution, but is in excellent agreement with the semi-analytical solution. The initial increase in coupling field intensity agrees verywell with both the analytical and semi-analytical solution. This was expectedsince this increase is caused by the probe field entering the medium, and there-fore before any significant distortion of the probe pulse has occurred.

Finally we show the solution for a strong (ε = 0.5), non-adiabatic (η = 0.2)probe pulse (figure 4.11) and the associated coupling field (figure 4.12). Againthe quantitative agreement between the numerical and analytical solutions ispoor, but the phenomenon of EIT is certainly still present even for strong probefields and the numerical solutions agree at least qualitatively with the analyticalcalculations.


−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−τ−3Tp) / T

p

Ωp /

Ωp,

0

Figure 4.9: Numerical solution (solid curve) for a strong (ε = 0.5), adiabatic (η =0.1) probe field, compared to the adiabatic (dotted curve) and non-adiabatic (dashedcurve) solutions. The weak probe approximation begins to break down in this case.

0 10 20 30 40 50 60 70 800.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

t / γba−1

|Ωc|2 /

|Ωc,

0|2

Figure 4.10: Numerical solution (solid curve) for the coupling field, compared to theanalytical solution (dotted curve) and the semi-analytical solution (dashed curve).The parameters are the same as in figure 4.9. The semi-analytical solution is inexcellent agreement with the numerical solution.


−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−τ−3Tp) / T

p

Ωp /

Ωp,

0

Figure 4.11: Numerical solution (solid curve) for a strong (ε = 0.5), non-adiabatic(η = 0.2) probe field, compared to the adiabatic (dotted curve) and non-adiabatic(dashed curve) solutions.

0 5 10 15 200.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

t / γba−1

|Ωc|2 /

|Ωc,

0|2

Figure 4.12: Numerical solution (solid curve) for the coupling field, compared to theanalytical solution (dotted curve) and the semi-analytical solution (dashed curve).The parameters are the same as in figure 4.11. The agreement between the numeri-cal and semi-analytical solution is not as good as in figure 4.10 because of the break-down of the adiabatic approximation.

C H A P T E R 5Dark state polaritons

So far we have restricted the discussion to cases in which the coupling laserintensity is time independent. Although we might imagine using EIT and ultraslow light as a quantum memory, the information being stored in the slowlypropagating probe field, such a setup is only of limited use as a high fidelityquantum memory because of the restraint (4.45) on the achievable probe pulsedelay.

We shall therefore study situations in which the coupling field is time depen-dent. Fleischhauer and Lukin [21, 22] have developed a quasi-particle theorythat provides a very elegant treatment of this situation. In this chapter we shallreview this theory and see how an efficient quantum memory can be realizedby switching off the coupling laser while the probe pulse is contained in themedium.

We shall also consider corrections to the theory in cases where the couplingfield changes quickly in time and compare the analytical solution with a fullnumerical solution of the problem.

5.1 Using dark states as a quantum memory

Our review of dark states in section 3.1.2 was based on a semiclassical descrip-tion wherein classical probe and coupling fields interact with a single lambdaatom. Since we are interested in quantum memory, such a description is inade-quate. To see how an efficient quantum memory may be realized by exploitingthe presence of dark states in lambda atoms, we need to consider a quantizedprobe field interacting with a large number of lambda atoms.

In the following we shall derive the expression given in [22] for the general-ized dark state in a system consisting of N identical lambda atoms interactingwith a quantized probe field and a classical coupling field. Armed with this ex-pression, we shall see how the quantum state of the quantized probe field canbe transferred to the atoms by adiabatically switching off the coupling field.

67

68 Chapter 5. Dark state polaritons

5.1.1 Many-atom HamiltonianProvided that the atoms do not interact among each other, the Hamiltonian forthe N atom system is given by a sum over single atom Hamiltonians

H = HF +N∑

j=1

(Hj

A + HjL

)(5.1)

where HF is the Hamiltonian for the free probe and coupling fields, HA is theHamiltonian for the free atom and HL describes the interaction between theatom and the fields1. The individual terms are given by

HF = hωpd†pdp + hωcd

†cdc (5.2a)

HjA = hωcbσ

jcc + hωabσ

jaa (5.2b)

HjL = −

(Ep + Ec

)·(dbaσj


). (5.2c)

The operators dp,c and their Hermitian adjoints are the annihilation/creationoperators for the probe and coupling fields, respectively. The field operatorsEp,c are given by

Ep,c(r, t) = Eap,c + Ec

p,c (5.3)

where

Eap,c(r, t) = i

√hωp,c

2ε0up,c(r)dp,c(t) Ec

p,c =(Ea

p,c

)†(5.4)

The mode function u is taken to be

up,c(r) =ep,c√

Veikp,cz, (5.5)

We also define spatially slowly varying atomic operators σµν by

σjba = σj

baeikpz, (5.6a)

σjca = σj

caeikcz. (5.6b)

Applying a rotating frame transformation U , given by

U = e−i(ωpd†pdp+ωcd†cdc+(ωp−ωc)σcc+ωpσaa)t, (5.7)

and invoking the rotating-wave approximation, we get the Hamiltonian

H ′ = −h

N∑

j=1

[∆σj

cc + δpσjaa −

(igpd

†pσ

jba + igcd

†cσ

jca + h.a.

)]. (5.8)

1Note that both fields are quantized. The transition to a semiclassical coupling field will be madelater. Note also that we are ignoring the coupling to the vacuum field modes.

5.1 Using dark states as a quantum memory 69

5.1.2 Number-state representationThe basis states of the N atom problem are product states of the states |a〉, |b〉 and|c〉. This makes the Hamiltonian (5.8) very cumbersome to work with. We canfacilitate the derivation by switching to the so-called number state representation[28]. The Hamiltonian in the number state representation is

H ′ = −h∆c†c− hδpa†a + h

(igpd

†pb†a + igcd

†cc†a + h.a.

). (5.9)

The operators a, b, c and their Hermitian adjoints are annihilation/creation op-erators for atoms in the states |a〉, |b〉 and |c〉, respectively. The basis states in thenumber state representation are Fock states, written as

|aibjck, np, nc〉 (5.10)

where i, j, k are the number of atoms in the corresponding state, and np,c are thenumber of photons in the probe and coupling field modes. In the following weshall adopt a notation in which we omit any state whose occupation number iszero.

5.1.3 Generalized dark statesWe are now ready to derive the expression for the dark states in the N atomsystem. We assume zero two-photon detuning (∆ = 0) and take the couplingfield to be classical by making the replacement gcd

†c → Ωc in the Hamiltonian

(5.9). The coupling laser Rabi frequency Ωc is assumed to be real. Under theseconditions the Hamiltonian becomes

H ′ = −hδpa†a + h

(igpd

†pb†a + iΩcc

†a + h.a.)

. (5.11)

Since the dark state does not contain the excited state |a〉, it must be of the form2

|D′, n〉 =n∑

k=0

αk|b(N−k)ck, n− k〉 (5.12)

where n is the total number of excitations in the system, defined as the sum ofthe number of probe photons and the number of atoms in state |c〉, and αk is anexpansion coefficient. Furthermore, the dark state must satisfy the equation

H ′|D′, n〉 = 0. (5.13)

This equation leads to a recursion relation for the expansion coefficients

αk+1 = − gp

Ωc

√(N − k) (n− k)

k + 1αk (5.14)

2We have omitted the number of coupling field photons in the number state since we are assum-ing that the coupling field is classical.


which is easily solved for αk

αk = α0

(− gp

Ωc

)k√

N !n!k! (N − k)! (n− k)!

. (5.15)

We now make the assumption that the number of atoms N is much larger thanthe total number of excitations n. With this approximation the expression forthe expansion coefficients becomes

αk = α0

(−gp

√N

Ωc

)k √n!

k! (n− k)!. (5.16)

To determine the first expansion coefficient α0 we make use of the fact that thedark state is normalized

n∑

k=0

|αk|2 = 1. (5.17)

This gives the following equation for α0

n∑

k=0

(n

k

)α2

0

(g2

pN

Ω2c

)k

= 1, (5.18)

where we have chosen α0 to be real. Comparing this equation to the binomialseries

n∑

k=0

(n

k

)xn

(y

x

)k

= (x + y)n (5.19)

we find that

α0 = cosn θ,gp

√N

Ωc= tan θ (5.20)

where the mixing angle θ is given by

cos θ =Ωc√

g2pN + Ω2

c

sin θ =gp

√N√

g2pN + Ω2

c

. (5.21)

The dark states of the system are therefore given by

|D′, n〉 =n∑

k=0

√n!

k! (n− k)!(− sin θ)k (cos θ)n−k |bN−kck, n− k〉. (5.22)

5.2 Polariton fields 71

5.1.4 Quantum memoryLooking at the expression for the dark state (5.22) we see that for very largecoupling field intensities (Ωc À gp

√N), the mixing angle θ → 0. In this case the

dark state is|D′, n〉 = |bN , n〉. (5.23)

Conversely, if Ωc = 0 the mixing angle θ = π/2 and the dark state is

|D′, n〉 = |bN−ncn, 0〉. (5.24)

By adiabatically switching off an initially strong coupling field, the mixing angleis rotated from 0 to π/2. According to the adiabatic theorem [17] the system willremain in the instantaneous eigenstate. If the initial quantum state of the systemis described by the density operator

ρi =∑n,m

ρnm|n〉〈m| ⊗ |bN 〉〈bN |, (5.25)

the system will consequently evolve according to∑n,m

ρnm|n〉〈m| ⊗ |bN 〉〈bN | → |0〉〈0| ⊗∑n,m

ρnm|bN−ncn〉〈bN−mcm|. (5.26)

It is clear that the quantum state of the probe field has been transferred to acollective excitation of the atoms. By turning the coupling field back on, theprocess is reversed and the stored quantum state returned to the probe field.Obviously, the initial number of probe photons needs to be smaller than thenumber of atoms to avoid saturation of the system.

5.2 Polariton fields

As in [21, 22] we now introduce two new quantum fields which are linear com-binations of the probe field and “atom field”

Ψ = cos θdp − 1√N

sin θb†c, (5.27)

Φ = sin θdp +1√N

cos θb†c. (5.28)

To examine the properties of the new fields we note that they satisfy the com-mutation relations

[Ψ, Ψ†

]= cos2 θ +

1N

sin2 θ(Nb − Nc

), (5.29a)

[Φ, Φ†

]= sin2 θ +

1N

cos2 θ(Nb − Nc

), (5.29b)

[Ψ, Φ†

]= sin θ cos θ

(1− 1

N

(Nb − Nc

)), (5.29c)


where Nb = b†b and Nc = c†c are the number operators for atoms in the states|b〉 and |c〉, respectively.

If the number density of excitations is much smaller than the number ofatoms, we see that the commutation relations (5.29) can be approximated by

[Ψ, Ψ†

] ' 1, (5.30a)[Φ, Φ†

] ' 1, (5.30b)[Ψ, Φ†

] ' 0. (5.30c)

Thus in the limit of a weak probe field, the two new fields obey bosonic com-mutation relations. It is also easy to verify that the number states generated byrepeated application of the creation operator Ψ† to the ground state

|D′, n〉 =1√n!

(Ψ†

)n

|bN , 0〉 (5.31)

are in fact the dark states of the system (H|D′, n〉 = 0), and are given by (5.22)in the weak probe field limit.

The number states generated by repeated application of the creation operatorΦ† to the ground state are

|B′, n〉 =1√n!

(Φ†

)n

|bN , 0〉. (5.32)

In the weak probe field limit they are given by

|B′, n〉 =n∑

k=0

√n!

k! (n− k)!(sin θ)n−k (cos θ)k |bN−kck, n− k〉. (5.33)

These states are called bright states because a system in such a state is suscepti-ble to excitation by the electromagnetic field. In the weak probe field limit thedark and bright states are orthogonal, as can easily be verified by making use ofthe commutation relation (5.30c).

The commutation relations (5.30) and the identification of the number statesgenerated by the new fields Ψ and Φ lead us to associate bosonic quasi-particleswith them. The quasi-particle associated with Ψ is called a dark state polariton(DSP), while the quasi-particle associated with Φ is called a bright state polariton(BSP).

5.2.1 Field equations for the polariton fields

In section 4.1 we solved the propagation problem in the case of a time inde-pendent coupling field. We shall now reformulate the propagation problem in


terms of the dark and bright state polariton fields, which will lead to a solutionvalid for a time dependent coupling field.

By inserting (4.23a) into (4.15a) the wave equation for the slowly varyingprobe field operator becomes

(∂

∂t+ c

∂

∂z

)Ep(z, t) =

gpNr

Ωc(t)

(∂

∂t+ Γbc

)σbc, (5.34)

where we have assumed that the coupling laser Rabi frequency is real and inde-pendent of z. Furthermore, we assume that the atomic density is constant.

We now define slowly varying dark and bright state polariton field operatorsas linear combinations of Ep and σbc

Ψ = cos θEp − sin θ√

Nrσbc, (5.35a)

Φ = sin θEp + cos θ√

Nrσbc. (5.35b)

The probe field operator Ep and the Raman coherence operator σbc can be ex-pressed in terms of Ψ and Φ

Ep = sin θΦ + cos θΨ, (5.36a)√Nrσbc = cos θΦ− sin θΨ. (5.36b)

By inserting the definitions (5.35) into the wave equation for the probe field(5.34) and into the expression (4.23a), we obtain two coupled differential equa-tions for the polariton fields

∂Ψ∂t

+ c cos2 θ∂Ψ∂z

= −θΦ− c sin θ cos θ∂Φ∂z

− Γbc

(sin2 θΨ− sin θ cos θΦ

)(5.37)

Φ =sin θ

g2pNr

[∂

∂t+ Γba

](tan θ

[∂

∂t+ Γbc

])(sin θΨ− cos θΦ) (5.38)

5.2.2 The adiabatic approximationThe two rather complicated equations (5.37) and (5.38) are greatly simplified byassuming that the coupling laser Rabi frequency, and thereby the angle θ, as wellas the probe field amplitude changes slowly in time. We shall also assume thatthe dephasing rate and the two-photon detuning are negligible (Γbc = 0). Thecharacteristic timescale on which the coupling laser Rabi frequency and probefield operator changes is denoted by T , and we define the small dimensionlessquantity

ζ =1

gp

√NrT

. (5.39)

Expanding the polariton fields in powers of ζ, we can solve the field equationsperturbatively. To zeroth order in ζ the BSP field operator is

Φ = 0. (5.40)


Inserting this into (5.37) we obtain the differential equation for the DSP operatorto the same order of approximation

∂Ψ∂t


= 0. (5.41)

This simple wave equation for the DSP operator can easily be solved by Fouriertransforming with respect to z. The solution is

Ψ(z, t) = Ψ(

z −∫ t

0

vg (t′) dt′, 0)

, (5.42)

where the instantaneous group velocity vg(t) is given by

vg(t) = c cos2 θ(t). (5.43)

From (5.36) we find the solutions for the probe field Ep and the Raman coherenceσbc in the adiabatic limit.

Ep(z, t) = cos θ(t)Ψ(

z −∫ t

0

vg (t′) dt′, 0)

, (5.44)

√Nrσbc(z, t) = − sin θ(t)Ψ

(z −

∫ t

0

vg (t′) dt′, 0)

. (5.45)

It is clear that the solution (5.42) preserves the shape and quantum state of theinitial DSP field, which is exactly what is required of an efficient quantum mem-ory. The group velocity of the DSP field is controlled by the intensity of thecoupling laser. In the limit of a strong coupling laser (Ωc À gp

√Nr), the group

velocity approaches the vacuum speed of light c and the DSP field has a purelyphotonic character which is evident from the definition (5.35). In the oppositelimit of a weak coupling laser (Ωc ¿ gp

√Nr), the group velocity approaches

zero and the DSP field is effectively a collective excitation of the atomic medium,often referred to as a spin-wave.

The quantum memory is realized by letting a probe pulse propagate intothe atomic medium with the coupling laser intensity sufficiently large to ensurethat the spectral width of the initial probe pulse is smaller than the transparencywidth. Once the probe pulse is completely contained in the medium, the cou-pling laser is adiabatically turned off, storing the quantum state of the probepulse in the atoms. The information is retrieved by turning the coupling laserback on, regenerating the probe pulse which then travels out of the medium.

As an example let us consider a dark state polariton which is deceleratedfrom the vacuum speed of light c to zero, and then reaccelerated back to c. Thetime dependence of the mixing angle θ is assumed to be given by3

cos2 θ(t) = 1 +12(tanh((t− 15Ts)/Ts)− tanh((t− 5Ts)/Ts)

), (5.46)

3In actual experiments, the initial group velocity of the DSP field is usually only a small fractionof c. The extreme example presented here is chosen for illustrative purposes.

5.3 Corrections to the adiabatic solution 75

where Ts is the characteristic switching time for the coupling laser.The initial DSP field is taken to have a Gaussian shape with a characteristic

width Lp = cTp

Ψ(z, 0) = Ψ0 exp

(−

(z − 3Lp

Lp

)2)

, (5.47)

where the initial polariton amplitude Ψ0 is given in terms of the initial probefield amplitude Ep,0 by

Ψ0 = Ep,0. (5.48)

The adiabatic solutions for the DSP amplitude, probe field and Raman coher-ence are plotted as functions of space and time in figure 5.1. We see that theprobe field is absorbed as the coupling laser is turned off, storing the informa-tion of the probe pulse in the medium as a stationary collective Raman coher-ence. As the coupling laser is turned back on, the probe field is regenerated asthe information is erased from the atoms.

5.3 Corrections to the adiabatic solution

As in chapter 4 we shall now investigate the validity of the various approxima-tions employed in obtaining the analytic solution (5.42). These approximationsare:

• The weak probe approximation.

• Negligible dephasing rate.

• Negligible retardation of the coupling field.

• The adiabatic approximation.

5.3.1 Validity of the weak probe approximationSince the analytic solution (5.42) was derived using the weak probe approxima-tion introduced in section 4.1.3, which assumes that the ratio of the probe andcoupling field Rabi frequencies is small, we need to ensure that this approxima-tion remains valid when the coupling laser is turned off.

In section 4.2.2 we found the population of state |c〉 to second order in theRabi frequency ratio ε (4.73a)

〈σcc〉 =|Ωp|2|Ωc|2 . (5.49)

If the initial density of probe photons is much smaller than the atomic density,we have 〈σcc〉 ¿ 1 and consequently the Rabi frequency ratio ε is always small,even when the coupling laser Rabi frequency approaches zero, and the weakprobe approximation remains valid.


0 5 10 15 200

0.5

1

tv g /

c

(a) Group velocity (b) DSP field

(c) Probe field (d) Raman coherence

Figure 5.1: Adiabatic storage and retrieval of a probe pulse. (a) Group velocity as afunction of time. (b) DSP amplitude normalized to Ψ0 as a function of z and t. (c)Probe field amplitude normalized to Ep,0. (d) Absolute value of Raman coherenceσbc in units of Ep,0/

√Nr. Time t is in units of Ts and position z is in units of Lp =

cTp where the temporal length of the pulse Tp = 57Ts. When the coupling laser

is turned off, the DSP field is brought to a halt as the probe field is absorbed andinformation is transferred to the atoms. The process is reversed as the couplinglaser is turned back on.

5.3.2 Effect of a small dephasing rateWe now consider the effect of a small but non-vanishing dephasing rate Γbc. Weassume that the dephasing rate is small enough to satisfy

|Γbc|gp

√Nr

¿ 1 (5.50)

so that to zeroth order we have again

Φ = 0. (5.51)

The equation of motion for the DSP field (5.37) to zeroth order becomes

∂Ψ∂t


= −Γbc sin2 θΨ, (5.52)


which has the solution

Ψ(z, t) = exp(−Γbc

∫ t

0

sin2 θ(t′)dt′)

Ψ(

z −∫ t

0

vg(t′)dt′, 0)

. (5.53)

From this solution we see that the effect of a small dephasing rate Γbc may beneglected provided that

t ¿ 1|Γbc| . (5.54)

The dephasing rate may in general be a complex number, and we see also thatthe effect of the real part of Γbc is to cause attenuation of the DSP field, while theimaginary part, due to a non-zero two-photon detuning, causes a phase shift.

The magnitude of the effect depends on the intensity of the coupling laserthrough the mixing angle θ. When the coupling laser is turned off, sin θ = 1 andthe effect is most pronounced. Conversely, when the coupling laser intensity isvery high, sin θ → 0 and the effect disappears. This is not surprising since in theadiabatic limit we have √

Nrσbc = − sin θΨ (5.55)

from which it is evident that when the coupling laser intensity is high, σbc ' 0and therefore the effect of the dephasing rate Γbc is negligible.

5.3.3 Retardation of the coupling laserThe assumption of a spatially constant coupling field is an approximation dueto the finite propagation velocity of the coupling laser. As we shall see in thefollowing, this approximation becomes invalid if the coupling laser intensity isvaried too rapidly.

In section 4.2.2 we found that to first order in ε the coupling field obeys thevacuum wave equation

∂Ec

∂t+ c

∂Ec

∂z= 0. (5.56)

The spatial change in the coupling field ∆Ec is now defined as

∆Ec = L∂Ec

∂z, (5.57)

where L is the length of the medium. Using the wave equation (5.56) we canreplace the spatial derivative with the time derivative of the coupling field sothat

∆Ec = −L

c

∂Ec

∂t. (5.58)

Introducing the characteristic timescale of coupling field switching Ts and thenormalized time t′ = t/Ts yields

∆Ec

Ec= − L

cTs

1Ec

∂Ec

∂t′. (5.59)


This gives us the desired estimate of the retardation of the coupling field:

∆Ec

Ec' − L

cTs. (5.60)

Thus in order for the retardation of the coupling field to be negligible we musthave

Ts À L

c. (5.61)

The above calculation applies to the case where the probe and coupling laserspropagate along the same axis. The obvious way to eliminate retardation of thecoupling field is to have the coupling laser propagate orthogonal to the probelaser. This works well for ultra cold gases or solid state media where atomicmotion is negligible. For thermal gasses, however, this solution is not avail-able because, as we saw in section 4.2.3, such a medium requires copropagatinglasers in order to eliminate the detrimental effects of atomic motion.

5.3.4 Non-adiabatic correctionsWe shall now investigate what happens when the coupling laser intensity changesrapidly in time. As in [22] we do this by keeping terms to higher order in ζ in theexpansion of the BSP field. However, the derivation outlined in [22] is flawed[29] and we shall present the correct derivation here.

We assume that the characteristic timescale T , introduced in (5.39), is com-parable to the timescale γ−1

ba , and that Γbc is small enough to be neglected. Tosecond order in ζ, the BSP field is then

Φ =sin θ

g2pNr

[Γba +

∂

∂t

] (tan θ

∂

∂t(sin θΨ)

), (5.62)

where Ψ is calculated to zeroth order. Using the fact that to zeroth order

∂Ψ∂t

= −c cos2 θ∂Ψ∂z

(5.63)

we arrive at the following expression for the BSP field:

Φ =sin θ

g2pNr

[∂

∂t(θ sin θ)Ψ− θ sin θ cos2 θc

∂Ψ∂z

− ∂

∂t(sin2 θ cos θ)c

∂Ψ∂z

+ sin2 θ cos3 θc2 ∂2Ψ∂z2

]+

Γba

g2pNr

[θ sin2 θΨ− sin3 θ cos θc

∂Ψ∂z

].

(5.64)

Inserting this expression into (5.37) we arrive at the wave equation for the DSPfield to second order in ζ

[∂

∂t+ c cos2 θ

∂

∂z

]Ψ = −A(t)Ψ + B(t)c

∂Ψ∂z

+ C(t)c2 ∂2Ψ∂z2

−D(t)c3 ∂3Ψ∂t3

(5.65)


where the time dependent coefficients are given by4

A(t) =(

Γba +12

∂

∂t

)(θ2 sin2 θ

g2pNr

), (5.66a)

B(t) =sin θ

g2pNr

(2θ

∂

∂t(sin2 θ cos θ)− 1

3∂2

∂t2sin3 θ

), (5.66b)

C(t) =(

Γba +12

∂

∂t

)sin4 θ cos2 θ

g2pNr

, (5.66c)

D(t) =sin4 θ cos4 θ

g2pNr

. (5.66d)

Since the coefficients depend only on time, we can solve the wave equation(5.65) by Fourier transforming with respect to z. The solution is

Ψ(q, t) = Ψ(q, 0) exp(−iqc

∫ t

0

(cos2 θ(t′)−B(t′))dt′)×

exp(−

∫ t

0

A(t′)dt′)

exp(−q2c2

∫ t

0

C(t′)dt′)

exp(

iq3c3

∫ t

0

D(t′)dt′)

(5.67)

From this solution we can see that A(t) describes homogenous losses due tonon-adiabatic transitions to the excited state |a〉, caused by the rapid switchingof the coupling laser intensity, followed by spontaneous emission. B(t) gives acorrection to the group velocity of the DSP field. C(t) leads to broadening of thepulse shape due to absorption of high spatial frequency components of the DSPfield. Finally, D(t) is responsible for a deformation of the pulse shape.

The homogenous losses A(t) and the group velocity modification B(t) aredue solely to the changing coupling laser intensity, while the non-homogenouslosses C(t) are due to both the finite length of the pulse as well as the non-adiabatic switching of the coupling laser intensity. The pulse shape deformationcaused by D(t) is due solely to the finite length of the pulse.

To find a limit on the characteristic switching time of the coupling field Ts,we demand that ∫ Ts

0

A(t)dt ¿ 1. (5.68)

Assuming that θ(0) = θ(Ts) = 0, the integral (5.68) becomes

γba

g2pNr

∫ Ts

0

(∂

∂tcos θ(t)

)2

dt. (5.69)

If the DSP pulse is accelerated/decelerated to/from a group velocity vg,0, thelimit on the switching time is estimated to be

Ts À γba

g2pNr

vg,0

c=

lavg,0

c2, (5.70)

4The B coefficient differs from the result given in [22] which is incorrect. The details of the calcu-lation are given in appendix B.


where we have introduced the absorption length la defined in chapter 3.Even with vg,0 ' c, this limit is extremely small which is desirable for quan-

tum memory purposes since this allows us to limit the medium length to thepulse length.

In order for the deformation of the pulse to be small, we must demand that

γbaq2c2

g2pNr

∫ Tf

0

sin4 θ(t) cos2 θ(t)dt ¿ 1, (5.71)

where Tf is the time required for the DSP pulse to travel the length of themedium L. Assuming that the group velocity of the DSP field is small, thatis cos2 θ(t) ¿ 1, we can approximate the integral by

γbaq2c2

g2pNr

∫ Tf

0

cos2 θ(t)dt =γbaq2cL

g2pNr

. (5.72)

Since the range of relevant spatial Fourier frequencies is determined by the spa-tial length of the pulse Lp, we can set q = L−1

p in (5.72). Introducing the opacityof the medium α, the criterion can be written as

L ¿ √αLp. (5.73)

By using the definition of the transparency width (3.82) we see that the criterion(5.72) is the same as the one we encountered in section 4.2.1: the initial spectrumof the probe pulse must be completely contained within the initial transparencywindow of the medium.

δωp ¿ δωtr. (5.74)

We could in principle calculate the non-adiabatic corrections using the methodemployed in section 4.2.1. However, as we shall see in section 5.4, the correc-tions due to rapid switching of the coupling laser are extremely small for real-istic switching times and medium parameters, and therefore we shall not carryout the rather cumbersome calculation.

5.4 Numerical solution

As in section 4.3 we compare the analytical solution to a full numerical solution.In particular, we shall focus on checking the validity of the adiabatic solution(5.42) when the coupling laser intensity changes rapidly.

Although it is possible to simulate the entire storage process in which theprobe pulse enters the medium and is subsequently stored by switching off thecoupling laser, it is not possible to directly compare such a numerical solutionto the analytical solution, since we assumed a constant atomic density in thederivation of the latter.

Instead, we simulate the retrieval of an initially stored probe pulse by rapidlyswitching on the coupling laser. This method will allow us to make a quantita-tive comparison of the numerical and analytical solutions.


5.4.1 Initial conditions and medium parameters

To calculate the initial conditions for the atomic variables σµν , we assume thatthe initial probe pulse is stored under perfect adiabatic conditions. Prior to stor-age, the DSP field is given by

Ep(z, t) = cos θ0Ψ(z, t), (5.75)

where c cos2 θ0 = vg,0 is the initial group velocity of the probe pulse. Since thepulse shape of the DSP field is unaffected by adiabatic switching of the couplinglaser, we can determine the Raman coherence after switch-off (at time t = 0) bythe relation √

Nrσbc(z, 0) = −Ψ(z, 0). (5.76)

Assuming that the initial probe field has a Gaussian profile of width Lp = vg,0Tp,the initial condition for the Raman coherence becomes

σbc(z, 0) = − Ep,0√Nr cos θ0

exp

(−

(z − 1

2L

Lp

)2)

, (5.77)

where Ep,0 is the initial amplitude of the probe pulse. The initial conditions forthe populations σcc and σbb can be calculated from σbc as we saw in section 4.2.2.We find

σcc(z, 0) =|Ep,0|2

Nr cos2 θ0exp

(−2

(z − 1

2L

Lp

)2)

. (5.78)

Assuming no population in the excited state |a〉, the ground state population is

σbb(z, 0) = 1− σcc(z, 0). (5.79)

The time dependence of the coupling laser Rabi frequency is chosen such that

cos2 θ(t) =vg,0

2c

(1 + tanh

(t− nTs

Ts

)), (5.80)

where n is a number chosen large enough to ensure that the coupling laser Rabifrequency and its derivative is essentially zero at t = 0.

The various decay rates as well as the vacuum Rabi frequencies must also bespecified. These parameters are kept the same for all the simulations presentedin this chapter. The decay rates are chosen to be

γca = γb = γc = γba γ = 2γba γbc = 0 (5.81)

and the vacuum Rabi frequencies are

g2pNr = g2

cNr = 106γ2ba. (5.82)


Adiabaticity parameter η 0.1Rabi frequency ratio ε 0.01Maximum group velocity vg,0/c 0.0001Length of medium L/vg,0Tp 8

Table 5.1: Parameters used in the numerical solution.

5.4.2 Adiabatic switchingWe shall first consider the case where the coupling laser is switched on fairlyslowly, and the probe and coupling lasers copropagate along the z axis.

The boundary condition for the coupling field follows from (5.80) with Ts =γ−1

ba and n = 5. The other relevant parameters are summarized in table 5.1.With these parameters, the condition for adiabatic switching (5.70) is very wellsatisfied, and the condition for negligible retardation of the coupling laser (5.61)is also well satisfied.

The numerical calculations of the BSP field (fig. 5.2), DSP field (fig. 5.3) andprobe field (fig. 5.4) are plotted as functions of z at time t = 5Ts and comparedto the analytic solutions.

The analytical solution for the DSP field is calculated from the zeroth ordersolution (5.42). Since the BSP field vanishes to zeroth order, we calculate theBSP field to lowest non-vanishing (second) order using (5.64). The analyticalsolution for the probe field is calculated from (5.36) using both the zeroth orderBSP field (Φ = 0) and also the second order BSP field.

As expected, the agreement between the numerical and analytical solutionsis excellent, and we see that the BSP field is indeed much smaller than the DSPfield. By comparing the magnitude of the individual terms in (5.64) we find thatthe main source of the BSP field in this case is the finite length of the probe pulseand not the switching of the coupling field.


0 1 2 3 4 5 6 7 8−1.5

−1

−0.5

0

0.5

1

1.5x 10

−5

z / vg,0

Tp

Φ /

Ψ0

Figure 5.2: Spatial profile of the BSP field at t = 5Ts during adiabatic retrieval of astored probe pulse. The numerical solution (solid curve) and the analytical solution(dotted curve) are nearly indistinguishable. Parameters are: Ts = γ−1

ba , η = 0.1,ε = 0.01, vg,0 = c/104, L = 8vg,0Tp.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z / vg,0

Tp

Ψ /

Ψ0

Figure 5.3: Spatial profile of the DSP field at t = 5Ts during adiabatic retrieval of astored probe pulse. The numerical and adiabatic solutions are indistinguishable.


0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z / vg,0

Tp

Ωp /

Ωp,

0

Figure 5.4: Spatial profile of the probe field at t = 5Ts during adiabatic retrievalof a stored probe pulse. The numerical (solid curve), adiabatic (dotted curve) andnon-adiabatic (dashed curve) solutions are indistinguishable.

5.4.3 Ultra fast switching

We shall now investigate the case where the coupling laser is switched on rapidly.Again the boundary condition for the coupling field is derived from (5.80) withTs = 0.01γ−1

ba and n = 50. All other parameters are given in table 5.1.Despite the shorter switching time, condition (5.70) for adiabatic switching

is still well satisfied. However, as we shall see, the retardation of the couplinglaser is no longer negligible in this case. For this reason we perform numericalsimulations for two different beam configurations: copropagating beams andorthogonal beams where the coupling laser Rabi frequency is independent of z.

As in section 5.4.2 we compare the numerical and analytical solutions. Forthe case of copropagating beams the results are shown for the BSP field (figure5.5), DSP field (figure 5.6) and probe field (figure 5.7).

The numerical solution for the DSP field is in perfect agreement with the adi-abatic solution (5.42). The BSP field, however, differs slightly from the analyticalsolution (5.64) but its magnitude is still very small compared to the DSP field.The fact that the BSP field is more sensitive to rapid switching of the couplingfield than the DSP field is not surprising, since the calculation of the BSP field isbased on an expansion in the small quantity ζ defined in (5.39). In the presentcase, ζ = 0.1 and thus we should not expect perfect agreement. On the otherhand the adiabaticity condition (5.70) for the DSP field is satisfied by a factor of108, and it is reasonable to expect perfect agreement between the numerical andadiabatic solutions for the DSP field.

The probe field differs markedly from the analytical solution. At first this


may seem surprising considering the excellent agreement between the numeri-cal and analytical solutions for the DSP field and the smallness of the BSP field.The discrepancy is due to the retardation of the coupling field. Because of theextremely short switching time, the coupling field at the position of the DSPpulse lags behind the value given by the boundary condition (5.80).

To investigate any transient behavior of the fields, we plot the temporal pro-file of the probe field at z = L in figure 5.8. We do indeed find a very smalltransient probe field as the coupling laser is switched on. This transient field,which propagates much faster than the revived probe pulse, is quickly dampedout before the restored probe pulse reaches the medium boundary and thereforedoes not present a problem.

The numerical and analytical solutions for the case of orthogonal beams areshown in figures 5.9, 5.10, 5.11 and 5.12. Notice that in this case the agreementbetween the numerical and analytical solutions for the probe field is much betterthan in the copropagating beam configuration due to the lack of retardation ofthe coupling field.

The simulations for both the copropagating and orthogonal beam configu-rations demonstrate the extreme “robustness” of the DSP field. This is highlydesirable for quantum memory purposes, since the retrieved probe pulse afterswitching is determined by the DSP field. The robustness of the DSP field thusimplies a high fidelity of the quantum memory.


0 1 2 3 4 5 6 7 8−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−4

z / vg,0

Tp

Φ /

Ψ0

Figure 5.5: Spatial profile of the BSP field at t = 50Ts during rapid retrieval of astored probe pulse in the copropagating beam configuration. The numerical solu-tion (solid curve) differs slightly from the analytical solution (dotted curve). Param-eters are: Ts = 0.01γ−1

ba , η = 0.1, ε = 0.01, vg,0 = c/104, L = 8vg,0Tp.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z / vg,0

Tp

Ψ /

Ψ0

Figure 5.6: Spatial profile of the DSP field at t = 50Ts during rapid retrieval of astored probe pulse in the copropagating beam configuration. The numerical andanalytical solutions are indistinguishable.


0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z / vg,0

Tp

Ωp /

Ωp,

0

Figure 5.7: Spatial profile of the probe field at t = 50Ts during rapid retrieval of astored probe pulse in the copropagating beam configuration. The numerical solu-tion (solid curve) differs from both the adiabatic (dotted curve) and non-adiabatic(dashed curve) analytical solutions due to the retardation of the coupling laser.

0.45 0.5 0.55 0.6 0.65 0.7 0.75−6

−4

−2

0

2

4

6x 10

−3

t / γba−1

Ωp /

Ωp,

0

Figure 5.8: Temporal evolution of the probe field at z = L during rapid retrieval ofa stored probe pulse in the copropagating beam configuration. The transient field isquickly damped out before the retrieved probe pulse reaches the medium boundary.


0 1 2 3 4 5 6 7 8−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−4

z / vg,0

Tp

Φ /

Ψ0

Figure 5.9: Spatial profile of the BSP field at t = 50Ts during rapid retrieval of astored probe pulse in the orthogonal beam configuration. The numerical solution(solid curve) differs slightly from the analytical solution (dotted curve).

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z / vg,0

Tp

Ψ /

Ψ0

Figure 5.10: Spatial profile of the DSP field at t = 50Ts during rapid retrieval of astored probe pulse in the orthogonal beam configuration. The numerical and ana-lytical solutions are indistinguishable.


0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

z / vg,0

Tp

Ωp /

Ωp,

0

Figure 5.11: Spatial profile of the probe field at t = 50Ts during rapid retrieval ofa stored probe pulse in the orthogonal beam configuration. The numerical solution(solid curve) differs slightly from the adiabatic solution (dotted curve), but is inexcellent agreement with the non-adiabatic solution (dashed curve).

0.45 0.5 0.55 0.6 0.65 0.7 0.75−8

−6

−4

−2

0

2

4

6

8x 10

−3

t / γba−1

Ωp /

Ωp,

0

Figure 5.12: Temporal evolution of the probe field at z = L during rapid retrievalof a stored probe pulse in the orthogonal beam configuration. The transient field isquickly damped out before the retrieved probe pulse reaches the medium boundary.

C H A P T E R 6Standing wave polaritons

in thermal gasses

So far we have considered EIT and light storage experiments in which the cou-pling field is a traveling wave field. Recently, however, an experiment has beenperformed with a standing wave coupling field [11]. In this experiment a probepulse is stored in the EIT medium, which consists of a thermal gas of 87Rbatoms, using the traveling wave light storage technique discussed in chapter5. The probe pulse is subsequently retrieved using a standing wave couplingfield, resulting in a stationary probe pulse.

Such stationary pulses could have potential applications for quantum infor-mation processing . Although the light storage technique discussed in chapter5 is an ideal candidate for quantum memory purposes, manipulating the storedinformation is difficult. Performing logic operations requires strong nonlinearinteractions between the excitations storing the information, which is not easyto achieve since the interactions between the atoms are very weak [30]. Fur-thermore, the absence of the probe field during storage prevents the use of non-linear optical interactions. Enhanced nonlinear optical interactions in the pres-ence of the usual traveling wave EIT has been predicted theoretically [31] andalso demonstrated experimentally [5], but the efficiency is limited by the non-vanishing group velocity of the light pulses, since strong nonlinear interactionsfor weak fields require long interaction time and tight confinement of the fields.The stationary light pulses created in the standing wave EIT experiment [11]are useful for quantum information processing [32], combining long interactiontime and tight confinement with a non-vanishing probe field.

A theoretical discussion of the standing wave EIT experiment [11] is givenby Zimmer et al. [33]. Unfortunately, this theory is incapable of treating timedependent coupling fields which is necessary in order to describe light storageand retrieval consistently.

Ideally we would like to formulate a theory based on the dark and brightstate polariton fields that worked so well in the traveling wave case. Unfortu-

91

92 Chapter 6. Standing wave polaritons in thermal gasses

nately, it has not been possible to formulate such a description due to the addedcomplication of treating standing wave fields. Inspired by the dark/bright statepolariton description of chapter 5, we shall nonetheless present a modificationof the theory given in [33] by introducing a DSP-like field. This modificationwill enable us to treat time dependent coupling fields in a consistent manner.

6.1 Heisenberg-Langevin equations

We consider a thermal gas of N lambda atoms interacting with probe and cou-pling lasers propagating parallel with the z axis. The two lower states |b〉 and|c〉 of the atoms are assumed to be nearly degenerate, such that the magnitudeof the wave vectors of the probe and coupling fields can be considered identical(kp ' kc = k).

The Hamiltonian for the N atom problem is

H = HF +N∑

j=1

(Hj

A + HjL + Hj

V

)(6.1)

where HF and HA describe the free electromagnetic field and atom, HL de-scribes the interaction of the atom with the probe and coupling fields, and HV


HF =∑m

hωma†mam (6.2a)

HjA = hωcbσ

jcc + hωabσ

jaa (6.2b)

HjL = −(

Ep + Ec

) · (dbaσjba + dcaσj

ca + h.a.)

(6.2c)

HjV = −EV ·

(dbaσj


)(6.2d)

As always, we introduce slowly varying field operators for the electromagneticfield. Since we are allowing for standing wave fields, we write the field op-erator as a superposition of two traveling wave fields propagating in oppositedirections

Ep,c(z, t) =√

hωp,c

2ε0Vep,cEp,c(z, t)e−iωp,ct + h.a. (6.3)

where the operators Ep,c are given by

Ep,c(z, t) = E+p,c(z, t)eikz + E−

p,c(z, t)e−ikz. (6.4)

The field operators E±p,c are the slowly varying field operators for the forward

and backward propagating components of the probe and coupling lasers, andep,c are the respective polarization vectors.

6.1 Heisenberg-Langevin equations 93

6.1.1 Fourier decomposition of the atomic operatorsAs in chapter 4, we introduce continuum atomic operators by summing over theindividual atoms in a small volume V . We also introduce slowly varying atomicoperators in both space and time. Because of the standing wave fields, this is alittle more involved than in the traveling wave case. It is necessary to expandthe continuum atomic operators σµν in a Fourier series, thereby introducing theslowly varying atomic operators σ

(n)µν . The slowly varying populations σ

(n)µµ are

defined by

σµµ(z, t) =∞∑

n=−∞σ(n)

µµ (z, t)einkz. (6.5)

The slowly varying optical coherences σ(n)ba and σ

(n)ca are defined by

σba(z, t) = e−iωpt∞∑

n=−∞σ

(n)ba (z, t)einkz, (6.6)

σca(z, t) = e−iωct∞∑

n=−∞σ(n)

ca (z, t)einkz. (6.7)

Finally, the slowly varying Raman coherence σ(n)bc is defined by

σbc(z, t) = e−i(ωp−ωc)t∞∑

n=−∞σ

(n)bc (z, t)einkz. (6.8)

Using the expansions of the atomic operators, we can write down Heisenberg-Langevin equations for the slowly varying operators in the RWA.

σ(n)aa = −i

(gpE

+†p σ

(n+1)ba + gpE

−†p σ

(n−1)ba

+ Ω+∗c σ(n+1)

ca + Ω−∗c σ(n−1)ca − h.a.

)− γσ(n)

aa + F (n)aa

(6.9a)

σ(n)bb = i

(gpE

+†p σ

(n+1)ba + gpE

−†p σ

(n−1)ba − h.a.

)+ γbσ

(n)aa + F

(n)bb (6.9b)

σ(n)cc = i

(Ω+∗

c σ(n+1)ca + Ω−∗c σ(n−1)

ca − h.a.)

+ γcσ(n)aa + F (n)

cc (6.9c)

σ(n)ba = i

(gpE

+p

(σ

(n−1)bb − σ(n−1)

aa

)+ gpE

−p

(σ

(n+1)bb − σ(n+1)

aa

)

+ Ω+c σ

(n−1)bc + Ω−c σ

(n+1)bc

)− Γbaσ

(n)ba + F

(n)ba

(6.9d)

σ(n)ca = i

(Ω+

c

(σ(n−1)

cc − σ(n−1)aa

)+ Ω−c

(σ(n+1)

cc − σ(n+1)aa

)

+ gpE+p σ

(n−1)cb + gpE

−p σ

(n+1)cb

)− Γcaσ(n)

ca + F (n)ca

(6.9e)

σ(n)bc = i

(Ω+∗

c σ(n+1)ba + Ω−∗c σ

(n−1)ba

− gpE+p σ(n−1)

ac − gpE−p σ(n+1)

ac

)− Γbcσ

(n)bc + F

(n)bc

(6.9f)


In deriving the Heisenberg-Langevin equations we have employed an approx-imation analogous to the RWA, omitting terms on the rhs. of (6.9) with rapidlyoscillating phase factors such as einkz . These terms correspond to processes thatdo not conserve momentum. We have also assumed that the coupling field canbe treated as a classical field with Rabi frequency

Ωc(z, t) = Ω+c (z, t)eikz + Ω−c (z, t)e−ikz. (6.10)

The effect of the noise operators Fµν is negligible in the adiabatic limit we shallconsider [24] and we therefore disregard them in the following.

By considering the structure of the Heisenberg-Langevin equations (6.9) andthe initial conditions for the atoms (all atoms in the ground state |b〉), we cansee that only components with odd values of n are non-vanishing for the opticalcoherences σba and σca, while the non-vanishing components of the populationsσµµ and the Raman coherence σbc have even values of n.

6.1.2 The weak probe approximationAs in chapter 4, we assume that the probe field is weak compared to the cou-pling field and solve the Heisenberg-Langevin equations perturbatively. To firstorder, the relevant Heisenberg-Langevin equations are

σ(n)ba = i

(gpE

+p δn,1 + gpE

−p δn,−1 + Ω+

c σ(n−1)bc + Ω−c σ

(n+1)bc

)− Γbaσ

(n)ba (6.11a)

σ(n)bc = i

(Ω+∗

c σ(n+1)ba + Ω−∗c σ

(n−1)ba

)− Γbcσ

(n)bc (6.11b)

6.1.3 Effect of atomic motionSo far we have not taken the effect of the thermal motion of the atoms into ac-count. In chapter 4 we saw how the effect of atomic motion can be includedin the Heisenberg-Langevin equations by writing a set of Heisenberg-Langevinequations for each velocity class of atoms. In the standing wave case consid-ered in this chapter, the Heisenberg-Langevin equations for each velocity classis simply given by (6.9) with the decay rates

Γba = γba − i(δp − nkvz), (6.12a)Γca = γca − i(δc − nkvz), (6.12b)Γbc = γbc − i(∆− nkvz), (6.12c)

where vz is the z component of the velocity of the atom.For the case of copropagating traveling wave laser fields considered in chap-

ter 4, we saw that the effect of atomic motion becomes negligible when the lowerstates |b〉 and |c〉 are nearly degenerate due to the very small difference betweenthe wave vectors of the probe field and the coupling field.

In the standing wave case considered here, the atomic motion is far fromnegligible and it is essential to include it in the description. It is argued in


[33] that the effect of thermal atomic motion can be adequately taken into ac-count by neglecting all components of the Raman coherence σ

(n)bc except for the

n = 0 component. The reasoning behind this approximation is that the n = 0component is created by two-photon Raman transitions in which the probe andcoupling field components involved are copropagating and thus not affected byatomic motion. All other components of the Raman coherence are created byRaman transitions involving counterpropagating probe and coupling laser fieldcomponents. Such processes are highly susceptible to atomic motion and thusdephase rapidly, as is also suggested by the very large dephasing rates Γbc inthe Heisenberg-Langevin equation for the corresponding operators.

Neglecting the n 6= 0 components of the Raman coherence implies that com-ponents of the optical coherence σba with −1 > n > 1 also vanish. With thisapproximation, the weak probe Heisenberg-Langevin equations become

σ+ba = igpE

+p + iΩ+

c σbc − Γbaσ+ba (6.13a)

σ−ba = igpE−p + iΩ−c σbc − Γbaσ−ba (6.13b)

σbc = i(Ω+∗

c σ+ba + Ω−∗c σ−ba

)− Γbcσbc (6.13c)

where the decay rates are the usual complex decay rates (3.31) without velocitydependence. We have also introduced the notation σ±ba for the n = ±1 compo-nents of the optical coherence, and σbc for the n = 0 component of the Ramancoherence.

6.1.4 Adiabatic elimination of the optical coherenceIn order to solve the Heisenberg-Langevin equations (6.13) we follow [33] andemploy a technique known as adiabatic elimination [26], which is very similarto the adiabatic approximation we used in chapter 4. Provided that the probeand coupling fields change slowly in time compared to the timescale γ−1

ba , theoptical coherences σ±ba can be assumed to be in a steady state. This steady stateis found by setting the time derivative of the optical coherence equal to zero inthe Heisenberg-Langevin equations (6.13). We find

σ+ba =

i

Γba

(gpE

+p + Ω+

c σbc

), (6.14a)

σ−ba =i

Γba

(gpE

−p + Ω−c σbc

). (6.14b)

Since the dephasing rate γbc ¿ γba we can insert the steady state solution forthe optical coherence (6.14) into the Heisenberg-Langevin equation (6.13c) forthe Raman coherence σbc. We get

σbc = − 1Γba

(gpE

+p Ω+∗

c + gpE−p Ω−∗c + Ω2σbc

)− Γbcσbc (6.15)

where we have introduced the total Rabi frequency Ω defined by

Ω =√|Ω+

c |2 + |Ω−c |2. (6.16)


Since the dephasing rate Γbc is quite small we cannot use adiabatic eliminationfor the Raman coherence since this gives too crude an approximation. Insteadwe proceed as in chapter 4 and introduce a characteristic timescale T for theevolution of the slowly varying operators and expand σbc in powers of the smallparameter (γbaT )−1 ¿ 1. To zeroth order, which corresponds to adiabatic elim-ination, we get

σbc = − 1Ω2

(gpE

+p Ω+∗

c + gpE−p Ω−∗c

). (6.17)

Here we have also assumed that the dephasing rate Γbc is small enough to satisfy|ΓbcΓba| ¿ Ω2 so that it may be disregarded to zeroth order.

Since we need a better approximation than the simple adiabatic elimination,we use the zeroth order solution to calculate the Raman coherence to first order.We find

σbc =Γba

Ω2

(Γbc +

∂

∂t

)[1

Ω2

(gpE

+p Ω+∗


)]

− 1Ω2

(gpE

+p Ω+∗


) (6.18)

In [33] it is shown by comparison with numerical calculations that the first orderexpression given above gives accurate results for realistic experimental param-eters. Additional correction terms can be calculated by going to higher order, aswe did in chapter 4 for the traveling wave case, but here we shall limit ourselvesto cases where the first order solution suffices.


The theory given in [33] deals directly with the probe field operators E±p and

is therefore forced to assume time independent coupling fields. Here we shallimprove upon this theory by introducing a DSP field Ψ, defined by

E±p (z, t) = cos θ(t)Ψ±(z, t), (6.19)

which will enable us to treat time dependent coupling fields. The angle θ isdefined by

tan θ(t) =gp

√Nr

Ω(t). (6.20)

The definition (6.19) is inspired by the dark/bright state polariton theory dis-cussed in chapter 5, in which the probe field operator is related to the DSP andBSP field operators by the expression (5.36a). In the adiabatic limit, we foundthat the BSP field vanished and the probe field operator became proportional tothe DSP field operator.


In order to find expressions for the components of the optical coherence σ±ba,we insert (6.18) into (6.14) and introduce the DSP field (6.19). This yields

√Nrσ

+ba =

iΩΓba

sin θκ−∗(κ−Ψ+ − κ+Ψ−

)

+iκ+

Ω

(Γbc +

∂

∂t

)(sin θ

(κ+∗Ψ+ + κ−∗Ψ−

)) (6.21a)

√Nrσ

−ba =

iΩΓba

sin θκ+∗(κ+Ψ− − κ−Ψ+)

+iκ−

Ω

(Γbc +

∂

∂t

)(sin θ

(κ+∗Ψ+ + κ−∗Ψ−

)) (6.21b)

where we have introduced the ratios of the Rabi frequencies of the forward andbackward propagating components to the total Rabi frequency of the couplingfield

κ± =Ω±cΩ

. (6.22)

6.2.1 Wave equation for the polariton field

The wave equations for the components of the probe field E±p in the slowly

varying amplitude approximation are

(∂

∂t+ c

∂

∂z

)E+

p (z, t) = igpNrσ+ba(z, t) (6.23a)

(∂

∂t− c

∂

∂z

)E−

p (z, t) = igpNrσ−ba(z, t) (6.23b)

By inserting the definition (6.19) of the DSP field into the wave equations (6.23)for the probe field components, we obtain a set of wave equations for the DSPfield

cos θ∂Ψ+

∂t+ c cos θ

∂Ψ+

∂z= igpNrσ

+ba + θ sin θΨ+, (6.24a)

cos θ∂Ψ−

∂t− c cos θ

∂Ψ−

∂z= igpNrσ

−ba + θ sin θΨ−. (6.24b)


We now insert the expressions (6.21) for the optical coherences σ±ba into the waveequations (6.24) which yields

(∂

∂t+ vg

∂

∂z

)Ψ+ = sin2 θκ−∗

(Γbc +

∂

∂t

)(κ−Ψ+ − κ+Ψ−

)

− sin2 θΓbcΨ+ − g2pNr

Γbacos2 θκ−∗

(κ−Ψ+ − κ+Ψ−

)

− 12

∂

∂t

(cos2 θ

)κ−∗

(κ−Ψ+ − κ+Ψ−

)(6.25a)

(∂

∂t− vg

∂

∂z

)Ψ− = sin2 θκ+∗

(Γbc +

∂

∂t

)(κ+Ψ− − κ−Ψ+

)

− sin2 θΓbcΨ− −g2

pNr

Γbacos2 θκ+∗(κ+Ψ− − κ−Ψ+

)

− 12

∂

∂t

(cos2 θ

)κ+∗(κ+Ψ− − κ−Ψ+

)(6.25b)

We have assumed that the ratios κ± are constant in time, whereas the total Rabifrequency Ω is allowed to be time dependent. We have also introduced thegroup velocity

vg(t) = c cos2 θ(t). (6.26)

We now introduce the normal modes [33] ΨS and ΨD defined by

ΨS = κ+∗Ψ+ + κ−∗Ψ− ΨD = κ−Ψ+ − κ+Ψ−, (6.27)

which are called the sum and difference normal modes, respectively. Exploitingthe fact that |κ+|2 + |κ−|2 = 1, the components Ψ± of the DSP field are given interms of the normal modes as

Ψ+ = κ+ΨS + κ−∗ΨD Ψ− = κ−ΨS − κ+∗ΨD. (6.28)

The wave equation for the sum normal mode is then

∂ΨS

∂t+

(|κ+|2 − |κ−|2)vg∂ΨS

∂z= −2κ+∗κ−∗vg

∂ΨD

∂z− sin2 θΓbcΨS (6.29)

while the wave equation for the difference normal mode is

cos2 θ∂ΨD

∂t− (|κ+|2 − |κ−|2)vg

∂ΨD

∂z= −2κ+κ−vg

∂ΨS

∂z

− 12

∂

∂t

(cos2 θ

)ΨD − g2

pNr

Γbacos2 θΨD (6.30)

Note that the wave equation for the difference normal mode contains a decayterm due to the time dependence of the total Rabi frequency Ω. We assume thatthe coupling field changes slowly in time such that

∂

∂t

(cos2 θ

) ¿ g2pNr

|Γba| cos2 θ. (6.31)

6.3 Solving the coupled wave equations 99

If the group velocity vg changes to/from vg,0 during the characteristic switchingtime Ts, we find that this condition is satisfied provided that

Ts À |Γba|g2

pNr, (6.32)

a condition which is easily satisfied for realistic experimental parameters. In thefollowing we shall therefore disregard the second term on the rhs. of equation(6.30).

6.3 Solving the coupled wave equations

To solve the coupled wave equations for the normal modes of the DSP field,we first note that the wave equation for the difference normal mode (6.30) con-tains a very large decay term with decay constant g2

pNr/Γba. We can thereforeproceed by adiabatic elimination of the difference normal mode. Discarding allderivatives of ΨD in (6.30) we find

ΨD(z, t) = −2κ+κ−Γbac

g2pNr

∂ΨS

∂z(6.33)

which we insert into (6.29) to get

∂ΨS

∂t+

(|κ+|2− |κ−|2)vg∂ΨS

∂z= 4|κ+|2|κ−|2vg

Γbac

g2pNr

∂2ΨS

∂z2− sin2 θΓbcΨS (6.34)

Equation (6.34) is very similar to the wave equation we solved in chapter 5. Itdescribes the propagation of the sum normal mode with a group velocity givenby (|κ+|2− |κ−|2)vg. It is clear that in the case of a standing wave coupling field(|κ+| = |κ−|), the sum normal mode remains stationary.

The first term on the rhs. of (6.34) is responsible for broadening of the pulseenvelope, an effect similar to the broadening of the DSP field we found whenwe considered the non-adiabatic corrections to light storage using a travelingwave coupling field in chapter 5. In the traveling wave case, the broadeningof the pulse was due to dispersion. This is not the case here, seeing that theterm on the rhs. of (6.34) vanishes in the traveling wave case (κ− = 0) whichis inconsistent with the result in chapter 5. Later we shall discuss the physicalorigin of this phenomenon and argue that the broadening of the pulse in thestanding wave case is a diffusive phenomenon.

To solve equation (6.34), we proceed as in chapter 5 by Fourier transformingwith respect to z. This yields the ordinary differential equation

∂ΨS

∂t= −

(iqvg

(|κ+|2 − |κ−|2) + 4q2vg|κ+|2|κ−|2 Γbac

g2pNr

+ sin2 θΓbc

)ΨS (6.35)


which is easily solved, the solution being

ΨS(q, t) = ΨS(q, 0) exp(−iq

(|κ+|2 − |κ−|2)r(t))×

exp(−4q2|κ+|2|κ−|2 Γbac

g2pNr

r(t))×

exp(−Γbc

∫ t

0

sin2 θ(t′)dt′)

(6.36)

where we have introduced the distance

r(t) =∫ t

0

vg(t′)dt′. (6.37)

6.3.1 Initial conditionsWe shall consider a case in which a probe pulse, propagating under the influenceof a traveling wave coupling field, is stored in the medium and subsequentlyretrieved by a standing wave coupling field.

Assuming that the standing wave coupling field is switched on at t = 0, weneed to find the initial condition for the sum normal mode ΨS(z, 0). In chapter5 we solved the problem of a probe pulse propagating under the influence ofa time dependent traveling wave coupling field. Assuming that the DSP fieldjust prior to switching on the standing wave coupling field is Ψ(z, 0), we find,under the assumption of adiabatic conditions, the initial Raman coherence ofthe medium σbc(z, 0) from (5.36b) to be

√Nrσbc(z, 0) = −Ψ(z, 0). (6.38)

From the zeroth order expression (6.17) we have√

Nrσbc(z, 0) = − (κ+∗Ψ+(z, 0) + κ−∗Ψ−(z, 0)

)= −ΨS(z, 0). (6.39)

The initial condition for the sum normal mode is therefore

ΨS(z, 0) = Ψ(z, 0). (6.40)

6.3.2 Solution with negligible diffusionSince it is not possible to invert the Fourier transform in (6.36) analytically for ar-bitrary initial conditions, we shall first consider cases in which the broadeningof the pulse is negligible. This implies that the argument of the second expo-nential function on the rhs. of (6.36) must be small. In the case of a standingwave coupling field (|κ+| = |κ−| = 1√

2) and negligible probe field detuning

(δp ¿ γba), the condition is

q2 γbac

g2pNr

∫ Tf

0

vg(t′)dt′ ¿ 1 (6.41)


where Tf is the time at which the standing wave coupling field is switched offagain. The width of the pulse in Fourier space ∆q ' L−1

p and we can thereforeset q = L−1

p in (6.41) which yields

∫ Tf

0

vg(t′)dt′ ¿ L2p

la. (6.42)

When this condition is satisfied, we can approximate the second exponentialfunction on the rhs. of (6.36) by 1, which yields

ΨS(q, t) = ΨS(q, 0) exp(−iq

(|κ+|2 − |κ−|2)r(t)), (6.43)

where we have disregarded the trivial decay of the DSP field due to the dephas-ing rate Γbc. We can now easily invert the Fourier transform which gives thesolution

ΨS(z, t) = Ψ(z − (|κ+|2 − |κ−|2) r(t), 0

). (6.44)

The difference normal mode is given by (6.33), but when the condition (6.42) issatisfied, it can be approximated as

ΨD(z, t) ' 0. (6.45)

The components Ψ± can then be found from (6.28). We find

Ψ+(z, t) = κ+Ψ(z − (|κ+|2 − |κ−|2) r(t), 0

), (6.46a)

Ψ−(z, t) = κ−Ψ(z − (|κ+|2 − |κ−|2) r(t), 0

). (6.46b)

Finally, we can find the components of the probe field E±p from (6.19).

E+p (z, t) = κ+ cos θ(t)Ψ

(z − (|κ+|2 − |κ−|2) r(t), 0

)(6.47a)

E−p (z, t) = κ− cos θ(t)Ψ

(z − (|κ+|2 − |κ−|2) r(t), 0

)(6.47b)

The Raman coherence σbc can be calculated from the zeroth order expression(6.17) which gives

√Nrσbc(z, t) = − sin θ(t)Ψ

(z − (|κ+|2 − |κ−|2) r(t), 0

). (6.48)

As an example, we take the initial condition for the DSP field to be

Ψ(z, 0) = Ψ0 exp(−(z/Lp)2

). (6.49)

The polariton amplitude Ψ0 is related to the initial probe field amplitude E0 by

Ψ0 =E0

cos θ0, (6.50)


where θ0 is determined by the Rabi frequency of the traveling wave couplingfield prior to storage. The sum normal mode of the DSP field after switching onthe standing wave coupling field is then

ΨS(z, t) = Ψ0 exp(−

(z − (|κ+|2 − |κ−|2)r(t)

)2

/L2p

)(6.51)

and the components of the probe field are

E+p (z, t) = κ+ cos θ(t)

cos θ0E0 exp

(−

(z − (|κ+|2 − |κ−|2)r(t)

)2

/L2p

), (6.52a)

E−p (z, t) = κ−

cos θ(t)cos θ0

E0 exp(−

(z − (|κ+|2 − |κ−|2)r(t)

)2

/L2p

). (6.52b)

The energy density of the probe field Up is given by [13]

Up(z, t) =hωp

V〈E†

pEp〉 =hωp

V

(|E+p |2 + |E−

p |2). (6.53)

Inserting the solution for the probe field components we get

Up(z, t) =vg(t)vg,0

Up,0 exp(−2

(z − (|κ+|2 − |κ−|2)r(t)

)2

/L2p

), (6.54)

where Up,0 = hωp|E0|2/V .We take the time dependence of the angle θ(t) to be given by the relation

cos2 θ(t) = cos2 θ0 tanh(

t

Ts

)t ≥ 0, (6.55)

where Ts is the characteristic switching time.Figure 6.1 shows the retrieval of the stored probe pulse by a standing wave

coupling field (κ+ = κ− = 1√2). The energy density of the probe field, normal-

ized to the energy density of the probe field prior to storage, is plotted as a func-tion of space and time. For convenience we have taken the length of the storedprobe pulse to be Lp = vg,0Ts. We see that the stored probe pulse is revived intoa stationary standing wave probe field with the amplitude determined by thetotal Rabi frequency of the standing wave coupling field.

We shall also study the situation in which the stored probe pulse is retrievedby a quasi-standing wave coupling field. By a quasi-standing wave field, wemean a coupling field of the form (6.10) with Ω+

c 6= Ω−c . We can also write thisfield as

Ωc(z, t) =(Ω+

c − Ω−c)eikz + 2Ω−c cos(kz). (6.56)

It is clear from this expression that when Ω+c 6= Ω−c , the coupling field can be

regarded as a superposition of a standing wave and a traveling wave. If Ω+c '

Ω−c , the traveling wave component is quite small and we refer to the field as aquasi-standing wave field.


Figure 6.1: Retrieval of a stored probe pulse with a standing wave coupling fieldunder conditions of negligible diffusion. The probe field energy density Up, nor-malized to the energy density prior to storage, is plotted as a function of z and t,where z is in units of the characteristic probe pulse length Lp = vg,0Ts and t is inunits of the characteristic switching time Ts. The stored probe pulse is revived intoa stationary standing wave probe field.

Figure 6.2 shows the retrieval of a stored probe pulse with the same initialconditions as in figure 6.1 but with κ+ =

√2/3 and κ− =

√1/3. It is clear that

the revived probe pulse is no longer stationary, but propagates in the positive zdirection with a group velocity of 1

3vg .

6.3.3 Solution including diffusion

To illustrate the effect of the diffusion term in (6.34), we shall now solve thisequation in the specific case in which the initial condition for the sum normalmode is given by

ΨS(z, 0) = Ψ0 exp(− (z/Lp)

2)

. (6.57)

We also assume that the detunings δp,c = 0 and that the dephasing rate γbc = 0.The Fourier transform of the initial condition (6.57) is

ΨS(q, 0) =√

π

2LpΨ0 exp

(−L2pq

2/4)

(6.58)

which, upon insertion into (6.36), yields

ΨS(q, t) =√

π

2LpΨ0 exp

(− (L2

p/4 + 4|κ+|2|κ−|2lar(t))q2

)×exp

(−i(|κ+|2 − |κ−|2) r(t)q

).

(6.59)


Figure 6.2: Retrieval of a stored probe pulse with a quasi-standing wave couplingfield (κ+ =

√2/3, κ− =

√1/3) under conditions of negligible diffusion. The probe

field energy density Up, normalized to the energy density prior to storage, is plottedas a function of z and t, where z is in units of the characteristic probe pulse lengthLp = vg,0Ts and t is in units of the characteristic switching time Ts. The revivedprobe pulse propagates with group velocity 1

3vg .

Inverting the Fourier transform can be done analytically in this case and we findthe solution

ΨS(z, t) =Ψ0Lp√

L2p + 16|κ+|2|κ−|2lar(t)

exp

(−

(z − (|κ+|2 − |κ−|2) r(t)

)2

L2p + 16|κ+|2|κ−|2lar(t)

)

(6.60)The difference normal mode is easily calculated from (6.33) and we find

ΨD(z, t) =4κ+κ−laLpΨ0(

L2p + 16|κ+|2|κ−|2lar(t)

) 32

(z − (|κ+|2 − |κ−|2) r(t)

)×

exp

(−

(z − (|κ+|2 − |κ−|2) r(t)

)2

L2p + 16|κ+|2|κ−|2lar(t)

).

(6.61)

Given the solution for the normal modes, the components Ψ± can be calculatedfrom (6.28).

Figure 6.3 shows a plot of the solution for the sum and difference normalmodes, as well as for the components Ψ±, as a function of z and r(t), given theinitial condition (6.57). The polariton amplitudes are normalized to Ψ0, while zand r(t) are in units of the initial pulse length Lp. The ratio of the absorptionlength la to the initial pulse length Lp, which determines the rate of diffusion,has been assigned the value la/Lp = 0.1 which roughly corresponds to the pa-rameters in the experiment by Bajcsy et al. [11].


(a) ΨS (b) ΨD

(c) Ψ+ (d) Ψ−

Figure 6.3: Retrieval of a stored probe pulse by a standing wave coupling field in athermal gas medium. Figure (a) shows the diffusion of the sum normal mode due tothe coupling to the difference normal mode shown in figure (b). Figures (c) and (d)show the plus and minus components of the DSP field, respectively. Ψ+ is shiftedslightly in the positive z direction, while Ψ− is shifted in the negative z direction.All fields are normalized to the initial amplitude Ψ0, while z and r(t) are in units ofthe initial pulse length Lp. The parameter la/Lp = 0.1

The diffusive broadening of the DSP field is clearly evident, and we alsonote that the components Ψ± are shifted slightly from the center in the posi-tive/negative z direction.

The excitation density D, defined as the sum of the probe photon density andthe density of atoms in state |c〉, is given by

D =1V

(Nr|σbc|2 + |E+

p |2 + |E−p |2

). (6.62)

Using the definition of the DSP field (6.19) and the normal modes (6.27), it iseasy to see that the excitation density is given by

D =1V

(|ΨS |2 + cos2 θ|ΨD|2) ' 1

V|ΨS |2. (6.63)


In the example we are considering, the excitation density is

D =D0L

2p

L2p + 16|κ+|2|κ−|2lar(t)

exp

(−2

(z − (|κ+|2 − |κ−|2)r(t))2

L2p + 16|κ+|2|κ−|2lar(t)

)(6.64)

where D0 = |Ψ0|2/V . Figure 6.4 shows a plot of the excitation density (6.64) asa function of z and r(t).

Figure 6.4: Plot of the excitation density D, normalized to D0, as a function of z andr(t) in units of the initial pulse length Lp.

The total number of excitations N is found by integrating the excitation densityD over the entire interaction region V .

N (t) =∫

VD(z, t)dV. (6.65)

Since we have assumed cylindrical symmetry, the integral becomes

N (t) = A

∫ ∞

−∞D(z, t)dz =

√π/2AD0L

2p√

L2p + 16|κ+|2|κ−|2lar(t)

, (6.66)

where A is the area of the cross section of the interaction region.1

The relative rate of change of the number of excitations is

∂N∂t

N (t)= − 8|κ+|2|κ−|2lavg(t)

L2p + 16|κ+|2|κ−|2lar(t)

. (6.67)

From this expression it is clear that the number of excitations in the system de-creases non-exponentially.

1In most experiments, the area of the cross section is determined by the beam diameter of thelasers.

6.4 The physical origin of the diffusion 107

6.4 The physical origin of the diffusion

Bajcsy et al. [11] explain the stationary nature of the retrieved probe field as aBragg reflection phenomenon [34]. In the nodes of the standing wave couplingfield, EIT is absent and the medium is absorbing. This periodic modulationof the susceptibility creates a band gap in the dispersion relation between thefrequency ωp and the wave vector k of the probe field, preventing the revivedprobe field from propagating in the medium.

Here we shall present a different physical picture of the standing wave situa-tion in which the diffusive behavior of the revived probe field can be understoodas a random walk phenomenon.

When the standing wave coupling field is switched on, probe field photonsare emitted with equal probability in the forward and backward propagatingmodes. The photons travel an average distance la with velocity vg until theystimulate a Raman transition, transferring an atom from the ground state |b〉 tothe state |c〉. The absorbed photon is subsequently reemitted, but due to thepresence of both forward and backward propagating components of the cou-pling field, it can be reemitted into both the forward and the backward propa-gating modes of the probe field with equal probability. This random walk pro-cess leads to a stationary probe field envelope which undergoes diffusion with adiffusion coefficient equal to the product of the mean free path between Ramantransitions and the velocity [35]. In our case the diffusion coefficient is thereforeD = lavg, which is exactly what is predicted by the theory presented in thischapter.

To understand the physical cause of the loss of photons predicted by thetheory, we first note that the effect vanishes in the traveling wave case (κ− = 0).The effect must therefore somehow be connected to the spatially rapidly varyingcomponents of the Raman coherence, which are associated with Raman transi-tions involving counterpropagating components of the probe and coupling fields.

It was argued that for the vast majority of atoms in a thermal gas, theseRaman transitions are highly suppressed due to atomic motion, while Ramantransitions involving copropagating components of the probe and coupling fieldsare not, and are therefore much more probable. Nevertheless, we speculate thatit is the presence of Raman transitions involving counterpropagating fields thatis the cause of the loss of photons in the standing wave case, since the associatedcomponents of the Raman coherence have large decay rates.

This claim is further substantiated by the fact that the decay rate (6.67) isinversely proportional to the square of the pulse length. A shorter pulse hasa wider frequency spectrum, and consequently contain frequency componentswhich are farther from resonance. These components are more likely to induceRaman transitions involving counterpropagating fields than the componentsthat are closer to resonance with Raman transitions involving copropagatingfields, leading to a greater loss rate.

The dissipative losses associated with the diffusion constitute a problem for


the quantum information processing applications suggested in [32], since thestored information deteriorates as excitations are lost. From (6.67) we see that toreduce the rate of losses, a weaker coupling field can be employed, but this inturn leads to a weaker stationary probe field which reduces the strength of thenonlinear optical interactions.

C H A P T E R 7Standing wave polaritons

in ultra cold gasses

In the previous chapter we studied light storage with a standing wave couplingfield in a thermal atomic gas, and showed how the standing wave pattern ofthe coupling field creates a stationary probe field in the medium. We arguedthat the thermal motion of the atoms causes a rapid dephasing of the spatiallyrapidly varying components of the Raman coherence, allowing us to disregardthese in the calculations.

In this chapter we shall consider light storage with a standing wave cou-pling field in a medium consisting of stationary atoms, such as an ultra cold gasor a solid state crystal. In this case we can no longer disregard the spatiallyrapidly varying components of the Raman coherence, and taking these into ac-count complicates matters somewhat.

As we did in the thermal gas case, we shall derive wave equations for theDSP field, and show that a stationary probe pulse can also be obtained in the ul-tra cold gas case. In addition, we will find that there are a number of interestingdifferences between the thermal gas and the ultra cold gas cases. One impor-tant difference is the absence of the diffusive broadening of the revived probepulse evident in a thermal gas medium. This implies that media comprisedof stationary atoms are better suited for the type of nonlinear optical interac-tions suggested in [32], since the dissipative losses associated with the diffusivebroadening are not present in such media.

7.1 Heisenberg-Langevin equations

We consider an ensemble of N non-moving lambda atoms interacting with probeand coupling lasers propagating parallel to the z axis. The two lower states |b〉and |c〉 of the atoms are assumed to be nearly degenerate, such that the mag-nitude of the wave vectors of the probe and coupling lasers can be consideredidentical (kp ' kc = k).

109

110 Chapter 7. Standing wave polaritons in ultra cold gasses

The Hamiltonian for the N atom problem is

H = HF +N∑

j=1

(Hj

A + HjL + Hj

V

)(7.1)

where HF and HA describe the free electromagnetic field and atom, HL de-scribes the interaction of the atom with the probe and coupling fields, and HV


HF =∑m

hωma†mam (7.2a)

HjA = hωcbσ

jcc + hωabσ

jaa (7.2b)

HjL = −(

Ep + Ec

) · (dbaσjba + dcaσj

ca + h.a.)

(7.2c)

HjV = −EV ·

(dbaσj


)(7.2d)

As in chapter 6, we introduce slowly varying field operators for the electromag-netic field. Since we are allowing for standing wave fields, we write the fieldoperator as a superposition of two traveling wave fields propagating in oppo-site directions

Ep,c(z, t) =√

hωp,c

2ε0Vep,cEp,c(z, t)e−iωp,ct, (7.3)

where the operators Ep,c are given by

Ep,c(z, t) = E+p,c(z, t)eikz + E−

p,c(z, t)e−ikz. (7.4)

The field operators E±p,c are the slowly varying field operators for the forward

and backward propagating components of the probe and coupling fields, andep,c are the respective polarization vectors.

As always, we introduce continuum atomic operators by summing over theindividual atoms in a small volume V , but unlike the thermal gas case of chapter6 we do not decompose the atomic operators into spatial Fourier components.Instead we introduce slowly varying atomic operators σµν defined by

σbb = σbb (7.5a)σcc = σcc (7.5b)σaa = σaa (7.5c)

σba = σbae−iωpt (7.5d)

σca = σcae−iωct (7.5e)

σbc = σbce−i(ωp−ωc)t. (7.5f)

Notice that the operators defined by (7.5) are slowly varying in time, but not inspace and thus contain all the spatial Fourier components.


The Heisenberg-Langevin equations for these operators in the RWA are

σaa = −i(gpE

†pσba + Ω∗cσca − h.a.

)− γσaa + Faa (7.6a)

σbb = i(gpE

†pσba − h.a.

)+ γbσaa + Fbb (7.6b)

σcc = i(Ω∗cσca − h.a.

)+ γcσaa + Fcc (7.6c)

σba = i(gpEp(σbb − σaa) + Ωcσbc

)− Γbaσba + Fba (7.6d)

σca = i(Ωc(σcc − σaa) + gpEpσ

†bc

)− Γcaσca + Fca (7.6e)

σbc = i(Ω∗cσba − gpEpσ

†ca


where we have assumed that the coupling field can be treated as a classical fieldwith Rabi frequency Ωc given by (6.10). Again we shall disregard the noiseoperators Fµν since we will be considering the adiabatic limit.

7.1.1 The weak probe approximationAs in chapter 4, we assume that the probe field is weak and solve the Heisenberg-Langevin equations perturbatively. To first order in ε the relevant Heisenberg-Langevin equations are

σba =1

iΩ∗c

(Γbc +

∂

∂t

)σbc (7.7a)

σbc = −gpEp

Ωc− i

Ωc

(Γba +

∂

∂t

)σba (7.7b)

Combining equations (7.7) we can obtain a differential equation for σbc

σbc = −gpEp

Ωc− 1

Ωc

(Γba +

∂

∂t

)[1

Ω∗c

(Γbc +

∂

∂t

)σbc

](7.8)

7.1.2 The adiabatic approximationIn order to solve equation (7.8), we employ the same adiabatic approximationthat we used in chapter 4. Introducing the characteristic timescale T of theslowly varying operators, we expand σbc in powers of (γbaT )−1. To zeroth orderwe find

σbc = −gpEp

Ωc. (7.9)

Inserting this expression into (7.7a), we find an expression for σba valid in theadiabatic limit

σba = − 1iΩ∗c

(Γbc +

∂

∂t

)(gpEp

Ωc

). (7.10)

By inserting the field decomposition (7.4) into the adiabatic expression for σba

(7.10) we get

σba =−gp

(Γbc + ∂

∂t

)

iΩ(1 + 2|κ+||κ−| cos(2kz + φ))

(E+

p eikz + E−p e−ikz

Ω

), (7.11)


where we have also introduced the time dependent total Rabi frequency Ω, de-fined by (6.16), and the constant ratios κ±, defined by (6.22). The phase angle φis defined by the relation

κ+κ−∗ = |κ+||κ−|eiφ. (7.12)


As in chapter 6, we now introduce the DSP field defined by (6.19). By insertingthe definition of the DSP field into (7.11) we get

√Nrσba =

−(Γbc + ∂

∂t

)

iΩ(1 + 2|κ+||κ−| cos(2kz + φ)

)(sin θ

(Ψ+eikz + Ψ−e−ikz

)). (7.13)

To derive wave equations for the components of the DSP field, we need to knowσ±ba. These two components can be found by writing (7.13) as a Fourier series.In particular, we have to find the Fourier series

11 + y cos x

=a0

2+

∞∑n=1

an cos(nx), (7.14)

where y = 2|κ+||κ−| and x = 2kz + φ. Note that y ≤ 1 which guarantees theexistence of the Fourier series except in the case of a standing wave couplingfield (y = 1). Fortunately, we can treat this case successfully by considering thelimit y → 1 at the end of our calculation.

Inserting the Fourier series into (7.13) we find

√Nrσba = − 1

iΩ

(a0

2+

∞∑n=1

an

2

(ein(2kz+φ) + e−in(2kz+φ)

))×

(Γbc +

∂

∂t

) (sin θ


)).

(7.15)

From (7.15) we see that σba can we written as

σba =∞∑

n=−∞σ

(2n+1)ba ei(2n+1)kz, (7.16)

and that the Fourier components σ±ba are given by

√Nrσ

+ba =

−12iΩ

(Γbc +

∂

∂t

) (a0 sin θΨ+ + a1e

iφ sin θΨ−)

(7.17a)

√Nrσ

−ba =

−12iΩ

(Γbc +

∂

∂t

) (a0 sin θΨ− + a1e

−iφ sin θΨ+). (7.17b)


We see that we only need to calculate the first two Fourier coefficients a0 and a1.These are given by

a0 =1π

∫ π

−π

dx

1 + y cosx=

2√1− y2

(7.18a)

a1 =1π

∫ π

−π

cosxdx

1 + y cosx= 2

√1− y2 − 1

y√

1− y2. (7.18b)

7.2.1 Wave equations for the polariton field

Inserting the adiabatic expression (7.17) for σ±ba into the wave equations (6.24)for the DSP field, we get the coupled wave equations

∂Ψ+

∂t+ c cos2 θ′

∂Ψ+

∂z= − sin2 θ′

[ΓbcΨ+ + seiφ

(Γbc +

∂

∂t

)Ψ−

− sθcos θ

sin θ

(yΨ+ − eiφΨ−

)] (7.19a)

∂Ψ−

∂t− c cos2 θ′

∂Ψ−

∂z= − sin2 θ′

[ΓbcΨ− + se−iφ

(Γbc +

∂

∂t

)Ψ+

− sθcos θ

sin θ

(yΨ− − e−iφΨ+

)] (7.19b)

where we have introduced a new angle θ′ defined by

tan θ′ =√

a0

2gp

√Nr

Ω=

√a0

2tan θ, (7.20)

as well as the constant

s =a1

a0=

√1− y2 − 1

y. (7.21)

Since we are considering the adiabatic limit in which the coupling field Rabifrequency changes slowly in time, we shall neglect the last term on the rhs. of(7.19) in the following.

7.2.2 Low group velocity limit

In the low group velocity limit cos2 θ ¿ 1 the wave equations take the simplerform

(Γbc +

∂

∂t

)Ψ+ + c cos2 θ′

∂Ψ+

∂z= −seiφ

(Γbc +

∂

∂t

)Ψ− (7.22a)

(Γbc +

∂

∂t

)Ψ− − c cos2 θ′

∂Ψ−

∂z= −se−iφ

(Γbc +

∂

∂t

)Ψ+. (7.22b)


By inserting (7.22a) into (7.22b) and vice versa, and inserting the Fourier coeffi-cients (7.18), we obtain a more convenient form of the coupled wave equations.For |κ+| ≥ |κ−| we have

(Γbc +

∂

∂t

)Ψ+ + |κ+|2vg

∂Ψ+

∂z= κ+κ−∗vg

∂Ψ−

∂z, (7.23a)

(Γbc +

∂

∂t

)Ψ− − |κ+|2vg

∂Ψ−

∂z= −κ+∗κ−vg

∂Ψ+

∂z. (7.23b)

In the case where |κ+| ≤ |κ−| the wave equations are(

Γbc +∂

∂t

)Ψ+ + |κ−|2vg

∂Ψ+

∂z= κ+κ−∗vg

∂Ψ−

∂z, (7.24a)

(Γbc +

∂

∂t

)Ψ− − |κ−|2vg

∂Ψ−


∂Ψ+

∂z. (7.24b)

We have introduced the group velocity vg = c cos2 θ in equations (7.23) and(7.24), and have also made use of the fact that in the low group velocity limit,cos2 θ′ '

√1− y2 cos2 θ.

7.3 Solving the coupled wave equations

We now turn to solving the coupled wave equations for the DSP field. We as-sume that |κ+| ≥ |κ−|. The other case can be solved in a completely analogousway. To solve (7.23) we perform a Fourier transform with respect to z whichyields

(Γbc +

∂

∂t

)Ψ+ = iqvg

(κ+κ−∗Ψ− − |κ+|2Ψ+

)(7.25a)

(Γbc +

∂

∂t

)Ψ− = iqvg

(|κ+|2Ψ− − κ+∗κ−Ψ+

)(7.25b)

where Ψ±(q, t) is the Fourier transform of Ψ±(z, t) with respect to z. On matrixform these equations can be written

(Γbc +

∂

∂t

)Ψ(q, t) = iqvg(t)AΨ(q, t), (7.26)

where

Ψ =[Ψ+

Ψ−

]and A =

[ −|κ+|2 κ+κ−∗

−κ+∗κ− |κ+|2]

. (7.27)

To solve the coupled differential equations, we diagonalize the matrix A suchthat A = CDC−1. According to elementary linear algebra [36], the diagonalmatrix D consists of the eigenvalues of A, and the columns of C are the corre-sponding eigenvectors.


The eigenvalues λ of A turn out to be

λ = ±β where β =√|κ+|2(|κ+|2 − |κ−|2). (7.28)

The diagonal matrix D is thus

D =[β 00 −β

](7.29)

and the corresponding matrix C is

C =1

|κ+|2 + β

[κ+κ−∗ |κ+|2 + β|κ+|2 + β κ+∗κ−

](7.30)

which has the inverse

C−1 =12β

[−κ+∗κ− |κ+|2 + β|κ+|2 + β −κ+κ−∗

]. (7.31)

The differential equation (7.26) can now be written as(

Γbc +∂

∂t

)Ψ(q, t) = iqvg(t)CDC−1Ψ(q, t). (7.32)

By introducing a new column vector χ defined by

χ =[χ1

χ2

]= C−1Ψ (7.33)

and multiplying equation (7.32) by C−1 from the left, we obtain the wave equa-tions for the new field χ

(Γbc +

∂

∂t

)χ(q, t) = iqvg(t)Dχ(q, t). (7.34)

Since D is diagonal, the differential equations for the two components χ1 and χ2

are decoupled and thus readily solvable. The solution is

χ(q, t) =

χ1(q, 0) exp

(iqβ

∫ t

0vg(t′)dt′

)

χ2(q, 0) exp(−iqβ

∫ t

0vg(t′)dt′

) e−Γbct. (7.35)

Inverting the Fourier transform yields

χ(z, t) =[χ1(z + βr(t), 0)χ2(z − βr(t), 0)

]e−Γbct (7.36)

where r(t) is the distance defined by (6.37).


Using (7.33) we finally arrive at the solution

Ψ+(z, t) =e−Γbct

2β

[(β + |κ+|2)Ψ+(z − βr(t), 0) + (β − |κ+|2)Ψ+(z + βr(t), 0)

+ κ+κ−∗(Ψ−(z + βr(t), 0)−Ψ−(z − βr(t), 0)

)]

(7.37a)

Ψ−(z, t) =e−Γbct

2β

[(β + |κ+|2)Ψ−(z + βr(t), 0) + (β − |κ+|2)Ψ−(z − βr(t), 0)

+ κ+∗κ−(Ψ+(z − βr(t), 0)−Ψ+(z + βr(t), 0)

)].

(7.37b)

7.3.1 Initial conditionsAs in the thermal gas case treated in chapter 6, we shall consider the case inwhich a probe pulse, propagating under the influence of a traveling wave cou-pling field, is stored in the medium and subsequently retrieved by a standingwave coupling field.

Assuming that the standing wave coupling field is switched on at t = 0,we need to find the initial conditions for the two components of the DSP fieldΨ±(z, 0). As discussed in chapter 6, the initial condition for the Raman coher-ence is given by √

Nrσbc(z, 0) = −Ψ(z, 0), (7.38)

where Ψ(z, 0) is the DSP field just prior to switching on the standing wave cou-pling field. Using (7.9) and the definition of the DSP field in the standing wavecase (6.19), along with the initial condition (7.38), we get

Ψ(z, 0)(κ+eikz + κ−e−ikz

)= Ψ+(z, 0)eikz + Ψ−(z, 0)e−ikz. (7.39)

From this expression we see that the initial conditions for the components of theDSP field Ψ±(z, 0) are

Ψ+(z, 0) = κ+Ψ(z, 0), Ψ−(z, 0) = κ−Ψ(z, 0). (7.40)

Inserting the initial conditions (7.40) into (7.37), we find the solution

Ψ+(z, t) =κ+

2

[(1 +

√|κ+|2 − |κ−|2

|κ+|

)Ψ(z − βr(t), 0)

+

(1−

√|κ+|2 − |κ−|2

|κ+|

)Ψ(z + βr(t), 0)

]e−Γbct

(7.41a)

Ψ−(z, t) =κ−

2

[Ψ(z − βr(t), 0) + Ψ(z + βr(t), 0)

]e−Γbct (7.41b)


7.3.2 Probe retrieval by a standing wave coupling field

To determine the solution for a standing wave coupling field, we let κ± → 1√2

in the solution (7.41). In this limit, β → 0 and we find the solution

Ψ+(z, t) =1√2Ψ(z, 0)e−Γbct, (7.42a)

Ψ−(z, t) =1√2Ψ(z, 0)e−Γbct. (7.42b)

Interestingly, no broadening of the pulse envelope is present in this solution,and the only decay is due to the dephasing rate γbc.

As an example, we take the initial condition for the DSP field to be

Ψ(z, 0) = Ψ0 exp(−(z/Lp)2

), (7.43)

where Lp is the characteristic length of the stored probe pulse. The polaritonamplitude Ψ0 is related to the initial probe field amplitude E0 by

Ψ0 =E0

cos θ0, (7.44)

where θ0 is determined by the Rabi frequency of the traveling wave couplingfield prior to storage.

The components of the retrieved probe field found from (6.19) are

E+p (z, t) =

1√2

cos θ(t)cos θ0

E0 exp(−(z/Lp)2

)e−Γbct, (7.45a)

E−p (z, t) =

1√2

cos θ(t)cos θ0

E0 exp(−(z/Lp)2

)e−Γbct. (7.45b)

The energy density of the retrieved probe field is

Up(z, t) =cos2 θ(t)cos2 θ0

Up,0 exp(−2(z/Lp)2

)e−2γbct, (7.46)

where Up,0 = hωp|E0|2/V .The time dependence of the angle θ is assumed to be given by

cos2 θ(t) = cos2 θ0 tanh(

t

Ts

)t ≥ 0, (7.47)

where Ts is the characteristic switching time.Figure 7.1 shows a plot of the energy density of the probe field as a function

of space and time, normalized to the energy density of the probe field prior tostorage. For simplicity, we have assumed zero dephasing (Γbc = 0) and takenthe characteristic length of the stored probe pulse to be Lp = vg,0Ts, wherevg,0 = c cos2 θ0 is the group velocity prior to storage.

As in the thermal gas case we see that the stored probe pulse is revived intoa stationary probe field, but we note that in the ultra cold gas case consideredhere, no broadening of the pulse is present.


Figure 7.1: Retrieval of a stored probe pulse with a standing wave coupling field.The probe field energy density, normalized to the energy density prior to storage, isplotted as a function of z and t, where z is in units of the characteristic probe pulselength Lp = vg,0Ts and t is in units of the characteristic switching time Ts. Thestored probe pulse is revived into a stationary probe field.

7.3.3 Probe retrieval by a quasi-standing wave coupling field

We shall now study the situation in which the probe field is retrieved by a quasi-standing wave coupling field.

In the previous chapter, we saw that in the thermal gas case a quasi-standingwave coupling field leads to a drift of the revived probe pulse in the direction ofthe stronger of the two coupling field components.

In the ultra cold gas case considered here, we shall find that the revivedprobe pulse instead splits into two parts. A stronger part which propagates inthe direction of the stronger of the coupling field components, and a weakerpart which propagates in the opposite direction.

Figure 7.2 shows the solution (7.41) with the same initial conditions as in fig-ure 7.1, but with κ+ =

√2/3 and κ− =

√1/3. The splitting of the revived probe

pulse is clearly evident in this case, indicating a qualitative difference betweenthe thermal gas and the ultra cold gas cases. The cause of this difference is thecoupling to the high spatial-frequency components of the Raman coherence σbc

in the ultra cold gas case.

7.3.4 Calculation of the Raman coherence

To calculate the Raman coherence of the atoms, we use the zeroth order expres-sion (7.9) for σbc

√Nrσbc = −gp

√NrEp

Ωc. (7.48)


(a) Ψ+(z, t) (b) Ψ−(z, t)

(c) |Ψ+|2 + |Ψ−|2 (d) |E+p |2 + |E−p |2

Figure 7.2: Retrieval of a stored probe pulse with a quasi-standing wave couplingfield (κ+ =

√2/3, κ− =

√1/3). Figures (a) and (b) show the polariton amplitudes

Ψ± normalized to Ψ0. Figure (c) shows the polariton density normalized to |Ψ0|2/Vand figure (d) shows the normalized probe field energy density. Time t is in units ofthe characteristic switching time Ts, position z is in units of the characteristic pulselength Lp = vg,0Ts. The revived probe field splits into two parts propagating in op-posite directions, the part propagating in the positive z direction being significantlystronger than the other.

By inserting the decompositions of the probe and coupling fields, as well as thedefinition (6.19) of the DSP field, we get

√Nrσbc(z, t) = − sin θ(t)

Ψ+(z, t)eikz + Ψ−(z, t)e−ikz

κ+eikz + κ−e−ikz. (7.49)

In the standing wave case, we insert the solution (7.42) into this expression tofind the solution for the Raman coherence

√Nrσbc(z, t) = − sin θ(t)Ψ(z, 0)e−Γbct. (7.50)

Interestingly, we see from this solution that only the spatially slowly varyingcomponent of the Raman coherence σ

(0)bc is present in the standing wave case,

despite the fact that the coupling to the rapidly varying components σ(n)bc is in-

cluded in the Heisenberg-Langevin equations.


In the more general case of a quasi-standing wave coupling field, we insertthe solution for the components of the DSP field (7.41) into (7.49). After a bit ofalgebra we get

√Nrσbc(z, t) = − sin θ(t)

(f(z, t) +

g(z, t)1 + κ−

κ+ e−2ikz

)e−Γbct, (7.51)

wheref(z, t) =

12(Ψ(z − βr(t), 0) + Ψ(z + βr(t), 0)

)(7.52)

and

g(z, t) =

√|κ+|2 − |κ−|2

2|κ+|(Ψ(z − βr(t), 0)−Ψ(z + βr(t), 0)

). (7.53)

Inserting the binomial series1

(1 +

κ−

κ+e−2ikz

)−1

=∞∑

n=0

(−κ−

κ+

)n

e−2inkz (7.54)

into (7.51), we get

√Nrσbc(z, t) = −1

2sin θ(t)

[Ψ(z − βr(t), 0) + Ψ(z + βr(t), 0)

+

√|κ+|2 − |κ−|2

|κ+|(Ψ(z − βr(t), 0)−Ψ(z + βr(t), 0)

)×∞∑

n=0

(−κ−

κ+

)n

e−2inkz

]e−Γbct.

(7.55)

From this expression we can find the Fourier components of the Raman coher-ence. The slowly varying component is

√Nrσ

(0)bc (z, t) = −1

2sin θ(t)

[(1 +

√|κ+|2 − |κ−|2

|κ+|)

Ψ(z − βr(t), 0)

+(

1−√|κ+|2 − |κ−|2

|κ+|)

Ψ(z + βr(t), 0)]e−Γbct.

(7.56)

For the rapidly varying components of the Raman coherence we find

√Nrσ

(−2n)bc (z, t) = −1

2sin θ(t)

√|κ+|2 − |κ−|2

|κ+|(Ψ(z − βr(t), 0)

−Ψ(z + βr(t), 0))(−κ−

κ+

)n

e−Γbct

(7.57)

1The convergence of the binomial series is assured since |κ−| < |κ+|.


and √Nrσ

(2n)bc (z, t) = 0 (7.58)

where, in both cases, n > 0.We see that in the quasi-standing wave case, the rapidly varying components

of the Raman coherence σ(2n)bc with negative values of n attain a small but non-

vanishing value, becoming progressively smaller with decreasing n. The rapidlyvarying components of the Raman coherence with positive values of n all vanish.This asymmetry in the Raman coherence is to be expected, since neither thecoupling field nor the revived probe field is symmetric in z.

7.4 Corrections to the adiabatic solution

When we considered traveling wave EIT and light storage in chapters 4 and 5,we found that the finite length of the probe pulse leads to broadening of thepulse envelope due to dispersion. In this section we shall investigate the sameeffect in the standing wave case by going to higher order in the adiabatic expan-sion of the Raman coherence. Interestingly, we shall see that this effect vanishesin the case of a pure standing wave, while it is still present in the quasi-standingwave case.

7.4.1 Non-adiabatic corrections

Our starting point is the differential equation (7.8) for the Raman coherence σbc.To first order in (γbaT )−1 we find

σbc = −gpEp

Ωc+

Γba

|Ωc|2∂

∂t

(gpEp

Ωc

), (7.59)

where we have assumed Γbc = 0 to simplify the calculations. Inserting thisexpression into (7.7a) and introducing the DSP fields defined in (6.19), we get

√Nrσba =

− sin θ

iΩ(1 + 2|κ+||κ−| cos(2kz + φ)

) ∂

∂t


)

+Γba

g2pNr

sin θ tan2 θ

iΩ(1 + 2|κ+||κ−| cos(2kz + φ)

)2

∂2

∂t2(Ψ+eikz + Ψ−e−ikz

)

(7.60)

where we have assumed that the coupling laser Rabi frequency changes slowlyenough to set θ = 0 in the equations.

As in section 7.2 we need to find the Fourier components σ±ba. To do this we


introduce the Fourier series

11 + y cos x

=a0

2+

∞∑n=1

an cos(nx) (7.61)

1(1 + y cos x)2

=d0

2+

∞∑n=1

dn cos(nx) (7.62)

where, as before, y = 2|κ+||κ−| and x = 2kz + φ. Inserting the Fourier seriesinto (7.60), we get

√Nrσ

+ba =

sin θ

2iΩ

(− ∂

∂t

(a0Ψ+ + a1e

iφΨ−)

+Γba

g2pNr

tan2 θ∂2

∂t2(d0Ψ+ + d1e

iφΨ−)) (7.63a)

√Nrσ

−ba =

sin θ

2iΩ

(− ∂

∂t

(a0Ψ− + a1e

−iφΨ+)

+Γba

g2pNr

tan2 θ∂2

∂t2(d0Ψ− + d1e

−iφΨ+)) (7.63b)

The Fourier coefficients a0,1 have already been calculated and are given by (7.18),while the Fourier coefficients d0,1 are given by

d0 =1π

∫ π

−π

dx

(1 + y cosx)2=

2(1− y2)3/2

, (7.64a)

d1 =1π

∫ π

−π

cos xdx

(1 + y cosx)2= − 2y

(1− y2)3/2. (7.64b)

We now insert the expressions (7.63) into (6.24) to obtain a set of coupled waveequations for the DSP fields

∂Ψ+

∂t+ c cos2 θ′

∂Ψ+

∂z= − sin2 θ′

[seiφ ∂Ψ−

∂t

− Γba

g2pNr

tan2 θ∂2

∂t2(s′Ψ+ + s′′eiφΨ−

)] (7.65a)

∂Ψ−


∂Ψ−

∂z= − sin2 θ′

[se−iφ ∂Ψ+

∂t

− Γba

g2pNr

tan2 θ∂2

∂t2(s′Ψ− + s′′e−iφΨ+

)] (7.65b)

where we have introduced the constants

s =a1

a0s′ =

d0

a0s′′ =

d1

a0. (7.66)


Once again we consider the low group velocity limit cos2 θ ¿ 1. With thisapproximation the wave equations simplify to

∂Ψ+

∂t+ c cos2 θ′

∂Ψ+

∂z= −seiφ ∂Ψ−

∂t+

Γba

g2pNr

tan2 θ∂2

∂t2(s′Ψ+ + s′′eiφΨ−

)

(7.67a)

∂Ψ−


∂Ψ−

∂z= −se−iφ ∂Ψ+

∂t+

Γba

g2pNr

tan2 θ∂2

∂t2(s′Ψ− + s′′e−iφΨ+

)

(7.67b)

To the same order of approximation, we can replace the second time derivativesof the DSP fields with the second time derivative of the zeroth order solution.Differentiating both sides of (7.67) with respect to t, and discarding derivativesof order greater than two, we get

∂2Ψ+

∂t2+ c cos2 θ′

∂

∂z

∂Ψ+

∂t= −seiφ ∂2Ψ−

∂t2(7.68a)

∂2Ψ−

∂t2− c cos2 θ′

∂

∂z

∂Ψ−

∂t= −se−iφ ∂2Ψ+

∂t2, (7.68b)

where we once again assume that the coupling laser Rabi frequency changesslowly. Using (7.67) we can solve for the second time derivatives of the DSPfield. We find

∂2Ψ±

∂t2=

1− y2

1− s2

(c cos2 θ

)2 ∂2Ψ±

∂z2, (7.69)

where we exploited the fact that in the low group velocity limit,

cos2 θ′ '√

1− y2 cos2 θ. (7.70)

Inserting (7.69) into (7.67), and assuming that |κ+| ≥ |κ−|, the coupled waveequations take the form

∂Ψ+

∂t+ |κ+|2vg

∂Ψ+

∂z= κ+κ−∗vg

∂Ψ−

∂z

+|κ+|2lavg√

1− y2

∂2

∂z2

(|κ+|2Ψ+ − κ+κ−∗Ψ−) (7.71a)

∂Ψ−

∂t− |κ+|2vg

∂Ψ−


∂Ψ+

∂z

+|κ+|2lavg√

1− y2

∂2

∂z2

(|κ+|2Ψ− − κ+∗κ−Ψ+) (7.71b)

To solve the coupled wave equations (7.71) we proceed as in section 7.3 byFourier transforming with respect to z. Written on matrix form, the Fouriertransformed wave equations become

∂Ψ∂t

= iqvg(t)AΨ(q, t). (7.72)


The matrix A is

A =[−a∗ b−b∗ a

](7.73)

wherea = |κ+|2(1 + iqξ), b = κ+κ−∗(1− iqξ) (7.74)

and

ξ =|κ+|2la√1− y2

. (7.75)

We now diagonalize A. The eigenvalues of A turn out to be

λ± = i|κ+|2ξq ± d (7.76)

whered =

√|κ+|2 (|κ+|2 − |κ−|2)− |κ+|2|κ−|2ξ2q2. (7.77)

The diagonal matrix D is thus

D =[λ+ 00 λ−

](7.78)

and the corresponding C matrix is

C =1

|κ+|2 + d

[b |κ+|2 + d

|κ+|2 + d b∗

](7.79)

which has the inverse

C−1 =12d

[ −b∗ |κ+|2 + d|κ+|2 + d −b

]. (7.80)

As before, we now introduce the field χ defined by

χ = C−1Ψ, (7.81)

which obeys the differential equation

∂χ

∂t= iqvg(t)Dχ. (7.82)

The solution to this equation is easily found to be

χ(q, t) =[χ1(q, 0) exp

(iqλ+r(t)

)χ2(q, 0) exp

(iqλ−r(t)

)]

, (7.83)

7.5 Comparison with the thermal gas case 125

Using the definition of the field χ, we find the solution for the DSP field compo-nents Ψ± to be

Ψ+(q, t) =12d

((bΨ−(q, 0)− (|κ+|2 − d)Ψ+(q, 0)

)exp

(iqλ+r(t)

)

+((|κ+|2 + d)Ψ+(q, 0)− bΨ−(q, 0)

)exp

(iqλ−r(t)

)) (7.84a)

Ψ−(q, t) =12d

((−b∗Ψ+(q, 0) + (|κ+|2 + d)Ψ−(q, 0))exp

(iqλ+r(t)

)

+(b∗Ψ+(q, 0)− (|κ+|2 − d)Ψ−(q, 0)

)exp

(iqλ−r(t)

)) (7.84b)

Inserting the initial conditions (7.40) for the DSP field and considering the limitκ± → 1√

2, corresponding to a pure standing wave coupling field, the solution

becomes

Ψ+(z, t) =1√2Ψ(z, 0), (7.85a)

Ψ−(z, t) =1√2Ψ(z, 0). (7.85b)

From this solution it is clear that the broadening of the pulse envelope due todispersion is absent in the case of a pure standing wave coupling field. The effectis present in the case of a quasi-standing wave coupling field. If we consider thelimiting case of a traveling wave coupling field (κ− → 0), we find the samedispersion term in the wave equation (7.71) that we found in section 5.3.4.

7.5 Comparison with the thermal gas case

Having developed the theory for light storage and retrieval with a standingwave coupling field in ultra cold gasses, we can now compare the results to thethermal gas case considered in chapter 6. In both cases, the retrieval of a storedprobe pulse with a standing wave coupling field results in a stationary probefield. However, there are a number of interesting differences between the twocases which we shall discuss here.

First of all, we saw in chapter 6 that the revived probe field undergoes adiffusive broadening in the thermal gas case. Associated with this broadeningis a loss of excitations from the system. As we have shown in this chapter, thiseffect is absent in the ultra cold gas case. This is a significant advantage overthe thermal gas case for quantum information processing purposes, since lossof excitations leads to deterioration of the stored information. An ultra cold gasor a solid state medium is thus ideally suited for the kind of nonlinear opticalinteractions suggested in e.g. [32].

Secondly, the qualitative behavior of a probe field retrieved by a quasi-stand-ing wave coupling field is different in the two cases. In the thermal gas case, the


probe pulse drifts in the propagation direction of the stronger of the two com-ponents of the coupling field, while in the ultra cold gas case, the revived probepulse splits into two components: a large component traveling in the directionof propagation of the stronger of the two components of the coupling field, anda smaller component traveling in the opposite direction.

From a mathematical point of view, the obvious difference between the ther-mal gas and the ultra cold gas cases is the absence of the coupling to the spatiallyrapidly varying components of the Raman coherence in the former case.

Attempting to give a qualitative physical picture of the ultra cold gas case,we first note that the random walk picture given for the thermal gas case is in-valid in the ultra cold gas case, seeing that the diffusion predicted by that modelis absent. Instead, the medium behaves as a perfect Bragg reflector in whichthe probe field cannot propagate at all. Presumably, the dispersion relation ofthe medium should have a band gap large enough to contain all the frequencycomponents of the probe pulse to avoid any diffusion.

The treatment of the ultra cold gas case in terms of a periodic medium willnot be discussed any further in this thesis, since such a treatment is non-trivialdue to the rapid frequency dependence of the susceptibility near two-photonresonance. Furthermore, the theory presented in this chapter, based on a per-turbative solution of the coupled Maxwell/Heisenberg-Langevin equations, isexpected to give a more accurate description. This is because a description basedon the susceptibility assumes that the atomic medium is in a steady state.

As of this writing, no experiments have been performed on standing wavelight storage and retrieval in an ultra cold gas or solid state medium. Hopefully,such an experiment will be performed in the future to test the predictions of thetheory presented here.

C H A P T E R 8Summary and outlook

Several aspects of EIT and light storage have been discussed in this thesis. Herewe present a summary of the results and discuss possible avenues for futureresearch.

8.1 Summary of the thesis

In chapter 4 we calculated non-adiabatic corrections to the adiabatic solutionfor a weak probe field propagating in an EIT medium, improving the accuracyof the analytical solution for realistic experimental parameters. We found thatthe finite width of the probe pulse leads to dispersive broadening, and also aslight asymmetry, of the pulse envelope. We also presented a novel approachto the phenomenon of adiabatons, which has the advantage of taking the ef-fect of spontaneous emission into account in a non-phenomenological way, anddescribed the physical mechanism behind this effect. The results of these ana-lytical calculations were found to be in good agreement with an exact numericalsolution of the coupled Maxwell/Heisenberg-Langevin equations. The numer-ical simulations were also used to investigate EIT for very short probe pulseswith intensities comparable to the coupling field intensity, and we found thatthe analytical solution still provides a reasonably good description of the phe-nomenon in this case. Finally, we gave an analytical treatment of the effect ofatomic motion on EIT, valid for realistic experimental parameters. It was foundthat the effect depends critically on the difference of the wave vectors of theprobe and coupling fields, suggesting that the inclusion of the effects of atomicmotion is essential when considering standing wave EIT and light storage in athermal gas medium.

Chapter 5 gave an introduction to the dark state polariton theory of Lukinand Fleischhauer [22] for light storage, along with a thorough discussion of thevalidity of the approximations used. The results of exact numerical simulationsinvestigating the validity and limits of the theory were also presented, demon-strating the amazing robustness of the light storage technique.

127

128 Chapter 8. Summary and outlook

In chapter 6, we improved the theory of Zimmer et al. [33] for light stor-age and retrieval using a standing wave coupling field in a thermal atomic gasmedium. By introducing a DSP-like field, we obtained a theory capable of treat-ing time dependent coupling fields. It was found that a stored probe pulse re-trieved by a standing wave coupling field generates a stationary probe pulsewhich undergoes diffusive broadening. We also discussed the physical mecha-nism behind the diffusion of the revived probe field, describing it as a randomwalk phenomenon and attributing the associated loss of excitations to the rapiddephasing of the spatially rapidly varying components of the Raman coherence.

In chapter 7, we presented a novel theory for light storage and retrieval us-ing a standing wave coupling field in EIT media consisting of stationary atoms,such as an ultra cold atomic gas or a solid state medium. We showed that thebroadening of the retrieved stationary probe pulse, evident in the thermal gascase, is absent in such a medium. We also calculated non-adiabatic corrections tothe solution, showing that the broadening of the probe pulse due to dispersionis suppressed when using a pure standing wave coupling field. We concludedthat such media would be ideally suited for quantum information processingapplications based on nonlinear optical interactions, since a tightly confined,stationary probe field can be maintained for long periods of time without losses.Finally, we found that in the case of a quasi-standing wave coupling field, a split-ting of the revived probe pulse occurs, an effect which is not seen in the thermalgas case.

8.2 Outlook

As with all good research, the work presented in this thesis not only providesanswers, but also raises a number of interesting new questions. Here we presentpossible applications of the work presented in this thesis and discuss variouspossibilities for future research.

As mentioned in chapter 7, no experiments involving standing wave EITand light storage in media composed of stationary atoms have been performed.Such experiments certainly seem feasible given the fact that several experimentsusing traveling wave lasers in ultra cold gases [6, 7] and solid state media [9, 10]have already been performed. In the experiment by Longdell et al. [10], coun-terpropagating probe and coupling lasers were used, and expanding the exper-iment to standing wave fields certainly seems possible. As already mentioned,such experiments would be well suited for demonstrating efficient nonlinear in-teractions and quantum information processing. A solid theoretical treatment ofsuch experiments would be valuable, and is provided by the theory presentedin chapter 7 of this thesis.

As for future theoretical work, a numerical solution of the coupled Maxwell/Heisenberg-Langevin equations in the case of standing wave fields could bevery interesting, not only for testing the accuracy of the theory presented inchapter 7, but also for investigating cases that cannot be handled by the theory,

8.2 Outlook 129

such as situations in which the Rabi frequencies of the forward and backwardpropagating components of the coupling field change in time independently ofeach other. Such a numerical simulation is far from trivial, since the presence ofstanding wave fields require a spatial grid spacing smaller than the wavelengthof the fields involved. Including atomic motion into such a numerical simu-lation could also be very interesting, testing the validity of the approximationemployed in chapter 6 of neglecting high spatial frequency components of theRaman coherence, but would of course make the numerical simulation an evenmore demanding task.

Undoubtedly, there are many other avenues of research not mentioned herethat could provide new insights and possibilities. Hopefully, the work pre-sented in this thesis can be useful on the road ahead.

A P P E N D I X AOperator relations in the

Heisenberg picture

Let X be any operator. In the Heisenberg picture the time evolution of thisoperator is given by the unitary time evolution operator U(t). If the operator isknown at time t = 0, the operator at any other time t is given by

X(t) = U†(t)X(t = 0)U(t). (A.1)

Now assume that at t = 0 the following operator relation holds,

C(0) = A(0)B(0). (A.2)

It follows from (A.1) that at time t we have

C(t) = U†(t)A(0)U(t)U†(t)B(0)U(t) = A(t)B(t), (A.3)

where we have used the fact that U U† = 1. We have thus shown that if theoperator relation (A.2) holds at time t = 0, it holds for all times.

131

A P P E N D I X BDerivation of the

non-adiabatic corrections

In this appendix additional details of the derivation of the non-adiabatic correc-tions presented in section 5.3.4 are given. Inserting the expression for the BSPfield to second order (5.64) into the wave equation (5.37) for the DSP field andassuming Γbc = 0, we get

∂Ψ∂t


= −A(t)Ψ + B(t)c∂Ψ∂z

+ C(t)c2 ∂2Ψ∂z2

−D(t)c3 ∂3Ψ∂t3

. (B.1)

The A coefficient is given by

A(t) =1

g2pNr

(θ sin θ

∂

∂t(θ sin θ) + Γba(θ sin θ)2

)

=1

g2pNr

(12

∂

∂t(θ sin θ)2 + Γba(θ sin θ)2

)

=(

Γba +12

∂

∂t

)θ2 sin2 θ

g2pNr

.

(B.2)

The B coefficient, which is the one that differs from the result given in [22], is

B(t) =1

g2pNr

(θ2 sin2 θ cos2 θ + θ sin θ

∂

∂t(sin2 θ cos θ)− sin2 θ cos θ

∂

∂t(θ sin θ)

)

=sin θ

g2pNr

(θ

∂

∂t(sin2 θ cos θ)− θ sin2 θ cos θ

).

(B.3)

Using the fact that

13

∂2

∂t2sin3 θ = θ sin2 θ cos θ + θ

∂

∂t(sin2 θ cos θ), (B.4)

133

134 Chapter B. Derivation of the non-adiabatic corrections

it is easy to see that the B coefficient can be written

B(t) =sin θ

g2pNr

(2θ

∂

∂t(sin2 θ cos θ)− 1

3∂2

∂t2sin3 θ

). (B.5)

Had the sign of the second term of the last line of (B.3) been the opposite, wewould have obtained the result given in [22], but this is not the case.

The C coefficient is given by

C(t) =1

g2pNr

(sin2 θ cos θ

∂

∂t(sin2 θ cos θ) + Γba(sin2 θ cos θ)2

)

=(

Γba +12

∂

∂t

)sin4 θ cos2 θ

g2pNr

(B.6)

and the D coefficient is given by

D(t) =sin4 θ cos4 θ

g2pNr

. (B.7)

A P P E N D I X CComputer programs

In this appendix the algorithm used to obtain a numerical solution of the cou-pled Maxwell/Heisenberg-Langevin equations will be described.The FORTRAN code for the program can be downloaded fromhttp://www.phys.au.dk/quantop/rymann/rymann.shtm

C.1 Dimensionless quantities

Dimensionless quantities are introduced to make the numerical solution as gen-eral as possible. The dimensionless probe and coupling fields E and F are de-fined by

Ωp(z, t) = Ωp,0E(z, t) Ωc(z, t) = Ωc,0F (z, t). (C.1)

The dimensionless time t′ and position z′ are defined by

t′ = γbat z′ =z

vg,0Tp, (C.2)

where vg,0 is the expected group velocity and Tp is the temporal length of theprobe pulse. The dimensionless quantities for these parameters, as well as forthe length L of the medium, are defined by

u =vg,0

cτp = γbaTp l =

L

vg,0Tp. (C.3)

The Rabi frequencies for the probe and coupling fields are measured in units ofγba

Ωc,0 = rcγba Ωp,0 = εΩc,0, (C.4)

and the vacuum Rabi frequency gp is given by

g2pNr = rvγ2

ba. (C.5)

The decay rates of the lambda atom are all given in units of γba and are denotedby gµ for the populations and gµν for the coherences.

135

http://www.phys.au.dk/quantop/rymann/rymann.shtm

136 Chapter C. Computer programs

The wave equations for the probe and coupling fields, written in terms of thedimensionless quantities, are

(∂

∂t′+

1uτp

∂

∂z′

)E(z′, t′) =

irv

εrcσba, (C.6a)

(∂

∂t′+

1uτp

∂

∂z′

)F (z′, t′) =

irv

rc, (C.6b)

and the dimensionless Heisenberg-Langevin equations are

σaa = iεrc(Eσ∗ba − c.c.) + irc(Fσ∗ca − c.c.)− gaσaa (C.7a)σbb = −iεrc(Eσ∗ba − c.c.) + gbσaa (C.7b)σcc = −irc(Fσ∗ca − c.c.) + gcσaa (C.7c)σba = iεrcE(σbb − σaa) + ircFσbc − gbaσba (C.7d)σca = ircF (σcc − σaa) + iεrcEσ∗bc − gcaσca (C.7e)σbc = irc(F ∗σba − εEσ∗ca)− gbcσbc, (C.7f)

where a dot denotes the derivative with respect to the dimensionless time t′.

C.1.1 Calculation of parameters

As mentioned in section 4.3, the temporal length of the probe pulse Tp and theRabi frequencies Ωp,0, Ωc,0 can be uniquely determined by specifying valuesfor the adiabaticity parameter η, the Rabi frequency ratio ε, the medium lengthL and the expected group velocity vg,0. The coupling laser Rabi frequency isdetermined from the expected group velocity

vg,0 =c

1 + g2pNr

Ω2c,0

. (C.8)

Rearranging this equation and introducing the dimensionless quantities yields

rc =√

rvu

1− u. (C.9)

The temporal length of the probe pulse is determined from the definition of theadiabaticity parameter

Td

Tp= η

√α. (C.10)

By inserting the definition of the probe delay Td and the opacity α, we get

τp =(1− u)2l

rvuη2. (C.11)

C.2 Discretization 137

C.2 Discretization

The Heisenberg-Langevin equations are propagated in time by a standard fourthorder Runge-Kutta algorithm and the wave equations are solved by a Lax-Wen-droff finite difference scheme [27]. An outline of the Lax-Wendroff scheme isgiven in the following.

The differential equations are solved on a grid consisting of N = 1000 grid-points in space and M = 10000 timesteps. The grid spacings are ∆z′ and ∆t′.The value of the probe field at each gridpoint is denoted Ej

i = E(i∆z′, j∆t′)and a similarly notation is used for the coupling field and atomic variables. Afinite difference approximation of the probe field wave equation (C.6a), accurateto second order in the grid spacing, is obtained by centering around i + 1

2 and j

Eji+1 − Ej

i

∆z′+ uτp

Ej+ 1

2i+ 1

2− E

j− 12

i+ 12

∆t′= uτpP

j

i+ 12, (C.12)

whereP =

irv

εrcσba. (C.13)

The intermediate values Ej+ 1

2i+ 1

2and E

j− 12

i+ 12

are calculated by a FSCT (ForwardSpace, Centered Time) finite difference scheme

Ej+ 1

2i+ 1

2= E

j+ 12

i − uτp∆z′

2∆t′

(Ej+1

i − Eji

)+

12uτp∆z′P j+ 1

2i (C.14)

Ej− 1

2i+ 1

2= E

j− 12

i − uτp∆z′

2∆t′

(Ej

i − Ej−1i

)+

12uτp∆z′P j− 1

2i (C.15)

and the intermediate values Ej+ 1

2i and E

j− 12

i are replaced by the average of theclosest gridpoints

Ej+ 1

2i =

12

(Ej+1

i + Eji

)E

j− 12

i =12

(Ej

i + Ej−1i

). (C.16)

A similar procedure is used for the intermediate values of P, and we finallyarrive at the update formula for the Lax-Wendroff scheme

Eji+1 = Ej

i − uτp∆z′

∆t′Aj

i +12uτp∆z′

(P j

i+1 + P ji

)(C.17)

where

Aji =

12

(Ej+1

i − Ej−1i − uτp

∆z′

∆t′

(Ej+1

i − 2Eji + Ej−1

i

)

+12uτp∆z′

(P j+1

i − P j−1i

)).

(C.18)

138 Chapter C. Computer programs

A similar equation can be derived for the coupling field. We get

F ji+1 = F j

i − uτp∆z′

∆t′Bj

i +12uτp∆z′

(Qj

i+1 + Qji

)(C.19)

where

Bji =

12

(F j+1

i − F j−1i − uτp

∆z′

∆t′

(F j+1

i − 2F ji + F j−1

i

)

+12uτp∆z′

(Qj+1

i −Qj−1i

)) (C.20)

andQ =

irv

rcσca. (C.21)

It is obvious that the update formulas (C.17) and (C.19) cannot be used for thelast timestep j = M . An update formula valid for j = M can be found from thediscretization

EMi+1 − EM

i

∆z′+ uτp

EMi+ 1

2− EM−1

i+ 12

∆t′= uτpP

Mi+ 1

2. (C.22)

Using the substitutions

EMi+ 1

2=

12

(EM

i+1 + EMi

)PM

i+ 12

=12

(PM

i+1 + P ji

)(C.23)

we find the update formula

EMi+1 =

11 + uτp

∆z′2∆t′

[EM

i +12uτp

(∆z′

∆t′AM

i+1 + ∆z′(PM

i+1 + PMi

))](C.24)

whereAM

i+1 = EM−1i+1 + EM−1

i − EMi . (C.25)

The fourth order Runge-Kutta method gives the polarizations P j+1i and Qj+1

i

from the values of the fields (Eji , F j

i ) and polarizations (P ji , Qj

i ) at the previoustimestep.

Starting from the boundary conditions for the probe and coupling fields (Ej0 ,

F j0 ) and the initial conditions for the atoms (P 0

i , Q0i ), the finite difference equa-

tions for the fields and the atoms can be solved self-consistently.

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Standing Wave Electromagnetically Induced Transparency

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