Collective Atomic Recoil in Ultracold Atoms: Advances and … · 2007. 3. 17. · 6 Entanglement...

Università degli Studi di Milano

Facoltà di Scienze Matematiche, Fisiche e Naturali

Dottorato di Ricerca in

Fisica, Astrofisica e Fisica Applicata

Collective Atomic Recoil

in Ultracold Atoms:

Advances and Applications

Coordinatore Prof. Rodolfo Bonifacio

Tutore Prof. Rodolfo Bonifacio

Tesi di Dottorato di

Mary Manuela Cola

Ciclo XVI

Anno Accademico 2002-2003

November 14, 2003 c© M M Cola 2003

In memory of Prof. Giuliano Preparata

If you do boast, consider this: you do not

support the root, but the root supports you.

(Rm. 11,18)

The figure upwards shows evidence for a Bose Einstein condensation of sodium atoms.

It is taken from PRL 75, 3969 (1995). The figure on the left shows evidence for a

collective atomic recoil in a BEC. It is a courtesy of M. Inguscio and coworkers.

Acknowledgments

First of all I wish to thank my tutor, Rodolfo Bonifacio, who gave me the possibility

to approach the interesting physics of collective phenomena.

Then I should sincerely thank Nicola Piovella, for his support and teachings during

these years, especially for what concerned the physics of CARL.

I also learned a lot from Matteo G.A. Paris: a great acknowledgment for his careful

aid. He introduced me to the physics of quantum optics and quantum information.

This let me to investigate fruitful topics in atom optics.

A sincere thank to my Referee Francesco S. Cataliotti for his punctual and concerned

correction of this thesis, and to Chiara Fort, Leonardo Fallani and Massimo Inguscio

for all the hours of collaboration we spent together.

I want to remember also other physicists with whom I often had stimulating discus-

sions about frontiers of science, Emilio Del Giudice, Enrico Giannetto and Marco

Giliberti.

Thanks to Stefano Olivares, Andrea R. Rossi, Alessandro Ferraro and Gabriele

Marchi. They keep cheerful the atmosphere of our group.

A special thought to Carlo, for his patience and his encouragement, and to my

friends Anna, Elisa and Federica for sharing the everyday difficulties.

Finally I remember Giuliano Preparata: in a difficult moment of my life his passion

for physics reminded me my passion for physics.

Contents

Introduction iv

1 Classical CARL 7

1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 The FEL limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 The undamped case . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 The damping case: adiabatic limit . . . . . . . . . . . . . . . . 15

1.3 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Experimental realizations . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Quantum CARL 23

2.1 First quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Linear regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


3 Quantum field theory 33

3.1 The CARL-BEC model . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Coupled-modes equations . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Linearized three-mode model . . . . . . . . . . . . . . . . . . . . . . . 40


4 Superradiant Rayleigh scattering and matter waves amplification 43

4.1 Directional matter waves produced by spontaneous scattering . . . . 44

4.2 Dicke superradiance and emerging coherence . . . . . . . . . . . . . . 47

4.3 Evidence for decoherence . . . . . . . . . . . . . . . . . . . . . . . . . 49

i

ii Contents

4.4 “Seeding” the superradiance . . . . . . . . . . . . . . . . . . . . . . . 51


5 Superradiant Rayleigh scattering from a moving BEC 57

5.1 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Experimental features . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Seeding the superradiance . . . . . . . . . . . . . . . . . . . . . . . . 65


6 Entanglement generation 71

6.1 The Hamiltonian model . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Spontaneous emission and small-gain regime . . . . . . . . . . . . . . 74

6.3 Solution of the linear quantum regime . . . . . . . . . . . . . . . . . . 76

6.4 Three mode entanglement . . . . . . . . . . . . . . . . . . . . . . . . 80

6.5 High-gain regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.5.1 The quasi-classical recoil limit ρ À 1 . . . . . . . . . . . . . . 836.5.2 The quantum recoil limit ρ ≤ 1 . . . . . . . . . . . . . . . . . 85

6.6 Atom-atom and atom-photon entanglement . . . . . . . . . . . . . . . 86


7 Radiation to atom quantum mapping 91

7.1 The entangled state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2 The Bell measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.3 The displacement operation . . . . . . . . . . . . . . . . . . . . . . . 95

7.4 The readout system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


8 Effects of decoherence and losses on entanglement generation 99

8.1 Dissipative Master Equation . . . . . . . . . . . . . . . . . . . . . . . 99

8.2 Solution of the Fokker-Plank equation . . . . . . . . . . . . . . . . . . 100

8.3 Evolution from vacuum and expectation values . . . . . . . . . . . . . 101

8.4 Numerical analysis for the relevant working regimes . . . . . . . . . . 103


Conclusion and Outlook 111

Contents iii

A General solution of the linear model 115

B Wigner functions 119

C Homodyne and multiport homodyne detection 123

C.1 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

C.2 Balanced homodyne detection . . . . . . . . . . . . . . . . . . . . . . 124

C.3 Double homodyne detection . . . . . . . . . . . . . . . . . . . . . . . 126

D Continuous variable teleportation as conditional measurement 129

D.1 Conditional quantum state engineering . . . . . . . . . . . . . . . . . 130

D.2 Joint measurement of two-mode quadratures . . . . . . . . . . . . . . 133

D.3 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

E Solution of the Fokker-Planck equation 137

Introduction

At the basis of most phenomena in atomic, molecular, and optical physics is the

dynamical interaction between optical and atomic fields.

In many ways, recently developed Bose Einstein Condensates (BECs) of trapped

alkali atomic vapors [1, 2] are the atomic analog of the optical laser. In fact, with

the addition of an output coupler, they are frequently referred to as “atom lasers”

[3]. Despite many interesting and important differences, the main similarity is that

both optical lasers and atomic BECs involve large numbers of identical bosons oc-

cupying a single quantum state. As a result, the physics of lasers and BECs involves

stimulated processes which, due to Bose enhancement, often completely dominate

the spontaneous processes which play central roles in the non-degenerate regime.

Just like the discovery of the laser led to the development of nonlinear optics, so

too the advent of BEC has led to remarkable experimental successes in the field of

nonlinear atom optics [4, 5, 6, 7, 8, 9].

The regimes of nonlinear optics and nonlinear atom optics, therefore, represent

limiting cases, where either the atomic or optical field is not dynamically independent

because it follows the other field in some adiabatic manner which allows for its

effective elimination. Outside of these two regimes the atomic and optical fields

are dynamically independent, and neither field can be eliminated. The dynamics of

coupled quantum degenerate atomic and optical fields in this intermediate regime is

the topic of “quantum atom optics”, namely that extension of atom optics where the

quantum state of a many-particle matter-wave field is being controlled, characterized

and used in novel applications. Some advances and applications in this field will be

the object of this thesis.

One of the more relevant system in quantum atomic optics is composed of a BEC

driven by a far off-resonant pump laser and coupled to a single mode of an optical

1

2 Introduction

ring cavity. The mechanism that lies below this kind of physics is the so-called

Collective Atomic Recoil Lasing (CARL) in his fully quantized version.

The CARL mechanism was originally proposed as a new mechanism for the

generation of coherent light [10, 11, 12]. It consist of three main ingredients: (1) a

gas of two-level atoms (the active medium) (2) a strong pump laser, which drives the

two-level atomic transition, and (3) a ring cavity which supports an electromagnetic

mode (the probe) counterpropagating with respect to the pump. Under suitable

conditions, the operation of the CARL results in the generation of a coherent light

field due to the following mechanism. First, a weak probe field is initiated by noise,

either optical in the form of spontaneously emitted light, or atomic in the form of

density fluctuations in the atomic gas which backscatters the pump. Once initiated,

the probe combines with the pump field to form a weak standing wave which acts

as a periodic optical potential. The center-of-mass motion of the atoms on this

potential results in a bunching, i.e. a modulation of their density, very similar to

the combined effects of the wiggler and the light field leads to electron bunching

in a Free Electron Laser (FEL) [13]. This bunching process is then seen by the

pump laser as the appearance of a polarization grating in the active medium, which

results in stimulated backscattering into the probe field. The resulting gain in the

probe strength further amplifies the magnitude of the standing wave field, generating

more bunching followed by an increase in stimulated backscattering, and so on. This

positive feedback mechanism give rise to an exponential growth of both the probe

intensity and the atomic bunching which leads to the perhaps surprising result that

the presence of the ring cavity turns the ordinarily stable system of an atomic gas

driven by a strong pump laser into an unstable system.

CARL effect was verified experimentally by Bigelow et al. in a hot atomic cell

[14]. Related experiments by Courtois et al. [15], using cold cesium atoms, and

by Lippi et al. [16], using hot sodium atoms, measured the recoil induced small-

signal probe gain, which was interpreted in terms of coherent scattering from an

induced polarization grating. However, these experiments missed a probe feedback

mechanism, which is necessary to see the long time scale instability which charac-

terizes the CARL. The first unambiguous experimental proof of the CARL effect

has been obtained only very recently [17] in a system of cold atoms in a collision-less

environment.

Introduction 3

In chapter 1 we review the essential conceptual framework of the original CARL

showing that self-bunching via an exponential instability can occur under very gen-

eral conditions. The original CARL theory considers the atoms as classical point

particles moving in the optical potential generated by the light fields. From an

atom optics point of view, this correspond to a “ray atom optics” treatments of the

atomic field, in analogy with the ordinary ray optics treatment of electromagnetic

fields. This description is valid provided that the characteristic wavelength of the

matter-wave field remains much smaller than the characteristic length scale of any

atom-optical element in the system. Such length, for the atomic field, is its De

Broglie wavelength, determined by the atomic mass and the temperature T of the

gas. The central atom-optical element of the CARL is the periodic optical potential,

which acts as a diffraction grating for the atoms, and has the characteristic length

scale of half the optical wavelength. Hence the classical description is valid provided

that the temperature is high enough so that the thermal De Broglie wavelength is

much smaller than the optical wavelength.

However, the spectacular recent advances in atomic cooling techniques makes

it likely that CARL experiments using ultracold atomic samples can and will be

performed in the next future. In particular, subrecoil temperatures can now be

achieved almost routinely. So CARL theory has been extended to this “wave atom

optics” regime [18]. In this regime matter-wave diffraction plays a dominant role in

the CARL dynamics. The main drawback of the semiclassical model is that, as it

considers the center-of-mass motion of the atoms as classical, it cannot describe the

discreteness of the recoil velocity, as has been observed in the experiment of Ref.[19]

for an atomic sample below the recoil temperature. So, to extend the model in the

region of ultracold atoms, a quantum mechanical description of the center-of-mass

motion of the atoms should be included. In chapter 2 we present a way to work

out this program simply performing a first quantization of the external variables of

the atoms, position and momentum. This simple model gives a description of all

the features of the considered system and in particular allows to define the main

different regimes. In the conservative regime (no radiation losses), the quantum

model depends on a single collective parameter, ρ, that can be interpreted as the

average number of photons scattered per atom in the classical limit. When ρ À 1,the semiclassical CARL regime is recovered, with many momentum levels populated

4 Introduction

at saturation. On the contrary, when ρ ≤ 1, the average momentum oscillatesbetween zero and ~~q, where ~q is the difference between the incident and the scatteredwave vectors, and a periodic train of 2π hyperbolic secant pulses is emitted. In the

dissipative regime (large radiation losses) and in a suitable quantum limit ρ À 1 ,a sequential superradiant scattering occurs, in which after each process atoms emit

a π hyperbolic secant pulse and populate a lower momentum state. These results

describe the regular arrangement of the momentum pattern observed in experiments

of superradiant Rayleigh scattering from a BEC [20].

In chapter 3 we derive a quantum field theory model of a gas of bosonic two-level

atoms which interact with a strong, classical, undepleted pump laser and a weak,

quantized optical ring cavity mode, both of which are as usual assumed to be tuned

far away from atomic resonances. Starting from the second-quantized hamiltonian

of the system, we will write an effective model for the time evolution of the ground

state atomic field operator and for the probe field operator (the CARL-BEC model),

adiabatically eliminating the excited state atomic field operator and including effects

of atom-atom collisions.

In chapter 4 we review the experimental situations, such as superradiant Rayleigh

scattering and matter waves amplification, that can be interpreted with the full

quantistic version of CARL model in the dissipative regime, where the radiation

emission is superradiant.[21]

In chapter 5 we analyze some experiment performed at European Laboratory for

Non-linear Spectroscopy (LENS) in Florence about superradiant Rayleigh scattering

from a moving BEC. This allows to investigate the influence of the initial velocity of

the condensate on superradiant Rayleigh scattering. The experiment gives evidence

of a damping of the matter-wave grating which depends on the initial velocity of

the condensate. We describe this damping in terms of a phase-diffusion decoherence

process, in good agreement with the experimental results. Moreover we analyze the

effect of seeding superradiance by a weak signal directed in the opposite direction

with respect to the pump laser.

One important consideration is to determine to which extent the quantum state

of a many-particle atomic field like a BEC can be optically manipulated. In the

single-particle case, the answer to this problem is known to a large extent. This is

the domain of atom optics [22], where a number of optical elements for matter waves

Introduction 5

have now been developed, including gratings, mirrors, interferometers, resonators,

etc. But these optical elements manipulate just the atomic field “density”, or at most

first-order coherence properties. However, Schrödinger fields possess a wealth of

further properties past their first-order coherence, including atom statistics, density

correlation functions. In chapter 6 we investigate the properties, such as quantum

fluctuations and entanglement, of the quantum system BEC-radiation in the linear

regime for a good cavity regime. We obtain new analytical results, calculating

explicitly the statistical properties for atoms and photons and evaluating the state

of the coupled BEC-light system evolved from vacuum. In the limit of undepleted

atomic ground state and unsaturated probe field, the quantum CARL Hamiltonian

reduces to that for three coupled modes. By calculating the exact evolution of the

state from the vacuum of the three modes we demonstrate that the evolved state

is a fully inseparable three mode Gaussian one. Moreover we show how this three

mode Gaussian state can provide a valuable source of atom-atom and atom-photon

entanglement.

Entanglement is a crucial resource in the manipulation of quantum information,

and quantum teleportation [23, 24, 25, 26, 27] is perhaps the most impressive ex-

ample of quantum protocol based on entanglement. It realizes the transferral of

(quantum) information between two distant parties that share entanglement. There

is no physical move of the system from one player to the other, and indeed the two

parties need not even know each other’s locations. Only classical information is ac-

tually exchanged between the parties. However, due to entanglement, the quantum

state of the system at the transmitter location (say Alice) is mapped onto a different

physical system at the receiver location (say Bob). In chapter 7 we show a scheme

to realize radiation to atom continuous variable quantum mapping, i.e. to teleport

the quantum state of a single mode radiation field onto the collective state of atoms

with a given momentum out of a BEC. The atoms-radiation entanglement needed

for the teleportation protocol is established through the CARL three linear model

studied in chapter 6, whereas the coherent atomic displacement is obtained by the

same interaction with the probe radiation in a classical coherent field.

In chapter 8 the results obtained in chapter 6 are extended to include the effects

of losses either due to the optical cavity or to the depletion of atomic modes. The

calculation are performed by means of the Master equation formalism and a system-

6 Introduction

atic comparison with respect to the ideal case is given. The results of this chapter

are promising and give indications for future experiments.

Chapter 1

Classical CARL

The CARL, a kind of hybrid between the FEL [13] and the ordinary laser, with

physical features common to both, was thought and presented like a source of tunable

coherent radiation. Its essential conceptual framework was first outlined in Ref.[10].

The ordinary laser and the FEL share an important physical trait: they gen-

erate electromagnetic waves through a noise-initiated process of self-organization.

In an ordinary laser, for example, the energy is stored initially as excitation of in-

ternal degrees of freedom of the active medium, while in a FEL it is brought into

the interaction region as translational kinetic energy of the incident electron beam.

The spectral character of laser light is constrained mainly by the gain profile of the

active medium, while in a FEL the frequency of the emitted radiation is assigned

by the speed of the incident electrons, and can be varied, in principle, over a very

wide range; hence, the FEL is intrinsically a widely tunable source. Furthermore,

the laser gain originates from the induced atomic polarization, under the constraint

that a suitable population inversion exists in the active medium; in a FEL, instead,

amplification of coherent radiation follows the spontaneous emergence of a suffi-

ciently large electron bunching, i.e. the appearance of a periodic spatial structure

in the form of a longitudinal grating on the scale of the electromagnetic wavelength.

Hence, light amplification in a FEL is the result of a coherent scattering process

from the grating structure created within the active medium, and it comes at the

expense of a recoil in the momentum of the individual electrons.

The physics of the FEL and of the atomic lasers are unified in the CARL. The

active medium now is a collection of two level atoms initial in their lower state and

7

8 Chapter 1. Classical CARL

exposed to a strong pump field. For appropriate values of the parameters the atoms,

through an exponential instability, can amplify a weak probe field counterpropagat-

ing with respect to the pump. As in the laser the active medium is characterized by

bound states which play a key role in the amplification process but do not posses a

population inversion. Common to the FEL, instead, is the existence of a reservoir of

momentum that can be transformed partly into radiation through a kind of cooper-

ative scattering. Furthermore, optical gain is initiated by the growth of a bunching

parameter. What happens is that the incident optical wave creates an atomic po-

larization wave in the medium. This polarization couples with the backscattered

radiation, creating a longitudinal self-consistent pendulum potential which traps

and than bunches the particles giving rise to a coherent scattering.

1.1 Equations of motion

The CARL model is based on the Hamiltonian of a collection of two-level atoms

interacting with a strong pump field and a weak optical probe counterpropagating

with respect to the pump. In addition to the internal atomic degrees of freedom,

which are typical of laser models, the CARL model take explicit account of the

center-of-mass motion. The explicit form of the Hamiltonian is

Ĥ = ~ω1â†1â1 + ~ω2â†2â2 + ~ω0

N∑j=1

Ŝzj +N∑

j=1

p̂2j2m

+i~

(g1â

†1

N∑j=1

Ŝ−j e−ik1ẑj + g2â

†2

N∑j=1

Ŝ−j eik2ẑj − h.c.

)(1.1)

where N is the number of atoms in the interaction volume V , ω1,2 = ck1,2 are

the carrier frequency of the probe and pump fields, respectively, k1 and k2 are the

corresponding wave numbers and ω0 is the atomic transition frequency when the

atoms are at rest relative to the observer. The couplings constants are defined as

gi = µ

[ωi

2~²0V

]1/2i = 1, 2 (1.2)

where µ is the modulus of the atomic dipole moment. Ŝzj and Ŝ±j are the standard

effective angular-momentum operators (in units of ~) describing the evolution ofthe internal degrees of freedom of the j-th atom so that Ŝzj measures one-half the

1.1. Equations of motion 9

difference between the excited and ground state populations of the j-th atom; ẑj

and p̂j denote, respectively, the position and momentum operators of the center of

mass of the j-th atom and â†i (i = 1, 2) are the photon creation operators of the

pump field (index 1) and of the probe field (index 2). The operators obey the usual

commutation relations:

[Ŝzi, Ŝ

±j

]= ±δijŜ±j ,

[Ŝ+i , Ŝ

−j

]= 2δijŜzi (1.3)

[ẑi, p̂j] = i~δij (1.4)[âi, â

†j

]= δij (1.5)

The Hamiltonian (1.1) admits two constants of the motion,

N∑j=1

p̂j + ~k1â†1â1 − ~k2â†2â2 = constant (1.6)

N∑j=1

Ŝzj + â†1â1 + â

†2â2 = constant; (1.7)

the first represents the conservation of the total momentum and the second the

conservation of the number of excitations. If we combine Eqs.(1.6) and (1.7) and

eliminate the number operator of the driving field, â†2â2, we can also write

N∑j=1

(p̂j + ~k2Ŝzj) + ~(k1 + k2)â†1â1 = constant (1.8)

whose obvious physical implication is that the expectation value of the number op-

erator for the probe field, â†1â1, can grow either as the result of a loss of internal

atomic energy or a decrease of the center-of-mass kinetic energy. This setting rep-

resents a generalization of the basic mechanism by which energy is produced in the

laser and in the FEL; in fact, the laser Hamiltonian does not involve the momentum

and position operators p̂j and ẑj, while the FEL Hamiltonian does not include the

angular-momentum operators, descriptive of the internal degrees of freedom of the

active medium.

In this chapter we analyze the dynamical evolution of the coherent atomic recoil

lasing within the framework of the standard semiclassical approximation. First we

construct the Heisenberg equations of motion for the relevant operators and map


the operator equation into their c-number counterparts in the usual factorized form

obtaining

dzjdt

=pjm

(1.9)

dpjdt

= −~k1g1a∗1Sje−ik1zj + ~k2g2a∗2Sjeik2zj + c.c. (1.10)

da1dt

= −iω1a1 + g1N∑

j=1

Sje−ik1zj (1.11)

dS−jdt

= −iω0S−j + 2g1a1Szjeik1zj + 2g2a2Szje−ik2zj (1.12)dSzjdt

= − (g1a∗1S−j e−ik1zj + g2a∗2S−j eik2zj + c.c.). (1.13)

where we have assumed that a2 is a constant real number. This program is ac-

complished by introducing the appropriate slowly varying variables ao1, ao2, and Sj

according to the definitions

a1(t) = a01(t) exp

{−i

[ω2 +

k1 + k2m

p̄(0)

]t

}(1.14)

a2(t) = a02 exp {−iω2t} (1.15)

S−j (t) = Sj(t) exp(−ik2(zj + ct)) (1.16)

where p̄(0) ≡ mv̄(0) is the average initial momentum of the atoms. Furthermore,we define the new position and momentum variables θj(t) and δpj(t)

θj(t) = (k1 + k2)

[zj − p̄(0)

mt

](1.17)

δpj(t) = pj(t)− p̄(0), (1.18)

and the population difference between the ground and excited states of the j-th

atom,

Dj(t) = −2Szj(t) (1.19)The required equations of motion take the form

dθjdt

=k1 + k2

mδpj (1.20)

d

dtδpj = −~k1g1a0∗1 Sje−iθj + ~k2g2a0∗2 Sj + c.c. (1.21)

da01dt

= iδ1,2a01 + g1

N∑j=1

Sje−iθj (1.22)


dSjdt

= i

[ω2c

δpjm

+ δ2,0

]Sj − g1a01Djeiθj − g2a02Dj − γ⊥Sj, (1.23)

dDjdt

=(2g1a

0∗1 Sje

iθj + 2g2a0∗2 Sj + c.c.

)− γ‖(Dj −Deqj ) (1.24)

where we have introduced the detuning parameters

δ2,1 = (k1 + k2)[v̄(0)− vr,1], (1.25)δ2,0 = k2[v̄(0)− vr,2], (1.26)

with

vr,1 =ω1 − ω2ω1 + ω2

c, vr,2 =ω0 − ω2

ω2c (1.27)

and added phenomenological decay terms to the polarization and population equa-

tions (Deqj = 1 because each atom is assumed to be in the ground state as it enters

the interaction region). Note that the two resonance conditions δ2,0 = δ2,1 = 0,

taken together, imply

ω1 =ω0

1− β0 , ω2 =ω0

1 + β0, (1.28)

with β0 = v̄(0)/c, i.e. they imply a resonance between the atomic transition fre-

quency ω0 and the Doppler-shifted frequencies of the probe and the pump beams.

As our final step we introduce the so-called universal scaling and cast the working

equations in dimensionless form. For simplicity we let k1 ≈ k2 ≡ k = ω/c, g1 ≈g2 ≡ g and, furthermore, define the dimensionless parameter

ρ′=

(g√

N

ωR

)2/3∝

(N

V

)1/3(1.29)

where ωR = 2~k2/m is the single-photon recoil frequency shift, the scaled time τ ,and the new dependent variables Pj and A1,2 according to the definitions

τ = ωRρ′t Pj =

δpj~kρ′

A1,2 =a01,2√Nρ′

(1.30)

where A2 is real for definiteness. The parameter ρ′

is a measure of the collective

effects; in fact g√

N is the collective spontaneous Rabi frequency of an ensemble of

N two-level atoms [28, 29].


With these definitions the final form of the CARL equations of motion is

dθjdτ

= Pj (1.31)

dPjdτ

= −A∗1e−iθjSj − A1eiθjS∗j + 2A2ReSj, (1.32)

dA1dτ

= iδA1 +1

N

N∑j=1

Sje−iθj (1.33)

dSjdτ

=i

2(Pj + 2∆)Sj − ρDj(A1eiθj + A2)− Γ⊥Sj, (1.34)

dDjdτ

= [2ρ(A′∗1 e

−iθj + A2)Sj + h.c.]− Γ‖(Dj −Deqj ), (1.35)

where the remaining parameters are defined as follow:

δ =δ2,1ωRρ

′ ∆ =δ2,0ωRρ

′ (1.36)

Γ⊥ =γ⊥

ωRρ′ Γ‖ =

γ‖ωRρ

′ . (1.37)

Eqs. (1.31-1.35) form a closed, self-consistent set of equations for the internal and

translational atomic degrees of freedom, coupled to the pump field A2 and the probe

field A1, whose amplification was the main objective of the original works on CARL.

In arriving at this result, as already mentioned, we have assumed k1 ≈ k2 ≡ k.If k1 6= k2, Eqs. (1.31-1.35) are still valid in the so-called Bambini-Ranieri frame[30, 31] moving with a velocity vr,1 [see Eq.(1.27)], where the transformed frequencies

coincide. We note that for a nearly resonant interaction, i.e. δ2,1 ≈ 0, it follows thatv̄(0) ≈ vr,1. We can distinguish two cases:

(a) Non relativistic particles [v̄(0) ¿ c]; in this case we have ω1 ≈ ω2 and ourequations are valid in the laboratory frame.

(b) Relativistic particles [v̄(0) ≈ c]; in this case it follows that

ω1 =c + vr,1c− vr,1ω2 =

1 + β01− β0ω2 ≈ 4γ

2ω2, (1.38)

where γ = (1 − β20)−1/2; hence ω1 can be considerably larger than ω2. Thus, thisformulation can also account for the dynamics of relativistic particles; of course, in

this case one needs an additional Lorentz transformation of Eqs.(1.31-1.35) back to

the laboratory frame. We will never deal here with this case because our purpose is

to show how the CARL model can be extended to the ultracold atoms regime.


Eqs. (1.34) and (1.35) are the optical Bloch equations, generalized for the in-

clusion of the atomic translational motion. In addition to the familiar detuning

term ∆Sj in the polarization equation (1.34), one may note the appearance of the

time dependent detuning contribution PjSj/2 resulting from the recoil suffered by

atoms under the action of the pump and probe fields. If we ignore the probe field

(A1 = 0), Eqs. (1.31-1.35) describe the usual cooling process for time long compared

to Γ−1⊥ and Γ−1‖ . If, instead, we set A2 = 0 and Sj = 1, for all j, the modified Eqs.

(1.31-1.33) reduce to the traditional FEL equations.

The structure of Eq.(1.33) indicates that the probe field A1 can be amplified

in the presence of an atomic polarization (but without the need of an initial pop-

ulation inversion) if the phase of the polarization has the appropriate value and

if the atomic positions are properly bunched. If the scaled position variables are

uniformly distributed between 0 and 2π, just as one has at the beginning of the evo-

lution, no macroscopic field source exists even if the atomic polarization variables

Sj are maximized for all values of j because

N∑j=1

e−iθj = 0 (1.39)

Eqs. (1.31-1.35) for a wide range of the parameters predict the development of an

exponential instability for the probe field and for the bunching parameter

B =

∣∣∣∣∣1

N

N∑j=1

e−iθj

∣∣∣∣∣ . (1.40)

The result of this instability is the growth of a macroscopic field and the spontaneous

creation of a longitudinal spatial structure in the initially uniform atomic beam with

a periodicity that matches the wavelength of the reflected field. This type of behavior

can be easily demonstrated by numerical integration of Eqs. (1.31-1.35)

We now want to show that, under certain approximations, the atomic degrees

of freedom can be eliminated and the CARL equations made formally identical to

the FEL-model equations with universal scaling [13, 32]. This allows the simple

description of a Hamiltonian instability leading to exponential growth of the probe

field and of the bunching parameter. It will be demonstrated both without and in

presence of atomic damping.


1.2 The FEL limit

1.2.1 The undamped case

Consider first the case in which Γ = 0 in the Bloch equations (this, in practice,

means that we take Γ 1. In this case, we can take the time average of the atomic variables

〈S1 = 0〉 and 〈S2〉 = −S0 where

S0 =1

2

Ω∆

Ω2 + ∆2. (1.44)

Note that S0 is maximized for ∆ = Ω, where S0 = 1/4. Upon substituting in Eqs.

(1.31-1.33) we obtain

dθjdτ

= Pj (1.45)

dPjdτ

= −S0(A∗e−iθj + Aeiθj) (1.46)

dA

dτ= −iδA + S0

N

N∑j=1

e−iθj (1.47)

1.2. The FEL limit 15

where we have defined A = iA1. Note that the time average has washed out the

absorptive part S1 of the polarization, leaving only the dispersive part S2, which

is antisymmetric in the detuning ∆. In particular, in Eq. (1.32), the radiation

pressure term 2A2> 1/Γ so that we can perform theadiabatic elimination of the polarization and population variables [11]. With simple

calculations, one can show that we obtain again the FEL equations (1.49-1.51) with

A substituted by√

2A and

S0 =1√2

Ω∆

Γ2 + ∆2 + Ω2(1.52)

This result can be obtained by again assuming |A2|2 >> |A1|2 and Pj


1.3 Linear stability analysis

Equations (1.49-1.51) admit two constants of motion:

〈p〉+ |A|2 (1.53)

and〈p2〉2

+ iS0[A∗〈e−iθ〉 − c.c] = H (1.54)

The first represents momentum conservation, the second defines the Hamiltonian

from which (1.49-1.51) can be derived. This Hamiltonian system is unstable.

A linear analysis of (1.49-1.51) leads to the identification of the conditions under

which the process of self-amplification of the spontaneous emission takes place. Let

us consider the initial conditions (where the system has an equilibrium solution) with

zero field, no spatial modulation of the beam of particles in which θj are randomly

distributed. In this initial situations∑

j e−imθj = 0 with m = 1, 2.

Equations (1.49-1.51) are generally valid also if the initial momentum has an

arbitrary distribution f(p0). In this case, we can label the particles by the initial

position θ0 and initial momentum p0 so that1N

∑Nj=1 e

−iθj is replaced by

〈e−iθ(θ0,p0,τ)〉 = 12π

∫ 2π0

f(p0)e−iθ(θ0,p0,τ)dθ0dp0 (1.55)

where we have assumed a uniform distribution for θ0. One can show that, by lineariz-

ing (1.49-1.51) around the initial situation above and using a Laplace transformation,

we find solutions of the form3∑

j=1

cjeiλjτ (1.56)

where cj depends on the initial conditions and λj are the roots of the cubic equation

λ +

∫f(p0)

(λ + p0)2dp0 = 0 (1.57)

In particular, for a “cold” case, i.e., if f(p0) is a Dirac delta function centered at a

value p0 = δ, the cubic equation takes the form

λ + (λ + δ)2 + 1 = 0. (1.58)

Note that for v0 = 0, we have δ = (ω2 − ω1)/ωRρ.

1.3. Linear stability analysis 17

Exponential growth, and thus, unstable behavior, results if the cubic Eq.(1.58)

has one real and two complex-conjugate roots. In this case, one of the imaginary

parts of the eigenvalues measures the exponential growth rate of the unstable solu-

tion.

For δ > δc = (27/4)1/3, all the roots are real and the system is stable, for

δ < δc the system is unstable and it shows a collective instability that leads to an

exponential growth of the probe-field intensity. In particular, for δ = 0 the gain is

maximum and the intensity grows as

|A|2 = e√

3τ = eGt (1.59)

where

G =√

3 ωRρ. (1.60)

This shows that the growth rate of the collective instability described by this Hamil-

tonian is governed by ωRρ. In general, the exponential gain depends on the detuning

parameters δ and ∆.

The collective recoil process of the system produces an atomic longitudinal grat-

ing on the scale of the wavelength. This can be measured by the behavior of the

“bunching” parameter . In fact, the bunching parameter is the source for the field in

(1.51) and its modulus can range from 0, when the particles are randomly distributed

in phase, to 1, when the particles are confined periodically to regions smaller than

the wavelength. When the collective instability develops, the emitted radiation cre-

ates a correlation between the particles, and the beam becomes strongly modulated

(self-bunching takes place). The time dependence of the bunching is also exponen-

tial and produces values that are almost unity. The strong bunching is due to the

fact that in the high-gain exponential regime, the particles are trapped in the closed

orbits of a pendulum phase space due the beat of the pump and of the self-consistent

scatter field which appears explicitly in (1.32). We stress that this behavior takes

place only for long times τ ≥ 1. In general, one must perform a careful analysis ofthe linear solution which has the form

A(δ, τ) =3∑

j=1

Cjeiλj(δ)τ (1.61)

where λj(δ) are the three roots of the cubic, and Cj are fixed by the initial conditions.

Here, we simply describe the results.


Figure 1.1: Gain as a function of δ for different interaction times τ . (a) τ = 1:

Small-gain Madey regime where G is an antisymmetric function of δ. (b) τ =

10: Characteristic symmetric shape of the gain curve in the high-gain exponential

regime. Note that the gain curve is orders of magnitude larger than for τ = 1.

For τ < 1, the time behavior is not exponential and one has the well known

small-gain Madey regime [33, 34] (see Fig. 1.1a). The gain

G(δ, τ) =[|A(δ, τ)|2]− |A0|2

|A0|2 (1.62)

is an oscillating function of τ , and for fixed τ is an antisymmetric function of δ,

which is positive (gain) for δ > 0, zero for δ = 0 and negative for δ < 0 showing

an absorptive behaviour. This gain is associated with the small bunching resulting

from the fact that the particles are not trapped, i.e. move on open orbits of the

pendulum phase space. However, when τ >> 1 and δ < δc, the particles becomes

trapped. As a consequence, the exponential behavior takes over and the gain G as a

function of δ changes shape as τ increases, and acquires a symmetric dependence on

δ, as shown in Fig. 1.1b. Its maximum at δ = 0 increases exponentially, as stated

before. Hence, the exponential regime is observable only if the interaction time is

sufficiently large so that τ >> 1.

The behavior of the probe field as a function of the normalized time τ is shown

in Fig. 1.2a. This is a numerical simulation of the exact set (1.31-1.33) under

1.4. Experimental realizations 19

Figure 1.2: (a): Probe field as a function of τ for the exact CARL equations with

Γ = 0, ∆ = 600, δ = 0; (b): Bunching parameter as a function of τ for the exact

CARL equations, using the parameters of a.

conditions of zero decay rate (Γ = 0). Fig. 1.2b shows the behavior of the bunching

parameter as a function of τ . Note the very high value of saturation of the modulus

of the bunching (≈ 0.8).

1.4 Experimental realizations

The signature of CARL is an exponential growth of a seeded probe field oriented

reversely to a strong pump interacting with an active medium. On the other hand,

atomic bunching and probe gain can also arise spontaneously from fluctuations with

no seed field applied. The underlying runaway amplification mechanism is particu-

larly strong, if the reverse probe field is recycled by a ring cavity.

After the theoretical proposal of CARL, experiments have been performed in

order to observe its peculiar features. As we saw the most striking effect due to the

CARL dynamics is the exponential growth of a seeded probe field oriented reversely

to the pump field. On the other hand, atomic bunching and probe gain can also

arise spontaneously from fluctuations with no seed field applied, particularly if the

runaway amplification mechanism is enforced recycling the reverse probe field by

a ring cavity. The first attempts to observe CARL action have been undertaken


Figure 1.3: Scheme of the experimental setup in Tbingen. A Ti-sapphire laser is

locked to one of the two counterpropagating modes (α+) of a ring cavity. The beam

αin− can be switched off by means of a mechanical shutter (S). The atomic cloud is

located in the free space waist of the cavity mode. The evolution of the interference

signal between the two light fields leaking through one of the cavity mirrors and the

spatial evolution of the atoms via absorption imaging are observed. Figure taken

from Ref. [17].

only in hot atomic vapors [14, 16]. In particular P.R. Hemmer, N.P. Bigelow and

coworkers [14] performed the first experiment in a strongly pumped atomic sodium

vapor without the introduction of a counterpropagating probe. These experiments

led to the identification of a reverse field with some of the expected characteristics.

However, the gain observed in the reverse field can have other sources [35], which

are not necessarily related to atomic recoil.

The first unambiguous experimental proof of the CARL effect has been obtained

only very recently [17] in a system of cold atoms in a collision-less environment.

In this experiment (see Fig.1.3) a high-Q ring cavity is pumped by a Ti-Sapphire

laser locked to one mode (α+) of the cavity. The85Rb atomic cloud is located in

the free space waist of the cavity mode with a magneto-optical trap, working at a

temperature of several 100µK. The reverse field α− has been monitored as the beat

signal between the field α− itself and the pump α+. In this experiment, in contrast

with the usual CARL model, the atoms are prepared already in a bunched state.

In Fig.1.4(a) we can see that oscillations appear on the beat signal, showing the

arising of the reverse field due to recoil effect even in the absence of a seed field .

Notice that the amplitude of the oscillations is rapidly dumped, however they are

still discernible after more than 1ms. Moreover as the interaction time between the

1.4. Experimental realizations 21

0 50 100 1500

2

4

6

t (µs)

Pbe

at (

µW) (a)

(b)

0 1 20

200

400

600

800

t (ms)

∆ω/2

π (k

Hz)

(c)1 mm

(f)

(e)

(d)

Figure 1.4: (a) Recorded time evolution of the observed beat signal between the

two cavity modes with N = 106 and P cav± = 2W. At time t = 0 the pumping of

the probe α− has been interrupted. (b) Numerical simulation with the temperature

adjusted to 200µK. (c) The symbols (X) trace the evolution of the beat frequency

after switch-off. The dotted line is based on a numerical simulation. The solid line is

obtained from numerical simulation with the assumption that the fraction of atoms

participating in the coherent dynamics is 1/10 to account for imperfect bunching.

(d) Absorption images of a cloud of 6×106 atoms recorded for high cavity finesse at0ms and (e) 6ms after switching off the probe beam pumping. All images are taken

after a 1ms free expansion time. (f) This image is obtained by subtracting from

image (e) an absorption image taken with low cavity finesse 6ms after switch-off.

The intracavity power has been adjusted to the same value as in the high finesse

case. Figure taken from Ref. [17].

pump and the atoms increases the detuning between probe and pump increases too

(see Fig. 1.4(c)). As a consequence, the collective recoil gives rise to a detectable

displacement of the atoms which has been indeed observed taking time-of-flight

absorption images of atomic cloud at various times (see Fig. 1.4(d),(e),(f)). Besides


these results, in experiment [17] a second set-up that differs from the original CARL

proposal has been used. The usual CARL dynamics never reaches a steady state and

the power of the reverse field decreases in time. Hence, in order to reach a stationary

regime, a friction force has been introduced through an optical molassa so that a

steady-state velocity of the atoms is reached when the velocity dependent dumping

force balances the CARL acceleration. As a consequence the reverse field too reaches

fixed detuning and amplitude, so that the lasing process becomes stationary. The

work on this subject to explain the experiment is still in progress [37].

1.5 Concluding remarks

By taking into account the translational degrees of freedom of the active medium,

we have described a mechanism that can lead to the exponential amplification of

a weak probe. Roughly speaking, we can interpret the process of amplification

as evolving in two steps: first, the external field creates a weak gain profile in

the frequency response of a collection of independent driven atoms and begins the

buildup of a spatial structure with the help of the atomic recoil; next the probe,

whose carrier frequencies lies within a selected gain region of the active medium,

undergoes exponential amplification. The role of the atomic recoil is essential to

this process: not only it is the cause of the emergence of the spatial grating pattern,

but it also reinforces the coherent growth of the signal to be amplified as energy is

transferred from the atoms to the probe field.

An alternative way of interpreting the probe amplification is to view it as the

reflection of the pump field from the moving grating pattern or as a kind of coherent

scattering from the bound states of the atoms.

We stress that even though we have demonstrated the amplification of a probe

signal, since the saturation value of the intensity and of the bunching is independent

of the initial value of the probe, the process can be initiated from spontaneous

emission noise.

Chapter 2

Quantum CARL

The realization of Bose Einstein condensation in dilute alkali gases [1, 2] opened the

possibility to study the coherent interaction between light and an ensemble of atoms

prepared in a single quantum state. For example, Bragg diffraction [38] of a BEC by

a moving optical standing wave can be used to diffract any fraction of the condensate

into a selectable momentum state, realizing an atomic beam splitter. In particular,

collective light scattering and matter-wave amplification caused by coherent center-

of mass motion of atoms in a condensate illuminated by a far off-resonant laser

[19, 39, 40] have been interpreted as superradiant Rayleigh scattering and can be

investigated using a quantum theory based on a quantum multi-mode extention of

the CARL model [21, 41, 42, 43]. The main drawback of the semiclassical model is

that, as it considers the center-of-mass motion of the atoms as classical, it cannot

describe the discreteness of the recoil velocity, as has been observed in the experiment

of Ref.[19]. The original CARL theory, which treats the atomic center-of-mass

motion classically, fails when the temperature of the atomic sample is below the

recoil temperature TR = ~ωR/kB, M is the atomic mass and kB is the Boltzmannconstant. So, to extend the model in the region of ultracold atoms, a quantum

mechanical description of the center-of-mass motion of the atoms must be included.

In this chapter we present a way to work out this program simply performing

a first quantization of the external variables θ and P of atoms [20]. Even if not

complete this model gives a simple description of all the features of the considered

system and in particular allows to define the main different regimes.

In the conservative regime (no radiation losses), the quantum model depends on

23

24 Chapter 2. Quantum CARL

a single collective parameter, ρ, that can be interpreted as the average number of

photons scattered per atom in the classical limit. When ρ À 1, the semiclassicalCARL regime is recovered, with many momentum levels populated at saturation.

On the contrary, when ρ ≤ 1, the average momentum oscillates between zero and~~q, and a periodic train of 2π hyperbolic secant pulses is emitted.

In the dissipative regime (large radiation losses) and in a suitable quantum limit

(ρ <√

2κ), a sequential superfluorescence scattering occurs, in which after each pro-

cess atoms emit a π hyperbolic secant pulse and populate a lower momentum state.

These results describe the regular arrangement of the momentum pattern observed

in the aforementioned experiments of superradiant Rayleigh scattering from a BEC.

2.1 First quantization

The atomic motion is quantized when the average recoil momentum is comparable

to ~~q where ~q = ~k2 − ~k1 is the difference between the incident and the scatteredwave vectors, i.e. the recoil momentum gained by the atom trading a photon via

absorbtion and stimulated emission between the incident and scattered waves. The

starting point of the following model is the classical model of equations (1.49-1.51)

derived in chapter 1

dθjdτ

= Pj (2.1)

dPjdτ

= − [Aeiθj + A∗e−iθj] (2.2)

dA

dτ= iδA +

1

N

N∑j=1

e−iθj (2.3)

for N two-level atoms exposed to an off-resonant pump laser, whose electric field has

a frequency ω2 = ck2 with a detuning from the atomic resonance, ∆20 = ω2 − ω0,much larger than the natural linewidth of the atomic transition, γ. The ‘probe

field’ has frequency ω1 = ω2 − ∆21 and electric field with the same polarization ofthe pump field. In the absence of an injected probe field, the emission starts from

fluctuations and the propagation direction of the scattered field is determined either

by the geometry of the condensate (as in the case of the MIT experiment [19], where

the condensate has a cigar shape) or by the presence of an optical resonator tuned

on a selected longitudinal mode.

2.1. First quantization 25

In order to quantize both the radiation field and the center-of-mass motion of the

atoms, we consider θj, pj = (ρ/2)Pj = Mvzj/~q and a = (Nρ/2)1/2A as quantumoperators satisfying the canonical commutation relations

[θ̂j, p̂j′

]= iδjj′

[â, â†

]= 1. (2.4)

With these definitions, Eqs.(2.1)-(2.3) are transformed into the Heisenberg equations

of motion

dθ̂jdτ

=2

ρp̂j (2.5)

dp̂jdτ

= −√

ρ

2N

[âeiθ̂j + â∗e−iθ̂j

](2.6)

dâ

dτ= iδâ +

√ρ

2N

N∑j=1

e−iθ̂j (2.7)

associated with the Hamiltonian:

Ĥ =1

ρ

N∑j=1

p̂2j + i

√ρ

2N

(N∑

j=1

â†e−iθ̂j − h.c.)− δâ†â =

N∑j=1

Hj(θ̂j, p̂j), (2.8)

where

Ĥj(θ̂j, p̂j) =1

ρp̂2j + i

√ρ

2N(â†e−iθ̂j − aeiθ̂j)− δ

Nâ†â (2.9)

We note that [Ĥ, Q̂] = 0, where Q̂ = â†â +∑

j p̂j is the total momentum in units

of ~q. In order to obtain a simplified description of a BEC as a system of Nnoninteracting atoms in the ground state, we use the Schrödinger picture for the

atoms (instead of the usual Heisenberg picture [44]), i.e.

|ψ(θ1, . . . , θN)〉 = |ψ(θ1)〉 . . . |ψ(θN)〉, (2.10)

where |ψ(θj)〉 obeys the single-particle Schrödinger equation,

i∂

∂τ|ψ(θj)〉 = Hj(θj, pj)|ψ(θj)〉. (2.11)

In this model we describe the scattered radiation field classically. Hence, considering

the corresponding c-number a of the field operator â (i.e. its expectation value),

eq.(2.7) yields:

da

dτ= iδa + g

N∑j=1

〈ψ(θj)|e−iθj |ψ(θj)〉. (2.12)


Let now expand the single-atom wavefunction on the momentum basis, |ψ(θj)〉 =∑n cj(n)|n〉j, where p̂j|n〉j = n|n〉j, n = −∞, . . .∞ and cj(n) is the probability

amplitude of the j-th atom having momentum −n~~q. Remembering that

[e±iθ̂i , p̂j ] = −δi,j e±iθ̂i and e±iθ̂j |n〉j = |n± 1〉j (2.13)

we obtainda

dτ= iδa + g

N∑j=1

c∗j(n + 1)cj(n). (2.14)

Introducing the collective density %̂ with matrix elements on the base {|n〉}

%m,n =1

N

N∑j=1

cj(m)∗cj(n)ei(m−n)δτ , (2.15)

a straightforward calculation yields, from Eqs.(2.12) and (2.15) , the following closed

set of equations:

d%m,ndτ

= i(m− n)δm,n%m,n+

ρ

2[A (%m+1,n − %m,n−1) + A∗ (%m,n+1 − %m−1,n)] (2.16)

dA

dτ=

∞∑n=−∞

%n,n+1 − κA, (2.17)

where δm,n = δ + (m + n)/ρ and we have redefined the field as A =√

2/ρNae−iδτ .

We have also introduced a damping term −κA in the field equation, whereκ = κc/ωRρ, κc = c/2L and L is the sample length along the probe propagation,

which provides an approximated model describing the escape of photons from the

atomic medium. In the presence of a ring cavity of length Lcav and reflectivity R,

κc = −(c/Lcav)lnR, as shown in the usual “mean-field” approximation [44].Eqs.(2.16) and (2.17) determine the temporal evolution of the density matrix

elements for the momentum levels. In particular, Pn = %n,n is the probability of

finding the atom in momentum level |n〉, 〈p̂〉 = ∑n n%n,n is the average momentumand

B =∑

n

%n,n+1 (2.18)

is the bunching parameter. Eqs.(2.16) and (2.17), as we will show in the next

chapter, are the equations for expectation values correspondent to those derived

2.1. First quantization 27

Figure 2.1: Classical limit of CARL for ρ À 1 in the case κ = 0. (a): |A|2 vs. τ asobtained from the classical eqs.(1)-(3) (dashed line) and from the quantum eqs.(7)

and (8) for ρ = 10 (solid line); (b): population level pn vs. n at the occurring

of the first maximum of |A|2, at τ = 12.4. The other parameters are δ = 0 andA(0) = 10−4. Figure taken from Ref. [20].

in a complete second quantized treatment which introduces bosonic creation and

annihilation operators of a given center-of-mass momentum [18].

For a constant field A, Eq.(2.16) describes a Bragg scattering process, in which

m− n photons are absorbed from the pump and scattered into the probe, changingthe initial and final momentum states of the atom from m to n. Conservation of

energy and momentum require that during this process ω1 − ω2 = (m + n) ωR, i.e.δm,n = 0. Eqs.(2.16) and (2.17) conserve the norm, i.e.

∑m %m,m = 1, and, when

κ = 0, also the total momentum 〈Q̂〉 = (ρ/2)|A|2 + 〈p̂〉.Fig. 2.1a shows |A|2 vs. τ , for κ = 0, δ = 0 and A(0) = 10−4, comparing the

semiclassical solution with the quantum solution in the semiclassical limit that

corresponds to ρ À 1: the dashed line is the numerical solution of Eqs.(2.1)-(2.3), fora classical system of N = 200 cold atoms, with initial momentum pj(0) = 0 (where

j = 1, . . . , N) and phase θj(0) uniformly distributed over 2π, i.e. unbunched; the

continuous line is the numerical solution of Eqs.(2.16) and (2.17) for ρ = 10 and

a quantum system of atoms initially in the ground state n = 0, i.e. with %n,m =

δn0δm0. We can notice that the quantum system behaves, with good approximation,

classically. Because the maximum dimensionless intensity is |A|2 ≈ 1.4, the constantof motion 〈Q̂〉 gives 〈p̂〉 ≈ −0.7ρ and the maximum average number of emitted


photons is about 〈â†â〉 ∼ Nρ. Hence in this limit the CARL parameter ρ canbe interpreted as the maximum average number of photons emitted per atom (or

equivalently, as the maximum average momentum recoil, in units of ~q, acquired bythe atom) in the classical limit. Fig.2.1b shows the distribution of the population

level Pn at the first peak of the intensity of Fig. 2.1a, for τ = 12.4. We observe

that, at saturation, twenty-five momentum levels are occupied, with an induced

momentum spread comparable to the average momentum.

2.2 Linear regime

Let us now consider the equilibrium state with no probe field, A = 0, and all the

atoms in the same momentum state n, i.e. with %n,n = 1 and the other matrix

elements zero. This is equivalent to assume the temperature of the system equal to

zero and all the atoms moving with the same velocity −n~~q, without spread. Thisequilibrium state is unstable for certain values of the detuning. In fact, by linearizing

Eqs.(2.16) and (2.17) around the equilibrium state, the only matrix elements giving

linear contributions are %n−1,n and %n,n+1, showing that in the linear regime the

only transitions allowed from the state n are those towards the levels n − 1 andn+1. Introducing the new variables Bn = %n,n+1 + %n−1,n and Dn = %n,n+1− %n−1,n,Eqs.(2.16) and (2.17) reduce to the linearized equations:

dBndτ

= −iδnBn − iρDn (2.19)

dDndτ

= −iδnDn − iρBn − ρA (2.20)

dA

dτ= Dn − κA, (2.21)

where δn = δ + 2n/ρ. Seeking solutions proportional to ei(λ−δn)τ , we obtain the

following cubic dispersion relation:

(λ− δn − iκ)(λ2 − 1/ρ2) + 1 = 0. (2.22)

In the exponential regime, when the unstable (complex) root λ dominates,

B(τ) ∼ ei(λ−δn)τ and, from Eq.(2.19), Dn = −ρλBn. The classical limit is recoveredfor ρ À 1 when κ = 0 or ρ À √κ when κ > 1 and δn ≈ δ, i.e. neglecting theshift due to the recoil frequency ωR. In this limit, maximum gain occurs for δ = 0,

2.2. Linear regime 29

with λ = (1− i√3)/2 when κ = 0 or λ = −(1 + i)/√2κ when κ > 1. Furthermore,|%n,n+1| ∼ |%n−1,n|, so that the atoms may experience both emission and absorbtion.This result can be interpreted in terms of single-photon emission and absorption by

an atom with initial momentum −n~~q. In fact, energy and momentum conservationimpose ω1 − ω2 = (2n ∓ 1)ωR (i.e. δn = ±1/ρ) when a probe photon is emittedor absorbed, respectively. Because in the semiclassical limit the gain bandwidth is

∆ω ∼ ωRρ À ωR when κ = 0 (or ∆ω ∼ κc À ωR when κ > 1) the atom can bothemit or absorbe a probe photon.

On the contrary, in the quantum limit the recoil energy ~ωR can not be ne-glected, and there is emission without absorbtion if |%n,n+1| ¿ |%n−1,n|, i.e.

Bn ≈ −Dn , λ ≈ 1ρ. (2.23)

This is true for ρ < 1 when κ = 0 with the unstable root

λ ≈ 1ρ

+δ′n

2− 1

2

√(δ′n)

2 − 2ρ (2.24)

(where δ′n = δn − 1/ρ), and for ρ <

√2κ when κ > 1 with

1),which are both less than the frequency difference 2ωR between the emission and

absorbtion lines. Hence, in the quantum limit the optical gain is due exclusively to

emission of photons, whereas in the semiclassical limit gain results from a positive

difference between the average emission and absorbtion rates. When κ = 0, the

resonant gain in the limit ρ < 1 is

GS = ωRρ

√ρ

2=

√3

8π

Ω02∆20

γ√

Neff , (2.26)

where γ = µ2k3/3π~²0 is the natural decay rate of the atomic transition, Ω0 isthe Rabi frequency of the pump and Neff = (λ

2/A)(c/γL)N is the effective atomic

number in the volume V = ΣL, where Σ and L are the cross section and the length

of the sample. When κ > 1, the resonant superfluorence gain in the limit ρ <√

2κ

is

GSF =ωRρ

2

2κ=

3

4πγ

(Ω0

2∆20

)2λ2

AN. (2.27)


Figure 2.2: Quantum limit of CARL for ρ < 1 in the case κ = 0. (a) |A|2 and (b)〈p〉 vs. τ , for ρ = 0.2, δ = 5, A(0) = 10−5 and the atoms initially in the state n = 0.We note that 〈p〉 = −(ρ/2)(|A|2 − |A(0)|2). Figure taken from Ref. [20].

The above results show that the combined effect of the probe and pump fields on a

collection of cold atoms in a pure momentum state n is responsible of a collective

instability that leads the atoms to populate the adjacent momentum levels n − 1and n + 1. However, in the quantum limit ρ < 1 when κ = 0 (or ρ <

√2κ when

κ > 1) conservation of energy and momentum of the photon constrains the atoms

to populate only the lower momentum level n− 1. This holds also in the nonlinearregime, as we have verified solving numerically Eqs.(2.16) and (2.17).

In the quantum limit above, the exact equations reduce to those for only three

matrix elements, %n,n, %n−1,n−1 and %n−1,n, with %n−1,n−1 +%n,n = 1. Introducing the

new variables Sn = Sn−1,n and Wn = %n,n − %n−1,n−1, Eqs.(2.16) and (2.17) reduceto the well-known Maxwell-Bloch equations [45]:

dSndτ

= −iδ′nSn +ρ

2AWn (2.28)

dWndτ

= −ρ(A∗Sn + h.c.) (2.29)dA

dτ= Sn − κA. (2.30)

When κ = 0 and δ′n = 0, if the system starts radiating incoherently by pure quantum-

mechanical spontaneous emission, the solution of Eqs.(2.28)-(2.30) is a periodic train

of 2π hyperbolic secant pulses [46] with

|A|2 = (2/ρ) Sech2[√

ρ

2(τ − τn)

], (2.31)

2.2. Linear regime 31

Figure 2.3: Sequential superfluorescent (SF) regime of CARL. (a) |A|2 and (b) 〈p〉vs. τ , for ρ = 2, δ = 0.5, κ = 10, and the same initial conditions of fig.2.2. Figure

taken from Ref. [20].

where τn = (2n + 1)ln(ρ/2)/√

ρ/2. Furthermore, the average momentum

〈p̂〉 = n + Th2[√

ρ

2(τ − τn)

]− 1 (2.32)

oscillates between n and n−1 with period τn. We observe that the maximum numberof photons emitted is 〈â†â〉peak = (ρN/2)|A|2peak = N , as expected. Fig. 2.2 showsthe results of a numerical integration of Eqs.(2.16) and (2.17), for κ = 0, ρ = 0.2 and

δ = 5, with the atoms initially in the momentum level n = 0 and the field starting

from the seed value A0 = 10−5. The intensity |A|2 and the average momentum 〈p̂〉

vs. τ are in agreement with the predictions of the reduced Eqs.(2.28)-(2.30).

In the superradiant regime, κ > 1, Eqs.(2.28)-(2.30) describe a single SF

scattering process in which the atoms, initially in the momentum state n, ‘decay’

to the lower level n − 1 emitting a π hyperbolic secant pulse, with intensity andaverage momentum

|A|2 = 14[κ2 + (δ′n)2]

Sech2[(τ − τD)

τSF

], 〈p̂〉 = n− 1

2

{1 + Th

[(τ − τD)

τSF

]}(2.33)

where τSF = 2(κ2 + δ′2n )/ρκ is the ‘superfluorescence time’ [28], the delay time

is τD = τSF Arcsech(2|Sn(0)|) ≈ −τSF ln√

2|Sn(0)| and |Sn(0)| ¿ 1 is the initialpolarization.

Figures 2.3a and b shows |A|2 and 〈p̂〉 vs. τ calculated solving Eqs.(2.16) and(2.17) numerically with κ = 10, ρ = 2, δ = 0.5 and the same initial conditions of Fig.


2.2. We observe a sequential SF scattering, in which the atoms, initially in the level

n = 0, change their momentum by discrete steps of ~~q and emit a SF pulse duringeach scattering process. We observe that for δ = 1/ρ the field is resonant only with

the first transition, from n = 0 to n = −1; for a generic initial state n, resonanceoccurs when δ = (1 − 2n)/ρ, so that in the case of Fig. 2.3a the peak intensityof the successive SF pulses is reduced (by the factor 1/[κ2 + (2n/ρ)2]) whereas the

duration and the delay of the pulse are increased. However, the pulse retains the

characteristic Sech2 shape and the area remains equal to π, inducing the atoms to

decrease their momentum by a finite value ~~q. We note that, although the SF timein the quantum limit (τSF = 2κ/ρ at resonance) can be considerable longer than

the characteristic superradiant time obtained in the classical limit, τSR =√

2κ, the

peak intensity of the pulse in the quantum limit is always approximately half of the

value obtained in the semiclassical limit (see Ref.[47] for details).


We have shown that the CARL model describing a system of atoms in their momen-

tum ground state (as those obtained in a BEC) and properly extended to include a

quantum-mechanical description of the center-of-mass motion, allows for a quantum

limit in which the average atomic momentum changes in discrete units of the photon

recoil momentum ~~q and reduce to the Maxwell-Bloch equations for two momen-tum levels. The behavior of the system is different for conservative and dissipative

regimes. The regular arrangement of momentum pattern observed in the superradi-

ant Rayleigh scattering experiments with BECs (see also chapter 4 for details) can

be interpreted as being due to the sequential superfluorescence scattering.

Chapter 3

Quantum field theory

In this chapter we derive a fully quantized model of a gas of bosonic two-level atoms

which interact with a strong, classical, undepleted pump laser and a weak, quantized

optical ring cavity mode, both of which are as usual assumed to be tuned far away

from atomic resonances. Starting from the second-quantized hamiltonian of the

system, we will write an effective model for the time evolution of the ground state

atomic field operator and of the probe field operator, adiabatically eliminating the

excited state atomic field operator and including effects of atom-atom collisions [48].

3.1 The CARL-BEC model

The second-quantized Hamiltonian of the system is

Ĥ = Ĥatom + Ĥprobe + Ĥatom−probe + Ĥatom−pump + Ĥatom−atom, (3.1)

where Ĥatom and Ĥprobe give the free evolution of the atomic field and the probemode respectively, Ĥatom−probe and Ĥatom−pump describe the dipole coupling betweenthe atomic field and the probe mode and pump laser, respectively, and Ĥatom−atomcontains the two-body s-wave scattering collisions between ground state atoms.

The free atomic Hamiltonian is given by

Ĥatom =∫

d3z

[Ψ̂ †g (z)

(− ~

2

2m∇2 + Vg(z)

)Ψ̂g(z)

+ Ψ̂ †e (z)(− ~

2

2m∇2 + ~ω0 + Ve(z)

)Ψ̂e(z)

], (3.2)

33

34 Chapter 3. Quantum field theory

where m is the atomic mass, ωa is the atomic resonance frequency, Ψ̂e(z) and Ψ̂g(z)

are the atomic field operators for excited and ground state atoms respectively, and

Vg(z) and Ve(z) are their respective trap potentials. The atomic field operators obey

the usual bosonic equal time commutation relations[Ψ̂j(z), Ψ̂

†j′(z

′)]

= δj,j′δ3(z− z′) (3.3)

[Ψ̂j(z), Ψ̂j′(z

′)]

= [Ψ̂ †j (z), Ψ̂†j′(z

′)] = 0, (3.4)

where j, j′ = {e, g}. The free evolution of the probe mode is governed by theHamiltonian

Ĥprobe = ~ck1Â†Â, (3.5)where c is the speed of light, k1 is the magnitude of the probe wave number k1, and

Â and Â† are the probe photon annihilation and creation operators, satisfying the

boson commutation relation [Â, Â†] = 1. The probe wavenumber k1 must satisfy

the periodic boundary condition of the ring cavity, k1 = 2π`/L, where the integer `

is the longitudinal mode index, and L is the length of the cavity.

The atomic and probe fields interact in the dipole approximation via the Hamil-

tonian

Ĥatom−probe = −i~g1Â∫

d3zΨ̂ †e (z)eiks·zΨ̂g(z) + H.c., (3.6)

where g1 = µ[ck1/(2~²0LS)]1/2 is the atom-probe coupling constant. Here µ isthe magnitude of the atomic dipole moment, and S is the cross-sectional area of

the probe mode in the vicinity of the atomic sample (where it is assumed to be

approximately constant across the length of the atomic sample).

In addition, the atoms are driven by a strong pump laser, which is treated classi-

cally and assumed to remain undepleted. The atom-pump interaction Hamiltonian

is given in the dipole approximation by

Ĥatom−pump = ~Ω2

e−iω2t∫

d3zΨ̂ †e (z)eik2·zΨ̂g(z) + H.c., (3.7)

where Ω is the Rabi frequency of the pump laser, related to the pump intensity I by

Ω2 = 2µ2I/~2²0c, ω2 is the pump frequency, and k2 ≈ ω2/c is the pump wavenumber.The approximation indicates that we are neglecting the index of refraction inside

the atomic gas, as we assume a very large detuning ∆20 = ω2 − ω0 between thepump frequency and the atomic resonance frequency.

3.1. The CARL-BEC model 35

Finally, the collision Hamiltonian is taken to be

Ĥatom−atom = 2π~2σ

m

∫d3zΨ̂ †g (z)Ψ̂

†g (z)Ψ̂g(z)Ψ̂g(z), (3.8)

where σ is the atomic s-wave scattering length. This corresponds to the usual s-wave

scattering approximation, and leads in the Hartree approximation to the standard

Gross-Pitaevskii equation for the ground state wavefunction (in the absence of the

driving optical fields).

We limit ourselves to the case where the pump laser is detuned far enough away

from the atomic resonance that the excited state population remains negligible,

a condition which requires that ∆ À γa. In this regime the atomic polarizationadiabatically follows the ground state population, allowing the formal elimination

of the excited state atomic field operator.

First we write the Heisenberg equation of motion for the field operators. The

commutation relation with the Hamiltonian are[Ψ̂g(z), Ĥprobe

]=

[Ψ̂e(z), Ĥprobe

]=

[Ψ̂e(z), Ĥatom−atom

]= 0 (3.9)

[Â, Ĥatom

]=

[Â, Ĥatom−pump

]=

[Â, Ĥatom−atom

]= 0 (3.10)

[Ψ̂g(z), Ĥatom

]=

(− ~

2

2m∇2 + Vg(z)

)Ψ̂g(z) (3.11)

[Ψ̂g(z), Ĥatom−probe

]= i~g1Â†Ψ̂e(z)e−ik1·z (3.12)

[Ψ̂g(z), Ĥatom−pump

]=

~Ω2

eiω2tΨ̂e(z)e−ik2·z (3.13)

[Ψ̂g(z), Ĥatom−atom

]=

4π~2σm

Ψ̂ †g (z)Ψ̂g(z)Ψ̂g(z) (3.14)[Ψ̂e(z), Ĥatom

]=

(− ~

2

2m∇2 + ~ω0 + Ve(z)

)Ψ̂e(z) (3.15)

[Ψ̂e(z), Ĥatom−probe

]= −i~g1Âeik1·zΨ̂g(z) (3.16)

[Ψ̂e(z), Ĥatom−pump

]=

~Ω2

e−iω2teik2·zΨ̂g(z) (3.17)[Â, Ĥprobe

]= ~ck1Â (3.18)

[Â, Ĥatom−probe

]= i~g1

∫d3zΨ̂ †g (z)e

−ik1·zΨ̂e(z) (3.19)

so the equation of motions read

i~dΨ̂g(z)

dt=

(− ~

2

2m∇2 + Vg(z) + 4π~

2σ

mΨ̂ †g (z)Ψ̂g(z)

)Ψ̂g(z)


+

(i~gsÂ†e−iks·z +

~Ω2

eiωte−ik·z)

Ψ̂e(z)

i~dΨ̂e(z)

dt= ~ω0Ψ̂e(z) +

(−i~g1Âeik1·z + ~Ω

2e−iω2teik2·z

)Ψ̂g(z) (3.20)

i~dÂ

dt= ~ck1Â + i~g1

∫d3rΨ̂e(z)e

−ik1·zΨ̂ †g (z) (3.21)

where we have dropped the kinetic energy and trap potential terms under the as-

sumption that the lifetime of the excited atom, which is of the order 1/∆, is so small

that the atomic center-of-mass motion may be safely neglected during this period.

For the same reason, we are justified in neglecting collisions between excited atoms,

or between excited and ground state atoms in the collision Hamiltonian (3.8).

We proceed by introducing the operators Ψ̂ ′e(z) = Ψ̂e(z)eiω2t and â = Âeiω2t,

which are slowly varying relative to the optical driving frequency. The new excited

state atomic field operator obeys then the Heisenberg equation of motion

i~d

dtΨ̂ ′e(z) = −~∆20Ψ̂ ′e(z) +

[~Ω2

eik2·z − i~g1âeik1·z]

Ψ̂g(z), (3.22)

We now adiabatically solve for Ψ̂ ′e(z) by formally integrating Eq. (3.22) under the

assumption that Ψ̂g(z) varies on a time scale which is much longer than 1/∆20. This

yields

Ψ̂ ′e(z, t) ≈1

∆20

[Ω

2eik2·z − ig1â(t)eik1·z

]Ψ̂g(z, t)

− 1∆20

[Ω

2eik2·z − ig1â(0)eik1·z

]Ψ̂g(z, 0)e

i∆t

+ Ψ̂ ′e(z, 0)ei∆20t. (3.23)

The third term on the r.h.s. of Eq. (3.23) can be neglected for most considerations

if we assume that there are no excited atoms at t = 0, so that this term acting on

the initial state gives zero. The second term may also be neglected, as it is rapidly

oscillating at frequency ∆20, and thus its effect on the ground state field operator

is negligible when compared to that of the first term, which is non-rotating. It is

useful to keep them temporarily to demonstrate that the commutation relation for

Ψ̂e(z) is preserved (to order 1/∆20) by the procedure of adiabatic elimination.

Dropping the unimportant terms, and then substituting Eq. (3.23) into the

equations of motion for Ψ̂g(z) and for Â, we arrive at the effective Heisenberg equa-

tions of motion for the ground state field operator and for the probe field operator

3.2. Coupled-modes equations 37

(CARL-BEC model)

i~d

dtΨ̂g(z) =

[−~22m

∇2 + Vg(z) + 4π~2σ

mΨ̂ †g (z)Ψ̂g(z)

+i~g(â†e−iq·z − âeiq·z)

+~(

Ω2

4∆20+

4∆20g2

Ω2â†â

)]Ψ̂g(z), (3.24)

i~d

dtâ = −~δ̃â + i~g


−iq·zΨ̂g(z), (3.25)

where we have introduced the new coupling constant g = g1Ω/∆20 that contains the

parameters of the pump field. The recoil momentum kick the atom acquires from

the two-photon transition is q = k1−k2, the detuning between the pump and probefields is δ21 = ω2−ω1 and the probe frequency is given by ω1 ≈ ck1, again assumingthat the index of refraction inside the condensate is negligible. The second term in

Eq. (3.24) is simply the optical potential formed from the counterpropagating pump

and probe light fields, and the last term gives the spatially independent light shift

potential, which can be thought of as cross-phase modulation between the atomic

and optical fields.

3.2 Coupled-modes equations

We assume that the atomic field is initially in a BEC with mean number of condensed

atoms N . Furthermore, we assume that N is very large and that the condensate

temperature is small compared to the critical temperature. These assumptions allow

us to neglect the non-condensed fraction of the atomic field. Thus this model does

not include any effect of condensate number fluctuations. We introduce creation

and annihilation operators for the atoms of a definite momentum p = n~q. So wesuppose we can write

Ψ̂(z) =+∞∑n=0

ĉnΦn(z) (3.26)

where Φ0(z) is the condensate ground state that satisfies the time independent

Gross-Pitaevskii equation

(~2

2m∇2 − Vg(z)− 4π~

2σ

mN |Φ0(z)|2

)Φ0(z) = 0. (3.27)


Φn(z) for n 6= 0 are the n-th side modes with momentum n~qΦn(z) = Φ0(z)e

inq·z. (3.28)

and ĉm are bosonic operators obeying the commutation relations

[ĉn, ĉ†n′ ] =

∫d3zΦ∗n(z)Φn′(z) = δnn′ [ĉn, ĉn′ ] = 0 (3.29)

We are assuming that the states Φn(z) form a complete orthonormal system. In

general this is non true, as the overlap integrals

〈Φn|Φm〉 =∫

d3zΦ∗n(z)Φm(z) =∫

d3z|Φ0(z)|2ei(n−m)q·z (3.30)are not zero for n 6= m and are not 1 for n = m. For most condensate sizes and trapconfigurations, however, these integrals are many orders of magnitude smaller than

unity. As a result, for typical condensate, the orthogonality approximation yields ac-

curate results. By properly taking into account the non-orthogonality of the atomic

field modes, it can be shown that the only surviving effect in the linearized theory

(see next section) is the modification of the atomic polarization term in the equa-

tion of motion for the probe field (3.25) to include a second scattering mechanism

in which a condensate scatters a photon without changing its center of mass state.

As a consequence of momentum conservation, this process is suppressed by a factor

〈Φn0|Φn0−1〉 relative to the process which transfers the atom from the condensate inthe state n0 to the side mode state n0− 1. Bose enhancement, on the other hand, isstronger for this transition by a factor

√N , because we now have N identical bosons

in both the initial and final states. Thus it is the product√

N〈Φn0|Φn0−1〉 whichmust be negligible if we have to make the orthogonality approximation.

From Eq. (3.26) the atomic field operator which annihilates an atom in the n-th

condensate side mode is defined

ĉn =

∫d3zΦ∗n(z)Ψ̂(z) (3.31)

Taking the derivative with respect to time and substituting Eq.(3.24) we obtain

i~d

dtĉn =

n2(~q)2

2mĉn + ~

( |Ω|24∆20

+g21

∆20â†â

)ĉn + i~g

(â†ĉn+1 − âĉn−1

)

+∑m

ĉm

∫d3zΦ∗n(z)e

inq·z(−~2∇2

2m+ V (z)

)Φ0(z)

+β∑

m,k,l

ĉ†mĉkĉl

∫d3zΦ∗nΦ

∗m(z)Φk(z)Φl(z) (3.32)

3.2. Coupled-modes equations 39

and inserting the Gross-Pitaevskii Eq. (3.27) finally

i~d

dtĉn =

n2(~q)2

2mĉn + ~

( |Ω|24∆20

+g21

∆20â†â

)ĉn + i~g

(â†ĉn+1 − âĉn−1

)

−βN∑m

ĉm

∫d3zΦ∗n(z) |Φ0(z)|2 Φm(z)

+β∑

m,k,l

ĉ†mĉkĉl

∫d3zΦ∗n+m |Φ0(z)|2 Φl+k(z) (3.33)

Substituting Eq. (3.26) in Eqs. (3.25) we obtain for the probe field operator

dâ

dt= i∆21â + g


−iq·z Ψ̂g(z). (3.34)

The source of the field equation (3.34) is the bunching operator

B̂ =


−iq·z Ψ̂g(z) (3.35)

If we consider an ideal condensate with a constant atomic density the ground state

is independent from position variables Φ0(z) = Φ0 = 1/√

V and eq.(3.33) takes the

simpler form

i~d

dtĉn =

n2(~q)2

2mĉn + ~

( |Ω|24∆20

+g21

∆20â†â

)ĉn + i~g

(â†ĉn+1 − âĉn−1

)

−βNV

ĉn +β

V

∑

m,k

ĉ†mĉkĉn+m−k (3.36)

and now the bunching operator is given by

B̂ =∞∑

n=−∞ĉ†nĉn+1 (3.37)

Generally in experiments that involves the CARL mechanism, like for example

superradiant Rayleigh scattering, the laser pulse is applied when the trap is com-

pletely switched off and the condensate is in expansion, so it can be interesting to

study the model when the effects of trap potential and of collisions are negligible.

In this regime we get to the following model for the coupled modes

d

dtĉn = −iωRn2ĉn + g

(â†ĉn+1 − âĉn−1

)(3.38)

dâ

dτ= i∆21â + g

∞∑n=−∞

c†ncn+1. (3.39)


We note that Eqs.(3.38) and (3.39) conserve the number of atoms, i.e.∑

n ĉ†nĉn = N ,

and the total momentum, Q̂ = â†â +∑

n nĉ†nĉn. Defining the operators %̂m,n = ĉ

†mĉn

from Eq.(3.38) we derive

d

dt%̂mn = iωR(m

2 − n2)%̂mn+g

{â (%̂m+1,n − %̂m,n−1) + â† (%̂m,n+1 − %̂m−1,n)

}(3.40)

Taking the expectation values for the operators, taking scaled variables and with

the substitution A =√

2/ρN〈â〉e−iδτ , Eqs. (3.40) and (3.39) are equivalent to Eqs.(2.16) and (2.17) introduced with first quantization in chapter 2. A more realistic

and complete model should take into account even effects of atomic decoherence

and cavity losses. We will see possible approaches to this problem in some of the

next chapters (see chapters 5 and 8), modifying the model in the proper way for the

considered situation.

3.3 Linearized three-mode model

From Eq. (3.40), we see that the first-order side modes are optically coupled to both

the condensate mode and to second-order side modes. For times short enough that

the condensate is not significantly depleted, the coupling back into the condensate

is subject to Bose enhancement due to the presence of ∼ N identical bosons in thismode. The coupling to the second-order side mode, in contrast, is not enhanced.

Hence for these time scales, the higher-order side modes are not expected to play a

significant role. These arguments suggest developing an approach where, assuming

that all N atoms are initially in the condensate mode n0 with momentum n0~q, thethree atomic field operators ĉn0 , ĉn0−1, and ĉn0+1 play a predominant role. Therefore,

we expand the atomic field operator as

Ψ̂g(z) = 〈z|Φn0〉ĉn0 + 〈z|Φn0−1〉ĉn0−1 + 〈z|Φn0+1〉ĉn0+1 + ψ̂(z), (3.41)

where the field operator ψ̂(z) acts only on the orthogonal complement to the sub-

space spanned by the state vectors |Φn0〉, |Φn0−1〉, and |Φn0+1〉. As a result, ψ̂(z)commutes with the creation operators for the three central modes.

With the assumption of negligible condensate depletion we can simply drop the

operator ĉn0 substituting it with its mean value 〈ĉn0〉 ≈√

Ne−in2τ/ρ. The system is

3.4. Concluding remarks 41

unstable for certain values of the detuning ∆. In fact, by linearizing Eqs.(3.38) and

(3.39) around the equilibrium state, the only equations depending linearly on the

radiation field are those for ĉn0−1 and ĉn0+1. Hence, in the linear regime, the only

transitions involved are those from the state n0 towards the levels n0−1 and n0 +1.With respect to scaled variables and introducing the operators

â1 = ĉn0−1ei(n20τ/ρ+∆τ) (3.42)

â2 = ĉn0+1ei(n20τ/ρ−∆τ) (3.43)

â3 = âe−i∆τ , (3.44)

Eqs.(3.38) and (3.39) reduce to the linear equations for three coupled harmonic

oscillator operators:

dâ†1dτ

= −iδ−â†1 +√

ρ/2â3 (3.45)

dâ2dτ

= −iδ+â2 −√

ρ/2â3 (3.46)

dâ3dτ

=√

ρ/2(â†1 + â2), (3.47)

with Hamiltonian

Ĥ = δ+â†2â2 − δ−â†1â1 + i

√ρ

2[(â†1 + â2)â

†3 − (â1 + â†2)â3], (3.48)

where δ± = δ ± 1/ρ and δ = ∆ + 2n0/ρ = (ω2 − ω1 + 2n0ωR)/ρωR. Hence, thedynamics of the system is that of three parametrically coupled harmonic oscillators

â1, â2 and â3 [49], which obey the commutation rules [âi, âj] = 0 and [âi, â†j] = δij

for i, j = 1, 2, 3. Note that the Hamiltonian (3.48) commutates with the constant of

motion

C = â†2â2 − â†1â1 + â†3â3. (3.49)We will solve exactly this model in chapter 6.


We have deduced an appropriate quantum field theory that extends into the ultra-

cold regime of BEC the CARL model, so that the unique coherence properties of

the condensates might be further understood and exploited by the interaction with


dynamical light fields. In the limit of no collisions and consider

Collective Atomic Recoil in Ultracold Atoms: Advances and … · 2007. 3. 17. · 6 Entanglement...

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Transcript of Collective Atomic Recoil in Ultracold Atoms: Advances and … · 2007. 3. 17. · 6 Entanglement...