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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
2.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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
3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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
5.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
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
6.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
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
7.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
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
8.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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
10 Chapter 1. Classical CARL
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)
1.1. Equations of motion 11
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].
12 Chapter 1. Classical CARL
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.
1.1. Equations of motion 13
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.
14 Chapter 1. Classical CARL
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
16 Chapter 1. Classical CARL
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.
18 Chapter 1. Classical CARL
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
20 Chapter 1. Classical CARL
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
22 Chapter 1. Classical CARL
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)
26 Chapter 2. Quantum CARL
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
28 Chapter 2. Quantum CARL
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)
30 Chapter 2. Quantum CARL
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.
32 Chapter 2. Quantum CARL
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).
2.3 Concluding remarks
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 = ~ck1†Â, (3.5)where c is the speed of light, k1 is the magnitude of the probe wave number k1, and
 and † are the probe photon annihilation and creation operators, satisfying the
boson commutation relation [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~g1†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)
36 Chapter 3. Quantum field theory
+
(i~gs†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
∫d3zΨ̂ †g (z)e
−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)
38 Chapter 3. Quantum field theory
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
∫d3zΨ̂ †g (z)e
−iq·z Ψ̂g(z). (3.34)
The source of the field equation (3.34) is the bunching operator
B̂ =
∫d3zΨ̂ †g (z)e
−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)
40 Chapter 3. Quantum field theory
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) + ↠(%̂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.
3.4 Concluding remarks
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
42 Chapter 3. Quantum field theory
dynamical light fields. In the limit of no collisions and consider