INTERACTIONS OF ELECTROMAGNETIC RADIATION WITH BOSE EINSTEIN CONDENSATES: MANIPULATING ULTRA–COLD...

January 16, 2013 8:55 WSPC/Guidelines-IJMPB S021797921330003X

International Journal of Modern Physics BVol. 27, No. 6 (2013) 1330003 (68 pages)c© World Scientific Publishing Company

DOI: 10.1142/S021797921330003X

INTERACTIONS OF ELECTROMAGNETIC RADIATION

WITH BOSE EINSTEIN CONDENSATES:

MANIPULATING ULTRA–COLD ATOMS WITH LIGHT

FEDERICA CATTANI

School of Mathematics, University of Southampton, Highfield Campus

Southampton, SO17 1BJ, UK

[email protected]

ARKADY KIM

Institute of Applied Physics, Russian Academy of Sciences,

603950 Nizhny Novgorod, Russia

[email protected]

MIETEK LISAK∗ and DAN ANDERSON†

Department of Earth and Space Sciences, Chalmers University of Technology,

SE 412 96 Gothenburg, Sweden∗[email protected]

†[email protected]

Received 26 November 2012Accepted 11 December 2012Published 16 January 2013

A review of models describing the interactions of ultra-cold atoms and laser light is given.Both semi-classical and fully quantum models are presented with particular attentiongiven to the introduction of local field effects. Some possible effects of self-localizationand guiding, consequences of such interactions, are discussed.

Keywords: Ultra-cold atoms; dipole interactions; mutual guiding; coupled nonlinearSchrodinger equations; solitons.

1. What and Why, the Structure of this Review

Electromagnetic radiation naturally interacts with matter. Much progress has been

achieved exploiting the interaction of magnetic fields with neutral atoms, which

has indeed become indispensible with the introduction of magneto-optical traps.

Besides, the interaction of the radiation with the atom internal degrees of freedom

is of fundamental importance, but we shall focus here on the effects of electric fields

on the external degrees of freedom of neutral atoms. With the aim of understanding

the reasons why these investigations are now attracting so much interest, we shall

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http://dx.doi.org/10.1142/S021797921330003X

mailto:[email protected]





F. Cattani et al.

start from a brief review of the original theoretical proposals and experimental

achievements, to move than to an overview of the mathematical description of the

physics of the interaction of laser light with ultra-cold atoms.

Even before the advent of quantum mechanics, Helmholtz presented us with the

tools to show how a dielectric undergoes infinitesimal deformations when subject

to the action of an electric field,1,2 showing how to calculate the force exerted by

such field on matter. In the 1960s the theory was applied to the first experimental

manipulations of neutral atoms with the aim of trapping, accelerating and guid-

ing the atoms, and we now have a much wider landscape of possible applications

(Sec. 2.1).

Essential to all the advancements has been the development in the 1970s and

1980s of theories and techniques to lower the atoms temperature, laser cooling being

one further application of light-atom interactions. In particular, the achievement of

Bose–Einstein condensation in the 1990s has opened the door to major advances

in applications and fundamental investigations. The landscape of applications is

extremely vaste, ranging from quantum enhanced metrology to atom interferome-

try and lithography. On the other hand, the importance of these systems from a

fundamental point of view cannot be underestimated. In fact, a Bose–Einstein con-

densate (BEC) provides the experimental realization of a matter wave: Quantum

mechanics on a macroscopic scale, atoms showing their wavelike nature (Sec. 2.2).

The nature of BECs thus leads to a second aspect of the story, of interest from

a more theoretical point of view. It is well-known that light propagating through

a medium can affect its properties and the propagation itself can in turn be modi-

fied. This idea is at the basis of the field of nonlinear optics.3 In analogy, one can

see the ultra-cold BEC atoms playing the role of coherent electromagnetic radia-

tion and laser radiation playing the role of the medium through which the matter

wave propagates. If atoms behave in the same way as laser light, how far can the

semi-classical models, known to work for laser radiation, be pushed to describe the

physical behavior of ultra–cold atoms? (Sec. 3). And what sort of phenomena can

be predicted by applying to these systems models valid for coherent electromagnetic

radiation? (Sec. 4).

2. The Action of Light on Atoms: Atoms as the Medium

The mechanical effect of electromagnetic radiation on a dielectric was studied al-

ready in the XIX century by Helmholtz1 and the basic ideas from the point of view

of classical electromagnetism are discussed in details in Landau’s textbook.4 This

mechanical action is now routinely employed to focus, steer, guide and accelerate

neutral atoms. However, given the importance of localizing atoms, a large effort in

the applications of the theory has been devoted to atom trapping, with a major step

forward made possible by the ability of lowering the atom temperature to regimes

where the trapping action of light is much more effective.

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Manipulating BECs with Light

2.1. From the early years to the 1990s: Deflecting, accelerating,

trapping neutral atoms

Progress in controlling matter with light was achieved already before the advent

of the laser, but the availability of a source of electromagnetic radiation with the

properties of coherence and monochromaticity which typically characterize laser

radiation, played an important role in giving impulse to this field. Analogously to

the action of the ponderomotive force on charged particles,5 laser radiation can

exert a mechanical force on neutral particles by inducing an electric dipole. The

fundamental mathematics have been reviewed by several authors,6–10 here we shall

only give a brief summary of the main physical points and we shall come back to a

more detailed discussion of the models in Sec. 3.

If a neutral atom possesses an electric dipole moment d, it is possible to exploit

the dipole interaction with an electric field E which, in its simplest expression, will

produce an interaction energy Vd given by Vd = −d ·E. The electric dipole quantum

operator d has vanishing elements if taken between states with the same parity but

the effect of light, exciting the atom, can set it into a superposition of states of op-

posite symmetry thus making the dipole interaction possible. The way this happens

is via exchange of linear momentum between the photons and the atoms which, not

considering the physics of the internal atomic degrees of freedom, results in a radia-

tion force that can be used to manipulate the external degrees of freedom, position

of the center of mass and linear momentum. The classical mathematical descrip-

tion of the nature of this interaction is embodied in the expression for the complex

atomic polarizability α which relates the dipole moment to the electric field as, in

the simplest approximation, d = αE. The dominant term in the dipole interaction

energy (proportional to the scalar product of d and E) will depend on the real part

of α and the intensity of the electric field, neglecting terms oscillating at twice the

frequency of the carrier wave. This gives rise to a dispersive force proportional to

the gradient of the electric field intensity,5,11 also called ponderomotive, striction or

dipole force. An absorptive contribution comes from the complex nature of α which

describes the scattering of photons from the atoms as they undergo absorption and

spontaneous emission. This gives rise to the radiation pressure force, a dissipative

force which, because of the symmetry of the spontaneous emission process, has a

major effect in the direction parallel to the wave vector of the electromagnetic wave.

The atom polarizability depends on the detuning between the frequency of the in-

cident radiation and that of the nearest optical transition of the atom, as can be

seen from a semi-classical model.8 A more accurate description shows that another

important parameter is the homogenous width of the transition and far-off reso-

nance can be related also to a frequency detuning larger than this width. For large

detunings, the dispersive force is the dominant term, whereas the scattering-related

radiation pressure dominates at resonance. Besides, the dispersive force changes its

nature depending on the sign of the detuning: Blue detuning, i.e., radiation fre-

quency higher than the resonance frequency, corresponds to a repulsive force and

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F. Cattani et al.

the atoms are pushed away from the region of high radiation intensity, while the

opposite happens for red detuning.

Classical results were obtained by exploiting the radiation pressure exerted by

light on matter. Lethokov demonstrated a “dragging effect” depending on the sign of

the detuning for atoms moving parallel to the wave front of a standing light wave.12

Ashkin discussed13 and measured14 both acceleration and deflection of micron-

size particles due to the action of a continuous laser. As pointed out by the same

author,15 the effect on the atom velocity is due to the radiation pressure, whereas the

dipole force acts in deflecting the particles, effects still important in today’s optical

tweezers used in biology.16 Focusing and steering of a beam of neutral sodium atoms

due to the transverse gradient dipole force exerted by a single-mode cw laser were

experimentally observed by Bjorkholm et al.17 These authors probed the effect of

the detuning, the change in nature of the dipole force depending on its sign and

the relative importance of the longitudinal radiation pressure force depending on

its magnitude. Standing waves came to play an important role as well: Letokhov’s

work in 196812 inspired the idea that neutral atoms could be trapped at the nodes

or antinodes of a strong standing wave depending on the sign of the detuning, the

origin of today’s optical lattices. Kazantsev found that these forces could be made

into velocity-dependent forces,18 an idea which will lead to optical molasses and

laser cooling. Investigations quickly moved to studying the possibility of trapping

and also slowing down neutral atoms.

The different ideas for atom trapping based on the interaction between light

and atoms can be classified according to the detuning of the radiation, from quasi-

resonant and near-resonant to far-off resonant, each regimes exploiting an aspect

of the induced dipole forces.19 Both the resonant radiation pressure and the dipole

force have been used for atom trapping but, for the scattering force of radiation

pressure to work, it was a case of overcoming an inherent instability20 which would

prevent the force from pointing inward from every point on a closed surface. It

was proposed to make use of time dependent fields21 or to exploit the internal

degrees of freedom of the atom via optical pumping and external fields22 in order

to avoid the obstacle. In particular, placing the atoms in an external magnetic

field, led to the nowaday routinely used magneto-optical trap. Realized on sodium

atoms for the first time in 1987,23 this trap uses a combination of magnetic forces

to modify the energy levels of the atoms thus allowing the optical forces to trap

and cool them, as will be briefly discussed in the next section. However, working

close to resonance involves optical excitations and strong perturbations of the atom

dynamics. Although, as we shall briefly discuss when introducing laser cooling,

these aspects can be made useful, theory and experiments on atom trapping have

moved also in the direction of employing the nonresonant dipole force.

The main difference from the practical point of view is that working off–

resonance keeps optical excitations low but yields a correspondingly weaker force,

giving rather shallow potential wells, so that the atom thermal energy can be enough

for the atoms to escape the trap. As noticed by Salomon et al.,24 trapping neutral

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atoms at the nodes (blue detuned radiation) or anti-nodes (red detuned radiation)

of a standing wave, requires that the atoms kinetic energy be low enough for the

dipole potential to overcome it. Similarly, if one wants to trap neutral atoms at

room temperature at the focus of a red-detuned laser beam, the detuning cannot

be too large and the intensity of the radiation has to be quite high, both points

leading to significant excitation. There is therefore a balance to be found between

the detuning (the larger it is the weaker the dipole trapping potential, but also

the weaker the excitations and spontaneous emission), and the atom temperature

(the weaker the trapping potential the lower the temperature must be to achieve

trapping).25 Once efficient methods to lower the atom temperature started to be

applied, not least the realization of radiation pressure magneto-optical traps, opti-

cal dipole traps based on the off-resonance dipole force given by red-detuned laser

radiation began to play a more important role.26 Their performance improved with

advances in cooling methods which allowed for larger detunings and, consequently,

a weakened role of radiation pressure due to spontaneous scattering and less diffu-

sive heating. From the trapping of about 500 atoms for a few milliseconds, traps

lifetimes reached 200 milliseconds in the 1990s27 with negligible heating losses, up

to hundreds of seconds at the beginning of the XXI century.28

Different geometries can then be envisaged. The simpler case of highly

anisotropic, quasi one-dimensional, trapping is realized at the focus of a single

Gaussian laser beam. The idea of an optical waveguide has been proposed29 and

realized with Rubidium atoms30 to channel the atoms along a hollow cylindrical

waveguide via the red detuned radiation of its fundamental mode. Both guiding and

cooling were observed for Rubidium atoms with a red detuned Gaussian laser beam

acting as waveguide.31 In this experiment, the absence of a waveguide for the laser

radiation allowed to show how a divergent beam led to cooling of the atoms. More

complex three-dimensional configuration of trapping are obtained at the crossing

of two or more beams creating almost isotropic configurations. Interestingly, with

these improvements in cooling and trapping, it was also found that the action of

photons on cold atoms can lead to induced photoassociation of atoms into excited

molecules.32 A further cause of concern from the point of view of the trap lifetime,

since excited dimers escape, this is also a reason of interest from the point of view of

spectroscopy, opening up an entirely different line of investigation on the collisional

properties of cold atoms.33–36

An alternative to dipole traps based on red-detuned laser light, is given by opti-

cal mirrors which exploit the repulsive nature of the dipole force with blue detuned

light. The challenge, as in the case of red detuning, is to produce intensity gradi-

ents strong enough that the dipole force can be effective notwithstanding the large

detuning. Evanescent waves have a strong decay within the range of a wavelength,

which makes them a good candidate for the gradient generation. Atom mirrors

based on evanescent waves were discussed37 and realized38 already in the 1980s

whereas in 1995, Sodium atoms were trapped by flat sheets of blue-detuned light.39

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Evanescent waves propagating inside cylindrical hollow fibers were proposed for a

cooling and channeling scheme.40 In a different approach, use is made of hollow

laser beams which are approximately doughnut-shaped41–43 and all-optical dark

toroidal traps have been realized.44 Similarly, results were obtained by using a hol-

low capillary fiber such that the blue-detuned light was confined to the internal

surface of the cylinder, with only the evanescent tail penetrating the hollow core

where the atoms were propagated.45 In all these configurations, which are techni-

cally more challenging than the simple red detuned beam configuration, the atoms

are trapped in the region with minimum light intensity and are thus less subject to

the inevitable perturbations due to the interaction with the laser.

Basically, the same principles are at work in standing-wave traps, either red

or blue detuned as discussed theoretically by Letokhov and Pavlin46 and later in-

troduced as optical lattices.47–49 Deflection and subsequent one-dimensional chan-

neling of slow Sodium atoms was observed by Balykin et al. for both signs of the

detuning50 with the dipole force dominating over radiation pressure. In these sys-

tems the atoms can be localized over lengths of the order of half a laser wavelength

and, in the case of red detuning, the laser radiation acts as a friction force and con-

tributes to farther slowing of the atoms. Periodic potentials and optical lattices51

allow for very high phase-space densities and their properties have been analyzed

experimentally since the beginning of the 1990s.52,53 Lattices in two54 and three di-

mensions55 have been realized as well and much work is being carried out following

the idea that a quantum bit of information could be stored by filling a far-detuned

lattice with one atom per site.56

Thus, one of the main lines of investigation aimed at manipulating atoms with

light focuses on guiding and trapping them. In general, optical trapping has many

attractive properties. The easily tunable parameters, the fact that the magnetic

field plays no role in the trapping, the possibility of trapping different species in the

same trap, or reaching long lifetimes and stability of the traps, are all characteristics

that can make optical trapping extremely interesting to study the physics of many-

body systems. It is a carefully aimed use of light-matter interactions that has made

optical trapping an experimental success.

2.2. From the 1990s to the XXI century: Cooling neutral atoms,

creating and trapping BECs

Many different problems had to be solved, from the finite lifetime of the trapped

atoms which shortened the life time of the traps and made their experimental use

problematic, instability of trapping configurations, see for instance the review in

Ref. 7. However, the true success of optical trapping was hindered mainly by the

temperature of the atoms. As long as the potential well depth is too small com-

pared to the thermal energy of the atoms, optical trapping can have only a limited

effect. In fact, given the low values of the off-resonance polarizability, the dipole

force tends to be too weak to trap thermal atoms. Tuning the radiation close to a

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resonance enhances the dipole force effect but also involves the complication of op-

tical excitations and heating, thus limiting the life time of the system. Lowering the

temperature of the atoms can make them more sensitive to the trapping effect of the

dipole force therefore, trapping and cooling have always been intrinsically coupled

concepts, they can both be obtained by manipulating neutral atoms with light57–59

and, it is quite obvious that any distinctions in the description of the two ideas are

rather artificial. In fact, it is the same resonant radiation pressure mentioned above

that was first suggested as a force to lower the atoms temperature.60 The basic

physics of a single-photon process involves the recoil effect imparted on moving

atoms which absorb counter-propagating photons, with a caveat which can be eas-

ily understood for simple two-level atoms: Due to the Doppler effect, atoms moving

towards the source of radiation will experience the laser frequency as up-shifted.

Therefore red detuned radiation shifts closer to resonance and becomes more effec-

tive, since the scattering cross-section is larger at resonance. For atoms moving in

the opposite direction, the red detuned laser frequency appears even farther from

resonance and therefore has weaker effect. The dominant effect will be the absorp-

tion of counter-propagating photons. In other words, the Doppler effects makes

radiation pressure velocity dependent. In presence of two counter-propagating laser

beams both red-detuned, absorption is always predominantly from the laser propa-

gating in the opposite direction of the atom, and this imbalance leads to a dissipative

force which slows the atoms down, i.e., laser cooling. It was suggested that isotropic

radiation could be used to apply these ideas,60 and the same physics was proposed

to cool and trap neutral atoms in a three-dimensional resonant standing wave and

for transverse cooling with two counter-propagating beams,61 or for longitudinal

cooling with a single counter-propagating beam.62 Making use of six beams sym-

metrically disposed yields a three-dimensional cooling scheme. Combined with a

precooling laser beam kept in resonance with the decelerating atoms so as to op-

timize the deceleration,62,63 the first optical molasses cooled and confined Sodium

atoms, the atommotion being damped by the effect of a “viscous fluid of photons”.64

The existence of a lower limit on the possible temperature due to the presence of

spontaneous emission was discussed quite early65,66 but a violation of the Doppler

cooling limit was demonstrated,67 also indicating that a larger detuning from the

resonance could be beneficial, and turned out to be one of the most striking effects

of the interplay between internal and external degrees of freedom.47,68 The first of

the assumptions to be found wanting was that of a simple two-level atom. Ground

state internal Zeeman sublevels were in fact playing an important role. As pointed

out by Cohen-Tannoudji,6 it is the interplay between external and internal degrees

of freedom which allowed to overcome the limits of laser cooling. For example, one

of several effects exploited in cooling schemes is related to the fact that the control

of the distribution of the populations among different Zeeman sublevels (optical

pumping),69–71 as well as light shifts and the potential energy perceived by the

atoms,72 depend on the laser polarization. In optical molasses created by beams of

different polarizations, there are polarization gradients because of beam interfer-

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ence, so the laser polarization undergoes spatial variations. Therefore populations

of different Zeeman sublevels and the potential energy due to the interaction with

light both depend on the position of the atoms and, if the atoms are in motion,

this can result in the further friction force of sub-Doppler cooling and Sisyphus

effect.73 Random single-photon recoil due to quantum fluctuations of the atoms

linear momentum was the second fundamental cooling limit discussed in the 1980s.

Again acting on the magnetic sublevel populations via optical pumping resulted in

“velocity-selective coherent population trapping”, a scheme in which the atoms are

prepared in a superposition of states such that absorption of light is impossible if

the atoms have zero velocity.74–76 Thus, optical pumping produces a superposition

of states for which absorption is suppressed and the trapping is effective only for

atoms at near zero velocity. Atoms with finite velocity absorbs light and are lost,

whereas atoms at zero velocity are trapped and cannot absorb light and acquire a

velocity. Further reductions in temperature for trapped atoms, were the trapping

allows for the required long interaction times, were discussed and experimentally

tested in the 1990s within an optical pumping scheme based on stimulated Raman

transitions between two hyperfine sublevels of the atoms ground state.77 As noted

by Cohen-Tannnoudji,57 no friction force is at work now and the effectiveness of the

scheme depends on the existence of magnetic sublevels. Blue detuned light can be

used for cooling as well,19 as proposed in the 1990s for traps based on evanescent

waves78–80 by exploiting a mechanism analogous to that of sub-Doppler cooling.

Advanced cooling and trapping techniques, a happy marriage of laser cooling

with magnetic trapping, have one of their most successful applications in the experi-

mental realization of Bose–Einstein condensation (BEC).81,82 Predicted in the 1920s

by Bose83 and Einstein84,85 as a consequence of the quantum nature of Bosonic

atoms, a BEC can be described, in the words of Pitaevskii,86 as “a spectacular

phenomenon, the condensation of atoms in their lowest quantum state”, resulting

in a macroscopic number of atoms collectively occupying the ground state. Laser

cooling on trapped atoms played an important role in the cold race for BEC in

dilute alkali atomic gases87–89 which had been thwarted by too large rates of inelas-

tic collisions. BECs were transferred from magnetic to dipole traps to demonstrate

efficient optical trapping,90 and were observed directly in optical traps.91 It was

realized that optical traps offer several advantages, for example the possibility of

manipulating the spin of trapped atoms or of confining atoms in arbitrary spin

states.19 Therefore, all-optical approaches have attracted a strong interest, the first

all-optical BEC being realized in 2001 at the crossing of two red detuned laser

beams92 and in an even simpler configuration with a single beam.93

3. Laser-BEC Interactions: Atoms as Light, Light as Medium

All the successes achieved with the manipulation of neutral atoms with light orig-

inate from the coupling of electromagnetic radiation and ultra-cold matter. The

ability of describing mathematically these processes is of fundamental importance

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for the understanding of the physics and the prediction of possible novel phenom-

ena. Here, we shall review ideas concerning matter waves as the propagating waves,

aiming at describing their dynamics coupled to that of the “medium” of electromag-

netic radiation through which they propagate. We shall show how a purely classical

description of the radiation propagating through a dielectric can shed light on the

physics of these systems. We shall than start from the point of view of quantum

theory and review the models that describe ultra-cold, dilute Bosons interacting

with electromagnetic radiation justifying the classical model and elucidating its

limitations. Finally, we shall investigate some possibilities of atom manipulation

granted by the atom-light interaction described by these equations.

3.1. When a change in the medium can affect the propagating

wave

Weakly interacting, dilute BECs in the zero temperature limit are described by

the Gross–Pitaevskii equation86,94,95 (GPE). The Kerr-like nonlinear term, in the

simplest s-wave approximation of weak interactions and low energy, describes the

two-body interactions of the atoms and can have either a focusing or a defocusing

nature depending on whether the atom interactions are attractive or repulsive. It

is exactly the existence of a regime of weak interactions for very dilute gases which

is of interest, allowing for a mathematical description completely analogous to the

nonlinear Schrodinger equation (NLSE) which describes electromagnetic radiation

propagating through a nonlinear medium.96,97 For the electromagnetic radiation,

the nonlinearity is the result of the dependence of the refractive index, or the polar-

ization, on the radiation intensity: Radiation propagating through a medium affects

its properties and this, in turn, has a back effect on the propagating radiation itself.

In fact, the mathematical structure of the two equations, GPE and NLSE, is exactly

the same. Self-focusing and self-trapping of light are well-known effects since the

1960s,98–100 and a lot has been done with BECs, including realizing most nonlinear

optics effects with cold atoms and the most spectacular realization of solitons both

dark and bright.101–103 As noticed by Bjorkholm in the 1970s, “[· · ·] the focusing of

atoms by light results from the same physical mechanism (momentum exchange)

responsible for self-focusing of light in atomic vapors”.17 Askar’yan discussed the

idea that high intensity radiation can induce gradients in the medium properties,

resulting in guiding of the radiation itself.5 This raises the question of how to de-

scribe the interaction between the propagating wave and the medium, the dynamics

of the atoms in a laser field becoming key for the understanding of the interactions.

The exact parallel between the two mathematical descriptions in the simplest ap-

proximation, suggests a switching of roles in which the atoms are the propagating

wave, and laser light constitutes the medium through which the quantum matter

wave propagates, mediating atom–atom interactions and giving rise to a new type

of nonlinearity.

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3.2. Modeling laser-BEC interactions: A purely semi-classical

description

In the framework of a classical discourse, the optical field maintains the role of the

propagating wave and the atoms that of dielectric medium. Classical electromag-

netism provides a description of the propagation of radiation through dielectrics

and shows how light can be seen as exerting a force on matter. It is interesting to

investigate the form of this force, the potential term it gives rise to and how it can

be used in the NLSE governing the dynamics of the medium atoms. In order to do

this, one must go back to Maxwell’s equations, the fundamental equations describ-

ing radiation-matter interactions. As well-known, instead of the microscopic exact

fields, introduce properties are already included in an average sense.104 Quanti-

ties are averaged over volumes infinitesimally small but still containing enough

molecules for the microscopic fluctuations due to the molecular structure to be ne-

glected. From Maxwell’s equations for the electric field, assuming that no magnetic

fields and no external charges or currents are present, we have

∇×E = 0 , (1)

∇ · E = 4πρ , (2)

where ρ is the average charge density of the medium. With no external charges, the

total charge of the dielectric is assumed to remain zero even when the dielectric is

immersed in an electric field, i.e., integrating over the whole volume of the medium,∫ρdV = 0, which implies, according to the divergence theorem, that the average

charge density can be expressed as the divergence of a vector P, conventionally

written as,

ρ = −∇ ·P . (3)

Outside the body, P = 0. Not only it can be demonstrated that P is related to

the surface charge density of the dielectric, it is also straightforward to show that

P is the dipole moment per unit volume of the dielectric, the classical dielectric

polarization, as discussed in Chap. 2 of Landau’s textbook.4 In fact, considering

the total dipole moment of the charges in the dielectric and using Eq. (3), basic

vector algebra leads to∫

Ω

rρ dV = −∫

Σ

r ·P ds+

∫

Ω

P dV , (4)

where the first integral on the right-hand side is over the dielectric surface Σ, and

the second one is over the dielectric volume Ω. The surface integral vanishes and∫

Ω

rρ dV =

∫

Ω

P dV . (5)

Therefore, Poisson equation (1) using (3) reads

∇ ·D = 0 , (6)

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where we have introduced the electric displacement vector D,

D(r, t) = E(r, t) + 4πP(r, t) . (7)

What the mathematics represents is the fact that matter can be considered as

composed of interacting particles embedded in vacuum. Such particles produce

a microscopic field with large local variations in the interior of the matter, and

any external field modifies the internal one. The macroscopic properties of matter

are derived by averaging the total field within it, which leads to a description

in terms of electric (and magnetic) dipole moments. For linear, isotropic media

with no permanent polarization, the induced polarization can be assumed, in the

simplest case of low field intensities (linear optics), to be linearly dependent on the

macroscopic electric field

P(r, t) = χE(r, t) , (8)

so that a dielectric constant ǫ can be defined for which

D(r, t) = E(r, t) + 4πP(r, t) = (1 + 4πχ)E(r, t) = ǫE(r, t) , (9)

with

ǫ = 1 + 4πχ . (10)

Notice that this is the point where one may introduce the well-known Kerr nonlinear

effect, i.e., a dependence of the polarization on the field intensity in first instance.

This would describe a change in the dielectric constant due to the laser intensity

and its back effect on the laser propagation. In the case we are discussing, the

electromagnetic field produces at a given position a certain amount of polarization

P, to a first approximation proportional to the field itself. Each infinitesimal volume

in the dielectric becomes then a source of a new secondary or scattered wavelet,

a field which depends on the polarization P. The scattered waves combine with

each other and with the microscopic field to give the total field. These relations are

of course only approximately valid, and the proportionality constants can in fact

depend on spatial and temporal coordinates.

The mean macroscopic field introduced here however is not the field experienced

by an atom inside a dielectric. The local, effective field, acting on a particle is the

result of the action of the macroscopic field E and of the internal field Eint due

to the polarization of its neighbors.105,106 This last field, investigated by Lorenz in

the XIX century, under the simplifying assumption of a symmetric cubic lattice, is

given by,107,108

Eint(r, t) =4π

3P(r, t) , (11)

so that, the effective field reads

Eeff(r, t) = E(r, t) +4π

3P(r, t) . (12)

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For a linear, homogeneous and isotropic material, the electric dipole moment p(r, t)

of a molecule depends linearly on the effective electric field through the causal

relation109

p(r, t) =

∫ t

−∞α(t− t′)Eeff(r, t

′)dt′ , (13)

where α denotes the complex polarizability of the medium. In terms of Fourier com-

ponents, with p(r, ω) = α(ω)Eeff(r, ω) and denoting with n the particles number

density, one finds from Eq. (8)

P(r, ω) = np(r, ω) = χ(ω)E(r, ω) , (14)

which gives for the electric susceptibility χ(ω)

χ(ω) =nα(ω)

1− 4π

3nα(ω)

. (15)

From now on, throughout the whole review, the atom density will be assumed to

be stationary.

One important point is that these equations relate the properties of the medium

to the macroscopic field, for which we have a description through Maxwell’s equa-

tions. It is now possible to derive an explicit expression for the force exerted by an

electric field onto a homogeneous dielectric. The force is related to the stress tensor

and the system free energy, see for example Ref. 4, Chap. 2. In a fluid at constant

pressure and temperature, assuming that the density spatial variations occur over a

length scale larger than one optical wavelength, it is found that such force depends

on spatial gradients of both the electric field and the dielectric constant ǫ as2,4,110

F = ∇[ |E|28π

n

(∂ǫ

∂n

)]

. (16)

This is a conservative force that can be derived as the gradient of a potential energy.

Pitaevskii showed that in a transparent dispersive medium the expression for the

stress tensor of a time-dependent electric field is the same as for a constant field

only averaged over time.111 For a stationary field of the form:

E = Re[A(r)e−iωLte] , (17)

where e is a unit vector denoting either linear or circular polarization, using the

expression found above for the dielectric constant (10) and (15), and denoting as

〈· · ·〉 the time average of a quantity, the potential acting on an atom reads,

Vdip = −1

2

〈|E|2〉α(

1− 4π

3nα

)2 . (18)

However, since absorption phenomena would be far too complicated to be

treated within this classical model, we shall assume that the effects related to the

imaginary part of the polarizability can be neglected and consider a real α ≃ αre

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with |αim| ≪ |αre|. In the second part of this section we shall show that this is

an acceptable approximation under the assumption that the laser frequency is not

resonant with any of the atoms transition frequencies. By Taylor-expanding the

expression for the susceptibility (15), it is found that the zero-order term is real

χ(0)(ω) = χ(0)re (ω) =

nαre(ω)

1− 4π

3nαre(ω)

, (19)

where χ(0)re denotes the zeroth-order real part of the susceptibility. Notice that,

from the Taylor expansion, it turns out that to first-order in αim there is only an

imaginary contribution to χ, i.e., χ(1) = i|χ(1)| and a new contribution to the real

part of the susceptibility appears only at second-order. Under this approximation,

the potential (18) including the local field effects reads

Vdip ≃ −1

2

〈|E|2〉αre(

1− 4π

3nαre

)2 . (20)

or, explicitly using (17),

Vdip ≃ −1

4

|A|2αre(

1− 4π

3nαre

)2 . (21)

We have denoted this potential as a dipole interaction since, by making use

of Eqs. (14) and (15), neglecting the local field effects, for a stationary field with

no assumptions on the absorptive process, one finds immediately that Eq. (18) is

equivalent to the usual dipole interaction energy:

Vdip ≃ −1

2〈(P/n) · E〉 . (22)

The equality is only approximate since local field effects have been neglected. The

physics of dielectrics has been studied in detail in classical electrodynamics and we

shall now try to make use of this knowledge. The induced dipole, within the limits

of linear models and isotropic media, oscillates at the frequency of the driving field

so that we can assume

P(r, t) = Re[P(r)e−iωLte] . (23)

The dipole potential, once the calculation of the time average in (22) has been

carried out explicitly, is seen to depend only on the real part of the susceptibility

χ = χre + iχim:

Vdip = −χre

4n|A(r)|2 = −2π

c

χre

nIL , (24)

where the electromagnetic field amplitude is related to its intensity as usual, IL =

c|A|2/(8π). Notice how the fast oscillating components disappear because of the

time average, which will reflect in the rotating wave approximation (RWA) of the

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quantum models introduced in the next sections. Notice also how the interaction

potential depends on the in-phase component of the dipole moment, as will be shown

also from the quantum description. Finally, notice how this expression coincides

with Eq. (21) provided the local field effects are discarded, i.e., χ ≃ nα. We shall

come back to the local field effects at the end of this section.

The assumption made earlier that we could consider only the real part of the

polarizability when deriving the dipole potential (20) can be now investigated. It is

well-known that a charge accelerated by a driving field absorbs energy as,

Pabs = 〈(P/n) · E〉 . (25)

Carrying out the same algebra as for Vdip,

Pabs =ωL

2

χim

n|A(r)|2 =

4πωL

c

χim

nIL . (26)

Notice that, within a purely classical framework, the power absorbed and conse-

quently re-emitted as dipole radiation is the closest one can get to the quantum

concept of atomic emission. Both the dipole force and the emitted power depend

on the field intensity (and its spatial gradients) and to understand the physics one

needs a model for the susceptibility. Classically, Lorentz introduced a damped os-

cillator model for the atom106,112: An electron in an atom is described as a charged

oscillator tied to the nucleus by the electrostatic interaction, its natural frequency

ωa corresponding to the optical transition frequency of the atom, and subject to the

Lorentz force (neglecting the magnetic term since we are in a deeply nonrelativistic

regime). Damping is due to the fact that an accelerating charge radiates and thus

loses energy. The oscillator equation of motion reads

x+ γx+ ω2ax = −eE(t)

m. (27)

It is important to notice that we are considering the average macroscopic field as

the driving force, thus discarding the local field effects discussed above. However, it

is straightforward, if slightly tedious, to substitute the driving field in (27) with the

local field E+(4π/3)P with P = −nex and carry out exactly the same calculation

as done in what follows. The main physical effect is a shift of the natural frequency

ωa and it will be discussed at the end of this section.

Working in Fourier components,

xω =−eEω/(m)

ω2a − ω2 − iωγ

(28)

and since Pω = −enxω = χEω = χA/2, it is found

χ(ω) =ne2/m

ω2a − ω2 − iωγ

. (29)

The classical radiative damping rate remains to be determined. It is known that an

accelerated particle (nonrelativistic) with charge q radiates a power given by the

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Larmor formula106

Prad =2q2x2

3c2. (30)

However, this process necessarily affects the particle dynamics since it decreases its

energy. One way of modeling this effect is by introducing a friction force oppos-

ing the particle motion, commonly called the classical radiation-reaction force. An

expression for such force is given by the Abraham–Lorentz formula106

Frad =2e2v

3c3= mτ v . (31)

This is then the friction force that should be inserted in the equation of motion, so

that the Abraham–Lorentz version of (27) reads

x− τ...x + ω2

ax = −eE(t)

m. (32)

Assuming that the driving field E(t) oscillates at frequency ω so that the particle

oscillates at ω as well and...x = −ω2x means that the equation can be rewritten as,

x+ τω2x+ ω2ax = −eE(t)

m. (33)

A comparison with (27) leads immediately to an estimate of the friction coefficient

for a particle oscillating at frequency ω

γ(ω) = τω2 =2e2ω2

3mc3. (34)

Of course, for the monochromatic field discussed here, ω = ωL. This gives all the

elements needed to write an explicit expression of both Vdip and Pabs. The real and

imaginary parts of the susceptibility are derived from Eq. (29) as

χre =3nc3γa2ω2

a

ω2a − ω2

L

(ω2a − ω2

L)2 + (ωLγ)2

≃ 3nc3γa2ω2

a

1

ω2a − ω2

L

=3nc3

2ω3a

(γa

ωa − ωL+

γaωa + ωL

)

, (35)

χim =3nc3γa2ω2

a

ω3Lγa/ω

2a

(ω2a − ω2

L)2 + (ωLγ)2

≃ 3nc3γa2ω2

a

ω3Lγa/ω

2a

(ω2a − ω2

L)2

=3nc3γ2a2ω2

a

ω3L

ω4a

(1

ωa − ωL+

1

ωa + ωL

)2

, (36)

where γa = γ(ωa). The approximate expressions are valid under the physically

interesting conditions of:

(1) large detuning |∆| = |ωL − ωa| ≫ γa (i.e., absorption processes negligible) and

(2) near resonance |∆| = |ωL − ωa| ≪ ωa, which ensures the validity of a two-level

model where only one natural frequency ωa is involved, see also Grimm et al.8

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It follows from (24) and (26) that

Vdip = −3πc2

2ω3a

(γa

ωa − ωL+

γaωa + ωL

)

IL(r) ≃3πc2

2ω3a

(γa∆

)

IL(r) , (37)

Γsc =3πc2

2~ω3a

(ωL

ωa

)3(γa

ωa − ωL+

γaωa − ωL

)2

IL(r) ≃3πc2

2~ω3a

(ωL

ωa

)3(γa∆

)2

IL,

(38)

where the scattering rate is defined as Γsc = Pabs/(~ωL). Here again the approx-

imate expressions are consistent with the previous assumption of near resonance

and still large detuning. As mentioned above, in order to simplify the physics de-

scribed in the quantum framework, it is interesting to investigate two-level atoms.

This requires that the laser frequency be tuned in such a way as to involve only

two levels and leave the others unperturbed, i.e., |∆| = |ωL − ωa| ≪ ωa and will

be discussed as the RWA. The immediate effect in terms of these semiclassic ex-

pressions is that the off-resonant denominators are so large that those terms can

be discarded. As pointed out in Ref. 8, it turns out that the dipole potential scales

as IL/∆ and the scattering rate as IL/∆2. This suggests the possibility of select-

ing the laser frequency so as to depress the effects of photon scattering (see also

the next section) which allows one to neglect the effect of χim/n. Since the Taylor

expansion of χ in terms of αim shows that the first imaginary contribution to χ

enters at order O(αim), neglecting it leads to neglecting terms at higher-orders as

well and what is left of χ under this approximation is the zero order term which is

real. This justifies the assumption made in writing Eq. (20). From the point of view

of classical electromagnetism, the propagation of light through a medium of atoms

with natural frequency ωa is described by the refractive index n2 = ǫ which in

principle is a complex quantity. The real part describes dispersion, the dependence

of the propagation velocity on the frequency. The imaginary part describes absorp-

tion, scattering of photons from a quantum mechanical point of view. Absorption

naturally leads to attenuation of the light intensity, maximum at resonance. This is

the classical physical effect neglected when assuming that the electric susceptibility,

and therefore the dielectric constant, is real.

If the local field effects are retained in the equation of motion, Eq. (27), and in

the definition of the dipole force, Eq. (22), the calculation proceeds along the same

lines. The dipole force is modified into a generalized expression:

V ′dip = −1

2

⟨

(P/n) ·(

E+4π

3P

)⟩

= −χre

4n|A(r)|2 − πn

3

|χ|2n2

|A(r)|24

. (39)

The equation of motion is modified into

x+ γx+ ω2ax = −eE(t)

m− e

m

4π

3P . (40)

With P = −nex and working in Fourier components, it is found that

xω =−eEω/(m)

ω20 − ω2 − iωγ

, (41)

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As anticipated, the main physical effect is a shift in the atoms natural frequency,

such that ω2a → ω2

a−ω2p/3 where ω2

p = 4πne2/m is the usual plasma frequency. This

leads to a modified expression for the susceptibility to use in (39). By maintaining

the same assumptions on the detuning as above, one arrives at an expression for the

susceptibility akin to the previous one but with the new shifted natural frequency

ω0 instead of ωa

χ′re ≃

3c3n

2ω30

γ0ω0 − ωL

. (42)

Here γ0 = γ(ω0). As discussed above, the contribution from the imaginary part of

the susceptibility in the generalized dipole potential can be neglected. The calcula-

tion is rather long, but it is quite easy to study the effect of a weak local field in a

low density medium by expanding in terms of the density and retaining only first

order terms in n, so that ω0 ≃ ωa − ω2p/(6ωa). Inserting the new expression of χ′

re

into the generalized dipole potential and finally, by making use of the expression for

γ, the generalized dipole potential including a weak local field effect (i.e., retaining

terms up to first-order in the density n) reads

V ′dip ≃ −

(a

4+

2π

3na2)

|A|2 , (43)

where we have introduced a = −e2/(2mωa∆). This expression is an agreement with

the weak field expansion of (21) if one identifies αre = a. This can be related to

the real part of the susceptibility with no local field effects since from (35), always

under the same assumptions as above,

χre

n≃ − e2

mωa∆= 2αre . (44)

From this relation, by using the correct quantum mechanical expression for the

coefficient γ in the expression of χre, see Eq. (91), it is found that

αre = − d2

~∆, (45)

where d denotes the quantum dipole matrix element between the two atom levels

considered.

The next step is the building of an equation for the atom dynamics or, in other

words, a propagation equation for the matter wave. One should insert the dipole

potential energy in the atom Hamiltonian and derive a Schrodinger equation for

the atoms. The final result is a system of two coupled equations:

(1) The Schrodinger equation for the atom evolution as determined by interatomic

interactions and by the force exerted by the optical field, i.e., a generalized

version of the NLS which includes the dipole interaction potential (21). Under

the assumption of a stationary state

Ψ(r, t) = Φ(r)e−iωat (46)

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with ~ωa = Ea = ~2k2a/(2m) the stationary Schrodinger equation reads

EaΦ = − ~2

2m∆Φ+

4π~2a

m|Φ|2Φ− α

4

|A|2(1− 4π

3 α|Φ|2)2Φ . (47)

(2) The equation for the optical field propagation, i.e., a wave equation, derived

from Maxwell’s equations, coupled to the atom dynamics because the dielec-

tric constant depends on the atom density through Eq. (15). From Maxwell’s

equations

−∇(∇ ·E) + ∆E− 1

c2∂2D

∂t2= 0 , (48)

with the dielectric constant defined by (10) and (15)

ǫ = 1 +4πα|Ψ|2

1− 4π3 α|Ψ|2 . (49)

where, consistently with the approximations discussed above, in particular large

detuning, α = αre but we shall drop the subscript. From ∇ · (ǫE) = 0 a sim-

plification follows with the reduction of the wave equation to a scalar equation

since

∇ · E ∼ ∇ǫ ·Eǫ

≃ 0 (50)

if either ∇ǫ ⊥ E or (1/LE) ≫ (1/Ln) where LE, Ln are respectively the charac-

teristic length scales of the electromagnetic field and of the density variations.

Notice that the second of these conditions is essential for the validity of the ini-

tial expression of the force (16), see Landau’s textbook.4 Under the assumptions

of stationary propagation

E = ReA(r)e−iωLte (51)

with ωL = kLc, the stationary laser equation becomes

∆A+ω2L

c2

(

1 +4πα|Φ|2

1− 4π3 α|Φ|2

)

A = 0 . (52)

In other words, the light induces a nonlinear modulation in the atom matter

wave and the refractive index depends on the atomic density. A last comment

can be made on the physical effects of the local field. The nature of the laser-

induced nonlinearity in the atom equation depends on the detuning of the laser

frequency with respect to the natural atomic frequency shifted by the nonlinear

dipole interaction (∝ α|Φ|2). The nonlinear term is

− α|A|2

4(1− 4π

3 α|Φ|2)2 ,

where α = −d2/(~∆) = −K/∆. Thus

− α|A|2/4(1− 4π

3 α|Φ|2)2 =

K∆|A|2/4(∆ + C|Φ|2)2 =

K∆|A|2/4[ωL − (ωa − C|Φ|2)]2 .

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Here C andK simply collect the various positive constants. If ∆ > 0, blue detuning,

ωL > ωa, the nonlinear term increases the effective detuning. If ∆ < 0, red detuning

ωL < ωa, the nonlinear term decreases the effective detuning and the resonance

becomes more dangerous. It must be kept in mind that this resonance is nonphysical

and results from the approximations made in the derivation of the model which

amount to neglecting part of the physics involved. A clear understanding of the

limitations of this classical model requires a deeper investigation.

3.3. Modeling laser-BEC interactions: Beyond the classical ideas

The theoretical basis of controlling the motion of atoms or molecules by means

of the radiation force of laser light was discussed by Kazantsev in 1975113 and by

Klimontovich and Luzgin in 1979.114 These authors predicted an effect of mutual

trapping due to the dipole–dipole interactions, achieved in a thermal gas propa-

gating coaxially to a light beam. These first models, which neglected the optical

field dynamics, set the basis to clarify the description of the light action on ultra-

cold atoms and to bridge the gap between classical and quantum models. Perhaps

reflecting the two approaches to many-body physics in quantum mechanics, there

seems to be two strategies to tackle the atom-light interaction question: One is

based on the introduction of the system density matrix, the second makes use of

second quantization theory. Both of them must start from the physical description

of the system, the Hamiltonian. The first models included only the dynamics of the

atoms, which allows for considerable simplifications. Besides, local field effects were

neglected. The coupled dynamics of the electromagnetic field was only later treated

self-consistently, here we shall try to follow the same lines of development and we

shall start by neglecting both the laser field dynamics and the local field effects.

3.3.1. Atom dynamics: An Hamiltonian and a force

Following the calculations reviewed by Balykin et al.,7 one can see how the concept

of a classical force acting on the atoms can be derived from a quantum model. Dif-

ferent descriptions, i.e., Hamiltonians, all perfectly equivalent, can be devised. As

pointed out in the textbook by Cohen-Tannoudji et al.,115 a multipole expansion

can be more convenient to describe atom interactions rather than working in the

Coulomb gauge, the multipolar Hamiltonian being obtained by exploiting the gauge

invariance of the theory. The minimal coupling Hamiltonian112 is unitarily trans-

formed via a canonical gauge transformation, better known as the Power–Zienau

transformation, generalization of the classical Goeppert–Mayer approach.116–118 Fi-

nally, the dipole interaction of an atom system with an electromagnetic field can

be described, in the simplest dipole approximation, by the interaction energy op-

erator,116–121

Hint = −D ·E , (53)

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where D is the atom electric dipole moment operator (notice the different notation

with respect to classical electromagnetism). In practice, moving from the velocity

gauge of the p ·A Hamiltonian (here A is the vector potential and p is the atom

kinetic moment) to the length gauge of the D · E one, leads to some important

simplifications in the mathematical description.122 For instance, atom states in

the velocity gauge are described by the generalize momentum p − A/c, which

involves the electromagnetic field whereas, after the Power–Zienau transformation,

the atoms are described by their kinetic momentum. Besides, in the length gauge one

does not have to deal with the cumbersome nonlinear A2 term. It is important to

notice that, in this nonrelativistic theory, all the unretarded terms in the interaction

cancel out.121

From a semi-classical point of view, Eherenfest theorem shows how the dipole

force can be calculated as the gradient of the interaction energy. Since

i~F = 〈[p, Hint]〉 ,we find

U = 〈V 〉 = −〈D〉 ·E (54)

F = −∇(〈D〉 · E) = −∑

i

〈Di〉 · ∇Ei . (55)

Several ideas are behind these steps. The very existence of a dipole moment is re-

lated to transitions between stationary states of the atoms with different parity, see

Eq. (59), which naturally involves quantum fluctuations. Besides, the localization

of a system, be it an atom, a molecule or a condensate, brings about quantum

fluctuations in its momentum. However, the concept of a force acting on an atom is

a classical concept, thus for it to be valid one must require that the quantum fluc-

tuations of the dipole moment be negligible, i.e., spontaneous decay effects must be

negligible, and that the atoms move quasi-classically.7 Under a second important

approximation, the electric field is assumed to be evaluated at the dipole position,

we are assuming the classical dipole approximation.123 This is justified as long as

the field does not vary significantly over the physical dimension of an atom, i.e., the

size of the atom can be neglected compared to the spatial variation length scale of

the laser field. A field that should be correctly calculated at an electron position

can be safely calculated at the nucleus position since the electron-nucleus distance

is of the order of the Bohr radius which is usually much smaller than any other

length scale, including the laser wavelength. As a second but equally important

consequence, the atom positions can be treated classically in the expression of the

field, substituting the quantum position operator with its expectation value. Fi-

nally, it is possible to start from the simplified case of two-level atoms instead of

treating multi-level systems by assuming that the laser frequency is nearly resonant

with one transition. The laser frequency detuning must be checked against the atom

natural frequencies in order to understand the physics involved in the interaction

and the limitations of the different models. As for the laser field, large number of

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photons justify a semi-classical treatment. Furthermore, as commonly done for laser

radiation, if the field is in a coherent state it can be treated as a complex number

neglecting the quantum nature of the field operators.124

3.3.2. Atom dynamics with the density matrix

Under the approximations discussed above, an explicit expression for the dipole

force requires knowledge of the induced dipole operator. Most of the physics ex-

plored, as discussed in the first part of the review, involves excitations of atoms

and spontaneous emission. Atom coherence is degraded and a strategy of choice to

describe statistical mixtures in quantum mechanics is based on the density matrix

operator. For the case of atom-light interactions where the laser field dynamics is

not considered, the atom density matrix is in fact the reduced operator obtained by

tracing over the electromagnetic field states. Within this formalism, for the induced

dipole operator one finds

〈D〉 = Tr(ρD) , (56)

where ρ is the atom density matrix operator and the stationary states of the un-

perturbed atoms are given by

|ψn〉 = |φn〉e−iEnt/~ , (57)

so that, if H0 is the unperturbed atom Hamiltonian,

H0|φn〉 = En|φn〉 , (58)

with ~ωmn = Em − En. For the dipole operator matrix elements (and analogously

for any other operator),

Dnm = 〈φn|D|φm〉 , (59)

with

Dnm = D∗mn . (60)

In this approach it thus follows that knowledge of the density matrix ρ is needed,

thus its Heisenberg equation of motion has to be written and solved therefore one

needs the system Hamiltonian. In the dipole approximation, this reads

H = H0 +Hint , (61)

where H0 is the Hamiltonian of the free unperturbed atoms and Hint describes the

dipole interaction with the laser field, Eq. (53).

The case we are interested in is that of a laser beam interacting with a sys-

tem of simple two-level atoms interacting with a quasi monochromatic, spatially

inhomogeneous, traveling electromagnetic field polarized along the direction e with

frequency ωL = kLc,

E = Re[A(r, t)ei(kLr−ωLt)e] = [E(+)(r, t)e−iωLt + E(−)(r, t)eiωLt]e . (62)

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The usual relation holds for the positive and negative frequency components of the

field, E(+)(r, t) = [E(−)(r, t)]∗. We shall consider only quasi monochromatic fields

under the assumption of the slowly varying envelope approximation so that the

amplitude temporal variations will be negligible with respect to the variations at

the carrier frequency ωL. For a perfectly monochromatic field the amplitude would

not depend on time.

Also, for two-level atoms, we shall denote as |g〉 and |e〉 the lower and the higher

energy state, respectively, and define

~ωa = ~ωe − ~ωg = Ee − Eg . (63)

In what follows, it will be assumed Eg = 0.

For the induced dipole moment in the two-level case, from Eq. (56)

〈D〉 = ρgeDeg + ρegDge + ρggDgg + ρeeDee (64)

= ρgeDeg + ρegDge , (65)

where cancellations occur because of symmetry. For the Hermitian dipole operator,

it can be further assumed that the off diagonal elements are real

Dge = D∗eg = 〈φg|D|φe〉 = d (66)

so that

〈D〉 = d(ρge + ρeg) . (67)

Introducing the Pauli transition operator S† defined as

S† = |e〉〈g| , (68)

one also finds that the dipole moment operator can be rewritten as

D = d(S† + S) . (69)

We are now in a position to construct the Heisenberg equation for the density

matrix,

i~ρ = [H, ρ] , (70)

where the Hamiltonian describes the free atom evolution and their interaction with

the electric field:

H = H0 +Hint = ~ωaS†S − d(S† + S) · E , (71)

and we have used the fact that Eg = 0 so that ωe = ωa. This Hamiltonian is a 2×2

matrix in the basis (|g〉, |e〉). Its diagonal elements give the unperturbed energy

eigenvalues Eg, Ee, while the off-diagonal ones, −d · E, are due to the coupling

between the two states which arises as a consequence of the interaction with the

laser field. Let us analyze the dipole interaction term in more detail:

Hint = −(Deg|e〉〈g|+D∗eg|g〉〈e|) · (E(+)e−iωLt +E(−)eiωLt) . (72)

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From now on, we shall focus on the case of a traveling wave, see Eq. (62). The Rabi

frequency which could be in general defined as,

Ω(r, t) =Deg ·E(−)(r, t)

~, Ω(r, t) =

Deg ·E(+)(r, t)

~, (73)

for a traveling wave reduces to

Ω(r, t) =d ·A(r, t)

2~, (74)

where the vectorA(r, t) now includes the polarization vector and the factor 2 comes

out of taking the real part in Eq. (62) so that

E(+)(r, t) → A(r, t)

2.

The exponential factor exp(ikLr) is left out of the slowly varying amplitude and will

be written explicitly. Notice that, assuming real amplitudes A as usual for traveling

waves, the Rabi frequency is now real. Introducing the short notation

θL = kLr− ωLt , (75)

one finds (we omit for brevity the dependence on spatial and temporal coordinates)

Hint = −~Ω(eiθL + e−iθL)|e〉〈g| − ~Ω(e−iθL + eiθL)|g〉〈e| . (76)

At this point the Rabi frequency is still a function of time through the dipole

moment and not knowing the time dependence it would be difficult to proceed

further. Moving to the interaction picture via the unitary transformation

U(t) = eiH0t/~ → |ψ′〉 = U |ψ〉 and H ′ = UHintU† , (77)

since we have for the atom unperturbed Hamiltonian H0 = ~ωa|e〉〈e|, one finds

from Hint

U = eiωat|e〉〈e| , (78)

H ′ = −eiωat|e〉〈e|[~Ω(eiθL + e−iθL)|e〉〈g|

+ ~Ω(e−iθL + eiθL)|g〉〈e|]e−iωat|e〉〈e| . (79)

By Taylor expanding the exponential operators

eiωat|e〉〈e||e〉〈g| = 1 + iωat|e〉〈e|e〉〈g|+ · · · = eiωat|e〉〈g| , (80)

|g〉〈e|e−iωat|e〉〈e| = |g〉〈e|e−iωat (81)

(all the other terms vanish). Thus

H ′ = −~Ω(eikLr−i∆t + e−ikLr+i(ωL+ωa)t)|e〉〈g|

− ~Ω(e−ikr+i∆t + eikr−i(ωL+ωa)t)|g〉〈e| . (82)

Here the detuning has been defined as

∆ = ωL − ωa . (83)

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In the spirit of the RWA, neglecting fast oscillating terms and keeping only nearly

resonant ones yields

H ′ ≃ −~Ωei(kLr−∆t)|e〉〈g| − ~Ωe−i(kLr−∆t)|g〉〈e| . (84)

This is of course quite delicate since it implies discarding one of the two counter-

propagating waves in the laser traveling field (62). In the framework of a second

quantized theory, this would be tantamount to neglecting interactions which do

not conserve the number of excitations. However, it can be expected that the fast

rotating terms will have a negligible effect when compared to the resonant ones,

the effect of the extra terms giving a slight modification of the final interaction

potential.125

Back to the Schrodinger picture through Eq. (77)

Hint ≃ −~ΩeiθL |e〉〈g| − ~Ωe−iθL |g〉〈e| . (85)

Finally, making use of (68) and recalling the definition of the Rabi frequency, the

interaction Hamiltonian reduces to

Hint ≃ −deg(S† · E(+)(r, t)e−iωLt + S ·E(−)(r, t)eiωLt) . (86)

The full Hamiltonian under the RWA thus is a 2× 2 matrix with elements

H = H0 +Hint =

(

Eg −~Ωe−iθL

−~ΩeiθL Ee

)

. (87)

Working out the commutator of the Heisenberg equation, the equations for the

density matrix components, also known as the optical Bloch equations,59,119,126,127

are given by

i~dρggdt

= −~Ωe−iθLρeg + ~ΩeiθLρge + i~γρee , (88)

i~dρeedt

= ~Ωe−iθLρeg − ~ΩeiθLρge − i~γρee , (89)

i~dρegdt

= ωaρeg − ~ΩeiθL(ρgg − ρee)− i~γ

2ρeg . (90)

In these equations, new phenomenological terms have been inserted to describe

spontaneous emission as an exponential decay59,108 with a constant rate γ/2. Here

the rate of spontaneous decay, related to Einstein’s coefficient A,128 is given by

γ =4d2ω3

a

3~c3. (91)

The fact that the induced dipole moment matrix element figures in this coefficient

underlines that the physics of spontaneous emission is related to excitations medi-

ated by the incident field. By assuming for the off-diagonal elements

ρeg = σegei(kLr−ωLt) (92)

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and considering the full expression of the convective time derivative

d

dt=

∂

∂t+ v

∂

∂r,

the equations are rewritten with no explicit time dependence:

i~dρggdt

= −~Ωσeg + ~Ωσge + i~γρee , (93)

i~dρeedt

= ~Ωσeg − ~Ωσge − i~γρee , (94)

i~dσegdt

= ~δσeg − ~Ω(ρgg − ρee)− i~γ

2σeg . (95)

Here

δ = ∆− kLv (96)

is the Doppler shifted detuning where the velocity dependence comes from the

convective time derivative and allows for velocity selective processes such as laser

cooling. It follows that

〈D〉 = d(σgee−i(kLr−ωLt) + σege

i(kLr−ωLt)) (97)

which in turn leads, from (55) to an expression for the radiation force on a two-

level atom as defined by the steady state off-diagonal elements of the atom density

matrix and written as the sum of two forces (in what follows the field amplitude

A(r) is assumed to be real as for a traveling wave):

F = (σgee−i(kLr−ωLt) + σege

i(kLr−ωLt))di∇[eiA(r)Re(ei(kLr−ωLt))] (98)

= (σgee−i(kLr−ωLt) + σege

i(kLr−ωLt))di

[

ei∇Aei(kLr−ωLt) + e−i(kLr−ωLt)

2

+ ikLeiAei(kLr−ωLt) − e−i(kLr−ωLt)

2

]

, (99)

that is

F ≃ (σge + σeg)d∇A(r)

2︸︷︷︸

Fgrad

+ ikL(σge − σeg)dA(r)

2︸︷︷︸

Frp

, (100)

with d = d ·e. Here, the RWA has been applied by keeping only the resonant terms

in the atom-laser coupling.

To have an explicit expression for the radiation force, the steady state off-

diagonal density matrix element is found by solving the stationary density matrix

equations of motion with the normalization condition ρgg + ρee = 1. The solution

gives

σeg = − Ω(r)(δ − iγ/2)

δ2 + 2Ω2(r) + γ2

4

, (101)

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and we remind that the detuning is now δ = ∆− kv. For the force:

Frp = ~kΓG2

1 +G2 +δ2

Γ2

(102)

and

Fgrad = −~δ

2

∇G2

1 +G2 +δ2

Γ2

. (103)

where γ = 2Γ is the scattering rate of the incident photons or the width of the

atoms ground state, and G2 is the adimensional saturation parameter

G2(r) =2Ω2(r)

Γ2=I(r)

IS(104)

with

IS =c

4π

(~Γ

d

)2

(105)

and as usual I(r) = (c/8π)A2(r), which shows the dependence of the two forces

on the laser intensity and its spatial gradient. As underlined by Kazantsev,113 the

radiation force is seen to be composed of two parts: A gradient force Fgrad, related

to variations over the length scale of A(r) and therefore requiring an inhomogeneous

laser beam with intensity gradients, analogous to the classical ponderomotive force

in continuous media; and a radiation pressure force Frp, related to variations over

the scale of λL. The presence of Γ in the expression of the radiation pressure force

stems out of the fact that this force originates in the momentum exchange due

to spontaneous emissions. Resulting from this dissipative process, this force always

acts in the longitudinal direction and can be exploited to modify the atoms velocity.

Notice that in (100) the wave vector is the result of a more general dependence of

this force on the gradient of the laser field phase and the radiation pressure depends

on the quadrature part of the dipole moment. On the other hand, the gradient

force, which depends on the in-phase part of the dipole moment (see also Sec. 3.2),

originates in a conservative process of stimulated emission of photons into states

with a different propagation direction but with the same energy. It is therefore

essential to have a localized pulse propagating, since this allows for the presence of

different plane wave components into which photons can be scattered, as it is shown

by its dependence on the spatial gradient of the laser intensity. Furthermore, the

gradient force changes its nature depending on the sign of the detuning, focusing

atoms to high intensity regions in the case of red detuning (∆ < 0), and pushing

them out of these regions in the case of blue detuning. We shall also discuss in the

next sections, see for instance (Sec. 3.3.3), how this force can be related to spatial

variations in the atoms energy levels due to the laser intensity.

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For small detunings δ ≪ Γ and for slow atoms, or in the lowest approximation,

assuming v = 0, Balykin7 shows that this radiation force leads to the potential

introduced by Ashkin15

Vgrad(r) =

∫ ∞

r

Fgr(v = 0)dr (106)

= −1

2~δ ln

(

1 +G2 +δ2

Γ2

)∣∣∣∣

∞

r

(107)

=1

2~δ ln

(

1 +G2 +δ2

Γ2

)

=1

2~δ ln

(

1 +δ2

Γ2

)

1 +

G

1 +δ2

Γ2

≃ 1

2~δ ln

1 +

G

1 +δ2

Γ2

. (108)

Whereas, for slow moving atoms and large detunings |δ| ≫ Γ,Ω by Taylor expand-

ing the logarithm as ln(1 + x) ≃ x

Vgrad(r) ≃1

2~Γ2

δG =

~

δΩ2(r) . (109)

From the expressions of the two forces, it is evident that the gradient force has

a dominant effect for large detunings. This reflects the fact that at large detunings

atoms are only weakly excited and spontaneous emission is a weak process. Large

detunings compared to the scattering rate are necessary since the probability of

excitation is anyway nonzero. If the radiation pressure force is at the heart of ma-

nipulating atoms with the aim of cooling them, because of its velocity dependence

at resonance, the conservative gradient dipole force with its focusing or defocusing

effect plays a major role in localizing and guiding matter waves.

3.3.3. Atom dynamics in the dressed-state picture

A different way of gaining insight into the physics of laser-light interactions and

in particular into the nature of the gradient dipole force, comes from applying

the dressed–atom picture to these systems as proposed by Dalibard and Cohen-

Tannoudji in the 1980s.129 The basic intuition is that for intense laser radiation the

atom-laser coupling is strong enough to suggest considering the atom-laser system

as a whole, i.e., looking for energy eigenstates of the combined system, the dressed

eigenstates, not of the atoms alone. The strength of the coupling is proportional

to the Rabi frequency, see for example the expressions of the dipole force, which is

proportional in turn to the laser intensity. The regime where a dressed-atom picture

is more suitable is thus given by Ω ≫ Γ. Considering a quantized description for

the laser field as well, the coupling happens in first instance between the atoms

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and the populated laser modes, to give rise to spontaneous emissions which couples

in second instance the atoms to the empty laser modes and is described in terms

of population transfers between dressed eigenstates. With the laser intensity space

dependent, the dressed states energies acquire a space dependence too. Dalibard

and Cohen-Tannoudji defined a force acting on each state as related to the gradient

of the energies and investigated how this force is related to the dipole gradient force

and how transitions between the dressed-states lead to fluctuations in the sign of

the dipole force which can be larger than the mean force itself and indeed have a

major effect for intense lasers.

As pointed out, in order to consider the eigenstates of the combined system, the

laser field is quantized as well and (62) is substituted by the corresponding second

quantization expression

E(r) =∑

λ

[ǫλ(r)aλ + ǫ∗λ(r)a

†λ] = E(+)(r) +E(−)(r) , (110)

where aλ, a†λ are the photons destruction and creation operators and now E(±)(r)

acquire an operator nature. In a laser, only one of the modes will be predominantly

occupied.

It follows that the unperturbed atom states are eigenstates of the free atom

Hamiltonian, see (71)

Ha = ~ωaS†S , (111)

analogously those of the free photons are eigenstates of the free field Hamiltonian

HL =∑

λ

~ωλa†λaλ , (112)

and the combined states are denoted as |g, n + 1〉, |e, n〉, where in addition to the

notation of the previous section, n and n+ 1 denote the number of photons in the

laser mode. The assumption is that of near resonance,

∆ ≪ ωL, ωa . (113)

Following the same steps as in the previous section, the interaction Hamiltonian

under the RWA is found to have the same form, see (86), albeit with the laser field

operators instead of the classical functions

Hint ≃ −deg(S† · E(+)(r, t) + S · E(−)(r, t)) . (114)

The Rabi frequency is now defined from the expectation value of the interaction

Hamiltonian as

Ω(r)eiφ(r) =2

~〈e, n|Hint|g, n+ 1〉 = −2

√n+ 1

d · ǫL(r)~

(115)

where the usual relations for creation and destruction operators acting on states

with n photons have been used. These relations also show that the only nonvan-

ishing matrix elements of the interaction Hamiltonian (114) are those connecting

states which differ for one photon. Under the RWA, the only interactions taken

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into account involve one-photon emission or absorption, two-photon processes are

neglected.

The full Hamiltonian written in the basis of the bare (unperturbed) states is

diagonalized to find the new eigenstates of the combined system, the space depen-

dent dressed states |±〉, combinations of the bare states, and their eigenenergies E±which are given by

E± = (n+ 1)~ωL − ~∆

2± ~Ω′(r)

2. (116)

The energy eigenvalues in the dressed state picture, apart from constant terms,

show a dependence on the laser amplitude and detuning through the effective Rabi

frequency given by

Ω′(r) =√

Ω2(r) + ∆2 . (117)

Besides, the two dressed states depend on the Rabi frequency with opposite signs.

Notice also that the request of a large effective Rabi frequency can be satisfied

equivalently by large detunings or intense laser fields. The dressed eigenstates can be

used as a basis and the whole theory developed for the optical Bloch equations can

be rewritten in this basis. The algebra is quite cumbersome but straightforward. The

starting point is the inversion of the expression of the dressed states as combinations

of the bare states so that the bare states are given as combinations of the dressed

ones. By retracing the steps taken in the previous section, the mean stationary

dipole force can be written as

Fgrad = −~∇Ω′

2(Π+ −Πm) , (118)

where Π± are the dressed states populations, i.e., the diagonal density matrix ele-

ments in the dressed state basis, and their coefficients are nothing but the gradients

of the energy eigenvalues (116) so that, in steady state

Fgrad = −∇E+Π+ −∇E−Πm . (119)

In the same fashion as for the optical Bloch equations discussed in Sec. 3.3.2,

the master equation in the dressed state basis can be solved to find an explicit

expression for Π± and consequently an explicit expression for the dipole gradient

force. The latter is found to be in agreement with the results of the optical Bloch

equations within the appropriate approximations. Given the physical meaning of

Π±, it is clear, as pointed out by Dalibard and Cohen-Tannoudji in Ref. 129, that

the mean dipole force of Eq. (119) is the average of two forces, −∇E±, the two

forces experienced by the two dressed eigenstates. Two points are important to

understand the dressed state interpretation. First, ascribing to −∇E± the nature

of a force, and therefore to the eigenvalues E± that of a potential, implies that

the two dressed states respond in an opposite way to the action of the laser: State

|+〉 is repelled (a low field seeking state) whereas state |−〉 is attracted to the high

laser intensity regions. Second, the resultant effect of the force will depend on the

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dressed state populations: If the high field seeking state is the most populated state,

the dominant effect will be attraction to high laser intensity regions. This connects

to the final expression of the dipole gradient force which depends on the detuning

since, from the exact solution of the master equation for Π±, it is found that the

high field seeking state |−〉 is more populated for negative detuning. If the two

populations are equal, which happens at perfect resonance for a stationary atom,

the force averages to zero.

3.3.4. Atom-laser dynamics: To the microscopic world and back

Armed with the knowledge of what can be expected from the study of the atom

dynamics under the effect of an incident electromagnetic field, we shall now delve

into a second quantization description which leads quite naturally to the introduc-

tion of local field effects and we shall aim at a microscopic model for the coupled

laser-BEC propagation to support the semi-classical model we have introduced to

start with.

Quantum field theory, which allows to consider the quantum statistics of the

particles, was applied to the problem of BECs interacting with coherent radiation

already at the beginning of the 1990s’ initially assuming the optical field as un-

affected by the atom dynamics. Exactly as for a description in terms of density

matrix or dressed-state, the starting point is the system Hamiltonian. Since the

coupling between atom states is due to the light field and is related to excitations

of the atoms, the Hamiltonian involves a quantized description of the laser field and

of the vacuum radiation field describing spontaneous emission. A model very much

similar to what we shall discuss hereafter, but with no spontaneous emission, local

field or collisional Kerr-like effects, was introduced by Zhang and Walls130 and used

to study the diffraction of two-level atoms from a standing wave. The simplifications

of the model allowed to identify the basic physics of laser–atom interactions and

the different behavior of Fermions and Bosons. Studying the effect of the radiation

fields of the atoms, lead Zhang to the description of how the exchange of photons

between atoms induced by the laser interaction can be seen as an atom–atom in-

teraction itself,131,132 a point of view that had already been discussed by Burns

et al. a few years earlier.133 The Hamiltonian here, although still considering ideal

cold atoms, included both the interaction with the incident laser field and with

the vacuum radiation field responsible for spontaneous emission. Possible nonlinear

effects such as self and cross-phase modulation due to the laser-induced atom–atom

interaction were predicted. For atoms propagating in a traveling laser beam with

Gaussian intensity profile, the longitudinal evolution of the incident laser field was

included in the model via an approximate solution of the wave equation, which

led to a Kerr-like nonlinearity for the atoms in the ground state and to the pre-

diction of atomic solitons.134,135 Another important step forward in clarifying the

description of atom-laser interactions was taken by Lewenstein et al. who intro-

duced an Hamiltonian and discussed the various possible approximations and the

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relation between the velocity gauge and the length gauge.122 Coupled equations

for the ground and first excited state were derived from a master equation, the

coupling mediated by light via the dipole interaction120,136 through a potential of

the form (109). These were used to study Thirring solitons for atoms in a resonant

standing wave after discarding noncoherent phenomena such as spontaneous emis-

sion. Alternative to the logic of writing an equation for the atoms interacting with

incident and vacuum radiation fields, a master equation for the atoms was derived

where the driving field was calculated as the incident field plus the field radiated

by all the atom dipoles (as in a local field model) and than inserted into the atom

equation.137 Along similar lines, Javanainen wrote an Hamiltonian including the

dipole coupling between ground and first excited state and for the electromagnetic

field an expression given by the sum of the free field and the field radiated by the

dipoles.138 Light scattering from the atoms was studied in this work, while a similar

Hamiltonian was introduced to study the effect of atom-light interactions on the

refractive index experienced by the propagating light.139

The first step thus is to write an Hamiltonian that can be used to describe the

physics one is interested in Refs. 121 and 140. We shall follow here the reasoning

presented by Krutitsky et al.140 who introduced a coupled system of equations for

the atom field and for the electromagnetic radiation, showing both the effect on the

light propagation due to the BEC and the effect on the BEC due to dipole interac-

tions. From the Hamiltonian, the Heisenberg equations for the field operators of the

ground state, the first excited state end the vacuum photons related to excitation

and emission of the atoms are derived. A solution for the vacuum photon operator

is found and used in the equation for the atom field operators. The dynamics of

the first excited state is decoupled from that of the ground state under specific

assumptions which leaves us with an equation for the ground state field operator

translated into complex numbers in the spirit of mean field theory. We shall now

go through these steps with a bit more details.

The starting point is a system which includes a two-level atom (ground and

first excited state) interacting with an external (classical) electromagnetic field.

The total Hamiltonian includes the Hamiltonian of the free atoms Ha, the Hamil-

tonian for the free photons Hvacuum (they are described by a vacuum field that

gives rise to spontaneous emission of the atoms), the Hamiltonian corresponding

to the dipole interaction between the atoms and the incident field Ha-laser (which

drives internal transitions for the atoms), the Hamiltonian corresponding to the

coupling between the atoms and the vacuum field Ha-vacuum (this should describe

an exchange of photons between the ground and first excited state of an atom and

consequent internal transitions), contact interactions (for example binary collisions

in the ground state of a BEC). These last interactions are neglected by some au-

thors on the ground that the saturation parameter can be quite large but may be

retained and included in the free atom Hamiltonian in the case of very large de-

tuning or very low laser intensities. Furthermore, this gives a Kerr-like nonlinear

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term which can play an important role if the atom density changes during the in-

teraction. The atom interaction with light occurs via the resonant dipole–dipole

interaction of the induced point dipoles and is mediated by the exchange of pho-

tons with annihilation operator ckλ (corresponding to plane waves in vacuum with

wave vector k and polarization λ).a Owing to the presence of light waves, internal

transitions of the atoms will be induced, this is the reason why internal levels must

be taken into account in the model, and only two levels are considered here under

the near resonance approximation discussed in the previous section. There is not

an Hamiltonian for the incident electromagnetic field and the equation governing

it is derived from Maxwell’s equation since this field is assumed to be classical. In

practice, the external electromagnetic field is fed into the system through the dipole

term. The Hamiltonian, comprising the terms described above,

H = Ha +Hvacuum +Ha-vacuum +Ha-laser , (120)

is known from quantum mechanics and we will give here only some intuitive justi-

fication for how it should look like.115 Here,

Ha =∑

i

[

− ~2

2m∇2

i + ~ωaS†i Si

]

(121)

=∑

i

[

− ~2

2m∇2

i + ~ωa|2i〉〈2i|]

, (122)

where S† = |2〉〈1| is the Pauli spin operator describing the transition from state 1

to state 2 and sums are over all the atoms with the provision that states of different

atoms are orthogonal. For a two level system as the one considered here, the two

states |1〉, |2〉 can be considered as a complete set. Besides the “internal energy”,

i.e., the energy related to the presence of the excited state, is described by

H0 =∑

i

~ωa|2i〉〈2i| =∑

i

~ωaS†i Si , (123)

so that

H0|1〉 = ~ωa|2〉〈2|1〉 = 0 ground state E1 = 0 , (124)

H0|2〉 = ~ωa|2〉〈2|2〉 = ~ωa|2〉 1st excited st. E2 = ~ωa . (125)

Hvacuum =∑

k,λ

~ωk c†k,λck,λ (126)

is the usual free field Hamiltonian now written for the vacuum field photons.

Ha-laser = −deg

∑

i

(Si + S†i )Einc(ri, t) (127)

aNotice that we shall now use the common notation for creation and annihilation operators inorder to avoid confusion with the notation later introduced for the classical equations.

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is the dipole Hamiltonian describing the interaction with the incident field. Under

the RWA

Ha-laser ≃ −deg

∑

i

(S†i · E(ri, t)

(+) + Si · E(ri, t)(−)) (128)

This is what introduced in Sec. 3.3.2 only with the explicit summation over all

the atoms: The incident field drives internal transitions described by Si, S†i . In

a full quantum field description, the incident field would be a quantum operator

as well and the first term could be interpreted as the transition from the excited

state to the ground state accompanied by the emission of a photon. The quantized

electromagnetic field is written as a mode superposition with a†e−ikx+iωt+aeikx−iωt

so that the part with negative frequency is connected to the creation operator.

Analogously, the second term would be the transition from the ground to the excited

state accompanied by the absorption of a photon. However, in this model, the

incident electromagnetic field is treated classically and E+, E− are just the positive

and negative frequency parts of the real field instead of creation and annihilation

photon operators.

Ha-vacuum = −~

∑

i

∑

k,λ

g∗k,λc†k,λe

−ikr(Si + S+i ) + H.c. (129)

describes the dipole interaction of the atoms with the vacuum field, thus the ex-

change of photons between the ground and the excited state. Here

g = i

√

2πωk

~Vdegek,λ (130)

gives the coupling strength atom-photons. This is exactly the same kind of interac-

tion Hamiltonian in dipole approximation as given above −d ·E but now the field is

the radiation field of the photons emitted and absorbed by the atoms. It is treated

as a quantum field and written with the usual mode expansion.115,118 Spontaneous

emission in this model will be described as resulting from the interaction of the

atoms with vacuum fluctuations of the electric field and it requires a quantum me-

chanical description of this field as well. This is one of the main differences from

what we have done previously introducing a phenomenological damping constant

into the optical Bloch equations. Therefore now there are two possibilities to have

internal transitions: Either because of the coupling to the incident field or because

of the spontaneous emission and absorption of photons.

Quantum statistics is taken into account in a second quantization formal-

ism122,141,142 by introducing the quantum field

|Ψ(r, t)〉 = ψ1(r, t)|1〉+ ψ2(r, t)|2〉 (131)

with the equal time commutators appropriate for a bosonic system (dropping the

index denoting the atom and keeping only the index denoting the state)

[ψj(r, t), ψl(r′, t)] = [ψ†

j(r, t), ψ†l (r

′, t)] = 0 , (132)

[ψj(r, t), ψ†l (r

′, t)] = δjlδ(r− r′) . (133)

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The different terms of the Hamiltonian in second quantization become:

Ha =

2∑

j=1

∫

dr ψ†j

(

− ~2

2m∇2

)

ψj +

∫

dr ψ†2~ωaψ2 , (134)

Ha-laser = −∫

drdeg · Einc(ψ†2ψ1 + ψ†

1ψ2) (135)

≃ −∫

dr(ψ†2deg · E(+)

inc ψ1 + ψ†1deg ·E(−)

inc ψ2) , (136)

Ha-vacuum = −~

∫

dr∑

k,λ

g∗k,λc†(t)k,λe

−ikr(ψ†1ψ2 + ψ†

2ψ1) + H.c. , (137)

Hcoll =Ug

2

∫

dr(ψ†2(r)ψ

†1(r)ψ1(r)ψ2(r) + ψ†

1(r)ψ†1(r)ψ1(r)ψ1(r)) . (138)

The atom-laser Hamiltonian has been simplfied under the RWA. The last term rep-

resents interatomic collisions with Ug = 4π~2a/m. Here, a is the scattering length

which characterizes these interactions under the assumptions of low energy and low

density for the shape-independent approximation to be valid. The introduction of

this binary collision Hamiltonian is central to the theory of dilute BEC95 and, for

the problem at hand, has been discussed by Lewenstein et al.122 in some detail. We

shall neglect collisions involving atoms in the excited state under the assumption

that excitation is rather weak.

Three coupled Heisenberg equations respectively for the ground state ψ1, the

first excited state ψ2 and the vacuum field photon operator ckλ are derived from

the total Hamiltonian as i~(∂f/∂t) = [f,H ]. Carefully applying the commutation

rules, we are left with

i~∂ψ1

∂t= − ~

2

2m∇2ψ1 − d · E−

incψ2 − ~

∑

k,λ


−ikrψ2

− ~ψ2

∑

k,λ

gk,λc(t)k,λeikr , (139)

i~∂ψ2

∂t= − ~

2

2m∇2ψ2 + ~ωaψ2 − d ·E+

incψ1

− ~

∑

k,λ


−ikrψ1 − ~ψ1

∑

k,λ

gk,λc(t)k,λeikr , (140)

i~∂ck,λ∂t

= ~ωkck,λ − ~g∗k,λ

∫

dre−ikr(ψ†1ψ2 + ψ†

2ψ1) . (141)

Equations (139) and (140) describe the dynamics of the quantum atom field in the

presence of a classical electromagnetic field and of the vacuum field. Equation (141)

describes the dynamics of the vacuum field due to spontaneous emission from the

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atoms: The incident field is propagating through a medium made of interacting

induced dipoles. The incident field causes these dipoles to emit radiation (notice

how g depends on d, see (130)), thus a given dipole does not react to the bare

incident field but to an effective field resulting from the incident field and the

radiation emitted by all other dipoles. So basically, this is the fuzzy quantum way

to say what the Lorentz–Lorenz model says in a classical language.

The equation for c is formally solved. This equation has the form c′(t) +

P (t)c(t) = Q(t), thus the solution is given by the sum of a solution of the ho-

mogeneous equation and a particular solution which can be found by the method

of Lagrange variation of constants:

ck,λ(t) = e−iωkt

[

ck,λ(0) + ig∗k,λ

∫ t

0

dt′∫

dr′e−ikr′+iωkt′

(ψ†1ψ2 + ψ†

2ψ1)

]

. (142)

Thus the dynamics of c is found to be given by the vacuum fluctuations of a sea

of free photons (first term) plus the effect of the radiation scattered by the atoms,

i.e., the result of the interaction between the field and the atoms (second term).

The solution for c(t) is then introduced into the two equations for ψ1 and ψ2,

under the RWA. We shall see how the local electromagnetic field naturally shows up

in these two equations and is related to the macroscopic field through the Lorentz–

Lorenz relation as usual.

Local field effects on the atom ground state. To see the effect of the scattered field

(the second term in c) let us use (142) into the equation for the ground state (139).

Writing only the terms of (139) involving c:

−~

∑

k,λ

g∗k,λc†(0)k,λe

−ikr+iωktψ2 − ~ψ2

∑

k,λ

gk,λc(0)k,λeikr−iωkt

− i~∑

k,λ

|gk,λ|2∫ t

0

dt′∫

dr′[e−ik(r−r′)+iωk(t−t′)(ψ†

1(r′, t′)ψ2(r

′, t′)

+ ψ†2(r

′, t′)ψ1(r′, t′))]ψ2(r, t)

− i~ψ2(r, t)∑

k,λ

|gk,λ|2∫ t

0

dt′∫

dr′[eik(r−r′)−iωk(t−t′)(ψ†

1(r′, t′)ψ2(r

′, t′)

+ ψ†2(r

′, t′)ψ1(r′, t′))] . (143)

In the third term of (143), ψ2(r, t) can be brought under the integral and, being

careful with the commutation rules, we have

−i~∑

k,λ

|gk,λ|2∫ t

0

dt′∫

dr′[eik(r−r′)−iωk(t−t′)(ψ†

1(r′, t′)ψ2(r

′, t′)ψ2(r, t)

+ ψ2(r, t)ψ†2(r

′, t′)ψ1(r′, t′))] . (144)

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Carrying out the same calculation for the second term and summing up, the last

two terms of (143) give

−i~∑

k,λ

|gk,λ|2∫ t

0

dt′∫

dr′[(eik(r−r′)−iωk(t−t′) − c.c.)ψ†

1(r′, t′)ψ2(r

′, t′)ψ2(r, t)]

− i~∑

k,λ

|gk,λ|2∫ t

0

dt′∫

dr′[eik(r−r′)−iωk(t−t′)ψ2(r, t)ψ

†2(r

′, t′)ψ1(r′, t′)]

+ i~∑

k,λ

|gk,λ|2∫ t

0

dt′∫

dr′[e−ik(r−r′)+iωk(t−t′)ψ†

2(r′, t′)ψ1(r

′, t′)ψ2(r, t)]

= ©1 +©2 +©3 . (145)

Now it is a matter of manipulating these terms. Krutitsky et al.140 show how

the dipole moment operator is naturally introduced in these equations. Here we

shall reproduce their calculations. For the first term of (145), since

(dek,λ)2 =

3∑

m,n=1

dmemk,λdne

nk,λ (146)

it turns out that

+i

~

∫ t

0

dt′∫

dr′∑

k,λ

2π~ωk

V

3∑

m,n=1

dmenk,λdne

mk,λ

× [(eik(r−r′)−iωk(t−t′) − c.c.)ψ†

1(r′, t′)ψ2(r

′, t′)ψ2(r, t)] (147)

which can be rewritten by using the commutators of the free electric field as

©1 = +i

~

∫ t

0

dt′∫

dr′3∑

m,n=1

dmdn[Em(r, t), En(r′, t′)]ψ†

1(r′, t′)ψ2(r

′, t′)ψ2(r, t) ,

(148)

(see for example the texbook by Cohen-Tannoudji115 CIII.1 and CIII.2, and the

reference book by Born and Wolf104 Sec. 2.2.3). Making use of the full expression

of such commutators,

[Em(r, t), En(r′, t′)] =

∑

l

2π~ωl

Vemen(e

ik·(r−r′)−iωl(t−t′) − c.c.) (149)

= i~c

(3RmRn

R2− δmn

)[δ′(R− cτ) − δ′(R+ cτ)

R2

− δ(R − cτ)− δ(R + cτ)

R3

]

−(RmRn

R2− δmn

)[δ′′(R − cτ)− δ′′(R+ cτ)

R

]

, (150)

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where R = r − r′, τ − t − t′, summed over m,n, integrating over time and finally

introducing the polarization operator

P(r, t) = degψ†1(r, t)ψ2(r, t) + H.c. = P+(r, t) + P−(r, t) (151)

the first term of (145) reads

©1 = −deg

∫

dr′[

1

c2([¨P+]n)n− [

¨P+]

R+

1

c

3([˙P+]n)n− [

˙P+]

R2

+3([P+]n)n− [P+]

R3

]

ψ2(r, t) .

Here, [P ] = P (r′, t−R/c) and n = R/R.

This last expression, following Born and Wolf, can be rewritten further and the

first term of (145) reads

©1 = −d

∫

dr′∇×∇× [P+]

Rψ2(r, t) . (152)

For the second term of (145), were it possible to invert the order of the matter

fields bringing ψ2 at the end, we would have an expression analogous to the struc-

ture treated for the integral ©1 . However, the fields do not commute since we are

integrating over r′ and t′ thus it is possible for the arguments of the field functions

to be the same somewhere in the domain of integration. Introduce τ = t − t′ andsplit the time integral as

∫ t

0

dt′ =

∫ t−ǫ

0

dt′ +

∫ t

t−ǫ

dt′ =

∫ ǫ

0

dτ +

∫ t

ǫ

dτ ,

where ǫ≪ t. Let us examine the two integrals resulting from ©2 .

©2 = −i~∫

dr′∑

k,λ

|gk,λ|2∫ ǫ

0

dτ [eik(r−r′)−iωkτ ψ2(r, t)ψ

†2(r

′, t− τ)ψ1(r′, t− τ)]

− i~

∫

dr′∑

k,λ

|gk,λ|2∫ t

ǫ


†2(r

′, t− τ)ψ1(r′, t− τ)].

(153)

Now, between ǫ and t the two fields ψ2, ψ†2 do commute. Thus

©2 = −i~∫

dr′∑

k,λ

|gk,λ|2∫ ǫ

0


†2(r

′, t− τ)ψ1(r′, t− τ)]

− i~

∫

dr′∑

k,λ

|gk,λ|2∫ t

ǫ

dτ [eik(r−r′)−iωkτ ψ†

2(r′, t− τ)ψ1(r

′, t− τ)ψ2(r, t)]

(154)

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For ǫ→ 0 (τ → 0) in the first integral t ≃ t′ and using [ψ2(r, t), ψ†2(r

′, t)] = δ(r−r′),we have

©2 = −i~∫

dr′∑

k,λ

|gk,λ|2∫ ǫ

0

dτ [eik(r−r′)−iωkτ δ(r− r′)ψ1(r

′, t− τ)]

− i~

∫

dr′∑

k,λ

|gk,λ|2∫ ǫ

0


2(r′, t− τ)ψ2(r, t)ψ1(r

′, t− τ)]

− i~

∫

dr′∑

k,λ

|gk,λ|2∫ t

ǫ


2(r′, t− τ)ψ1(r

′, t− τ)ψ2(r, t)] .

(155)

In the second integrand ψ2 and ψ1 commute, thus the second integrand is equal to

the third

©2 = −i~∫

dr′∑

k,λ

|gk,λ|2∫ ǫ

0

dτ [eik(r−r′)−iωkτδ(r− r′)ψ1(r

′, t− τ)]

− i~

∫

dr′∑

k,λ

|gk,λ|2∫ t

0


2(r′, t− τ)ψ1(r

′, t− τ)ψ2(r, t)]

(156)

or

©2 = −i~∑

k,λ

|gk,λ|2∫ ǫ

0

dτe−iωkτ ψ1(r, t− τ)

− i~∑

k,λ

|gk,λ|2∫

dr′∫ t

0


2(r′, t− τ)ψ1(r

′, t− τ)ψ2(r, t)] .

(157)

Notice that the second integral in ©2 has a structure analogous to that of ©1 .

The third term of (145), ©3 can be summed to the second integral of ©2 to give

exactly the same structure of ©1 .

At this point, one can see that the last two terms of (143) give

©1 +©2 +©3 = −deg

∫

dr′∇×∇× [P+]

Rψ2(r, t)− deg

∫

dr′∇×∇

× [P−]

Rψ2(r, t) − i~

∑

k,λ

|gk,λ|2∫ ǫ

0

dτe−iωkτ ψ1(r, t− τ) . (158)

All this is valid under the RWA, under which assumption the Hamiltonian has been

derived, see also Sec. 3.3.2. With ψ2 = φ2e−iωat, the term involving [P+] can be

neglected, essentially because it oscillates at 2ω. The last term of (158) can be

calculated in the limit t → ∞, ǫ → ∞, ǫ ≪ t by extending it as∫∞0e−iωtdt =

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∫∞−∞H(t)e−iωtdt (H is the step function) and it vanishes in the limit of V → ∞confirming that the ground state has no decay or Lamb shift. In conclusion, defining

the local field as

E±eff(r, t) = E±

in(r, t) +

∫

dr′∇×∇× [P±]

R, (159)

where [f ] = f(r′, t−R/c), the ground state equation reads

i~∂ψ1

∂t= − ~

2

2m∇2ψ1 − d · E−

eff ψ2

− ~

∑

k,λ



∑

k,λ

gk,λc(0)k,λeikr−iωkt . (160)

Local field effects on the atom first excited state. To see the effect of this scattered

field (the second term in the solution for c) let us use (142) into the equation for

the first excited state (140). The procedure runs as for the ground state equation.

For the two terms involving c we end up with exactly the same result as in (158)

only with ψ2 in place of ψ1 and viceversa:

−d

∫

dr′∇×∇× [P+]

Rψ1(r, t) − d

∫

dr′∇×∇× [P−]

Rψ1(r, t)

− i~∑

k,λ

|gk,λ|2∫ ǫ

0

dτ [e−iωkτ ψ2(r, t− τ)] . (161)

Now, being τ = t− t′ and under the rotating wave assumption

ψ2(r, t− τ) = φ2(r, t− τ)e−iωa(t−τ) (162)

the last integral becomes

− i~∑

k,λ

|gk,λ|2∫ ǫ

0

dτe−i(ωk−ωa)τ φ2(r, t− τ)e−iωat . (163)

Considering t→ ∞, ǫ→ ∞, ǫ≪ t,

−i~∑

k,λ

|gk,λ|2∫ ∞

0

dτe−i(ωk−ωa)τ φ2(r, t)e−iωat

= −i~∑

k,λ

|gk,λ|2[

πδ(ωk − ωa)− iP(

1

ωk − ωa

)]

ψ2(r, t)

= ~(δ − iγ/2)ψ2(r, t) (164)

which gives a Lamb shift ~δψ2 and a decay rate −i~γ/2ψ2.

These two terms do not vanish in the case of the excited state equation since even

though the coefficient g vanishes in the limit V → ∞, the two terms multiplying it

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can still explode at resonance. Finally the equation for the excited state reads

i~∂ψ2

∂t= − ~

2

2m∇2ψ2 + ~(ωa + δ − iγ/2)ψ2 − d · E+

effψ1

− ~

∑

k,λ



∑

k,λ

gk,λc(0)k,λeikr−iωkt .

(165)

The local field. Notice that the second term in the definition of the local field (159) is

exactly what expected in classical electromagnetism for the electric field produced

by linear electric dipoles: Classically particles react to the incident field as dipoles

thus emitting dipole fields which will determine (a) the effective force acting on any

other dipole and (b) the average measurable field. Considering an electromagnetic

wave propagating through an homogeneous isotropic nonmagnetic medium, the field

E′j , acting on the jth dipole is given by

E′j = Einc +

∑

l

Ejl

︸︷︷︸

. (166)

contribution from all other dipoles

The field produced by the lth dipole is the typical field produced by a dipole in

vacuum, E = ∇×∇× [p]/R. Thus

Ejl = ∇j ×∇j ×pl(t−Rjl/c)

Rjl, (167)

with Rjl = |rj − rl|. Considering the dipole distribution as a continuum, the

dipole moments become continuous functions and so does the density: p = p(r, t),

n = n(r, t), and for the total electric dipole per unit volume P = np = nαE′.Substituting Ejl into E′ and going over to a continuous distribution

E′(r, t) = Einc +

∫

dr′∇×∇× P(r′, t−R/c)

R. (168)

which gives the connection between effective and incident field. Bowden and Dowl-

ing108 demonstrated how this local field is indeed related to the macroscopic field

for which Maxwell’s equations are written. Assuming that the local field is given

by the macroscopic field and an internal field due to action of the dipoles within

a cubic wavelength E′ = E + Eint, Bowden and Dowling proved that, even for

time-dependent fields, one has,

E′(r, t) = E(r, t) +4π

3P(r, t)

(see also Sec. 3.2). It follows that the two equations for ψ1 and ψ2 are now coupled

to Maxwell’s equations via the monochromatic macroscopic electromagnetic field.

Decoupling of the atom equations under adiabatic approximation. A step suggested

now by most authors is to move to the reference frame rotating with ωL since

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the atom equations are coupled to the macroscopic average field assumed to be

monochromatic at ωL:

E+macro(r, t) = A(r, t)e−iωLt , (169)

ψ2(r, t) = φ2(r, t)e−iωLt . (170)

Moving to a new reference frame involves the transformation U = eiωL|2〉〈2|t and

|ψ′〉 = U |ψ〉 or|ψ′〉 = ψ1|1〉+ ψ2e

iωLt|2〉 → ψ′1 = ψ1 , ψ′

2 = ψ2eiωLt ,

while, from the exctinction theorem, we can assume that the macroscopic field has

the monochromatic time dependence ∝ e−iωLt induced by the incident monochro-

matic field.

Defining

Ω+(r) =2d ·A(r)

~, Ω−(r) =

(Ω+(r)

)∗(Rabi frequency), (171)

G1 = −~

∑

k,λ


−i(kr−ωkt+ωLt)φ2 − ~φ2∑

k,λ

gk,λc(0)k,λei(kr−ωkt−ωLt)

= − ~[Γ†1φ2 + φ2Γ2] (noise term), (172)

G2 = −~

∑

k,λ


−i(kr−ωkt−ωLt)ψ1 − ~ψ1

∑

k,λ

gk,λc(0)k,λei(kr−ωkt+ωLt)

= − ~[Γ†2ψ1 + ψ1Γ1] (noise term), (173)

∆ = ωL − ωa − δ detuning, (174)

Krutitsky et al. arrive to two stochastic equations for the ground and first excited

state:

i~∂ψ1

∂t= − ~

2

2m∇2ψ1 −

~

2Ω−φ2 −

4π

3d2φ†2φ2ψ1 +G1 , (175)

i~∂φ2∂t

= − ~2

2m∇2φ2 − ~(∆ + iγ/2)φ2 −

~

2Ω+ψ1 −

4π

3d2ψ†

1ψ1φ2 +G2 . (176)

The noise terms describe the effects of vacuum fluctuations (spontaneous emission

and absorption) on the atom quantum fields, and Γ are noise operators with random

correlations

〈Γl(r, t)Γj(r, t′)〉 = 0 , (177)

〈Γ†l (r, t)Γj(r, t

′)〉 = 0 , (178)

〈Γl(r, t)Γ†j(r

′, t)〉 =

G−(t− t′, r− r′) j = l = 1 ,

G+(t− t′, r− r′) j = l = 2 ,

0 j 6= l ,

(179)

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G±(τ, r− r′) =∑

k,λ

|gk,λ|2e−i(ωk±ωL)τ+ik(r−r′) (180)

is the photon Green function.

The equations for ψ1 and ψ2 can now be decoupled via the adiabatic elimination

of the excited state.9 The equation for ψ2, which includes both the Lamb shift and

a decay rate γ due to spontaneous emission, is solved within the adiabatic approx-

imation (large detuning ∆ ≫ γ) and the solution is then inserted into the equation

for ψ1 so that an equation for the ground state atom field is obtained coupled to

the local field. If the detuning is large enough, as discussed in the previous sec-

tions, spontaneous emission gives a negligible contribution. In this limit it can be

assumed that the excited state dynamics can be neglected. In more detail, consider-

ing Eq. (176) for the first excited state, the first assumption consists in saying that,

with the ground state fully occupied, in the region of significant condensate density

the term ∇2φ2 can be neglected with respect to ∆φ2 (large detuning hipotesis).

The same is assumed in case there are other terms of the form φ†2φ2.

i~∂φ2∂t

≃ −~

(

∆+ iγ

2

)

φ2 −~

2Ω+ψ1 −

4π

3d2ψ†

1ψ1φ2 +G2 . (181)

A local detuning can be defined as,

∆loc(r, t) = ∆

(

1− 4π

3αψ†

1(r, t)ψ1(r, t)

)

, (182)

where α = −(d2/~∆), see Eq. (45), so that

∂φ2∂t

+ P (t)φ2 = Q(t) , (183)

P (t) = −i[

∆loc + iγ

2

]

, (184)

Q(t) =i

2Ω+ψ1 −

i

~G2 (185)

which can be solved as

φ2(r, t) = ei∫

t

0(∆loc(r,s)+iγ/2)ds

∫ t

0

dsQ(s)e−i∫

s

0(∆loc(r,s

′)+iγ/2)ds′ . (186)

Assuming that ∆loc is a very slow function of time

φ2(r, t) ≃ ei(∆loc(r,t)+iγ/2)t

∫ t

0

dsQ(s)e−i(∆loc(r,s)+iγ/2)s , (187)

i.e.,

φ2(r, t) ≃ei∆T t

−i∆T

∫ t

0

ds g(s)f ′(s) , (188)

with ∆T = ∆loc(r, s) + iγ/2, g = Q(s), f ′ = −i∆T e−i∆T .

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Integration by parts leads to

φ2 ≃ −Ω+(r, t)ψ1(r, t)

2[

∆loc + iγ

2

] +G2(r, t)

~

[

∆loc + iγ

2

] . (189)

Inserting this result into the equation for the ground state, with G[G1, G2] collecting

all the terms involving the noise functions G1, G2, one finds

i~∂ψ1

∂t= − ~

2

2m∇2ψ1 +

~

4∆

|Ω+|2(

1− iγ

2∆

)

(

1− 4π

3αψ†

1ψ1

)2

+( γ

2∆

)2ψ1 + G[G1, G2] . (190)

Here G = VRψ1 is nothing but a random potential related to the vacuum fluctua-

tions, with vanishing average 〈VR〉 = 0:

VR = ~∆− iγ/2

∆2l + γ2/4

[1

2(Γ2 + Γ†

1)Ω+ +

1

2(Γ†

2 + Γ1)Ω− + |Γ†

1 + Γ2|2]

. (191)

The structure of this equation is akin to the equation found via semi-classical rea-

soning with the exception of the missing decay rate which goes to eliminate the

otherwise unphysical singularity. In our model, all noncoherent terms and terms

related to spontaneous emission will be neglected on the grounds that we want to

analyze the coherent behavior of the system with very large detuning (= very small

spontaneous emission = very weak incoherent term). As pointed out by Chebotayev

et al.,144 this condition and the corresponding adiabatic elimination of the dynamics

of the excited state are intimately related to maintaining and transporting spatial

coherence.

The dipole operator. From (151)

P(r, t) = dψ†1(r, t)ψ2(r, t) + H.c. = P+(r, t) + P−(r, t) (192)

it follows that

P+(r, t) = dψ†1(r, t)ψ2(r, t) (193)

=αψ†

1ψ1

1− 4π

3αψ†

1ψ1 + iγ

2∆

A(r)e−iωLt + F [G2] = χE+(r, t) , (194)

where the stochastic term will again be neglected.

Finally, in the spirit of the BEC mean field theory, a macroscopic occupation of

the ground state allows for the ground state field operator to be substituted by a

C-number95,143 so that ψ†1ψ1 → |ψ1|2.

As for the electromagnetic field, it will be coupled to the atom fields via the

dipole operator. It has to be clear that the local field approximation used to relate

the incident field to Maxwell’s mean field is necessarily limited with respect to the

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exact macroscopic integral relations that can be discussed146 and atom correlations

are not considered exactly in (194) and consequently in the model equations.147

Within the limits of mean field theory and under the assumptions discussed

above, i.e., neglecting the stochastic terms, the atom equation (190) coincides with

Eq. (47) which was derived from a purely semi-classical reasoning, while the laser

equation is given by (52).

4. What Is It All For? The Coupled Propagation Problem,

Mutual Guiding and Structure Formation

Not only the interactions and forces discussed up to this section are responsible

for all the effects mentioned at the beginning of this review, from laser cooling to

trapping and deflecting atoms. The coupled atom-laser equations, by describing

the interplay of light and matter, can also shed light on the basic physics of this

process. Wallis,121 who arrived to the coupled equations showing clearly how and

when atom density correlations are neglected, proposed this model to study a BEC

in a gravito-optical trap where the atoms are subject to the gravitational field and

a blue-detuned evanescent laser wave which acts as a mirror to prevent the atoms

from falling. He showed the nature of the modifications on both atoms and light

due to the coupling in different parameter regimes. The coupled equations have

been applied by Krutitsky et al. to the study of the optical properties of ultra cold

atoms and to describe the diffraction of atoms from a strong standing wave.140,145

This is indeed the most typical example in which the wave nature of matter is

brought to light and electromagnetic radiation acts as the medium which scatters

the matter wave. One of the most interesting results is the fact that nonlinear effects

on both atoms and light can be governed via the dipole–dipole interactions. Besides,

the transverse laser dynamics was shown to be affected in a rather important way

by the work of Saffman and coworkers. These authors148 studied the evolution of

atoms and laser under the development of the modulational instability, a very well-

known instability that affects plane waves propagating through nonlinear media.

Under appropriate conditions, an homogenous field whose dynamics is described

by the nonlinear Schrodinger equation, is unstable, ripples and breaks down into

filaments.149 By numerically studying the dynamics described by the coupled atom-

laser equations, Saffman showed that this instability leads to filamentation for the

copropagating atomic beam and discussed the possibility of mutual trapping.150

This model was also extended to noncondensed atoms which do not show the long

range spatial coherence characteristic of BECs.151

Taking a step back from Saffman’s investigations to a simpler configuration, it is

interesting to study the effects of coupled propagation under stationary assumptions

from the point of view of the formation of localized structures. Structure formation

could be considered as signatory of nonlinear systems,152 and the long range coher-

ence of a BEC brings about new possibilities. Besides, localization and guiding for

the atom density have an obvious interest from the point of view of applications. If

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atoms behave like a wave, their propagation can be described via a refractive index

determined by light. Indeed, the stationary Schrodinger equation (47) is formally

analogous to the optics Helmholtz’s equation. It was discussed in 2005 how the shift

in the atom energy levels caused by the interaction with laser light and the subse-

quent dipole interaction with light can be used to change the phase of the atoms

wavefunction to the aim of focusing matter waves.153 This is not altogether differ-

ent from what happens in nonlinear optics where the Kerr nonlinearity produces a

modulation on the phase of the electromagnetic field which can lead to geometrical

deflection and focusing.96 The difference is that for the atoms the nonlinearity is

not only the simple Kerr-like term of the nonlinear Schrodinger equation but is

also mediated by the presence of light and vice versa for the light. This allows to

search, for instance, for a regime of mutual guiding, where copropagating atomic

and laser beams affect the mutual transverse dynamics leading to mutually local-

ized structures that could even be akin to solitons. In order to do this we shall use

the mean-field equations derived within a semiclassical framework keeping in mind

that, due to limitations discussed, these will be only ideal results.

4.1. The coupled equations

We shall report here the coupled mean field equations as derived in Sec. 3.2 for ease

of reference. Assuming a stationary state for the atoms

Ψ(r, t) = Φ(r)e−iωat (195)

and for the field

E = ReA(r)e−iωLte (196)

with ~ωa = Ea = ~2k2a/(2m) and ωL = kLc, the stationary Schrodinger equation

for the atoms reads

EaΦ = − ~2

2m∆Φ+

4π~2a

m|Φ|2Φ− α

4

|A|2(1− 4π

3 α|Φ|2)2Φ (197)

and the stationary laser equation becomes

∆A+ω2L

c2

(

1 +4πα|Φ|2

1− 4π3 α|Φ|2

)

A = 0 . (198)

In the atom equation we have now the interatomic collision potential as well Vcoll =

4π~2a/m|Φ|2. Here a is the scattering length and in what follows we shall consider

a > 0, repulsive interactions.

These equations can be rewritten in adimensional variables as

µ2Φ = −∆Φ + β|Φ|2Φ− s|A|2(

1− s|Φ|2)2 Φ , (199)

∆A+

(

1 +3s|Φ|2

1− s|Φ|2

)

A = 0 , (200)

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where lengths are normalized with respect to the laser wavelength as r = rkL, for the

atom waveffunction |Φ| = |Φ|/Φ∗ where (4π|α|/3)Φ2∗ = 1 defines a critical density,

for the laser |A| = |A|/A∗, with m|α|A2∗/(2~

2k2L) = 1, s = sign(α) = −sign(∆),

µ = ka/kL and β = 6as/(k2L|α|). Unless otherwise stated, we shall consider β ≃ 38

and µ = 1 and the tilde will be dropped.

4.2. Stationary localized solutions

A first question is whether mutual localization is possible as a consequence of the

dipole–dipole interactions. An answer to this question depends on the existence of

stationary localized solutions to the model equations. Clearly, the only possibility

to avoid the natural broadening of the atom wavefunction and the diffraction of the

laser field is to counteract the diffractive tendency due to the kinetic energy and to

the repulsive collisions with the dipole interaction.

4.2.1. Stationary localized solutions with a constant laser field

The simplest thing to do is to ignore the back action of the atoms on the laser

field and assume that one can consider E = const.154 The question being answered

is then whether the action of a homogeneous laser field can be enough to induce

localization on the atoms. From the low density limit, it is evident that for the

dipole interaction to overcome the repulsive collisional term, the laser intensity

must exceed a threshold:

|A|2 > |Ath|2 =β

2. (201)

Equation (199), with a constant laser field and considering a real amplitude Φ

and assuming planar geometry where everything depends only on x, admits a first

integral

1

2

(dΦ

dx

)2

− β

4|Φ|4 + s|A|2Φ2

2(1− sΦ2)+ EΦΦ

2 = C . (202)

This is equivalent to the Hamiltonian of a particle with position Φ under the effective

potential

V = −β4|Φ|4 + s|A|2Φ2

2(1− sΦ2)+ EΦΦ

2 (203)

and a localized solution, which corresponds to Φ, dΦ/dx → 0 for x → ∞ requires

C = 0. Thus the only possibility for a localized solution is given by V → 0− when

Φ → 0+ at x→ +∞, where it is found that

V ∼ EΦΦ2 +

s|A|22

Φ2 < 0 → EΦ < Ec = −s|A|2

2. (204)

With these information in mind, a study of the phase portrait of the system, i.e., of

the dependence of Φ′ on Φ confirms the existence of localized solutions for laser

intensities exceeding the threshold, see Fig. 1 from Ref. 154. A localized solution

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0 0.2 0.4 0.6−2

−1

0

1

2

ψ

ψ ′

(a)

0 0.2 0.4 0.6−0.2

−0.1

0

0.1

0.2

ψ

ψ ′

(c)

0 0.5 1−1

−0.5

0

0.5

1

ψ

ψ ′

(d)

0 0.2 0.4−2

−1

0

1

2

ψ

ψ ′

(b)

A

A

A B

Fig. 1. Phase portrait for Eq. (199). (a) and (c) are calculated with |A|2 > |Ath|2 for α > 0

and α < 0 respectively. (b) and (d) show the two analogous cases but with |A|2 < |Ath|2 where

no localized solution can be found. Trajectories from −A to A in (b), (c), (d) represent darksoliton-like solutions. The separatrix around the equilibrium point B in (b) represents a periodicsolution with amplitude always finite as a perturbation of a homogeneous background.

starts at x → −∞ close to 0 and with vanishing first derivative, moves to x =

0 where it reaches it maximum amplitude again with vanishing first derivative,

and proceeds to x → +∞ where it tends to 0. This in terms of a phase portrait

corresponds to a separatrix trajectory such as the curves passing through the origin

seen in Fig. 1(a) and 1(c). The only cases where such a separatrix is always realized

are those for |A|2 > |Ath|2 for both signs of α. With α > 0 it is found that

a separatrix trajectory of the kind shown in Fig. 1(a) is realized also for laser

intensities below the threshold provided the number of atoms exceeds a critical

number. This critical number corresponds to a ground state solution with energy

exactly equal to Ec. However, the case of red detuning is found to be unstable

according to the stability criterion of Kolokolov and Vakhitov.155 According to

this stability analysis, the ground state is unstable against collapse when the total

number of atoms is an increasing function of the energy. By calculating numerically

the solutions to (199) it is found that this is exactly the case for red detuning.

Blue detuning, i.e., α < 0 leads to a stable configuration due to the saturation-type

nonlinearity. This is a clear signal of a break down of the model used which cannot

account for the strong focusing effect possible in the case of red detuning.

4.2.2. Stationary localized solutions as an effect of mutual guiding

Let us now investigate the effect of the coupling to the laser field by considering

the back action of the atoms on it. As before, the first question is whether mutual

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localization is possible within the model (197) and (198).156 To answer this, one

must investigate the existence of stationary modes localized in the transverse di-

rection, analogous to Kerr spatial solitons,

ψ(r) = φ(r⊥) exp(ihφz) , (205)

A(r) = a(r⊥) exp(ihz) . (206)

These must satisfy the fully stationary equations, in adimensional variables and

dropping the tilde:

∇2⊥φ− βφ3 +

sa2φ

(1− sφ2)2− 2µ2κφφ = 0 , (207)

∇2⊥a+

3sφ2a

1− sφ2− 2κa = 0 , (208)

and, if they exist, they describe long distance mutual guiding. Further to the normal-

ization introduced above, we have here κφ = (h2φ/k2a− 1)/2 and κ = (h2/k2L− 1)/2.

The two field amplitudes a(r), φ(r) are real amplitudes and, for simplicity, we shall

consider only one transverse dimension, r⊥ = x assuming that the system has a

very large extension in y and can be considered as homogeneous in that direction.

In addition, we shall consider µ = 1 and β ∼ 38 corresponding to a detuning of

∼ 100Γ for 87Rb atoms. If these states can be found, then it is worth investigating

how the system reaches them.

Similarly to what done above, by integrating the two equations over x, a first

integral is found which plays the role of the Hamiltonian from which they can be

derived:

1

2

(dφ

dx

)2

+1

2

(dχ

dx

)2

+Π(φ, χ) = C , (209)

Π(φ, χ) = −µ2κφφ2 − κχ2 − β

4φ4 +

3s

2

χ2φ2

1− sφ2, (210)

where c is an integration constant. The laser field amplitude has been redefined as

χ(x) = a(x)/√3 to show clearly the analogy between the Hamiltonian (209) and

the Hamiltonian for a fictitious particle of unit mass moving in a two-dimensional

space (spanned by the coordinates φ, χ with x playing the role of time) under the

potential Π(φ, χ). A mutually localized solution requires φ, χ, dφ/dx, dχ/dx → 0

for x → ∞, i.e., C = 0. Denoting the maxima of the two wavefunctions as φ0, χ0

and remembering that their first derivatives must vanish at these maxima, the

Hamiltonian gives a relation between the two peak values:

χ20 =

φ20(1− sφ20)(βφ20 + 4µ2κφ)

2[sφ20(3 + 2κ)− 2κ]. (211)

The search for mutually localized solutions leads to several conditions on the pa-

rameters of the problem. First of all, the potential Π has a singularity at 1−sφ2∗ = 0

and limits the subsequent analysis to the regime 1 − sφ2 > 0 which, in physical

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terms, limits us to densities lower than the critical one corresponding to φ2∗ = 1.

The laser equation (208) can be considered as a stationary Schrodinger equation

for χ(x) and a localized solution requires a trapping potential, i.e., s = +1. This

can be easily seen by expanding Eq. (208) to first-order in the atom density or,

alternatively, assuming a localized bell-shaped solution for φ2, and requiring that

the potential felt by the laser be a trapping potential given that 1− sφ2 > 0. Fur-

thermore, the trapping potential felt by the laser is induced by the atoms and we

are aiming at a localized solution for the atoms wavefunction, therefore the trap-

ping potential acting on the laser vanishes as x→ ±∞. This means that a localized

solution for the laser requires a negative eigenvalue, i.e., κ > 0. The same reasoning

holds for the atom equation, i.e., κφ > 0. Finally, in the parameter regime s = +1,

β > 0, 1 − sφ2 > 0, κφ, κ > 0, Eq. (211) implies that the existence of a real value

for χ0 requires 1 > φ20 > φ2min = 2κ/(3 + 2κ). This is nothing but the result of

the laser-atom interplay: The interatomic interaction induced by the laser has an

attractive nature for red detuning (s > 0) to counteract the natural tendency of the

atoms to broaden under the effect of their transverse kinetic energy and repulsive

interactions (β > 0), while the atoms provide a focusing refractive index which

counteracts the natural diffraction of light in vacuum. It is this two-way relation

that allows one to find mutually localized solutions with the same characteristic

width for both atom and laser wavefunctions (single scale solutions) or with the

laser width larger than the atom ones (multiple scale solution). In the opposite

case, the laser would experience an almost constant refractive index and nothing

would prevent it from diffractive spreading so that, after a short propagation, its

strength would not be enough to keep the atoms together.

Low density regime. These two kinds of solutions can be found analytically in the

low density limit where Eqs. (207) and (208) reduce to

φ′′ + φ(−2µ2κφ − βφ2 + sa2 + 2a2φ2) ≃ 0 , (212)

a′′ + a(−2κ+ 3sφ2) ≃ 0 . (213)

For the single scale solution, assuming a solution for the laser proportional to that

of the atoms as a(x) = ηφ(x), Eqs. (212) and (213) in the low density limit, i.e., re-

taining only linear terms in the atom density, reduce to two focusing nonlinear

Schrodinger equations for φ which must be identical (thus η =√3 + sβ) and yield

the classical soliton solution

φ(x) = 2

√sκ

3sech(

√2κx) ,

a(x) =√3 + sβφ(x) , (β > −3) .

(214)

In the case of multi-scale solutions where the laser width is expected to be much

larger than the atom one, the last term in the atom equation (212) can no longer

be dropped, it does not scale anymore as ∼ φ4. Furthermore, one can replace a(x)

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with its peak value a0 = a(0) in the atom equation, since the laser amplitude varies

very slowly with respect to the atom wavefunction. This leads to another soliton

solution for the atoms

φ(x) =

√

2µ2κφ − sa20a20 − β/2

sech

[√

|β/2− a20|β/2− a20

(sa20 − 2µ2κφ)x

]

. (215)

The laser being wider than the atoms, it mostly sees an atom wavefunction negli-

gibly small. Outside a central region of the order of the atom width, in the laser

equation one can neglect the atom density altogether, and the solution will be given

by

a(x) = a0e±√2κx . (216)

From the effective boundary conditions, if the atom wavefunction extends from

−Lφ to +Lφ,

da

dx

∣∣∣∣Lφ

− da

dx

∣∣∣∣−Lφ

≃ −3sa0

∫ Lφ

−Lφ

φ2dx (217)

recalling that the integration limits can be extended to infinity given the vanishing

atom wavefunction out of the interval (−Lφ,+Lφ), and calculating the integral

exactly, it is found that

κ =9

2

2µ2κφ − sa20(a20 − β/2)2

. (218)

The condition that κ must be positive from Eq. (218), combined with the condition

that the square root terms in Eq. (215) must be real, require that a20 > β/2, which

limits the multi-scale structure to the high field regime.

In this simpler case, it is possible to find approximate analytical solutions via

the variational method.157 The Lagrangian corresponding to Eqs. (207) and (208),

is given by L = 1/2(dφ/dx)2 + 1/2(dχ/dx)2 − Π. Gaussian trial functions of the

form φ = A exp(−(x2/2a2)) and χ = B exp(−(x2/2b2)) are used, with A, a, B, b as

variational parameter to be determined. Inserting these into the Lagrangian yields

an averaged Lagrangian, L =∫Ldx, that depends on the parameters of the trial

function (P = A,B, a, b). Variations with respect to these parameters,

∂L∂P

=∂

∂x

∂L∂(∂xP )

, (219)

give a system of four equations, which determines the parameters. An example

of such a solution is given in Fig. 2, from Ref. 156, compared to the numerically

obtained full solution. The agreement between a variational solution and the exact

one depends on how good the trial functions are, how close they are to the exact

solution.

General case. Outside the low density regime, the general solution to the coupled

system (207), (208) can be found only numerically, for instance via the shooting

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0 10 20 300

0.05

0.1

0.15

x/λL

φ,

χ

Fig. 2. (Color online) Comparison between results obtained using the numerical shooting methodand the approximate variational approach for the atom wavefunction (dotted line) and the laseramplitude (solid line). The variational results are the two green dashed lines. The chosen pa-

rameters are: κ = 10−3, kφ = 0.0905, β = 38.4281 and µ = 1. All quantities dimensionless,normalization as in the text and the coordinate x is measured in units of laser wavelength λL.

0 2 40

0.005

0.01

0.015

0.02

x/λL

φ2

κ = 6 x 10−3

0 2 40

0.2

0.4

0.6

κ = 6 x 10−3

x/λL

χ2

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

x/λL

φ2

κ = 0.6

0 0.5 10

0.5

1

x/λL

χ2

κ = 0.6

Fig. 3. Examples of stationary solutions numerically calculated. κ as indicated in the plots,κφ = 6×10−4 (solid line), kφ = 6×10−1 (dotted line). All quantities dimensionless, normalizationas in the text and the coordinate x is measured in units of laser wavelength λL.

method. For fixed values of the eigenvalues κ and κφ, the method requires to make a

guess on the peak laser amplitude and atom density, solve the two coupled equations

which are now ordinary differential equations and vary the guess until the solution

found is a localized, single hump solution for both atom and laser. Examples of

localized solutions found for different values of κφ, κ are shown in Fig. 3, from

Ref. 156.

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The solutions are seen to become narrower and more peaked for increasing

values of κ and κφ but no solution is found with the laser intensity distribution

more narrow than the atom density distribution, thus confirming what deduced

earlier.

One important point is that single-hump solutions are not the only stationary

solutions that can be found. Indeed multipeak solutions were found numerically158

by varying the parameters in the equations.

The stability question. It must be underlined that having found localized stationary

solutions does not guarantee that they can be realized. If they proved to be unstable,

such solutions could never be of physical importance. The system (225) and (226)

corresponds to the Hamiltonian density,

H =1

2|∇⊥ψ|2 +

1

2|∇⊥a|2 +

β

4|ψ|4 − s

2

|a|2|ψ|2(1 − s|ψ|2) . (220)

In addition to the Hamiltonian, H =∫ +∞−∞ Hd2r⊥, the system also admits other

constants of motion, the total momentum and the two Manley–Rowe integrals

N =

∫ +∞

−∞|ψ|2d2r⊥ , P =

∫ +∞

−∞|a|2d2r⊥ . (221)

The Hamiltonian is not limited, neither from below or above and therefore infor-

mation about the stability of the solutions can be drawn by the beam propagation

method, i.e., by propagating the solutions subject to small perturbations. This will

be discussed shortly for the general case but in the low density limit, useful informa-

tion can be obtained analytically by neglecting the denominator in the dipole–dipole

term:

H ≃∫ +∞

−∞

(1

2|∇⊥ψ|2 +

1

2|∇⊥a|2 +

β

4|ψ|4 − s

2|a|2|ψ|2

)

d2r⊥ ,

where for the last term:∫ +∞

−∞|a|2|ψ|2d2r⊥ ≤ N · P , (222)

which means that for the case of β > 0 the Hamiltonian is limited from below. For

systems with more than one transverse dimension, as is well-known in nonlinear

optics,159 a collapsing regime may be reached. It will be shown in the next sec-

tion that, within this model, this can happen even in the case of one transverse

dimension if the full denominator of the dipole–dipole term is taken into account.

With the Hamiltonian limited from below, it is possible to find a class of solutions

that minimize H and a Lyapunov analysis provides in this case a clear indication

that single-scale structures are stable. In fact, applying the method of Lagrangian

multipliers with

H = H + λφN + λεP

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the complex variation of H yields exactly the equations for single-scale solitons

provided λφ = µ2κφ and λε = κ, i.e., the single-scale soliton solutions minimize

the functional H . Since this functional consists of integrals of motion [(220) and

(221)] only, it can be treated as Lyapunov functional. Thus, for all perturbations we

have (dH/dz) < 0, which proves the stability of the single-scaled soliton solutions

against small perturbations. Numerical simulations of the evolution according to

the propagation equations (225) and (226) show different stability scenarios. For

relatively low values of κ, κφ, the stationary structures are robustly stable, prop-

agating effectively unchanged even if slightly perturbed initially. For higher values

of κφ and still relatively low κ the stationary structures during propagation evolve

towards different stationary structures, still mutually localized but with lower peak

atom density and laser intensity. Further increasing κ, the structures show a clear

sign of instability leading to yet another scenario in which a minimal increase in N0

leads to collapse while a decrease leads to broadening and total loss of localization

of the structures. The fact that these results are derived within the paraxial approx-

imation restricts the stability domain of the solutions. It is known from nonlinear

optics results,160 that the extra terms concur to make the structures stable be-

yond the limits found under paraxial approximation, physically because of stronger

diffraction effects for narrow structures.

4.3. Coupled propagation and the effects of mutual guiding

Since there are hints that stable stationary mutually localized solutions can be

found, it is important to understand how they are realized.161 We shall assume a

slowly varying envelope approximation for both atoms and laser field by writing (in

dimensional variables for clarity)

Φ(r) = ψ(r⊥, z) exp(ikaz) , (223)

A(r) = a(r⊥, z) exp(ikLz) , (224)

where r⊥ denotes the dimensions transverse to the propagation direction z and

requiring ∂2f/∂z2 ≪ k2ff (here f represents either the atoms or the laser field).

The coupled system of Eqs. (197) and (198), can then be written in normalized

variables as (dropping the tilde)

iµ∂ψ

∂z= −1

2∇2

⊥ψ +1

2β|ψ|2ψ − s

2

|a|2(1− s|ψ|2)2ψ , (225)

i∂a

∂z= −1

2∇2

⊥a−3s

2

|ψ|2a1− s|ψ|2 . (226)

Notice that the laser equation cannot be considered linear exactly because of the

coupling to the atom density. The aim is to understand how the transverse density

and intensity distributions change with propagation under the effect of the dipole

coupling for an atom beam copropagating with a laser beam along z.

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4.3.1. Initial seeds for the evolution

Being interested in the coupling effect, we shall consider an initially flat top laser

intensity profile by assuming a super-Gaussian for a(x, 0).

ψ(x, 0) = ψ0e(−x2/2d2

a) , (227)

a(x, 0) = a0e−(x2/2d2

L)g , (228)

where g is the super-Gaussian parameter (g = 10 in the simulations). This ensures

that, initially, there is no gradient force on the atoms due to spatial variations of

the laser intensity. The only effects in the very first stages of propagation are due

to density gradients, i.e., to the spatial variation of the refractive index for the laser

and to the spatial variation of the density as mediated by the presence of the laser

for the atoms. The first thing to be expected for the laser is linear diffraction in the x

direction. At this stage, the laser amplitude can be described in terms of its Fourier

components proportional to exp[i(hx− kz)] where the dispersion relation between

k and h determines the propagation velocity in the x direction as vx = −∂k/∂h|h0.

Here h0 is the central wave number in x, which is assumed to be zero. What is

important is that

dvxdh

= −∂2k

∂h2

∣∣∣∣h0

> 0 , (229)

i.e., parts of the pulse with h > h0 have a higher propagation velocity along x and

vice versa (we are in a regime of anomalous dispersion). In fact, Taylor-expanding

k = k(h0) + ∂k/∂h|h0(h − h0) + 1/2∂2k/∂h2|h0

(h − h0)2 + · · · . and applying the

transformation ik ↔ i∂/∂z, from the expansion for k we obtain an equation for

the laser amplitude which corresponds exactly to our (226) if ∂2k/∂h2|h0< 0. It is

then necessary to investigate whether certain parts of the pulse in the initial stage

develop a higher h. By describing the laser amplitude in the initial stages in terms

of amplitude and phase

a(x, z) = |a(x, z)|eiϕ(x,z) , (230)

it is possible to find an expression for ϕ(x, z). In fact, inserting this expression into

the laser equation (226), separating real and imaginary part and neglecting the

second derivative since the amplitude profile is flat, one finds

∂ϕ

∂z=

3

2

s|ψ|21− s|ψ|2 , (231)

which can be integrated under the assumption that the initial dependence of the

atom density on z is weak, |ψ(x, z)|2 ≃ |ψ(x, 0)|2, and gives

ϕ(x, z) ≃ 3

2

sz|ψ|21− s|ψ|2 . (232)

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Thus, the presence of the atoms translates into a chirp on the laser transverse wave

number96 creating the instantaneous wavenumber

hc =∂ϕ

∂x=

3sz

2

∂

∂x

( |ψ|21− s|ψ|2

)

=3sz

2(1− s|ψ|2)2∂|ψ|2∂x

. (233)

Parts of the pulse localized around higher peaks of the chirp function (233), i.e., with

higher h, will move with higher velocities in the direction of positive x. The behavior

of the chirp function depends on the sign of the detuning as can be seen from Fig. 4,

from Ref. 161. Not only the chirp effect is slightly depressed in the blue detuning

case, there is also a qualitative difference: For red detuning, during the initial stages

of propagation, the part of the laser pulse centered around the positive peak in the

chirp function will move faster to the right, towards the center of the pulse. The

part of the pulse centered around the negative peak of hc will slow down and get

closer to the center of the pulse. Therefore in correspondence of the two peaks in

the chirp function, the laser intensity will be depleted in favor of an increase in the

center. The opposite occurs in the blue detuning case. This can be clearly seen by

numerically simulating the initial stages of propagation, see Fig. 5 from Ref. 161,

starting with the same initial conditions but opposite sign of the detuning. This

initial change in the laser profile, due to the atom density transverse gradients,

drives a dynamical reaction on the atom density since the potential now felt by

the atoms has been modified and clearly the response depends on the sign of the

detuning.

An approximate perturbative analysis of the model equations in the low density

limit complements the reasoning in terms of chirp effect, introduced above for the

∆ < 0

chirp

func

tion

x

∆ > 0

x

Fig. 4. Qualitative behavior of the chirp function (233), solid line, for an initial Gaussian atom

density distribution, dashed line. Left: Red detuning, i.e., sign(α) > 0; right: Blue detuning,i.e., sign(α) < 0.

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−20 0 20x/λ

L

−20 0 200

2

4x 10

−4 ∆ < 0

|ψ(x

)|2

−20 0 20

|a(x

)|2

x/λL

−20 0 200

2

4x 10

−4 ∆ > 0

Fig. 5. Atom density and laser intensity (detail of flat top) for red detuning (left) and bluedetuning (right) after a very short propagation (z = 0.0012λL). Dotted line: Initial intensitydistribution. Normalization as given in the text.

initial laser evolution, by showing what happens initially to the atom density.162

Expanding the denominators up to terms linear in the atom density, separating am-

plitude and phase as a(x, z) = A(x, z) exp(iθ(x, z)), ψ(x, z) = B(x, z) exp(iφ(x, z))

and real and imaginary part, Equations give

∂φ

∂z=

1

2

[

1

B

∂2B

∂x2−(∂φ

∂x

)2]

+A2

2−B2

(β

2−A2

)

,

∂B2

∂z= − ∂

∂x

(

B2 ∂φ

∂xi

)

,

∂θ

∂z=

1

2

[

1

A

∂2A

∂x2−(∂θ

∂x

)2]

+3

2B2,

∂A2

∂z= − ∂

∂x

(

A2 ∂θ

∂x

)

.

(234)

Considering a perturbative expansion F (x, z) = F0(x) + F1(x)z + F2(x)z2 and

G(x, z) = G1(x)z+G2(x)z2 up to second-order in z where F stands for the functions

A and B while G stands for θ and φ and the zeroth-order terms are the initial

functions allows to investigate the initial stages of propagation z ≪ 1. Identifying

powers of z, a solution is obtained for the amplitudes:

A2 = A20(x)

[

1− 3

2

B20(x)

d2a

(2x2

d2a− 1

)

z2]

, (235)

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B2 = B20(x)

[

1 +1

2

β′(x)

d2aB2

0(x)

(2x2

d2a− 1

)

z2 − z2

2d4

]

, (236)

where β′(x) = β − 2A20(x). This solution has the features discussed above: The

laser intensity profile peaks in the center and at the same time two troughs are

created on each side of the rising peak (consequence of the chirp effect induced

by the atoms). The atom density profile shows a depression due to the nonlinear

collisional defocusing and two humps are created on both sides of the depression.

This again can be seen as the seed of the subsequent evolution.

The evolution on the shoulders of the laser, assuming as we shall do dL ≫ dato guarantee a constant laser intensity over the atoms width, is initially perfectly

linear. The generation of modulations is followed by the loss of the intensity which

is not trapped since there are no atoms on the wings.163,164

4.3.2. Localized structure formation as an effect of mutual guiding

The single peak structure. The red detuning case can thus lead to mutual

localization:161 The initial flat top laser undergoes a central rippling because of

the effect of the atom density gradients. This enhances the focusing effect experi-

enced by the atoms in the central region. A higher peak density will result in a

higher refractive index and consequent stronger focusing on the laser. A possibility

for balancing of these effects comes from the diffractive terms in the atoms equa-

tions: Kinetic energy and repulsive atom collisions acquire more importance with

the focusing of the atoms and the growth of the peak atom density. This can lead

to the formation of mutually localized structures. The interaction can also result in

a catastrophic collapse, if the diffractive effects are not strong enough to counteract

the nonlinear focusing. In general the outcome of the process will depend on the

initial condition and can be studied numerically. A case leading to mutual and sta-

ble guiding is shown in Fig. 6 from Ref. 161. The inset in Fig. 6 shows the potential

initially acting on the atoms: On the length scale of the laser width dL ≫ da there

is a trapping potential, whereas in the center the dipole interaction with the flat

top laser is of no consequence and the most important terms is due to the repulsive

collisions. Atoms tend to spread away from the central region with a decrease in the

peak density as shown in Fig. 7 from Ref. 161. This decrease in the peak density is

accompanied by the previously discussed modifications in the laser profile so that

stronger focusing dipole interactions occur simultaneously to a weakening of the

repulsive collision effect. Characteristic relaxation oscillations consequence of this

interplay are seen during the approach to the final stationary state, see Fig. 7.

The two peak structure and jet emission. Increasing the initial laser intensity

or atom peak density leads to a more violent transition dynamics but to quali-

tatively similar results unless a threshold value in the initial peak atom density

(which depends on the initial laser intensity) is exceeded. In this case the initial

collisional repulsive barrier is strong enough to expel atoms from the central re-

gion. The atoms having a relatively high peak density can focus the laser and two

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−60 −40 −20 0 20 40 600

1

2

3

4

5

x 10−4 z/λ

L = 1.02e+005

x/λL

|ψ(x

)|2

−60 −40 −20 0 20 40 600

0.02

0.04

0.06

x/λL

|a(x

)|2

0

0

x/λL

Vψ

Fig. 6. Stationary atom density and laser intensity after 1.02 × 105λL of propagation along zfor run (a). Dotted lines: initial distributions. For this run: da = 5λL, dL = 40λL, initial peak

atom density 5.81× 10−18 m−3 (corresponding to ψ0

2= 3.41× 10−4), initial peak laser intensity

0.0153 mW/cm2 (corresponding to a02 = 0.0181), βcoll = 38 (corresponding for instance to adetuning of 100 times the decay rate for 87Rb atoms). The inset shows the potential initially feltby the atoms. Normalization as in the text.

0 2 4 6 8 10

x 104

0

2

4

6

x 10−4

z/λL

|ψ(x

= 0

)|2

0 2 4 6 8 10

x 104

0

0.02

0.04

0.06

0.08

0.1

z/λL

|a(x

= 0

)|2

Fig. 7. Peak atom density and laser intensity during propagation along z for the same parametersof Fig. 6. Normalization as in the text.

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symmetrically placed bell-shaped structures are formed symmetrically propagating

outward unchanged thereon, suggestive of optical effects such as soliton ejection.165

Structure emission, differently from what usually studied in optics,166,167 here

is not due to engineering of the waveguide or the trap but to the interaction of

the newly born structures which happens through the tails overlap. Once these

structures are formed, further propagation is seen to lead to different possible sce-

narios: (1) outward motion of the two peaks moving farther apart from each other

(the jet emission scenario) with or without a central remnant structure; (2) inward

motion of the two peaks resulting in coalescence into a single central peak; or (3) in-

ward motion of the two peaks resulting into a bound state with the two structures

oscillating about the central position.162

−100 −50 0 50 100

−10

−5

0

5x 10

−4z(λ

L) = 1.17 x 10 3

x (λL)

(a)

−100 −50 0 50 100

−10

−5

0

5

x 10−4

z(λL) = 2.05 x 10 3

x (λL)

(b)

−100 −50 0 50 100

−2

−1

0

1x 10

−3z(λ

L) = 2.8 x 10 3

x (λL)

(c)

−100 −50 0 50 100−1.5

−1

−0.5

0

0.5

1x 10−3

z(λL) = 5.61 x 10 3

x (λL)

(d)

−100 −50 0 50 100−1.5

−1

−0.5

0

0.5

1x 10

−3z(λ

L) = 8.71 x 10 3

x (λL)

(e)

−100 −50 0 50 100

−1.5

−1

−0.5

0

0.5

1x 10

−3z(λ

L) = 1.85 x 10 4

x (λL)

(f)

Fig. 8. Details of the process of structure formation for |ψ(x, z)|2. Here ψ0 = 0.06645 (n0 =7.51 × 1019 m−3), da = 5λL, dL = 8da, a0 = 0.1346 corresponding to an initial peak laserintensity of 0.0153 mW/cm2. Solid line: atom wavefunction, dotted line: laser-induced potentialacting on the atoms (divided by 10 to make the figure more easily readable). The propagationdistance is indicated on the plots. All other parameters as specified in the text. All quantitiesnormalized as in the text, x, z measured in units of λL.

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First of all, the formation of two lateral structures can be predicted via the

perturbative analysis introduced earlier on in this section. It was described how an

initial central depression for the atom density can be expected with the formation

of lateral modulations. The subsequent evolution depends on the balance between

focusing dipole interactions and diffractive terms, i.e., for fixed widths, on the ratio

between peak atom density and laser intensity. Provided these opposing forces are

not strongly out of balance, some atoms will remain in the central region while some

atoms, pushed away from the center, will start to broaden away but will still feel the

wide laser trapping effect. If enough atoms escape, the density of the lateral modula-

tions will be high enough to focus the laser radiation and create two lateral localized

laser structures acting as secondary traps for the escaping atoms. These atoms will

undergo oscillations inside the newly formed traps which meanwhile adjust to the

atoms focusing action. Once complete mutual localization is achieved and the laser

has lost completely the initial super-Gaussian shape, the lateral structures under

the repulsion exerted from the central peak start to move away, see Fig. 8 from

Ref. 161, which shows snapshots of the evolution for ψ0 = 0.06645 and a0 = 0.1346.

These results are suggestive of the possibility of soliton steering. The properties of

the structures ejected (peak density, velocity, and number of jets) depend on the

initial conditions: Changing the initial atom peak density at fixed laser intensity,

jets are emitted at different angles with respect to the propagation direction and

with different peak densities and peak laser intensities, as can be seen from Fig. 9

which shows jet positions for a few different cases. This last figure also shows the

anomalous behavior of the structures emitted starting from ψ0 = 0.0668. They ini-

tially move clearly inward before being ejected. For growing initial peak density,

there seems to be a stronger central trapping capable of attracting the lateral peaks

toward the center. Thus, slightly higher initial peak densities, ψ0 = 0.0669 for the

0 1 2 3x 10

4

0

50

100

150

200

z(λL)

Jet p

ositi

on

a

b

c

d

e

Fig. 9. (Color online) Jet positions for different initial values of the atom peak density ψ0. Theidentification of the different lines from top to bottom is made at the very end of the propagationlength. (a) ψ0 = 0.052 (fourth line from top), (b) ψ0 = 0.054 (second line from top), (c) ψ0 =

0.0662 (last line from top, slowest jet), (d) ψ0 = 0.0664 (third line from top), (e) ψ0 = 0.0668 (firstline from top, fastest jet). All quantities normalized as in the text, z measured in units of λL.

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−80 −60 −40 −20 0 20 40 60 800

1000

2000

3000

4000

5000

0

1

2

3

4

5

6

7x 10

−3

x/λL

z/λL

|ψ(x

)|2

−80 −60 −40 −20 0 20 40 60 800

1000

2000

3000

4000

5000

0

0.02

0.04

0.06

0.08

0.1

0.12

x/λL

z/λL

|a(x

)|2

Fig. 10. Two peak structures formed starting from initial peak density n0 = 1.44 × 1020 m−3.All other parameters are the same as for Fig. 6. Normalization as in the text.

Fig. 11. (Color online) Contour plot of the evolution of the atom wavefunction for ψ0 = 0.196(n0 = 6.54 × 1020 m−3) showing a characteristic oscillatory behavior. All quantities normalizedas in the text, x, z measured in units of λL.

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same a0 = 0.1346 as before, lead to the formation of two lateral structures which

then merge again into a single-hump central structure. One explanation is that

the stronger initial collisional repulsion leaves a weaker central peak which is not

enough to eject the oscillating lateral structures. A second point is that for higher

initial densities, the two lateral structures tend to form closer to the central peak

where the laser is peaking, giving a stronger trapping effect which can be enough

to determine the merging of the atom structures, see Fig. 9 from Ref. 162. For

higher initial peak densities, ψ0 = 0.092 for the same a0 = 0.1346, no central peak

is left and most of the atoms are gathered in the laterally ejected structures, see

also Fig. 10 from Ref. 161.

However, the role of soliton-like interaction is shown in Fig. 11, from Ref. 162,

where, for ψ0 = 0.092 for the same a0 = 0.1346, a structure is excited very similar

to the bound system observed for optical solitons in which two pulses perform an

oscillatory motion by bouncing back and forth in their own potential well.168–170

The collapsing case. Increasing the initial laser intensity above a threshold value

(which depends on the initial atom peak density) leads to a collapse-like evolution.

The initial effect of the dipole interaction in the central region is too strong for the

diffractive terms to counteract it and the atom peak density increases monotonously

towards the critical value φ2∗ = 1 (corresponding to n∗ = 1.7×1022 m−3), see Fig. 12

from Ref. 161. The dependence of such threshold on the peak atom density is also

justified by the resonant denominator in the atom equation: The higher the peak

density, the closer to zero the denominator and the less reliable the calculations.

Numerically the propagation step required to maintain the error within given limits

decreases to zero and physically the models break down since at such high densities

all the effects that have been neglected will come into play.

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

n/n*

a 02 (m

W/c

m2 )

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

z/λL

|ψ(x

= 0

)|2

(a)

(b)

(c)(d)

Fig. 12. Left: Minimum initial laser intensity at which collapse was numerically observed fordifferent values of the initial atom density n0 given with respect to the critical one n∗. Right:Peak atom densities during propagation for varying initial laser intensities. For all cases ψ0 = 0.1,n0 = 1.7 × 1020 m−3 and (a) a0 = 1, I0 = 0.847 mW/cm2, (b) a0 = 1.8, I0 = 1.52 mW/cm2,(c) a0 = 1.975, I0 = 1.67 mW/cm2, (d) a0 = 10, I0 = 8.47 mW/cm2. Normalization as in thetext.

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5. Conclusions. What It Is Really All For?

We have thrown a look at one of the most exciting developments in physics, the

possibility of manipulating matter with light, which is not an ideal theory anymore

thanks to sources of both coherent electromagnetic radiation and coherent matter

waves. A review such as this one is necessarily going to be very reductive, given

the fantastic vitality of this research field. We have focused on phenomena which

are at the borders between nonlinear optics and atomic physics but approaches,

interests and goals are much broader, think for instance of the recent advances in

electromagnetic induced transparency, slow light, atomic wave amplification, only

to mention a few of the issues we have not touched upon. In this review, after

the basics of the mathematical modelization of laser-BEC interactions were intro-

duced, the lion’s share was taken by guiding and mutual localization phenomena

leading to soliton-like structures for both atoms and light, albeit with important

limits due to the approximations in the model. These structures are of interest

not only because of their properties of self-localization and robust propagation but

also in relation to the creation of meta-lenses and comoving potentials to refocus

atom waves.171 Besides, the phenomena described by the basic models, lend them-

selves to the implementation with matter waves of quintessentially optical feats,

such as all-optical switching and directional couplers, or to extensions and refine-

ment of tools such as the optical tweezers. Processes such as atom lithography172

and atom interferometry173 require a very precise and delicate control of matter

waves. Matter waves must be precisely guided or coherently split and separated

over macroscopic distances, for which dipole interactions are an interesting candi-

date174 and guided atomic soliton-like structures could play a role. Atom-surface

studies would take great advantage from precise guiding of low energy atoms which

do not penetrate the surface.175 Guiding of atoms with red detuned light is exper-

imentally achieved31,176,177 and, from the point of view of light, nonlinear effects

at low light intensity have been experimentally realized by exploiting the dipole

interaction.178 Systems such as BECs could be extremely sensitive to the coher-

ence property of light and it has been predicted that collapse in a metastable BEC

can be initiated because of laser intensity fluctuations.179 Advanced waveguides for

matter waves will play a fundamental role in atom optics applications, see for in-

stance Ref. 180 and references therein. For both applied and fundamental reasons,

methods to generate correlated atomic fields, such as oppositely directed beams,

are of great importance181 and again the basic dipole interactions could play a role.

From the point of view of the back effect of atoms on light, it has been predicted182

that optical lattices could undergo significant distortion which would be quite an

important factor in high precision quantum measurement. Even the field of single-

photon nonlinear optics could take advantage of interactions with matter waves

which could develop the necessary nonlinear effects at microwatt power levels.183

Never before we have had the possibility of testing the wave nature of matter in

such detail.

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INTERACTIONS OF ELECTROMAGNETIC RADIATION WITH BOSE EINSTEIN CONDENSATES: MANIPULATING ULTRA–COLD...

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