2 August 1999

Ž .Physics Letters A 259 1999 67–70www.elsevier.nlrlocaterphysleta

Spectrum of light scattered from a ‘deformed’ Bose–Einsteincondensate

Stefano Mancini a, Vladimir I. Man’ko b

a Dipartimento di Fisica and Unita INFM, UniÕersita di Milano, Via Celoria 16, I-20133 Milano, Italy` `b P.N. LebedeÕ Physical Institute, Leninskii Prospekt 53, Moscow 117924, Russia

Received 25 May 1999; accepted 3 June 1999Communicated by V.M. Agranovich

Abstract

The spectrum of light scattered from a Bose–Einstein condensate is studied in the limit of particle-number conservation.To this end, a description in terms of deformed bosons is invoked and this leads to a deviation from the usual predictspectrum’s shape as soon as the number of particles decreases. q 1999 Published by Elsevier Science B.V. All rightsreserved.

PACS: 03.65.Fd; 03.75.Fi; 42.50.Ct

The recent achievements of Bose–Einstein con-Ž .densate BEC with a gas of atoms confined by a

w xmagnetic trap 1 has stimulated renewed interest inthe question as to what signatures Bose–Einsteincondensation imprints in the spectrum of light scat-

w xtered from atoms in such a condensate 2,3 .As well known, to deal with the dynamics of BEC

gas the Bogolubov approximation in quantumw xmany-body theory 4 is an efficient approach, in

which the creation and annihiliation operators forcondensated atoms are substituted by c-numbers.One shortcoming of this method is that the totalatomic particle-number may not be conserved afterthe approximation. Or a symmetry may be broken.

w xTo remedy this default, Gardiner 5 suggested amodified Bogolubov approximation by introducingphonon operators which conserve the total atomicparticle number N and obey the bosonic commuta-tion relation in the case of N™`. In this sense, this

phonon operator approach gives an elegant infiniteatomic particle-number approximation theory forBEC taking into account the conservation of the totalatomic number.

Along this line, the case of finite number ofw xparticle has been recently investigated 6 , and the

algebraic method of treating the effects of finiteparticle number in the atomic BEC has been devel-oped. It results a physical and natural realization of

w xthe quantum group theory 7 in the BEC systems,w xwhose possibility was already suggested in 8 ,

thought in a different manner.Here, we shall use the deformed algebra to study

the response of a condensate with finite number ofatoms to the laser light and focus our attention onsteady-state excitation.

We consider a system of weakly interacting Bosegas in a trap and a classical radiation field interactingwith these two-level atoms, where b†, b denote the

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00400-4

( )S. Mancini, V.I. Man’korPhysics Letters A 259 1999 67–7068

creation and annihiliation operators for the atoms inthe excited state; a†, a, the creation and annihiliationoperators for the atoms in the ground state. Theseoperators satisfy the usual bosonic commutation rela-tions. The Hamiltonian of the model reads

† † ) †Hs"√ b bq" g t b aqg t ba , 1Ž . Ž . Ž .

Ž .where g t is a time-dependent coupling coefficientŽ .for the classical laser field coupled to those two

states with level difference "√ . Usually, the timeŽ . Ž .dependence of g t is given by gexp yiV t , with

V being the frequency of the laser beam.Note that, with the above Hamiltonian, the total

atomic particle number Nsb†bqa†a is conserved.In the thermodynamic limit N™`, the Bogolubov

w xapproximation 4 is usually applied, in which theladder operators a†, a of the ground state are re-placed by a c-number N , where N is the number( c c

Ž .of the initial condensated atoms. As a result, Eq. 1becomes the Hamiltonian of a forced harmonic oscil-lator.

Moreover, we have to consider the bath of photonmodes, beside the classical driving field, so that the

w xtotal Hamiltonian will be 3

† †Hs"√ b bq" N g t b qh.c.Ž .( c

† †q V c c q" N j k b c qh.c. ,Ž .(Ý Ýk k k c kk k

2Ž .

Žwhere c represent radiation modes of frequencyk. Ž .V which constitute the bath and j k is the cou-k

pling coefficient pertaining to the internal atomicstates.

Now, by eliminating the heat-bath variables, inview of the Markov approximation, in the case of

Ž . w xHamiltonian 2 , it is possible 9 to obtain a quan-tum stochastic differential equation that describes thedynamics of the b mode in the Heisenberg picture

E b t syiDb t y ig N yG b tŽ . Ž . Ž .(t c

'q 2 G b t , 3Ž . Ž .in

where Ds√yV , and G is the damping rate.w xRoughly, the latter is given by Gsg N 3 , where( c

w x Ž .g is the one-atom linewidth 9 . Finally, b t is thein

vacuum noise operator

² † X : ² X :b t b t s b t b t s0 ,Ž . Ž . Ž . Ž .in in in in

² † X : Xb t b t sd ty t . 4Ž . Ž . Ž . Ž .in in

Ž . w xThe solution of Eq. 3 is well known 9 , and in thesteady-state regime it becomes

yig N( c² :b t 'bs , 5Ž . Ž .Gq iD

'2 Gd b v s b v , 6Ž . Ž . Ž .in

Gq iD

Ž .where the semiclassical approximation b t sbqŽ . Ž . Ž .d b t has been used. In 6 , d b v is the Fourier

Ž .component of the operator d b t .The spectrum of the light scattered from the

atoms is given by the correlation of the operators†Ž . Ž . w xb t and b t 3 . Hence, in the steady state, the

² †Ž . Ž X .:spectrum of fluctuations d b v d b v resultsŽ . Ž .zero everywhere, by virtue of 6 and 4 . This

means that in the long time limit, only the equal timecorrelations survive.

Let us now come back to the Bogolubov approxi-w xmation 4 . It destroys the symmetry of Hamiltonian

Ž .1 , i.e., the conservation of the total particle numberw xis violated because N, H /0. Then, to preserve the

property of the initial model, it is possible to deter-w xmine the following phonon operators 5

1 1† † †Bs a b , B s ab . 7Ž .' 'N N

w xThese operators obey a deformed algebra 6 . In fact,a straightforward calculation leads to the followingcommutation relation

w † x †B , B s1y2hb b , 8Ž .where we have introduced a small operator parame-ter hs1rN, which for sufficiently large number ofatoms is considered as c-number. The algebra de-

Ž .fined by Eq. 8 belongs to the f-deformed algebraw x10 , where in general the deformed operator isrelated to the undeformed one through an operatorvalued function f as

Bsbf b†b . 9Ž . Ž .

( )S. Mancini, V.I. Man’korPhysics Letters A 259 1999 67–70 69

In our particular case, we have

† †(f b b s 1yh b by1 , 10Ž . Ž . Ž .and for small deformation we get

h†Bfb 1y b by1 . 11Ž . Ž .

2

Ž .With the above in mind, the total Hamiltonian 2should be rewritten as

† †'Hs"√ b bq" N g t B qh.c.Ž .† †'q V c c q" N j k B c qh.c. .Ž .Ý Ýk k k k

k k

12Ž .

Since now N is a conserved quantity, we can con-Žsider it as a c-number we suppose that it coincide

with N , i.e., all the atoms are initially in the con-c. Ž .densate . Essentially, Eq. 12 describes the damped

dynamics of a deformed oscillator. This is a ratherw xcumbersome problem to deal with, as shown in 11 .

Here, we simplify the treatment with the followingargumentations: in the second term of r.h.s. of Eq.Ž .12 , the nonlinear character of B must be taken intoaccount, since it is evidenciate by the radiation-fieldamplitude g; instead, in the last term of r.h.s. of Eq.Ž .12 , such nonlinear character can be neglected dueto the weak-coupling assumption with the heat bath.Hence, the resulting effective Hamiltonian, in a framerotating with the laser frequency, is

† †'H s" Db bq" N g b qbŽ .eff

'N gh† 2 † 2 †y" b b qb b q V c cŽ . Ý k k k2 k

† †'q" N j k b c qbc , 13Ž . Ž .Ý k kk

where we have expressed B in terms of b by meansŽ .of Eq. 11 .

The nonlinear quantum stochastic differentialw xequation 9 describing the dynamics of the b-mode

Ž .is now derived from Eq. 13

'N gh2 †E b t s i b t q2b t b tŽ . Ž . Ž . Ž .Ž .t 2

'y iDb t y ig N yG b tŽ . Ž .'q 2 G b t . 14Ž . Ž .in

< < < < < <Fig. 1. A plot of the quantity b y b as a function of N.`

Values of the parameters are: Ds0 and g s2.5g . Furthermore,w x w xarg b sarg b sp r2 ;N.`

Ž .It obviously reduces to linear Eq. 3 as soon ash™0.

The steady state value of the field is given by thesolution of the following equation

'N gh22 '< <0s i b q2 b y iDby ig N yGb .Ž .

215Ž .

Of course, the solution of the above equation will beŽ . Ždifferent from that of Eq. 5 we refer to the latter as

.b , but they approach each other as soon as N`

increases, as can be seen in Fig. 1.The dynamics of the small fluctuations is given

by† 'E d b t sAAd b t qBBd b t q 2 G b t ,Ž . Ž . Ž . Ž .t in

16Ž .where

)'AAsyiDyGq i N gh bqb , 17Ž . Ž .'BBs i N ghb . 18Ž .

In this case, the solution takes the form

1)w xd b v s ivyAA b vŽ . Ž .� in

J vŽ .qBB b† v , 19Ž . Ž .4in

where

< < 2 < < 2 2 )J v s AA y BB yv y iv AAqAA . 20Ž . Ž . Ž .Ž . Ž .Finally, the spectrum, by means of Eqs. 19 , 20

Ž .and 4 , reads

< < 2BBX X†² :S v s dv d b v d b v s .Ž . Ž . Ž .H 2< <J vŽ .

21Ž .

( )S. Mancini, V.I. Man’korPhysics Letters A 259 1999 67–7070

Fig. 2. The spectrum S as a function of v and N. Values ofparameters as in Fig. 1.

Ž .It is shown in Fig. 2 as a function of N besides v .Ž .It trivially vanishes for h™0 i.e., N™` , other-

wise it gives a signature of finite number of particles.More precisely, we may see that it shows a central

Ž .peak typical of the Lorentian shape by decreasingthe number of particles. On the other hand, it wouldbe interesting to study the transition from this struc-

w xture to the characteristic Mollow triplet 12 of asingle trapped atom. Unfortunately, our approxima-tions are no longer valid for very few atoms, and oneshould devise a technique to solve completely theproblem or investigate it numerically. This is a planfor the future study.

Summarizing, we have seen that the particle-num-ber conservation in BEC requires a deformation ofthe bosonic field, hence the introduction of nonlin-

w x Žearity 10 , which may lead in the limiting case of.small number of particles to observable effects on a

probe light field. Beyond the oversimplified modelused, we retain the measurement of the light spec-trum in presence of few condensed atoms a promis-ing experimental challenge. On the other hand, theuse of a BEC with small number of atoms would be

w xthe subject of next generation experiments 13 .The same aim could be pursued in elementary

particle field as well. In fact, the BEC may also

describe the final state of pions in high-energy-w xheavy-ion collisions 14,15 .

Acknowledgements

V.I.M. is grateful to Russsian Foundation forBasic Research under the Project No. 99-2-17753.

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