Stat isti cal Physics of Bose{ Ei nste in...

Vol . 100 (2001) A CT A PHY SIC A POLON IC A A Suppl ement

St at isti cal P hy sics of B ose{ Ei nste in

C on densation

M . Gaj da a an d K. R z¨ âew sk ib

a In st itute of Physics, Pol ish Ac ademy of Sciences

al . Lotni k§w 32/ 46, 02-668 W arsaw, Polandb Center for Theo reti cal Physi cs, Pol ish Aca demy of Sciences

al . Lotni k§w 32/ 46, 02-668 W arsaw, Poland

(Received Novem ber 15, 2001)

W e rev iew a pr ogress i n understandi ng of statisti cal prop erties of a quan -

tu m degenerate Bose gas. W e show how the Ma xw ell demon ensemble helpsto compute Ûuctuations of the Bose{ Eins tein condensate of an ideal Bosegas according to the micro canoni cal ensemble. T hen, we review a metho d ofmeasuring these Ûuctuations. U sing a solubl e model of interacting Bose gas

w e also stress the imp ortance of higher- order correlation functions. Finall y,w e review our novel computational metho d of studying an interacting Bosegas near its critical temp erature.

PAC S numb ers: 03.75.Fi, 05.30.Jp

1. I n t rod uct io n

It to ok seventy years to real ize exp erim enta lly the Bose{ Ei nstein condensa-ti on, a qui ntessential ly quantum phenom enon discovered theoreti cally in 1924 [1, 2].It wo uld not happen if not for the pro gress in cooling and tra ppi ng of neutra lato m s. The Nobel Com mittee has recognized i ts im pact in 1997 [3]. The seminalexp eriments tha t pro duced a Bose{Ei nstein condensate (BEC) in di lute rubi di umand sodi um gases[4, 5], perf orm ed in 1995, bro ught a No bel Pri ze to E. Co rnel l ,C. W ieman, and W . Ketterl e in 2001.

The di scovery has let to an avalanche of papers, both experim ental and the-oreti cal, devoted to stati stical and dyna m ical properti es of the new quantum stateof m atter. Even the stati stical properti es of an ideal Bose gas cal led for revi siti ng.

(7)

8 M . Gajda, K. Rz¨ âewski

The textb ook theo ry of Bose{Ei nstein condensati on was devel oped under stan-dard assumpti ons of the therm odyna mic l imi t, the spati al hom ogeneity, and thegrand canoni cal stati sti cal ensembl e. In the exp eriment, the num ber of ato m s isÙnite and varies between several tho usand and several m i ll ion, far from inÙnity .The gas is conÙned by a harm onic potenti al , so the density changes from point topoint. Fi nal ly, the system is nearl y perfectly decoupled from the envi ronm ent ; inparti cul ar, i t does not exchange parti cles wi th an outsi de reservoi r | the schemeim pl ied by grand canonical ensembl e.

In thi s paper, we shal l revi ew the stati stical pro perti es of the condensed Bosegas wi th parti cul ar emphasis on our own wo rk. In Sec.2 we expl ain on a very simpl eexam ple the ro le of Bose symm etry pri nci ple in the bui ld-up of the m acroscopicpopul ati on in the non-degenerate ground state of the tra ppi ng potenti al . In Sec. 3the micro canonical theo ry of the harm onically, bound ideal Bose gas is sketched.T o thi s end, we revi ew a noti on of the Ma xwel l demon ensembl e. In Sec. 4 we revi ewa metho d of inv estigating the Ûuctua ti ons of the condensate. In Sec. 5 the ro le ofhi gher-order correl ati on functi ons is stressed, based on an exactl y solubl e m odelof elastical ly intera cti ng bosons. Fi na l ly, in Sec. 6 a novel m etho d of studyi ng thestati onary dyna m ics of the weakl y intera cti ng Bose gas based on the hi gh-energysoluti ons of the ti m e-dependent Gross{Pi taevski i equati on is expl ained.

2. W h y b oson s bu nch t oget her ?

Qua ntum parti cl es are indi stinguishable. Let us ta ke two states and twoparti cl es. There are two ways in whi ch cl assical parti cles m ay be placed in thesetwo states. Two , because classical parti cles m ay be labeled. Quantum identi calparti cl es have just one way. They cannot be labeled. Am ong quantum parti cles,these wi th an integer spin, bosons, do not m ind to be al l pl aced at a single quantumstate, whi le hal f integ er spi n parti cles, ferm ions, need one state for each. Ato ms arebui ld of ferm ions: electro ns, proto ns, and neutro ns. Neutra l ato ms have as manyelectrons as proto ns in the nucl eus. Thus the pari ty of the num ber of neutro nsdeterm ines the stati sti cs of the who le ato m . Thi s is unusua l . Mo st global , lowenergy properti es of m atter depend on the cloud of electrons surro undi ng thenucl eus. Typi cal ly only nucl ear reacti ons and the hi gh-energy processesinvol ve thenucl eus. Here, at very low tem peratures, gros features depend on the compositi onof the nucl eus!

T o i l lustra te the power of indi stigui shabi l i t y let us consider a set of N par-ti cles movi ng in a one-dim ensional harm onic bi nding potenti al characteri zed bythe angular frequency ! . Let us assume further tha t the system is described bythe canonical ensemble wi th tem perature T as i ts contro l parameter. Hence, thedensity matri x of the system ta kes a form

£ b ;c l =¿

PN

i = 1a

+

ia i

Z b; cl ( N ; ¿); (2 .1)

Stati stica l Physics of Bose{ Ei nstein Condensat ion 9

both for bosons and classical parti cl es. Z b; cl ( N ; ¿) i s the canoni cal parti ti on func-ti on. We have intro duced here an abbrevi a ti on ¿ = exp (À ñh! =k T ) , where k i s theBol tzm ann constant. The physi cs depends now on the character of the parti cles.The classical parti ti on functi on resul ts f rom the independent summ atio n over al lconÙgurati ons for each parti cl e

Z c l =

ê1X

n = 0

¿n

! N

=

˚1

1 À ¿

ÇN

; (2 .2)

whi ch is just the N -th power of the single-parti cle parti ti on functi on. No correla ti onis present in the classical ideal gas. Summ ati on for bosons must be restri cted. Inparti cul ar, out of N ! conÙgura ti ons wi th each parti cle at di ˜erent level only one ispresent for bosons. It is obvi ous tha t the indi sti ngui shabi l it y is assured i f we sumover the restri cted range for each index

Z b =

1X

n N =0

¿n NÂ Â Â

1X

n 2 = n 3

¿n 2

1X

n 1 = n 2

¿n 1 =

NY

n = 1

1

1 À ¿n: (2 .3)

No w, for each classical parti cl e, the pro babi li ty of being in the ground state is ¿

and in any place above is rem aini ng 1 À ¿. Theref ore, the pro babi li ty of Ùndingexactl y n parti cles in the ground state out of N i s given by

P n ;c l( N ; ¿) =

˚N

n

Ç

¿n (1 À ¿) N À n : (2 .4)

The m ean numb er of parti cles in the ground state and i ts variance as a functi onof tem perature are shown for N = 1 0 0 0 in Fi g. 1 as dashed l ines. Thi s is to becontra sted wi th the bosonic result, shown wi th sol id l ines, whi ch again requi resrestri cted summatio ns

P n ;b ( N ; ¿) =1

Z b

1X

j n À

¿ j n 1 . . .

1X

j 1 j 2

¿ j 1 = ¿n

N À nY

j n

(1 À ¿j ) : (2 .5)

10 M. Gaj da, K. Rz¨âewski

The contra st is stri ki ng. W hi le both system s occupy the ground state at zerotem perature, the m acroscopi c occupati on of thi s state for bosons occurs al readyat tem peratures, at whi ch there is enough energy to l i ft a ll parti cles to the exci tedstates. Al l thi s is solely due to quantum stati stics and also takes place in highernum ber of dim ensions.

The reader expecting a ri gorous phase tra nsi ti on i .e. a nonanalyti c behavi orof physi cal quanti ti es may be disappointed. Two rem arks are in order. Fi rst: wedeal here, as in exp eriments, wi th a Ùnite sampl e. Everythi ng m ust be smooth. Sec-ond: i t is a one-dim ensional system , so in the absence of (long-range) intera cti onswe do not expect a pha se tra nsiti on, even for N going to inÙni ty .

In the next secti on we are going to present results for three- dim ensionalharm oni c tra ps.

3. Th e M ax we l l d em on en sem ble

A standard textb ook appro ach to Bose{ Einstei n condensati on of an idealBose gas is based on the grand canoni cal ensemble [6] intro duced by Gibbs. Thetwo contro l param eters here are the tem perature T and the chemical potenti alñ . The latter determ ines a mean numb er of parti cl es in the system. The systemis assumed to be in conta ct wi th the reservo ir of energy, whi ch determ ines i tstem perature, and wi th the reservo i r of parti cles, whi ch determ ines i ts chemicalpotenti al. A beauty of thi s ensembl e stems from the fact tha t the grand canoni calparti ti on functi on is expl icitl y given by the single parti cle energies " i :

Ë ( T ; ñ ) =Y

" i

1

1 À eÀ " i = k T z; (3 .1)

where the fugaci ty z i s given by

z = exp( ñ= k T ) : (3 .2)

Specifyi ng the f orm ula for the three- di mensional isotro pic harm oni c oscil lato r,accounti ng f or the degeneracy of its levels, we get

Ë ( ¿; z ) =

1Y

n =0

ç1

1 À z ¿n

Ñ( n +1 )( n +2)

2

: (3 .3)

Useful as a to ol to obta in the tem perature dependence of the mean numb er ofquanta in the condensed state, the grand canonical ensemble fai ls when used forthe stati stics of the condensed fracti on. It has long been kno wn [7] tha t i t predi ctsvari ance of the occupati on of the ground state to be of the order of i ts m ean valueeven at zero temperature. But in the experim ent the num ber of ato m s in a tra p isstri ctl y conserved. Theref ore at zero tem perature, in the absence of intera cti ons,al l ato m s shoul d be in the ground state wi th Ûuctua ti ons vanishing. Hence, m orerestri cted ensembles are better suited f or the Ùnite sam ple, tra pp ed situa ti on. Thecanoni cal Ûuctua ti ons were Ùrst computed by Pol i tzer [8]. W e, on the other hand,

Statist ical Physi cs of Bose{Ei nstein Condensat ion 11

devoted our attenti on to m icrocanoni cal ensemble. Its contro l param eters are theto ta l energy of the system E and the numb er of atom s N . The relevant parti -ti on functi on is the degeneracy of the (E ; N ) subspace, whi ch wi l l be denotedby À ( N ; E ). The m icrocanonical parti ti on functi on is related vi a power series ex-pansion to the grand canoni cal one wi th the canoni cal parti ti on functi on being aninterm ediate step

Ë ( z ; ¿) =

1X

N =0

z N Z (N ; ¿) =

1X

N =0

z N

1X

E =0

¿E À ( N ; E ): (3 .4)

No w, the m icrocanoni cal parti ti on functi on in a natura l way spli ts into a sum ofdegeneracy factors of the sam eto ta l energyÊ E but wi th di ˜erent numb er of ato msin the ground state (or com plementa ry num ber of ato ms in al l excited states):

À ( N ; E ) =

NX

N e x = 1

À ex ( N ex ; E ) : (3 .5)

The probabi l i ty distri buti on, whi ch carri es al l inf orm ati on about the stati sti cs ofthe condensate, is

P ( N ex ; E ) =À ex ( N ex ; E )

À ( N ; E ): (3 .6)

Its m om ents m ay be com puted vi a di ˜erenti ati on f rom a generati ng functi on

Y ( z ; E ) =

1X

N e x = 1

z N e x À ex ( N ex ; E ) : (3 .7)

In parti cul ar,

N N = N ex =

˚

z@

@z

Ç

Y ( z ; E ) z (3 .8)

and

£ N = £ N ex =

˚

z@

@z

Ç

Y ( z ; E ) z : (3 .9)

The form ul a (3.5) has a form of a parti ti on functi on of a new stati sti cal ensemble[9].Li ke the grand canoni cal one, i t has an unspeciÙed numb er of ato m s whi le l ike them icrocanoni cal one it has a wel l -deÙned energy. It describes the part of the system ,whi ch is excited. It is coupl ed to the condensate, serving as a reservo i r of theparti cl es, all at energy zero. W e coined a term Ma xwel l demon ensemble for thi ssitua ti on. If the exci ted ato m s were connected to a standard parti cle reservo ir, asmart demon wo uld have to be on guard letti ng only zero energy ato m s in and out.As we see,summ atio n in (3.5) has been extended to inÙnity . In our appro xi mati onwe trea t the condensate as inÙnitel y large.


One m ore step up to deÙne the next generati on functi on easily yi elds some-thi ng f amil iar

1X

E =1

¿E Y ( z ; E ) = Ë ex ( z ; ¿) ; (3 .10)

the grand canonical parti ti on functi on of the exci ted states, tha t is the pro duct (3 .3)wi th m issing n = 0 term . The expansion (3.8) is best achi eved vi a the Ca uchyinteg ra l

Y ( z ; E ) =1

2 ¤ i

IË ex ( z ; ¿)

¿E +1d¿; (3 .11)

where the integ rati on conto ur runs around the origin of the com plex ¿ plane.For E ƒ 1 the steepest descent m etho d may be used. Al l we need is asym pto ti cform ula for Ë ex , or rather i ts logari thm . Thi s may be easily found. In fact, them ost detai led analysis of thi s functi on was given in [10]. Here we just need i tsleading high tem perature term

ln Ë ex ( z ; ¿) = À

g 4 ( z ¿)

( ln ¿) 3; (3 .12)

where the g4 i s the Bose functi on

g n ( x ) =

1X

k =1

x k

k n: (3 .13)

The rest is j ust the appl icati on of the saddl e point appro xi mati on and the di ˜er-enti ati on (3.6) and (3.7). The resul ti ng expressions are givi ng the mean num berof condensed ato m s and i ts variance as a functi on of N and E . To pass to a m oreconventio nal variabl e | the tem perature | one may use ei ther i ts micro canonicaldeÙniti on or, as a shortcut, just the expression known from the grand canoni calform al ism

E = 3 ±( 4 )

˚k T

ñh!

Ç4

; (3 .14)

where ± denotes the R iemann zeta f uncti on. The results are

h N i = N

"

1 À

±(3 )

N

˚k T

ñh!

Ç #

; (3 .15)

wi th the cri ti cal tem perature | the one f or whi ch the mean num ber of condensedato m s becom es zero, and wi th the v ariance of thi s numb er equal to

£ N =

˚k T

ñh!

Ç ç

±(2 ) À

3 ± (3 )

4 ±(4 )

Ñ

: (3 .16)

The above form ula holds (f or large N ) al l the wa y to the cri ti cal tem perature. Itsvalue at the cri ti cal tem perature gives the m axi mal Ûuctuati ons. As we see thi s ispro porti onal to N . Theref ore the vari ance i tsel f is proporti onal to i ts square root.


In stati sti cal physi cs such Ûuctua ti ons are cal led norm al . Note tha t the vari ance,apart f rom i ts maxim al value, does not depend on the num ber of ato m s. Thi s isthe result of extendi ng to inÙni ty the sum in (3.5) or, m ore physi cal ly, the resul tof the condensate being macro scopical ly popul ated.

The Ùnite sam ple resul ts for the m ean v alue are shown in Fi g. 2, whi le suchresul ts for the vari ance are shown in Fi g. 3. The exact data were obta ined bym eans of the powerf ul , itera ti ve num erical algori thm found by Gajda et al . [11].

Fig. 2. Mean numb er of condensed atoms for the micro canoni cal ensemble of the ideal

Bose gas in three- dimensi onal harmonic spherical trap (red ). Blue curve is the asymptotic

result.

Fig. 3. Variance of the numb er of condensed atoms for the ideal Bose gas of 1000 atoms.

Show n are results for three maj or statistical ensembles. N ote non- physical character of

the grand canonical ensemble Ûuctuation s.

As we close thi s section let us m entio n an interesti ng l ink to the num bertheo ry . In one-dim ensional harm oni c oscil lato r, the microcanoni cal parti ti on func-ti on is just a num ber of parti ti ons of a given integ er E into a sum of N sum-m ands. In fact, the wel l known generati ng functi on for thi s numb er of parti ti onsis just (2.3) [12] and the gamma i tsel f for not- to o-large argum ents is avai lable inM a th emat i ca .

The correcti ons to our Ûuctua ti ons f orm ula (3.14), whi ch are due to weakintera cti ons, were considered by several groups [13{ 16]. The result seems to depend


on the assumpti ons m ade and, in our vi ew, a pro blem remains unsolved. W ecomeback to the ro le of intera cti ons in Sec. 5.

4. Ho w t o ob ser ve Ûu ct uat ions of t he co n den sat e?

In the previ ous secti on we have described the micro canoni cal sta ti stics of theideal Bose gas conÙned in the harm oni c potenti al. The experim ental veri Ùcati onof these and other, com peti ng predi cti ons is scarce.

W hi le the mean numb er of condensed ato m s N 0 as a functi on of tempera-ture has been m easured in several laboratori es [17], from the exp erimenta l point ofvi ew very l i ttl e is known about Ûuctuati ons of BEC. The m ain source of inform a-ti on concerni ng hi gher-order correl ati ons com es from the studi es of the depleti onof BEC due to inelasti c tw o-body and three- body col l isions [18]. Thi s way thesquare and the cub e of the popul ati on of condensed state were estim ated. Theexp erimenta l results have unequi vo cal ly rul ed out the therm al Ûuctua ti ons in thecondensate im pl ied by the grand canoni cal ensemble. Preci sion of tho se m easure-m ents is, however, unabl e to disti ngui sh between sub-Poissoni an and PoissonianÛuctua ti ons. The latter present in the coherent state, is often mentio ned in con-necti on wi th the Bogolyub ov appro xi mati on [19].

In one of our recent papers [20] we have pro posed to use a scatteri ng of a shortweak non-resonant laser pul sesas a m eans of probi ng the BEC stati stics. Before weshortl y describe the m etho d and the results, a rem ark concerni ng intera cti ons is inorder. Our study of the stati sti cs of the ideal gas was based on enum erati on of al lpossible conÙgura ti ons of the system whi ch are compati bl e wi th the constra intsim posed by tw o contro l parameters | the energy and the parti cl e num ber. Noexp erimenter can make a lot of condensates wi th identi cal E and N . In real isticexp eriment, the shot to shot Ûuctua ti ons of these param eters wo uld inevi ta bl ym ask the stati stics we have computed.

W hat can be done instead, is a ti me series: interro gati on of the system atequal ti me interv als and a bui ld-up of the histogram of the frequenci es, and a givenoccupati on of the condensate is encountered. Thi s metho d, on the one hand worksonl y for the intera cti ng gas (the ideal gas does not change) and on the other hand,needs the ergodic hyp othesis (noto ri ously hard to veri fy) to be equivalent to theensemble appro ach.

The proposed m etho d of expl oring the stati stical properti es of the condensateis based on the scatteri ng of the series of short l ight pul ses. The Ûuctua ti ons ina condensate occur at the ti mescale given by the intera to mic col lisions, thus topro be the Ûuctua ti ons the ti m e delay between consecuti ve pul ses shoul d be of theorder of m i ll iseconds. The di stri buti on of the numb er of pho tons scattered into agiven sol id angle shoul d be m easured. Out of the di stri buti on of the numb er ofscattered photo ns we m ay com pute the m ean and i ts vari ance. It is assumed tha tthe pul se of l ight is weak and far detuned in order to avo id heati ng the condensateduri ng the intera cti on. The pul se of l ight should be also su£ cientl y short in ti m e.


It should sati sfy two condi ti ons of techni cal nature. It should be so short tha t theato m s have no ti me to m ove far in the harm onic tra p duri ng the intera cti on. Itshould be also much shorter tha n the l i feti me of the atom ic tra nsi ti on to justi f yneglect of the sponta neous emission losses.

W ith these sim pl iÙcati ons, one can Ùrst solve the Hei senberg equati ons sat-isÙed by the ato mic vari ables dri ven by the short pul se, and in the next stage,compute the scattered l ight Ùeld, trea ti ng the oscil lati ng ato mic di poles as a givensource of radi ati on. No back acti on of the radiated l ight on the m oti on of ato m icdi po les is incl uded. Thi s is a neglect of sponta neous emission.

The resul ti ng creati on and anni hi lati on operato rs of scattered photo ns arequadra ti c functi onals of the ato m Ùeld (the tra nsi ti on di pole m oment is quadrati c).As we have al ready mentio ned, we need to compute the vari ance of num ber of scat-tered photo ns. Thi s is quadra ti c in the photo n operators, thus of the eighth order inato m ic Ùeld. The com puta ti onal compl exit y, both analyti c and num erical is there-fore signi Ùcant. The results are summ arized in Fi g. 4. W eha ve plotted the angulardi stri buti on of suita bl y norm alized vari ance of the numb er of scattered photo ns.The calcul ati ons are perform ed for 1000 nonintera cti ng ato m s at the tem peratureof 0 : 3 T c. We com pare here the resul ts for the grand canonical ensembl e, desig-nated as therm al , wi th the Poissoni an stati sti cs of the coherent state and wi th them icrocanoni cal Ûuctuati ons. It is evident tha t the m etho d proposed shoul d easilydi scriminate not only between the non-physi cal therm al Ûuctuati ons and the restbut also between Poissoni an and micro canonical predi cti ons.

Fig. 4. A ngular distributi on of the variance of the numb er of scattered photons is

show n. N ote that not only the thermal but also the coherent states results are clearly

disting ui shed. I t is imp ossibl e, how ever, to distingu ish betw een the micro canon ical and

the decorrelated results.

R egretf ul ly, as indi cated in the Ùgure, micro canonical sta ti sti cs is so narro wtha t it canno t be disti nguished f rom nonÛuctua ti ng, Fock state results, cal led\ decorrelated " in the Ùgure. One woul d have to deal wi th even smal ler condensatesto reveal thi s di ˜erence.

The experim ents probing the stati sti cs rem ain a chal lenge. Perhaps them etho d outl ined here wi l l be of some help. In the next section we turn our


attenti on to yet another aspect of the stati stics | the spacial correla ti onfuncti ons.

5. Co h er en ces | a sm oki n g gun f or a Bo se{ Ei nst ei n co n den sat i on

In previ ous sections we have shown tha t a Bose{Ei nstein condensate mightbe vi ewed as a giant matter wa ve of parti cles occupyi ng the sam e quantum state.Such a pi cture cannot be auto mati cal ly extended to an intera cti ng system . Thelatter shoul d be described by an N -body wa ve functi on (or density m atri x) andsing le-parti cle states of a tra pping potenti al do not have much of a physi cal sig-ni Ùcance. The cri teri on for the Bose condensati on requi res more deta i led analysis.T o a large extent, thi s goal has been compl eted m any years ago by Penro se, On-sager, and Yang [21{ 23]. Let us im agine for a m oment tha t an exact N -bodywa ve functi on is given expl icitl y, and a problem in hand is, whether the systemshows Bose{Ei nstein condensati on. A conventio nal wi sdom gives a hint: a pi ctureof a giant m atter wa ve shoul d be sti l l v alid even in a case of intera cti ng system .Mo re preci sely, a Bose{Ei nstein condensate shoul d exhi bi t macro scopic quantumphenom ena | phenom ena tha t are inevi ta bl y related to large scale quantum cor-relati on.

T o i l lustra te basic concepts and subtl ety of issue of a Bose{Ei nstein con-densate we consider an exactl y solvable model of parti cles tra pped by an externa lharm oni c potenti al and intera cti ng by oscil lato ry forces [24] described by the fol -lowi ng Ham i lto ni an:

H =1

2

NX

i =1

( p 2i

+ x2i

) +1

2

˚! 2

À 1

N

Ç X

i ; j

( x i À x j ) ; (5 .1)

where i i s a positi on of the i -th parti cle and i is i ts m omentum . The parameter! determ ines an intera cti on strength; 0 < ! < 1 corresponds to repul sive, whi le! > 1 to attra cti ve forces. Al l v ariables are expressed in uni ts of harm oni c potenti alof the tra p. The assumed form of tw o-body intera cti ons is not physi cal because ofinÙni te range of two -body forces. However, from a point of vi ew of cri teri on f or aBose condensati on i t m ight be qui te useful . The ground state of the Ham i lto ni anin a sim ple case of N = 2 is

` ( ; ) =

˚1

¤

Ç =

exp

ç

À

1

4( + )

Ñ± !

¤

² =

exph

À

!

4( À )

i: (5 .2)

The wave functi on Eq. (5.2) is a to ta l ly sym metri c product of two term s: thecenter- of-m assone, whi ch is a ground state of externa l tra p, and the relati ve coor-di nate term whi ch is a ground state of harm onic potenti al of frequency ! . Thi s f ormreveals a generic feature of any N -body intera cti ng system . The center-of-massex-periences the externa l potenti al onl y. The relati ve degree of freedom wave functi onis tra nslati onal ly invaria nt. The two -parti cl e eigenstate becom es a pro duct of the


ground state wa ve functi on of externa l potenti al , only i f ! = 1 . Thi s signiÙesnon- intera cti ng case. A generalizati on of Eq. (5.2) to an arbi tra ry numb er of par-ti cles leads to the fol lowing expression for the ground state:

` ( f x j g ) =

˚1

¤

Ç d= 4

exp

˚

À

12

x2C M

Ç ±!

¤

² d ( N À 1 ) = 4

È exph

À

!

2

±X¿ j À

2C M

² i; (5 .3)

where we used a short- hand nota ti on f i g = ( i ; . . . ; N ), and

C M = (1 =p

N )

NXj

i s an N -body center-of-m ass coordi nate (no te an unco nventi onal norm al izati on).One m ight naivel y thi nk tha t the ground state of a Bose system m ust be Bosecondensed. W e m ight, theref ore, ask the questi on: whi ch is thi s m acroscopi cal lyoccupied state?

T o answer thi s question we shoul d thi nk for a m oment about a possibledetecti on and relate di ˜erent m easurem ents to quantum m echanica l observabl es.Fortuna tel y, in the early sixti es(of the last century) Glaub er discussedsim i lar issuein detai l [25]. He developed the coherence theo ry for a systemati c descripti on ofthe stati stical pro perti es of l ight. Thi s is nothi ng else but higher-order coherences,whi ch al low to disti ngui sh a therm al from a laser l ight. Our ta sk is in fact sim i lar:we wa nt to disti ngui sh a cl assical system from a Bose{Ei nstein condensate. Andwe want to identi fy these experim ental ly observed features whi ch are uni que to aBose condensate.

Needl ess to say, properti es of a quantum system depend on a m easurem entperform ed. The sim plest one consists of a num erously repeated detecti on of a sing leparti cl e. Acco rdi ng to quantum m echani cs, the pro babi l i ty density of detecti ng aparti cl e at is equal to diagonal elements, = , of a one-parti cle reduceddensity matri x

£ ( ; ) = Trf g

[ ` Ê ( ; ; . . . ; N ) ` ( ; ; . . . ; N ) ] : (5 .4)

One-parti cle m easurement is, however, not the only detecti on, whi ch can be per-form ed on a m any -body system . One can also simul taneousl y m oni to r a num ber ofparti cl es in a single shot of a cam era. In such a case an observed parti cle conÙgura-ti on f s g represents a sing le real izati on of the quantum system . Thi s conÙgura ti onis a stati sti cal vari able. It is di stri buted accordi ng to a di agonal part of s -parti clereduced density m atri x (hi gh-order correl ati on functi on) £ s ( f s g ; f s g ) , deÙnedanalogously to (5.4). The joint detecti on can be related to a condi ti onal m easure-m ent | a detecti on of a parti cle at some positi on s pro vi ded tha t sim ul ta neouslyother parti cles are detecte d at prescri bed space-points f s À g . The relati on


£ ( s ) ( xs ; f x s À 1 g ; ys ; f xs À 1 g )

= £ (1 =s À ( s ;s

j f s Àg )£ sÀ ( f s À

g ; f s Àg ) (5 .5)

deÙnescondi ti onal one-parti cle density matri x. Ob vi ously, i f parti cles are indepen-dent, then

£ j s À ( s ; s j f s À g ) = £ ( s ; s )

and

£ s ( f s g ; f s g ) =

s

j

£ ( j ; j ) : (5 .6)

Such a separati on of s-order correl ati on functi on signiÙesthe s -order coherence inthe system. If the system is coherent, then joint and num erousl y repeated detecti ongives the same resul t. In the opp osite case both detecti on schemes might givedi ˜erent answers. We wi l l m ake use of thi s observati on soon.

Being equipped wi th the basic knowl edge of a detecti on pro cess and therelated correl ati on functi ons, we are ready now to analyze the m odel system. D ueto a sim ple form of the wa ve functi on (5.3) al l relevant reduced density m atri cescan be evaluated exactl y. Mo reover, they can be easily di agonal ized, i .e. broughtto the form

£ ( ; ) =

j

Ñ j ' Ê

j ( ) ' j ( ) : (5 .7)

A physi cal m eani ng of Eq. (5.7) is qui te obvi ous. The one-parti cle subsystemis in a mixed state being a stati sti cal mixture of one-parti cle \ orbi ta ls" ' i ( )

wi th pro babi l iti es Ñ i being relati ve occupati ons of the i -th state. D iagonali zati onpro vi des us wi th a natura l and very conveni ent one-parti cle basis. In our case, ina l im it o f large N , a dom inant eigenvalue is

Ñ =2

1 +p

1 + N ç À + N À ç À

=

; (5 .8)

where we intro duced an \ intera cti on strength" param eter ç related to the frequency! by ! = N ç . Surpri sing ly, i f j ç j < 1 , then Ñ ¤ 1 . One-parti cle subsystem is ina pure state even i f every parti cl e intera cts wi th a ll remaini ng ones! From thevi ewp oint of one-parti cle m easurements the system is undi sti ngui shable from asystem of nonintera cti ng parti cl es | al l of them occupyi ng the sam e state

' ( ) =!

¤

=

exp À

1

2:

In such a situa ti on the one-parti cle matri x separates

£ ( ; ) ¤ ' Ê ( ) ' ( ) : (5 .9)

In Fi g. 5 we pl ot relati ve popul ati on of the highest occupied state as a functi onof intera cti on strength ç f or di ˜erent Ùnite values of parti cle numb er N . No te a


Fig. 5. Fraction of condensed atoms as a function of the interaction strength ç for

di ˜erent numb er of particles ; N = 10 3 | green line; N = 10 5 | red line; N = 10 7 |

blue line.

sym m etry wi th respect to a \ sign" of intera cti ons: ç < 0 for repul sive, and ç > 0

for attra cti ve forces. The larger numb er of parti cl es, the m ore rapidly dom inanteigenvalue vani sheswi th increasing ç .

The m odel system is a perfect candi date for a Bose{Ei nstein condensatein a region of ç < 1 . Separabi l i ty of one-parti cl e density matri x justi Ùes a pic-ture of m acroscopi c occupati on of one single-parti cle state. The system can beassigned a macro scopic wa ve functi on ' ( ) whi ch furni shes a basis for the cele-bra ted Gross{Pi ta evski i equati on and Bogolyub ov appro ach [26]. Separabi l ity , ina language of the coherence theo ry , signi Ùes the Ùrst-order coherence. Interf er-ence fringes wi th 100% vi sibi l i ty wi l l be observed in a tw o-slit Young interf erenceexp eriment perform ed on such a system . But thi s feature cannot be exclusivel yattri buted to Bose condensates. Interf erence experim ents have been successful lyperform ed also wi th therm al , but su£ ci entl y monochro mati c, ato mic beams [27].If the Ùrst-order coherence were the onl y quantum feature of Bose condensates,they would be real ly poor condensates. Genuine condensates, simi larl y as genuinelasers, ought to show higher-order coherences. In the case studi ed here i t can beshown [28, 29] tha t in the region of ç < 1 the system shows also higher-ordercoherences,

£ ( ; ) = £ ( ; ) ; (5 .10)

provi ded tha t s N . Thi s inequal i ty is an im porta nt l im ita ti on. The coherentstate of the electrom agneti c Ùeld is coherent in al l orders.

The studi ed exampl e teaches us another importa nt lesson. A ground stateof intera cti ng Bose system is not necessari ly a Bose{Ei nstein condensate. At leasti f the Ùrst-order coherence is considered. If one perf orm s a Young exp erimentand col lects data from num erousl y repeated one-parti cle detecti on, then no in-terf erence wi l l be observed. A reason for thi s decoherence ori ginates in existenceof two drasti cal ly di ˜erent length scales. Am pl i tude of a zero point oscil lati ons


of the center- of-m ass is Ùxed because is related to the externa l potenti al . Onthe contra ry , spati al spreading of relati ve coordi nates depends on the intera cti onstreng th and parti cle numb er. For stro ng repul sive forces i t exceeds signi Ùcantl ythe center-of-massspreading. In a case of attra cti ve forces the system is aggregatedover a di stance m uch smal ler tha n quantum uncerta inty of the center-of-m ass. Aspati al extensi on of the system revealed in a one-parti cl e measurement is relatedto larger of the two length scales. But the Ùrst-order coherence length is l imi tedto smaller of them . Tha t is why the system does not show coherences on them acroscopi c scale comparable to its size.

Ca n one, however reveal som e coherence in a joint detecti on of a large num -ber of parti cles? Let us consider the sim pl est case of a condi ti onal m easurem ent| positi on of a parti cle is col lected only when simul ta neously another one is be-ing observed at a given locati on x0 . Corresp ondi ng reduced density m atri x can beevaluated in the m odel . Its diagonal part is a probabi l i ty density of condi ti onaldetecti on and i t is shown in Fi g. 6 for a case of attra cti ve intera cti ons ç = 1 : 3

(bl ue l ine). For a com pari son the red line shows one-parti cle pro babi li t y density

Fig. 6. One-particle probabili ty density (in dimension less units); single- parti cl e detec-

tion | red line; conditiona l detection of a particle pro vided that simultaneously another

one have been found at x 0 = 0 :00 3 | blue line. Total numb er of particles N = 104 , and

interaction strength .

£ ( ; ) whi ch is spread wi dely over a large di stance. Thi s spati al extensi on de-Ùnes a region where the Ùrst parti cl e can be e˜ecti vel y detected. W e assumed tha tthe parti cle has been f ound at = 0 : 0 0 3 . As the Ùrst parti cl e is detected at somepoint, the second one must be (at the sam e ti me) relati vely close. Because of astro ng attra cti on, parti cl es are correl ated. Co ndi ti onal detecti on is a practi cal re-al izati on of a measurement perform ed in the center-of-mass fram e in a case whenpositi on of the latter has large quantum uncerta inty . The center-of-m ass lengthscale is eliminated in a sing le real izati on of a quantum system . W hat remains i t isonl y a relati ve degree of freedom spreading whi ch determ ines a di stance on whi chthe system shows the coherence. Ana lysis of eigenvalues of corresp ondi ng condi -ti onal density matri x supp orts thi s interpreta ti on. There exists single dom inant


eigenvalue equal (i n large N lim it) to

Ñ 0 =

˚2

1 +p

2

Ç3 = 2

:

Such system is a Bose{Einstei n condensate despite tha t i t cannot be assigneda macro scopic wave functi on.

Si tua ti on is com pletel y di ˜erent i f parti cles repel each other. The joint mea-surement does not reveal any coherences i f ç < À 1 . The larger length scale iscomm on to al l N À 1 relati ve degrees of freedom and cannot be el im inated by ajoint m easurement. In our m odel , stro ng repul sive two -body intera cti ons lead toinevi ta ble destructi on of the Bose{Ei nstein condensate even in the ground stateof a bosonic system .

6. In t eract i ng Bo se{ Ei n st ein co nd ensat e at Ùn i t e t em per at u res

Theo reti cal descripti on of a stati onary Bose{Ei nstein condensate is a qui tecompl icated and subtl e problem even at zero tem perature. One can easily im agine,theref ore, tha t a study of a dyna mical behavi or of the system, in parti cul ar at Ùnitetem peratures, is a real challenge. The best suited language for tha t ki nd of studi esis a language of Ùeld theory . N -parti cl eBose system is being assigned a bosoni c Ùeldoperato r ^ˆ ( x ) tha t destro ys a parti cle at positi on x and obeys standard bosoniccomm utati on relati ons wi th i ts conj ugated partner ^ + ( x ) creati ng a parti cle at x :

[ ^ˆ ( ) ; ^ˆ ( )] = £ ( À ) : (6 .1)

N -parti cl e Ha mi l to nian can be expressed in term s of the Ùeld operator

H =

Z^ˆ ( )H ( ) ^ˆ ( ) +

1

2

Z Z^ˆ ( ) ^ ( ) U ( À ) ^ ( ) ^ˆ ( ) ; (6 .2)

where H ( ) i s a sing le parti cle Ha mi l to nian whi le U ( À ) i s a two -body intera c-ti on potenti al . In real isti c situa ti ons of Bose condensates of di lute ato m ic v aporsin m agneti c tra ps the two -body intera cti on is appro xi mated by a conta ct potenti al

U ( À ) = g £ ( À ) ; (6 .3)

where g characteri zesthe atom { atom intera cti on in the low-energy, s -wa ve appro x-im ati on. A Hei senberg equati on orig inati ng from the Ham i l toni an (6.2) acqui res aform

iñh@

@t^ = H ^ + g ^ˆ ^ˆ ^ˆ : (6 .4)

A ful l operator soluti on of (6.4) is im possibl e. There are di ˜erent appro aches toa ti m e evoluti on of an intera cti ng condensate. They lead to num erical algori thm s,whi ch are hard to implem ent [30, 31]. No ne of them are in fact capable to handlea real isti c case of a large num ber of parti cles at a relati vel y hi gh tem perature. Butthi s is onl y the Ùrst obstacle whi ch one encounters whi le studyi ng an intera cti ng


Bose system at Ùnite tem peratures. It is not easy to extra ct relevant physi cal in-form ati on l ike, for exam ple, a m ean occupati on and Ûuctua ti ons of a condensate| even havi ng obta ined the Ùeld operato r. It obvi ously requi res an uni que identi -Ùcati on of a condensed part of the system . W e have al ready discussedthi s del icateissue in the previ ous section. A genuine condensate can be assigned a condensatewa ve functi on whi ch is an eigenvector corresponding to a dominant eigenvalue ofa one-parti cle density m atri x. Thi s m atri x can be expressed wi th the help of Ùeldoperato rs

h ^ˆ + ( x) ˆ ( y ) i = N £ (1 ) ( x; y) ; (6 .5)

and the avera ge is ta ken in the ini ti al state of many-b ody system . In a general case,a di agonalizati on of a density m atri x is needed in order to determ ine physi cal lyrelevant quanti ti es. There is however, one parti cul ar situa ti on when sym metry ofa pro blem is of great help. Thi s is the case of a gas tra pped in a box wi th periodicbounda ry condi ti ons. The sym metry of the conÙning potenti al uni quely enforces aform of eigenstates of the one-parti cle density matri x regardl essof intera cti on. Thedensity matri x must be peri odic and depend only on a di ˜erence of i ts argum ents

£ (1 ) ( x; y ) = £ (1 ) ( x À ) = Ñ f ( ) f ( ) : (6 .6)

The above condi ti on indi cates tha t eigenfuncti ons are simpl y plane wa ves

f ( ) =1

Lexp( À i Â ) ;

and the wa ve vector ta kes quanti zed values = (2 ¤ =L ) wi th n = 0 ; Ï 1 ; Ï 2 ; . . .

( i = x ; y ; z ) , and L being a length of the cubi c box. A stati onary state of aBose{ Einstei n condensate corresponds, theref ore, to = state of the box. Thi sis the reason why qui te often a Bose condensati on is being tho ught of as a con-densati on in a m omentum space. If we expand the Ùeld operato r in the basis of abox eigenfuncti ons

^ˆ ( ) = f ( ) a ; (6 .7)

then we save the e˜o rt of diagonal izati on a density m atri x, simulta neously gaininga di rect insi ght into physi cal quanti ti es.The operato r a ( a ) anni hi lates (creates)a parti cl e in the state f ( ) , and n = a a counts parti cles occupyi ng thi s state.Using an adv anta ge of the periodic sym metry we lim i t our study to a condensatetra pped in a periodic box. In thi s case a one-parti cl e Ha mi lto nian is

H =2 m

;

and dyna m ical equati on for the operato rs a (i n the intera cti on pi cture) acqui rethe fol lowing form :

a = À ig exp[2i( À )( À ) t ]a a a : (6 .8)

The two -body intera cti on energy g and ti m e t are expressed in uni ts of Â =

and § = ñh= Â respect ivel y. Ob vi ously, som e appro xi matio ns are neces-

sary i f one wants to solve the nonl inear operato r equati ons (6.8). A semiclassical


appro xi m atio n intro duced in [32, 33] is parti cul arl y wel l sui ted for a descripti on ofthe system at tem peratures below the cri ti cal one, except the region cl ose to theabsolute zero. The appro xi matio n consists in replaci ng al l creati on and anni hi lati onoperato rs by c-numb ers a kk ! a kk . Such repl acement is legi ti mate i f a popul ati onof a given mode is greater tha n i ts quantum Ûuctua ti ons, i .e. i f n kk = a Ê

kk a kk > 1 .At tem peratures close to zero it is onl y k = 0 m ode for whi ch the latter condi ti onis f ulÙlled. Quantum Ûuctua ti ons in exci ted m odes are im porta nt. At hi gher tem -peratures, however, many modes are m acroscopi cal ly popul ated and then therm alÛuctua ti ons dom inate quantum ones.

The semicl assical appro xi m atio n leads to a set of coupled nonl inear equa-ti ons, whi ch have to be solved num erical ly. Dyna m ical equati ons (6.8) can bevi ewed as equati ons for coupl ed macroscopical ly occupi ed \ m ean Ùelds" a kk . Ourappro xi m atio n is an extensi on of a Bogolyub ov appro ach [26]. Thi s appro ach isval id at low tem peratures because i t assumes tha t onl y k = 0 mode is macroscop-ically occupied and theref ore onl y the \ condensed" m ode, described by a com plexÙeld, is ta ken into considerati on. Evi dentl y, for Ùnite system at higher tem per-atures other modes can be trea ted in the sam e fashion. Let us observe tha t ourappro xi m ate dyna m icsconserves both the tota l energy and the num ber of parti cles.It corresp onds theref ore, to a genuine m icrocanonical descripti on.

Our Ùrst goal is to describe the equi l ibri um state of an intera cti ng system .W e assume tha t our dyna mics reaches a steady state for any \ typi cal " ini ti al con-di ti on. Thi s assumpti on is very im porta nt and should be careful ly tested [33, 34].One-di mensional version of thi s dyna m ics is com pletel y integ rable [35], however,in higher numb er of dim ensions or in a case of m ore realisti c tra ppi ng poten-ti als, the evoluti on m ay be chaoti c. W e start the evoluti on choosing the ini ti al\ Bose{ Einstei n-l ike" di stri buti on of popul ati ons j a kk j . In determ ining thi s distri -buti on, we neglect al l intera cti ons, thus the ini ti al condi ti on evi dentl y does notcorrespond to the equi l ibri um di stri buti on of the intera cti ng system . Mo reover,there is sti l l a freedom in selecting phases of the coupl ed Ùelds. In our appro ach,each m ode is assigned an ini ti al , random ly chosen, phase. Any subsequent dy-nam ics depends on ini ti al phasesand, in a sense, describes a single real izati on ofthe quantum system . Mi cro canonical expectati on values woul d requi re the averageover the ini ti al phases. Instea d of doing so, we tra ce the system for a su£ cientl ylong ti m e and compute ti m e avera ges. In fact, ti m e avera gesrather tha n ensembl eones are m easured exp erimenta l ly. In addi ti on, the two are equal i f a system isergodic.

In our three- dim ensional calculati ons we assumed tha t N = 1 0 0 0 0 0 rubi di um( Rb) ato m s are tra pp ed into a cubi c box of a size equal to the Tho masÀ Fermiradi us of the same system tra pped in a harm oni c tra p of frequency 2 ¤ È 8 0 Hz. Thetwo -body intera cti on strength is g =Â = 2 : 5 2 È 1 0 À . W eperform ed our calcul ati onswi th 729 m odes. Further increase in a numb er of m odes does not lead to anysubsta nti al modiÙcati ons of Ùnal results. W e tra ced the evoluti on of the system on


a ti mescale equal appro xi m atel y to one second. The calcul ati ons show tha t af terfew m i l liseconds the system reaches a state of a dyna m ical equi l ibri um regardl essthe choice of ini ti al phases. The occupati on of the k = 0 m ode stabi l izes at somevalue, and on a larger ti m escale i t only Ûuctua tes around tha t value, Fi g. 7.

Fig. 7. C ondensate occupation as a function of time for total energy p er particle E =ñh =

53 9 H z.

Fig. 8. Phase- space portraits | imaginary versus real part of complex amplitud es

corresp ondi ng to di˜erent mo des of the system.

In Fi g. 8 we present a typi cal \ phase-space portra i ts" of some com plex am pl i -tudes a kk . The Ùgure depicts im aginary versus real part of am pl itudes corresp ondi ngto k = (0 ; 0 ; 0 ) , k = (0 ; 0 ; 2 ) , and to = (0 ; 0 ; 4 ) modes. D i˜erent points corre-spond to di ˜erent instances of ti me. Two im porta nt features coul d be observed.Fi rst of al l , we stress a stri ki ng di ˜erence between = 0 m ode (a condensate) andhi gher modes. The am pl itude of condensate mode acqui res a steady- state valueand i ts relati ve Ûuctuati ons are smal l . On the contra ry , relati ve Ûuctua ti ons of


hi gher modes reach 100%. The second observati on is tha t a phases of al l com plexÙelds a kk vari es uni form ly over the who le accessible range between zero and 2 ¤ .Fi gure 8 provi des a strong indi cati on tha t the dyna m ics studi ed here is ergodicand the system vi sits al l accessibl e states whi ch are al lowed by conservati on ofenergy and parti cle numb er.

In Fi g. 9 a m ean popul ati on (bl ue l ine) of the intera cti ng Bose{Ei nsteincondensate and i ts Ûuctua ti ons (red l ine) are presented as a functi on of the to ta lenergy of the system . Both curves are smooth and do not show any singulari tycharacteri sti c of a phase tra nsiti on. Thi s behavi or is not surpri sing because oursystem is Ùnite. W eobserve tha t a condensate popul ati on decreaseswi th the energyand Ùnal ly di sappears at the energy close to E =ñh = 2 0 0 0 Hz. The Ûuctua ti ons growwi th energy and reach thei r m axi mal value at E =ñh = 1 4 5 0 Hz. Let us observe tha ti t is also an inÛection point of a mean condensate popul ati on. In a vi cini ty oftha t energy the system undergoes rapid changes. Thi s energy is a Bose{Ei nsteincondensati on energy.

Fig. 9. C ondensate occupation (in blue) and Ûuctuations (in red) versus total energy

p er particle.

The metho d presented here is very powerf ul . A caref ul reader, however,m ight be confused. By replacing creati on and anni hi lati on operato rs by c-numb ersa +

kk ; a kk ! a Ê

kk ; a kk we simulta neously substi tuted the Ùeld operato r by a \ classicalwa ve functi on"

^ˆ ( x ) !

p

N ˆ ( x) = a f ( ) ; (6 .9)

and the set of equati ons (6.8) is in f act nothi ng else but the celebrated Gross{Pi ta-evski i equati on wri tten in the basis of plane waves. Thi s equati on tra di ti onal lydescribes a pure condensate at zero tem perature. The corresp ondi ng one-parti cledensity matri x

£ ( ; ) = ˆ Ê ( ) ˆ ( ) ; (6 .10)

pro jects onto a pure state ˆ ( ) whi ch is macro scopically occupi ed by N ato m s.W hat about our argum ents tha t i t is j a j whi ch gives popul ati on of a condensate?


Ha ve we been cheati ng so far? Certa inl y not! W e owe, however, an expl anati on.And again, the detai led analysis of a detecti on process helps to resolve the issue.

W e adm i t tha t in fact we are solvi ng the Gross{Pi ta evskii equati on. Butthe ini ti al sta te is f ar from equi l ibri um . The system , theref ore, is stro ngly devel -opi ng, and the wa ve functi on ˆ ( x ) vari es rapidl y in ti m e. Co mpl ex am pl i tudesof di ˜erent m odes oscil late very fast, even when the evoluti on reaches a stage ofdyna mical equi libri um. Needl ess to say, a detecti on pro cess is not insta nta neous.Opti cal metho ds are used to moni to r an ato m ic Bose{ Einstei n condensate | aCCD cam era m oni tors a l ight passing thro ugh a condensate. An expositi on ti m e,§ , varies from few m icroseconds up to several hundreds of microseconds thus wha tis being observed i t is a ti m e average of a diagonal part of a one-parti cle reduceddensity matri x

h £ (1 ) ( x ; y; t ) i t = h ˆ Ê (x ; t ) ˆ ( y ; t ) i t =1

N

X

kk ; kk

h a Ê ( t )a ( t ) i t f Ê ( ) f ( ) : (6 .11)

Thi s ti me average \ spoils" a puri ty of dyna micall y evolvi ng state. The detecti onintro duces a coarse-gra ined ti m e. R apidl y oscil lati ng phasesof di ˜erent mean Ùeldsaverage to zero on tha t ti m escale, and only diagonal elements survi ve

h a Ê ( t ) a (t ) i t = £ ; h n ( t ) i : (6 .12)

Mo reover, i f the system reaches a state of dyna m ical equi l ibri um , avera ged occu-pati ons of di ˜erent m odes do not depend on ti m e (they depend onl y on the to ta lenergy and the num ber of parti cl es) and one-parti cle density matri x acqui res astati onary form


h £ (1 ) ( x ; y) i t =X

kk

h n kk i t

Nf Ê

kk ( x) f kk ( ) : (6 .13)

Equati on (6.13) furni shes a basis for our m etho d.It is obvi ous tha t the sam e m etho d can be used in a m ore real istic case

of a condensate tra pped in a harm oni c potenti a l. By avera ging the pro duct of ati m e-dependent soluti on of the Gross{Pi ta evski i equati on ˆ Ê ( ; t ) ˆ ( ; t ) over aÙnite observati on ti me we obta in the one-parti cle density m atri x. In Fi g. 10 theone-parti cle density of intera cti ng Bose gas tra pped in m agneti c tra p is shown.One can easily see a bro ad therm al cloud and a sharp peak of the Bose condensedato m s on to p of i t. D o not conf use the Ùgure wi th an experim enta l resul t. Thi sis the theo reti cal Ùgure obta ined wi thi n our appro ach! It stro ngly resembl es theresul ts of real exp eriments. And, as i t is done in experim ents, by identi fyi ng thetwo com ponents (the condensate and the therm al cloud) i t is possible to determ inea condensate popul ati on and a temperature. W e are currentl y wo rki ng on theappl icati on of the above m etho d to the quanti ta ti ve analysis of the properti es ofthe ul tra cold Bose gas, tra pp ed in the harm oni c potenti al just below the cri ti caltem perature [36].

The results presented here were obta ined in col laborati on wi th J. Mo stowski ,M. Za¤uska-Kotur, Z. Idzi aszek, and K. G§ra l . M. G. tha nks the State Comm itteefor Scienti ÙcR esearch, grant 2P03B07819, for support, whi le the research of K. R.wa s sponsored by the subsidy from the Founda ti on for Pol ish Science.


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[34] M. J. Da vis, S.A . Morgan, K . Burnett, 160402 (2001).

[35] F.M. Izrailev, B. V . C hiriko v, 57 (1966) [30 (1966)] .

[36] K . G§ral, M. Gaj da, K . Rz¨âew ski, in preparation.

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