Non-diffusive phase spreading of a Bose-Einstein condensate at finite temperature

0.48 MB·English

by  Alice Sinatra

Non-diffusive phase spreading of a Bose-Einstein condensate at finite temperature
A. Sinatra and Y. Castin
Laboratoire Kastler Brossel, Ecole Normale Sup´erieure,
24 Rue Lhomond, 75231 Paris Cedex 05, France
E. Witkowska
7 Institute of Physics, Polish Academy of Sciences,
0 Aleja Lotnik´ow 32/46, 02-668 Warszawa, Poland
0
2 We show that the phase of a condensate in a finite temperature gas spreads linearly in time at
longtimesratherthaninadiffusiveway. Thisresultissupportedbyclassical fieldsimulations,and
n
analyticalcalculationswhicharegeneralizedtothequantumcaseundertheassumptionofquantum
a
J ergodicityinthesystem. Thissuper-diffusivebehaviorisintimatelyrelatedtoconservationofenergy
duringthefree evolution of thesystem and to fluctuations of energy in the prepared initial state.
8
1
PACSnumbers: 03.75.Kk,03.75.Pp

]
r
e I. INTRODUCTION time)[7,8,9,10]oraballisticbehavior(variancegrowing
h
quadratically in time) [11]. We study this problem first
t
o with aclassicalfield model[12, 13, 14, 15, 16, 17], where
Phase coherence is one of the fundamental properties
t. exact numerical simulations can be performed. We then
a of Bose-Einstein condensates. It is also a key feature in
explainthenumericsanalytically,andextendtheanalyt-
m the present developments of the researchon condensates
ical approach to the quantum case.
which, ten years after the first experimental realization,
-
The important result that we obtain is that the vari-
d go in the direction of integrating this powerful tool into
n other branches of physics, of which metrologyand quan- anceofthe phaseincreasesquadraticallyintime. Thisis
o atvariancewiththepredictionofphasediffusionfromthe
tum information are two promising examples [1].
c “quantumoptics”opensystemapproachesof[7,8,9,10]
[ The problem of the condensate phase dynamics due
assumingthecondensatetoevolveunderthe influenceof

to atomic interactions at zero temperature has been an-
3 Langevin short memory fluctuating forces. Our predic-
alyzed by different authors in theory [2] and in experi-
v tion results from two ingredients, (i) the system is pre-
4 ment [3, 4, 5]. It is now well understood that an ini- pared in an initial state with an energy fluctuating from
4 tially prepared relative phase between two condensates one experimental realization to the other, here sampling
6 will spreadin time due to the correspondinguncertainty
the canonical ensemble, and (ii) the system is isolated
0 in the relative particle number as the relative phase and
in its further evolution and therefore keeps a constant
1
therelativeparticlenumberareconjugatevariables. The
6 energy. As we shallsee,the combinationof these twoin-
phasedynamicsofatwocomponentcondensateinrealis-
0 gredientspreventssometemporalcorrelationfunctionsto
tic situations including harmonic traps, non stationarity
/ vanishatlong times. Our predictionqualitativelyagrees
t and fluctuations in the total number of particles was an-
a with the one of[11], but notquantitatively,as we obtain
m alyzed in [6], where a comparison to the experiments of a different expression for the long time limit of the vari-
[4] is also performed. An important conclusion was that
- ance of the phase over the time squared. This difference
d the zero temperature theory could not account for the
is due to the fact that we take into accountergodicity in
n coherencetimesobservedinexperiment,whichraisesthe
thesystemresultingfromtheinteractionsamongBogoli-
o question of the role of the non-condensed fraction.
ubov modes such as the Beliaev-Landauprocesses.
c
: In this paper we address the fundamental problem of In section II we present the classical field model; nu-
v
phasespreadingofaBose-Einsteincondensateinafinite merical predictions for this model are presented in sec-
i
X temperature atomic gas. In order to obtain simple and tion III, and analytical results reproducing the numerics
r general results, we consider the ideal case of a spatially at short or long times are givenin section IV. These an-
a uniform condensate at thermodynamic equilibrium, and alytical results are extended to the case of the quantum
we assume that one has access to the first order tem- field in section V. We conclude in section VI.
poral correlationfunction a†(t)a of the component a
h 0 0i 0
of the atomic field in the condensate mode. In real life,
the situation is more complex: the atoms are trapped
II. THE CLASSICAL FIELD MODEL
in harmonic potentials, and the measurement of phase
coherence is a delicate procedure, usually relying on the
interference between two condensates [4]. In the litera- In this section we develop a classical field model that
ture two well distinct predictions exist for the long time has the advantage that it can be exactly simulated nu-
spreading of the condensate phase at finite temperature, merically. This will allow us to understand the physics
either a diffusive behavior (variance growing linearly in governing the spreading of the condensate phase and to
2
test the validity of various approximations, paving the We adjust the dispersion relation E˜ in order to repro-
k
way to the quantum treatment. duce the Bose law for the occupation numbers of the
We consider a lattice model for a classical field ψ(r) quantum field in the Bose-condensed regime:
in three dimensions. The lattice spacings are l , l , l
1 2 3 1 k T
tahloengvotluhme ethorfeethdeiruecntiitoncsellofinsptahceelaanttdiced.V W=el1eln2cll3osies eβh¯2k2/2m−1 = EB˜k . (7)
the atomic field in a spatial box of sizes L , L , L and
1 2 3 For all modes with large occupation number E˜
volumeV =L1L2L3,withperiodicboundaryconditions. ¯h2k2/2m, while the occupation of modes kwit≃h
The discretized field has the following Poisson brackets
¯h2k2/2m k T, whose quantum dynamics is not well
B
≫
δ approximated by the classical field model anyway, is ex-
i¯h ψ(r ),ψ∗(r ) = r1,r2 (1)
1 2 ponentially suppressed as in the quantum theory.
{ } dV
Inthe interactingcase,onecouldadaptthesametrick
where the Poissonbracketsaresuchthatdf/dt= f,H of a modified dispersion relation, by including the fact
{ }
for a time-independent functional f of the field ψ. The that the relevant spectrum is not h¯2k2/2m but the Bo-
field ψ may be expanded over the plane waves goliubov spectrum [18]. The resulting E˜ would now
k
startgrowing exponentially with k when the Bogoliubov
eik·r
ψ(r)= a , (2) energy [(h¯2k2/2m)(2ρg+h¯2k2/2m)]1/2 reaches k T.
k B
k √V In the classical field model we restrict our analysis to
X
the regime k T ρg so that at energies of the order of
where k is restricted to the first Brillouin zone, k B ≫
α ∈ kBT, the Bogoliubov energy is dominated by the kinetic
[−π/lα,π/lα[ where α labels the directions of space. term ¯h2k2/2m. One can then simply use in the Hamil-
We assume that, in the real physical system, the total tonian the modified dispersion relation E˜ as given by
number of atoms is fixed, equal to N. In the classical k
Eq.(7). This is what we did in the simulations of this
field model, this fixes the norm squared of the field:
paper, so that the classical field ψ evolves according to
the non-linear equation [19]:
dV ψ(r)2 =N. (3)
| |
Xr i¯h∂ ψ = k T exp β ¯h2 ∆ 1 +g ψ(r,t)2 ψ.
t B
Equivalently the density of the system − 2m − | |
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
(8)
N
In practice this equation is integrated numerically with
ρ= (4)
V the FFT splitting technique.
We then introduce the density and the phase of the
is fixed for each realizationof the field. The evolutionof
condensate mode
the field is governed by the Hamiltonian
H = E˜ a∗a + g dVψ∗(r)ψ∗(r)ψ(r)ψ(r), (5) a0 =eiθ N0. (9)
k k k 2
k r In what follows, we concentrapte on three physical quan-
X X
tities: the condensate amplitude correlationfunction
whereE˜ isthedispersionrelationofthenon-interacting
k
waves,andthebinaryinteractionbetweenparticlesinthe ha∗0(t)a0(0)i, (10)
real gas is reflected in the classical field model by a field
self-interaction with a coupling constant g = 4π¯h2a/m, the condensate atom number correlation function
where a is the s-wave scattering length of two atoms.
δN (t)δN (0) where δN =N N , (11)
0 0 0 0 0
Ingeneral,weexpectthepredictionsofaclassicalfield h i −h i
modelto be cut-off dependent, i.e. the predictions ofour and the variance of the condensate phase change during
model may depend on the lattice spacings l . We use t:
α
herearefinementtotheusualclassicalfieldmodel,which
makes it cut-off independent for some observables like Varϕ(t)= ϕ(t)2 ϕ(t) 2 where ϕ(t)=θ(t) θ(0).
h i−h i −
(12)
the condensate fraction, a quantity expected to play an
Theaveragesaretakenoverstochasticrealizationsofthe
important role here. An obvious example of a quantity
classicalfield,asthe initialfieldsamplesathermalprob-
whichwill remaincut-offdependent is the meanvalue of
ability distribution.
the Hamiltonian H in thermal equilibrium.
Let us consider first the non-interacting case (g = 0)
in presence of a condensate. For a thermalized classical
III. CLASSICAL FIELD: NUMERICAL
field the occupation numbers of the excited plane wave
RESULTS
modes are given by the equipartition formula
ha∗kaki= kEB˜T . (6) 70W0¯he2/cmonVsi2d/e3riangaasboofxNof=no4n×co1m05maetnosmursawbliethsqρugar=e
k
3
lengths to guarantee efficient ergodicity in the system, In figure 1, we show the real part of the amplitude
in the ratio L2 : L2 : L2 = √2 : (1+√5)/2 : √3. We correlation function of the condensate a∗(t)a (0) as a
1 2 3 h 0 0 i
choose the number of the lattice points in a tempera- function of time, for a temperature T = 0.17T , where
c
ture dependent way, such that the maximal Bogoliubov T is the critical temperature of the ideal gas. The zero-
c
energy [(h¯2k2/2m)(2ρg+h¯2k2/2m)]1/2 on the lattice is temperature evolution eiρgt/h¯ is removed so that the os-
equal to 3k T. cillations in the figure are due to the above mentioned
B
To generate the stochastic initial values of the classi- effect µ(T) = ρg. Due to the finite temperature in the
6
cal field we proceed as follows. (i) For each realization, system,thecorrelationfunctionofthecondensateampli-
we generate a non condensed field ψ (r) at temperature tude is smeared out at long times.
⊥
T in the Bogoliubov approximation as explained in [20].
In practice we generate complex numbers b for each a
{ k} > 1.0
vector k on the grid according to the probability distri- N0
bution <
P(bk)= π1kǫ˜BkT e−(|bk|2ǫ˜k/kBT) (13) a(0) > } / 0 00..05
where ǫ˜k = [E˜k(E˜k +2ρg)]1/2. With a set of {bk} for a *ã(t) 0-0.5
given realization we build the non condensed field < -1.0
ψ (r)=eiθ b U˜ eik·r +b∗V˜ e−ik·r (14) Re { 0 1 2 3 4 5
⊥ k=6 0(cid:18) k k √V k k √V (cid:19) t [2p mV2/3/ h ]
X
0.20
where the initial value of the condensate phase θ is ran- b
domly chosen with the uniform law in [0,2π[, and where N>0 0.15
tahree greivaelnambypltihtueduessuUa˜lk,BV˜okg,onliourbmovaltizheedorays, Uh˜ek2re−wV˜ikt2h=th1e, > } / < 00..0150
modified dispersion relation, so that 0) 0.00
a(0-0.05
U˜ +V˜ = E˜k 1/4 . (15) *ã(t) 0-0.10
k k E˜k+2ρg! e { < --00..2105
(ii) We create the classical field with the constraint that R 14 15 16 17 18 19 20
the total number of atoms N is fixed: t [2p mV2/3/ h ]
a
ψ(r)= 0 +ψ (r) (16)
⊥ FIG.1: (Coloronline)Realpartofthecondensateamplitude
√V
correlationfunction(10)normalizedtoitst=0value,anddi-
where a =√N N eiθ, N is the number of non con- videdbythezerotemperatureevolutioneiρgt/h¯,asafunction
0 ⊥ ⊥
densed atoms, − oftime: intheverticalaxislabel,a˜0(t)standsfora0(t)eiρgt/h¯.
(a)Shorttimesbehaviorand(b)longtimesbehavior. Insolid
N = dV ψ (r)2 . (17) line from an average over 500 solutions of Eq.(8), in dashed
⊥ | ⊥ | line (red) theBogoliubov approximation (40). Herethe tem-
r
X perature is k T = 3077.3h¯2/mV2/3 = 0.1711T , where T
B c c
(iii) We let the field evolve for some time interval with is the critical temperature k T =(2π¯h2/m)(ρ/ζ(3/2))2/3 of
B c
the Eq.(8) to eliminate transients due to the fact that theideal gas, thenumberof particles is N =4 105 andthe
×
theBogoliubovapproximationusedinthesamplingdoes couplingconstantissuchthatthezero-temperaturechemical
potential is ρg=gN/V =700¯h2/mV2/3.
notproduceanexactlystationarydistribution. Afterthis
‘thermalization’ period we start calculating the relevant
observables, as ψ evolves with the same Eq.(8). Correspondinglythestandarddeviationoftheconden-
Firstweinvestigatethemeancondensatephasechange satephase changeincreaseswith time, as weshowin fig-
ϕ (t). We find a linear dependence with time, with a ure 2 for five different values of the temperature, up to
h i
slope slightly different from the value ρg/¯h naively ex- T = 0.65T . In all cases, at long times, we observe a
c
−
pected, e.g. from the zero temperature Gross-Pitaevskii quadratic growth of Varϕ contrarily to the phase diffu-
equation. The slope difference is temperature dependent sion behavior t predicted in the literature [7, 8, 9, 10].
∝
and is expected physically to correspond to the discrep- To complete the physical picture, we show in figure
ancy between the zero temperature chemical potential 3 the correlation function of the condensate atom num-
ρg and the actual finite temperature one µ(T). This we ber (11). At very short times, see the beginning of the
shallconfirmusingBogoliubovtheoryinSec.IV(seealso curves in Fig.3a, the simulation (square symbols) con-
[21]). firmsthe Bogoliubovprediction(dashedoscillatingline);
4
1.2 2.0
3.0 a b a
>
2.4 0.9 ) 1.6
0
1/2j[Var ] 011...628 1/2j[Var ] 00..36 dN(t) N(00 01..82
d<
0.0 0.0 5 0.4
0 5 10 15 20 0 1 2 3 4 -0
1
t [2p mV2/3/h] t [2p mV2/3/h] 0.0
0.0 0.1 0.2 0.3 0.4 0.5
1.2
c 1.2 d t [2p mV2/3/h]
0.9 0.9 1.2 b
1/2j[Var ] 00..36 1/2j[Var ] 00..36 (0) >0 01..80
N
0.6
0.00.0 0.5 1.0 1.5 2.0 0.00.0 0.5 1.0 1.5 2.0 dN(t) 0 0.4
t [2p mV2/3/h] t [2p mV2/3/h]
d< 0.2
5
FIG. 2: (Color online) Standard deviation of the condensate -0
1 0.0
phase change ϕ(t) (12) as a function of time for (a) T =
0 5 10 15 20 25 30
0.08245T (lowercurve)andT =0.1711T (uppercurve),(b)
c c
T = 0.29467T , (c) T = 0.453T , (d) T = 0.6473T . Thick t [2p mV2/3/h]
c c c
solid line (black): numerical solution from the classical field
model Eq.(8) averaged over 500 realizations. Thin solid line FIG. 3: (Color online) Correlation function of the conden-
(red): a linear fit. The parameters N and ρg have the same sateatom number(11). (a) Short times, (b)long times. The
values as in Fig.1. classical field results are obtained from an average over 500
solutions of Eq.(8); they are represented by symbols in (a)
and a solid line in (b). The dashed lines (red) are the Bo-
goliubovapproximations(21)(oscillating line)and(22)(hor-
at long times, see Fig.3b, the correlation function drops
izontal line) in (a), and only (22) in (b). The dashed-dotted
to a value significantly smaller than the Bogoliubov pre-
line (purple) is the Gaussian model. For clarity in (b) we
diction (fast oscillations are not shown in the figure); a
washedoutfastoscillationsinthesimulationresultandinthe
key point is that this long time value of the correlation Gaussian model, by averaging over consecutive points over
function of the condensate atom number is not zero. a time width 0.45mV2/3/¯h. The horizontal dashed-dotted-
One may fear at this stage that the classical field dottedline(blue)in(a)and(b)istheergodiclongtimelimit
model is missing some source of damping in the dynam- prediction,describedinsectionIV. TheparametersN,T and
ρg havethe same values as in Fig.1.
ics of the system. However it is a well established fact
that the classical field model is able to simulate damp-
ing processes, including the finite temperature Beliaev-
Landau processes [22, 23, 24, 25], since the interaction
among the Bogoliubov modes is included in this model
[12,13,15,16,20,21,26,27,28,29]. Morequantitatively the phase ϕ˙ does not decorrelateat long times, although
wenowcheckthatthedampingtimesduetotheBeliaev- it is a function of the b ’s; similarly, the fluctuations of
k
Landau processes in the simulation are much shorter thenumberofcondensateatomsδN ,whicharefunctions
0
than the evolution times considered here. To this end, of the b ’s, do not decorrelate at long times.
k
weextractfromthe simulationsthe temporalcorrelation
functionshb∗k(t)bk(0)iandh|bk|2(t)|bk|2(0)i−h|bk|2i2,ob- This paradoxwill be explainedinSec. IV, andquanti-
tained by projecting the classical field over the corre- tative predictions for long times behavior of the conden-
sponding Bogoliubov mode and averagingover many re- sate atom number correlation function and of the vari-
alizations. We show these correlation functions for the anceofthecondensatephasechangewillbederived. An-
lowest energy Bogoliubov mode and for an excited Bo- ticipating these analytical results, we show in Fig.5a the
goliubov mode in Fig.4. longtimelimitof(Varϕ)1/2/tasafunctionofT/T ,from
c
Wecomethenintoaparadox. Ononeside,thevarious theresultsoftheclassicalfieldsimulations,butalsofrom
Bogoliubov oscillators b decorrelate at long times. On thepredictionsoftheBogoliubovapproximationEq.(38),
k
the other side, the variance of the phase change ϕ of andoftheergodictheoryofSec.IV. Infigure5bweshow
the condensate varies quadratically at long times, which thesameresultsandpredictionsfortheasymptoticvalue
implies, as we shall see in Sec. IV, that the derivative of of the condensate atom number correlationfunction.
5
500 1.2
2b|k a a
r 400 1.0
r
o
C
and | 230000 1/2]/t 00..68
2b| k100 jVar 0.4
rr | [ 0.2
o 0
C
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0
t [2p mV2/3/h]
T / T
8 c
2b|k b 5.0
orr 6 ) > b
C 0 4.0
(
d | 4 N0
2b| ank 2 d N(t) 0 23..00
r | d<
or 0 5
C -0 1.0
1
0.0 0.1 0.2 0.3 0.4 0.5
0.0
t [2p mV2/3/h] 0.0 0.2 0.4 0.6 0.8 1.0
T / T
FIG. 4: (Color online) Correlation function of bk 2 (black c
| |
solid line) and modulus squared of the correlation func-
FIG. 5: (Color online) (a) Long time limit of (Varϕ)1/2/t,
tion of bk (green dashed line), as obtained from 2000
realisations of the classical field simulations, (a) for the in units of ¯h/mV2/3, as a function of the temperature over
mode k = (0,0,2π/L ) and (b) for the mode k = theideal gas critical temperature. (b) Long timelimit of the
3
(2π/L ,10π/L ,4π/L ). We define the correlation function condensateatomnumbercorrelationfunction,asafunctionof
1 2 3
ofaquantityX asCorrX = X∗(t)X(0) X 2. Thepur- T/Tc. Insquaresymbolsthenumericalresultsoftheclassical
ple dashed-dotted line is an ehxponentialif−un|chtioin| of t, given field simulation (averaged over 500 realizations), in dashed
byEq.(24). It reproduceswell thesimulation results, and we line (red) the Bogoliubov prediction and in dashed-dotted-
have checked that the agreement is good for all vectors k ly- dotted (blue) theergodic prediction. The parameters N and
inginanarbitrarilychosenplaneink-space. Theparameters ρg havethe same values as in Fig.1.
N, T and ρg havethe same values as in Fig.1.
A. Correlation function of the condensate atom
number
IV. CLASSICAL FIELD: ANALYTICAL
RESULTS
Asthetotalnumberofparticlesisfixed,itisequivalent
to calculate the correlation function of δN in Eq.(11)
0
The general procedure used here to obtain analytical and of the number of non-condensed particles N⊥. In-
results is the following. First one expresses the quan- jecting the expansion Eq.(14) for the time dependent
tity of interest (the number of condensate atoms or the non-condensed field ψ⊥ over the Bogoliubov modes into
time derivative of the condensate phase) in terms on the Eq.(17) we obtain
amplitudes b of the field ψ over the Bogoliubov modes,
k
N (t)= U˜ b (t)+V˜ b∗ (t)2. (19)
⊥ | k k k −k |
k=6 0
e−ik·r X
b (t) = dV U˜ e−iθ(t)ψ (r,t)
k k ⊥
√V Bogoliubov theory: In Bogoliubov theory interaction
r
X
eik·r among the Bogoliubov modes is neglected so that at all
+V˜k √V eiθ(t)ψ⊥∗(r,t) (18) times
bk(t)=bk(0)e−iωkt with ωk =ǫ˜k/¯h. (20)
where ψ is the component of ψ orthogonal to the con-
⊥
densatemode. Secondoneevaluatesthecorrelationfunc- As Wick’s theorem applies for the initial thermal dis-
tions of products of b in various physical limits. tribution we obtain for the correlation function of the
k
6
condensate atom number: space [30]. In the Bogoliubov approximation for the en-
ergy,thismeansthattheb (t)’ssampletheunnormalized
k
hδN0(t)δN0(0)iBog = n˜2k |U˜k2eiωkt+V˜k2e−iωkt|2 probability distribution
kX=6 0 h
+(U˜kV˜k)2|eiωkt+e−iωkt|2 (21) P∞({bk})=δE− ˜ǫkb∗kbk (25)
i k=6 0
where n˜ = k T/ǫ˜ is the Bogoliubov mean occupa- X
k B k  
tion number of a mode for the classical field. At very where the Bogoliubov energy E is fixed by the initial
short times, a good agreementof the Bogoliubov predic- value of the field:
tion with the simulation is observedin Fig.3a. Smearing
outthetermsoscillatingrapidlyatBohrfrequencies2ωk, E = ǫ˜kb∗k(0)bk(0). (26)
we obtain a prediction directly comparable to the coarse
k=6 0
X
grained numerical result of Fig.3b:
First,foragiveninitialconditionofthefield,wecalcu-
hδN0(t)δN0(0)iBognonosc = (U˜k2+V˜k2)2n˜2k. (22) late the expectation value of N⊥(t) as given by Eq.(19)
over the ergodic distribution Eq.(25), which is equiva-
k=6 0
X
lent to the temporal average of N (t) over an infinite
⊥
This amounts to considering the correlation function of time interval. The terms of the form bb or b∗b∗ have
a zero mean, since the phases of the b ’s are uniformly
N⊥non osc(t)= (U˜k2+V˜k2)b∗k(t)bk(t), (23) distributed over 2π, according to Eq.(2k5). To calculate
k=6 0 theexpectationvalueoftheb∗bterms,itisconvenientto
X
introduce rescaled variables
deduced from (19) by eliminating the oscillating terms
such as bkb−k. As can be seen in Fig.3b, Bogoliubov ǫ˜ 1/2
k
theory fails at long times. Note that in the thermody- Bk = bk. (27)
E
namic limit, where the above sum is dominated by the (cid:18) (cid:19)
low k terms, one may approximate V˜ U˜ , so that
k ∼ − k According to Eq.(25) the real parts and the imaginary
Eq.(22) is roughly half of the t = 0 value of Eq.(21); in
parts of all the B are uniformly distributed over the
k
other words, it is approximately half of the variance of
unit hypersphere in a space of dimension 2 , where
the condensatenumber. Inthe numericalresultofFig.3, M
= V/dV 1 is the number of Bogoliubov modes so
the correlationfunction drops by much more than a fac- M −
that we obtain B 2 = 1/ where the overline stands
tor 2. | k| M
for the average over the ergodic distribution (25). As a
Gaussian theory: ApossibleapproachtoimproveBogoli-
consequence the ergodic average of N is
ubov theory consists in assuming that the b are Gaus- ⊥
k
sian variables with a finite time correlation due to the 1 E
Beliaev-Landau mechanism: N⊥ = (U˜k2+V˜k2)ǫ˜ . (28)
k
Mk=6 0
X
b∗(t)b (0) 2 =n˜2e−2γk|t| (24)
|h k k i| k Notethatthisergodicaveragedependsonthet=0value
whereγ iscalculatedwithtime dependentperturbation of the b ’s via (26).
k k
theoryincludingthediscretenatureofthespectrumasin Second, we average the product N N (0) over the
⊥ ⊥
[20]. This amounts to weighting eachterm of Eq.(22) by thermalcanonicaldistributionfortheinitialvaluesb (0).
k
exp( 2γ t). Thisassumptionissupportedbynumerical This gives the long time limit of the correlationfunction
k
− | |
evidence for a single mode, see Fig.4, and by an analytic of the number of condensate atoms:
derivation in the thermodynamic limit for one or two
2
modes, see Appendix A. Nevertheless, the resulting pre-
1
diction for the correlation function of N0, while looking hδN0(t→+∞)δN0(0)iergo =  (U˜k2+V˜k2)n˜k .
promising at short times, see Fig.3a, is in clear disagree- M k=6 0
X
mentwiththesimulationatlongtimes,seeFig.3b. Since  (29)
the assumptionofalongtime decorrelationofb∗(t)with Thispredictionisingoodagreementwiththesimulations
k
b (0) is physically reasonable, one may suspect that the at long times, see Fig.3b for a fixed value of the temper-
k
Gaussianhypothesisisnotaccuratewhenalargenumber ature, and Fig.5b as a function of temperature. Note
of modes are involved as for the correlation function of that, accordingto Schwartz inequality, the ergodic value
N . This is indeed the case, as we now show. is lower than the coarse grained Bogoliubov prediction
0
Ergodic theory: A systematic way to calculate the long Eq.(22), as was expected physically.
time limit of the correlation function is to assume that This clearly shows that the existence of infinite time
the non-linear dynamics generated by Eq.(8) is ergodic: correlationsin the number ofcondensate atoms is a con-
at long times, the b (t)’s for a given realization of the sequence of the conservation of energy during the free
k
field explore uniformly a fixed energy surface in phase evolution of the system.
7
To understand the failure of the Gaussian model, we whichis afunction ofthe b ’s only accordingto Eq.(14).
k
give the ergodic prediction of the long-time limit of the This leads to
correlation function of the Bogoliubov mode occupation
a˙ g
numbers n = b 2, i¯h 0 = ρg+ dV Λ(r)2+ Λ(r)2
k k
| | a V | |
0 r
X (cid:2) (cid:3)
n˜kn˜k′ g Λ∗(r)Λ2(r)
nk(t + )nk′(0) ergo nk nk′ = . (30) + dV . (35)
h → ∞ i −h ih i M √V r √N −N⊥
X
This long-time value is non-zero,contrarilyto the Gaus-
The real part of the above equation gives ¯hθ˙, which is
sian model prediction. One may argue that the value −
also ¯hϕ˙.
Eq.(30) tends to zero in the thermodynamic limit, so −
Restrictingtoaweaknon-condensedfraction,wedrop
that the error in the Gaussianmodel looks negligible for
the cubic terms in Eq.(35), to obtain [33]
a large system. However, in calculating the correlation
fbulencstuimonoovferatmheacBroogsocolipuibcoqvumanotditeys asupcpheaarss,Nso⊥t,haatdothue- ¯hϕ˙ ρg 1 g dV [Λ(r)+Λ∗(r)]2
≃ − − 2V
small deviations Eq.(30) from the Gaussian model pre- r
X
diction sum up to a macroscopic value. In other words, = ρg 1 g (U˜ +V˜ )2 b +b∗ 2. (36)
in the calculation of a given correlation function, one is − − 2V k k | k −k|
k=6 0
not allowed to take the thermodynamic limit before the X
end of the calculation. It turns out that the productsb b generateoscillating
k −k
terms which do not contribute to a coarse grained time
average. It is thus useful to define
B. Variance of the condensate phase change g
¯hϕ˙ = ρg (U˜ +V˜ )2 b 2. (37)
non osc k k k
− − V | |
k=6 0
To reproduce the approach of the previous subsection X
for the phase, one should express the phase change ϕ(t) Bogoliubov theory: Byusing(20)andWick’stheoremwe
of the condensate amplitude a0 as a function of the bk’s. calculate the correlation function of Eq.(36). By tempo-
It turns out that the quantity easily expressed in terms ral integration we obtain the variance of the condensate
of the bk’s is the time derivative ϕ˙. The variance of ϕ is phase change
then related to the correlationfunction of ϕ˙:
C g 2 sin2ω t
(Varϕ) = (U˜ +V˜ )4n˜2 t2+ k .
t t Bog ¯hV k k k (2ω )2
Varϕ= dτ dτ′ (τ τ′ ) (31) (cid:16) (cid:17) kX=6 0 (cid:20) k (cid:21)
Z0 Z0 C | − | (38)
Qualitatively Bogoliubov theory correctly predicts a
where time translational invariance in steady state im- quadratic growth of the variance of ϕ at long times. As
poses fora classicalfield that depends onlyon τ τ′ :
we show in Fig.5a, however, it is not fully quantitative:
C | − |
itdoesnotreproducethe value ofthe dephasingrateob-
(τ τ′ )= ϕ˙(τ)ϕ˙(τ′) ϕ˙(τ) ϕ˙(τ′) . (32) tained from the simulations. This is not surprising as in
C | − | h i−h ih i
the full non linear theory the b ’s interact and do not
k
If (τ) 0 fast enough when τ then Varϕ grows follow Eq.(20).
C → → ∞
linearly in time. On the other hand, if (τ) has a non- To be complete, we also give the Bogoliubov approxi-
C
zero limit at long times, then Varϕ grows quadratically mationforthecorrelationfunctionofthecondensateam-
in time [31]. plitudea . Neglectingthe fluctuationsofthe modulusof
0
Toexpressϕ˙ intermsofthebk’s,wewritetheequation a0, one can set
of motion for a :
0
a∗(t)a (0) N e−iϕ(t) . (39)
h 0 0 i≃h 0ih i
i¯ha˙0 = i¯hg{a0,H}=∂a∗0H Dropping the oscillating terms in bkb−k and b∗kb∗−k in
= ψ∗(r)ψ2(r) (33) ϕ˙(t), which give a small contribution, we get
√V
r
X 1
a∗(t)aˆ (0) N .
where we used ∂a∗0ψ∗(r) = 1/√V obtained from Eq.(2). h 0 0 iBog ≃h 0ikY=6 01+ih¯gV(U˜k+V˜k)2n˜kt
We split ψ as in Eq.(16); we eliminate the condensate (40)
amplitude in the resulting expression for a˙ /a (i) by The resulting expression is plotted as a dashed line in
0 0
using a 2 = N N , where N is a function of the Fig.1 against the result of the simulation.
0 ⊥ ⊥
| | −
bk’s, see Eq.(19), and (ii) by introducing the field [32] Gaussian theory: If we add by hand a decorrelation of
the b ’s and assume Gaussian statistics, we get a diffu-
k
Λ(r)=e−iθψ (r) (34) sive spreading of the condensate phase change, with the
⊥
8
variance of ϕ growing linearly at long times: cut-off,thatisonthenumberofBogoliubovmodes in
non osc
M
thesimulation,asisapparentonEqs.(29,42). Inthissec-
(Varϕ) = g 2 (U˜ +V˜ )4n˜2 e−2γkt−1 + t ,tion,wethereforeadaptthepreviousphysicalreasonings
Gauss ¯hV k k k 2γ2 γ to the quantum field case.
(cid:16) (cid:17) kX=6 0 (cid:20) k k(cid:21)
(41)
in clear contradiction with the numerical simulations.
A. The quantum model
This prediction corresponds to a correlation function
C
vanishing at long times, whereas the correct correlation
function has a finite limit, see Fig.6. We use a straightforward generalization of the classi-
Ergodic theory: as in subsection IVA we calculate the cal field lattice model, taking here for simplicity a cubic
longtimevalueofthecorrelationfunctionforϕ˙ usingthe lattice, as discussed in [20, 34, 35]. The bosonic field ψˆ
ergodic assumption. The varioussteps of the calculation evolves according to the Hamiltonian
are rigorously the same as in Sec. IVA and lead to
¯h2k2 g
2 H = 2m aˆ†kaˆk+ 20 dV ψˆ†ψˆ†ψˆψˆ, (43)
(τ + ) = g 2 1 (U˜ +V˜ )2n˜ . Xk Xr
ergo k k k
C → ∞ ¯hV  
(cid:16) (cid:17) M kX=6 0 whereaˆkannihilatesaparticleofwavevectorkinthefirst
 (42) Brillouinzone. Thedispersionrelationofthewaveisnow
Thispredictionisinexcellentagreementwiththesimula- the usual one. The total number of atoms is fixed, equal
tions: itgivesthecorrectasymptoticvalueof ,seeFig.6, to N. The coupling constant g depends on the lattice
and from the asymptotic expression Varϕ C (+ )t2 spacing l in order to ensure a 0l-independent scattering
≃ C ∞
it gives the correct values of the long time limit of length for the discrete delta interaction potential among
(Varϕ)1/2/t, see Fig.5a, as a function of temperature. the particles [34, 35]:
g
0.025 V(r1−r2)= dV0 δr1,r2. (44)
0.020 Since weconsiderhere the weaklyinteractingregime,we
can restrict to a lattice spacing much larger than the
non osc 0.015 sgc=att4eπr¯hin2ga/lemn.gth a so that g0 is actually very close to
) 0.010
t( To be able to use Bogoliubov theory as we did in the
C
classical field reasoning, we restrict to the low temper-
0.005
ature regime T T with a macroscopic occupation of
c
≪
the condensate mode. We thus neglect the possibility
0.000
that the condensate is empty, which allows us to use the
0 5 10 15 20 25 30
modulus-phase representation of the condensate mode:
t [2p mV2/3/h]
FIG. 6: (Color online) Correlation function Cnon osc of the aˆ0 ≃eiθˆ Nˆ0 (45)
quantityϕ˙non osc defined in Eq.(37) calculated from thesim- q
ulation (square symbols; the solid line is a guide to the eye), where Nˆ = aˆ†aˆ and where θˆ is a Hermitian ‘phase’
orusingtheGaussiantheory(purpledashed-dottedlinegoing 0 0 0
operator obeying the commutation relation
to zero at long times). In dashed line (red) the Bogoliubov
prediction. Indashed-dotted-dottedline(blue)thelongtime
[Nˆ ,θˆ]=i. (46)
prediction (42) of the ergodic theory. The parameters T, N 0
and ρg havethesame valuesas in Fig.1.
This allowsto consider the correlationof the condensate
atom number fluctuation δNˆ Nˆ Nˆ but also the
0 0 0
≡ −h i
variance of the condensate phase change ϕˆ(t) θˆ(t)
≡ −
V. QUANTUM TREATMENT: ANALYTICAL θˆ(0), as we did for the classical field.
RESULTS
So far the classical field model was very useful in re- B. Correlation function of the condensate atom
vealingthephysicalprocessesgoverningthelongtimebe- number
haviorofthe phaseandatomnumber fluctuations in the
condensate. Howeveritisnotafullyquantitativetheory, To predict the correlationfunction of δNˆ , we use Bo-
0
as the long time limits of the correlation functions con- goliubov theory at short times and the quantum analog
sidered here depend on the precise choice of the energy of the ergodic theory at long times.
9
In the number conserving Bogoliubov theory [32, 36], quantity, equal to the real part of the non-symmetrized
written here for a spatially homogeneoussystem, one in- correlation function. The time coarse grained version
troducesthefieldconservingthetotalnumberofparticles of the prediction (53) is obtained by averaging out the
oscillating terms, which amounts to considering the cor-
Λˆ(r) e−iθˆψˆ⊥(r) (47) relation function of the temporally smoothed operator
≡
number of non-condensed particles
wherethenon-condensedfieldψˆ isobtainedbyproject-
⊥
ing out the component of the field ψˆ on the condensate Nˆnon osc U2+V2 ˆb†ˆb +V2 . (55)
mode. The field Λˆ then admits the modal expansion on ⊥ ≡ k k k k k
the Bogoliubov modes kX=6 0h(cid:0) (cid:1) i
eik·r e−ik·r Quantum Ergodic theory: Discarding from the start the
Λˆ(r)= ˆbkUk√V +ˆb†kVk √V (48) oscillating terms in Nˆ⊥, as in (55), we face here the
k=6 0 problemof calculating the long time limit of A(t)A(0) ,
X h i
where A is a linear function of the Bogoliubov mode oc-
where the real amplitudes U , V , normalized as U2
k k k − cupation numbers,
V2 =1, are given by the usual Bogoliubov theory,
k
¯h2k2/2m 1/4 A= γkˆb†kˆbk. (56)
Uk+Vk = 2ρg +h¯2k2/2m . (49) kX=6 0
(cid:18) 0 (cid:19)
As the quantum state of the system is given by the Bo-
Since the total number of particles is fixed to N, it is
goliubovapproximationEq.(51), wemay inject aclosure
equivalent to consider the fluctuations of Nˆ or of the
0 relation in the Bogoliubov Fock eigenbasis:
number of non-condensed atoms
Nˆ⊥ =Xr dVΛˆ†(r)Λˆ(r). (50) hA(t)A(0)i = ZB1og e−β k6=0ǫknk γknk
This, together with the expansion (48), expresses Nˆ⊥ as n{XnkA}(t) Pn , kX=6 0 (57)
a function of theˆb ’s. ×h{ k}| |{ k}i
k
Theequilibriumstateofthesystemisapproximatedin where the sum is taken over all possible integer values
the canonical ensemble by the Bogoliubov thermal den- of the occupation numbers, not to be confused with the
sity operator mean occupation numbers (54).
ρˆBog(T)= 1 e− k6=0ǫkˆb†kˆbk/kBT (51) eleTmheentnoonf-Aex(tp)l,icwithpicihecme aoyf tbheisreeixnptreerspsrieotnedisatshefomlloawtrsi:x
Z
Bog
P
where the normalization factor ZBog is the Bogoliubov h{nk}|A(t)|{nk}i=Tr [Aσ(t)] (58)
approximation for the partition function, and where we
have introduced the Bogoliubov spectrum wherethedensityoperatorσ,initiallyapurestateinthe
Bogoliubov Fock basis,
¯h2k2 ¯h2k2 1/2
ǫ = +2ρg . (52)
k 0 σ(0)= n n (59)
2m 2m k k
(cid:20) (cid:18) (cid:19)(cid:21) |{ }ih{ }|
Bogoliubov theory: In the Bogoliubov approximation for evolves during t with the full Hamiltonian H. We know
the time evolution,theˆb merelyaccumulateaphase,at that this evolution involves Beliaev-Landau processes
k
the frequency ω = ǫ /¯h, similarly to the classical field that will spread σ over the various Fock states n′ .
k k |{ k}i
case. From Wick’s theorem one then obtains This evolution is complex. But we need here the long
time limitonly,inwhichwemayassumethatanequilib-
1
δNˆ (t),δNˆ (0) = n¯ (n¯ +1)(U2+V2)2 rium statistical description is possible. Since the system
2h{ 0 0 }iBog k k k k
k=6 0 is isolated during its evolution, we take for σ(t + )
+2U2V2Xcos(2ω t)[n¯2 +(n¯ +1)2(]53) the equilibrium density operator in the microca→noni∞cal
k k k k k
ensemble [37], and we calculate the expectation value of
where Awithσ(t + )aswedidfortheclassicalfieldmodel.
→ ∞
Thecalculationcanbedoneinthethermodynamiclimit.
1
n¯k(T)= (54) As shown in the Appendix B, one can calculate to lead-
exp(ǫ /k T) 1
k B − ing order in this limit the difference between canonical
is the mean occupation number of the Bogoliubov mode and microcanonical averages.
k. Note that we have considered here the so-called sym- Here the microcanonical ensemble has an energy E =
metric correlationfunction (as X,Y standsfor the an- EBog+ ǫ n , where EBog is the ground state Bo-
{ } 0 k=6 0 k k 0
ticommutatorXY+YX oftwooperators)whichisareal goliubov energy. We introduce the effective temperature
P
10
T suchthatthemeanenergyinthecanonicalensemble C. Correlation function of the time derivative of
eff
at temperature T is equal to E, the condensate phase
eff
0= H (T ) E = ǫ [n¯ (T ) n ] (60)
h Bogi eff − k k eff − k Asintheclassicalfieldcase,wefirstlookforanexpres-
kX=6 0 sion of the first order time derivative of the condensate
phase operator θˆin terms of the amplitudes of the field
where ... standsforanaverageinthecanonicalensem-
h i Λˆ on the Bogoliubov modes. Taking as a starting point
ble and H is the Bogoliubov Hamiltonian. Using the
Bog
in Heisenberg picture
results of Appendix B one gets
A¯(E) A (T )= 1k T2 hAi′ ′(T ) (61) i¯hdθˆ=[θˆ,H], (65)
−h i eff −2 B eff H ′ eff dt
(cid:18)h Bogi (cid:19)
whereA¯(E)isthemicrocanonicalaverageofAatenergy we split the quantum field in a condensate part and a
E and where the apex ′ stands for derivation with re- non-condensed part,
spect to temperature. We further use the fact that, in
aˆ
the thermodynamic limit, for typical values of the occu- ψˆ(r)= 0 +ψˆ (r), (66)
⊥
pation numbers n , T weakly deviates from the physi- √V
k eff
cal temperature T. We calculate T by expanding (60)
eff andweinsertthissplittingintheexpressionofH. Using
up to secondorder in T T [38]. Evaluating (61) with
this value of T , keepineffg−terms up to the relevant order the modulus-phase representationof aˆ0 and the commu-
eff tationrelationEq.(46),weobtain,usingaˆ†aˆ =Nˆ Nˆ ,
[38], gives the desired result 0 0 − ⊥
h{nk}|A(t→+∞)|{nk}i=hAi ¯h dθˆ = g0 Nˆ 1 + dVΛˆ†Λˆ
A ′ 1 A ′ ′ − dt V " − 2 r #
+ kX=6 0ǫk(nk−n¯k)hHhBoigi′ + 2(cid:18)hHhBoigi′(cid:19) + g0 dV ΛˆXNˆ0+1/2 Λˆ +h.c.
× hPk=6 0hǫHk(Bnokgi−′n¯k)i2 −kBT2 (62) + 2Vg0Xr dV qN1ˆ0(NΛˆˆ0†Λˆ+21+)h.c. . (67)
2√V " Nˆ #
r 0
whereallthecanonicalaveragesarenowevaluatedatthe X
p
physical temperature T [39].
The quantity (Nˆ + 1/2)/ Nˆ (Nˆ +1) is actually
It remains to inject this expression into Eq.(57). The 0 0 0
resulting average over nk leads to the long time value of 1+O(1/Nˆ02) so it can to a hqigh accuracy be replaced
the correlation function: by unity. Furthermore, as we did in the classical field
model, we now keep the leading terms in Λˆ, under the
A ′ 2
A(+ )A(0) A 2 = h i VarH (63) assumption of a weak non-condensed fraction. We can
h ∞ i−h i (cid:18)hHBogi′(cid:19) Bog also replace θˆby ϕˆ under the temporal derivative, since
2 θˆ(0) is time independent. We obtain [33]
γ ǫ n¯ (n¯ +1)
k=6 0 k k k k
= (6,4)
hP k=6 0ǫ2kn¯k(n¯k+1)i ¯hdϕˆ g0 Nˆ 1 + dV Λˆ†Λˆ + 1Λˆ2+ 1Λˆ†2 .
whereweusedWicktheoremanPdthepropertydn¯k/dT = − dt ≃ V " − 2 Xr (cid:18) 2 2 (cid:19)#
ǫ n¯ (n¯ +1)/k T2 [40]. Using Schwartz inequality, one (68)
k k k B
canshowthatthislongtimevalueofthecorrelationfunc- Taking the expectation value of this expression over
tion is less than its zero time value γ2n¯ (n¯ +1). To the thermalstateinthe Bogoliubovapproximationleads
k k k
be complete, we present an alternative derivation of our toanexpressioncoincidingwiththevalueofthechemical
prediction (64) in the Appendix C, Pbased on results ob- potentialpredictedbyEq.(103)of[35],whichincludesin
tained in [37]. We also note that the quantum ergodic a systematicwaythe firstcorrectionto the pure conden-
calculationdirectly leadstoapredictionofthelongtime sate prediction ρg0 [41]:
limitforthecorrelationfunctionoftheBogoliubovmode
occupation numbers, see Eq.(A17). g 1
Replacing in Eq.(64) the coefficients γk by their ex- µ(T)= V0 N − 2 + (Uk+Vk)2n¯k+Vk(Uk+Vk).
pressionfromEq.(55), γk =Uk2+Vk2, weobtainthe long kX=6 0
time value of the condensate atom number correlation  (69)
function in the quantum ergodic theory. Note that, in At this order of the expansion, this analytically shows
the thermodynamic limit, this long time value scales as that ¯h dθˆ/dt is the chemical potential of the system.
the volume V, whereas the t=0 value scales as V4/3. We n−owhturn tio various predictions for the symmetrized