Table Of ContentTQO-ITP-TUD/01-2011
Canonical ensemble of an interacting Bose gas:
stochastic matter fields and their coherence
Sigmund Heller and Walter T. Strunz
Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, D-01062 Dresden, Germany
(Dated: January 14, 2011)
Wepresentanovelquantumstochasticevolutionequationforamatterfielddescribingthecanoni-
calstateofaweaklyinteractingultracoldBosegas. Intheidealgaslimitourapproachisexact. This
1 numerically very stable equation suppresses high-energy fluctuations exponentially, which enables
1 us to describe condensed and thermal atoms within the same formalism. We present applications
0
togroundstateoccupation andfluctuations,densityprofileofgroundstateandthermalcloud,and
2
groundstatenumberstatistics. Ourmainaimarespatialcoherencepropertieswhichweinvestigate
n throughthedetermination of interferencecontrast and spatial densitycorrelations. Parameters are
a taken from actual experiments[1].
J
3 PACSnumbers: 05.30.Jp,67.85.-d,02.50.Ey
1
] Equilibrium fluctuations in ultracold gases reveal de- totheevolutionofthefieldWignerdistributionisworked
s
tailedinformationaboutstatesandphasesofinteracting out in an approach by Stoof and co-workers [18]. Care
a
g many-body quantum systems [2]. Recent experiments has to be taken with respect to the white noise driving
- permit to control ultracold quantum gases in a hitherto these equations. Exact methods based on the positive
t
n unknownprecisionandtoinvestigatetemperaturedepen- P-representationareused by Drummondand co-workers
a dentquantitieslikethethermaldensityandgroundstate [19]. It is possible to use this approach for 3D systems;
u occupancy [3], spatial correlation functions [4–7] density still,thelong-timenumericalsolutionhastobeexercised
q
fluctuations [8] or interference contrast [1]. with caution. We see the strength of our approachin its
.
t Inthisworkwedeterminethe canonicalstateofanin- unified applicability to a vast number of different phe-
a
m teracting ultracold Bose gas. A novel quantum stochas- nomena: from ground state fluctuations to properties of
tic evolution equation for a c-number field ψ(x) is pre- the thermal cloud, to coherence properties and contrast
-
d sented such that canonical quantum statistical expecta- in Bose gas interferometry. For the latter we obtain nice
n tion values can be replaced by an ensemble mean over agreementwith experiments of the Schmiedmayer group
o these stochastic matter fields. The equation allows to thatmaybewelldescribedbyLuttingerliquidtheory[1]
c determine coherence properties and other relevant ob- or by a stochastic phase model [20].
[
servables;itisbasedonamean-fieldtypeapproximation Twopropertiesofournovelequationshouldbeempha-
1 and strictly valid for the non-interacting case. sized: first,unlikeinourpreviousattempt[21],theequa-
v Most theoretical descriptions of interacting ultracold tion is not norm preserving. Still, the norm fluctuations
7
Bose gases at finite temperature are based on grand are small compared to those of related stochastic equa-
1
canonicalstatistics [2,9]. For actualexperiments involv- tions used for grand canonical simulations. Secondly, as
6
2 ing a fixed and finite number of particles, however, a in [21], ultraviolet cutoff problems do not appear due to
. canonical description is natural. In studying the role of theuseoftheGlauber-SudarshanP-function: effectively,
1
the chosen ensemble, attention so far has been paid to ourtreatmentleadstospatiallycorrelatednoise,unphys-
0
1 ground state number fluctuations [10, 11]. While for the icallylargemomentumkicksaresuppressed. Theseprop-
1 ideal gas canonical and grand canonical ensemble give erties afford a very stable numerical solution of full 3D
: vastlydifferentpredictions[12],this ceasesto be truefor problems, using arbitrary trap potentials. Due to lack
v
interactinggasesinthe thermodynamicallimit[13]. Our of space and its current interest, we here concentrate on
i
X work is basedon canonicalstatistics rightfrom the start 1D gas interference. Still, we emphasize that we are also
r andallows us to notonly investigate occupationfluctua- able to treat full 3D gases within our approach[22].
a tions but also spatial coherence properties. We propose to use the stochastic (Ito) equation
Manystochasticfieldmethodsexistforthedescription
1 N
ofBosegasesatfinitetemperature. Allofthesearebased dψ = (Λ+i)H Λ He−H/kT ψ dt
on grand canonical statistics, and nicely overviewed and | i −~(cid:18) − ψ ψ (cid:19)| i
h | i
compared in [9, 14]. In the truncated Wigner approach, 2Λ
theevolutionofthefieldWignerfunctionalisdetermined +r ~ He−H/kT|dξi (1)
p
approximatelyfromasamplingoverrandominitialfields
whosedynamicsisgivenbytheGross-Pitaevskiiequation for a c-number matter field ψ(~x,t) = ~xψ(t) to deter-
h | i
[15,16]. Basedonaquantumkinetictheoryandasepara- mine all equilibrium properties of a weakly interacting
tionofcondensedandnon-condensedpart,Gardinerand Bose gas of N particles at arbitrary temperature T (k is
co-workers derive a stochastic Gross-Pitaevskii equation Boltzmann’s constant). Throughout, we will refer to (1)
[17]. Withasimilarresult,afunctionalintegralapproach as the stochastic matter field equation (SMFE) for finite
2
temperature. Crucially, the operator H is the effective (1) with (2) is just mean field Gross-Pitaevskii theory
(mean-field) one-particle energy operator and is again expected to give good results. Clearly,
the fluctuations we describe are of thermal origin; quan-
~p2 ψ(~x,t)2 tum fluctuations are taken into account to some extent
H = +V(~x)+g(N 1) | | (2)
2m − ψ(t)ψ(t) through the use of the P-representation. Note also that
h | i
it is of crucial importance to use the current, stochas-
suchthatequ. (1)mayalsobeseenasastochasticGross- tic ψ(x,t) in (2), such that on average H =
(N−1)
Pitaevskiiequation. As usual,V(~x)denotesthe trappo- h i
~p2/2m+V(~x)+g ψˆ†(x)ψ(ˆx) . Thequalityofchoice
tential,theinteractionparametergisproportionaltothe h i(N−1)
(2) was tested by solving equation (1) for a two mode
s-wavescatteringlengtha andmisthemassofaBoson.
s system and comparing with numerically exact quantum
The parameterΛappearingin(1) is aphenomenological
results over a wide temperature range.
damping rate that sets the time scale for transition to
equilibrium. Its appearanceassquarerootwiththe fluc-
tuations reflects a fluctuation-dissipation-relation. The 200 20
fluctuating partisdrivenby complexItoincrements dξ
with dξ dξ = 11dt, dξ∗ dξ = 0. Note, however t|hati 150 15
the o|periahtor| √He−H|/kTihacts| on the noise, effectively hni0100 n∆010
leading to spatially correlated noise [21]. 50 5
Before we show the versatility and accuracy of the
0 0
SMFE in applications later, let us sketch how we ar- 0 0.25 T0./5Tc 0.75 1 0 0.25 T0./5Tc 0.75 1
rive at (1). Our aim is to determine mean values 2e+05
1000
... =tr[...ρ ] with the canonical density operator
h iN N ρˆN = 1 e−Hˆ/kTΠˆN (3) 3−hnii)0 5000 4−hnii)01,51ee++0055
ZN hn(0-500 hn(0 50000
in second quantization with Hamiltonian Hˆ, canon- 0
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
ical partition function Z , and projector Πˆ = T/Tc T/Tc
N N
n n onto the N-particle subspace.
k k
|{ }ih{ }|
PnPk=N FIG. 1: Ground state occupation (top, left), its variance
Normally-ordered matter field correlation functions are (top,right) ∆n0 = h(n0−hn0i)2i, third(bottom, left) and
expressed in terms of functional phase space integrals p
fourth(bottom, right)centralmomentsasafunction of tem-
[21], for instance perature(scaled withthecritical temperatureofthethermo-
dynamic limit T = ~ωN1/3/kζ(3)1/3) for an ideal 3D Bose
1 c
ψˆ†(~x)ψˆ(~x′) = [ψ]ψ∗(~x)ψ(~x′)W (ψ), gas of 200 particles in a harmonic trap. The data obtained
N N−1
h i CN Z D withtheSMFE(blackplussigns)iscomparedwiththeresults
(4) of the recursion relation (red solid line).
with the weight functionals W (ψ) =
N
1 ψ ψ N e−hψ|ψiP(ψ), where P denotes the
N!h | i We convince ourselves of the validity of (1) by first
Glauber-Sudarshan P-function [23] of state e−Hˆ/kT considering an ideal Bose gas of 200 particles in a 3D
Z
and C = [ψ] W (ψ). Note that second (or higher) harmonic trap. In Fig. 1 results for ground state oc-
N N
order correRlaDtions require the use of WN−2 (or lower cupation, its variance and further centered moments are
index) in expression (4), while C remains. compared with exact results for the canonical ensemble
N
Fortheidealgascase(g =0),weprovethattheSMFE obtained from a recursion relation [25].
(1) corresponds to a Fokker-Planck equation [24] whose As a first application to the interacting case in Fig.
stationary solution is just the weight functional W (ψ). 2, the density profile (green solid line) of a 87Rb quasi-
N
Thus,equilibriumexpectationvaluesofthecanonicalen- 1D gas of 20240 atoms in a trap with frequencies ωz =
semble are obtained from propagating equ. (1) and av- 2π 9Hz and ω⊥ = 2π 32Hz at a temperature of
× ×
eraging. In practice, we use a long-time-average over a 185nK is shown (we use 1D coupling constant g1D =
single trajectory ψ(x,t). 2~ω⊥as). The blue dashed line is the contribution with
The SMFE (1) is exact for an ideal gas; interactions off-diagonallongrangeorder(ODLRO)whosewavefunc-
can be included with great success: we propose to use tion ψ0(z) is obtained from a diagonalization of the full
the single equation (1) with (2), containing the current ρ(z,z′) = ψˆ†(z)ψˆ(z′) applying the Penrose-Onsager
N
h i
stochastic mean field energy, to describe all properties criterion [26]. Moreover, a histogram of the stochastic
of weakly interacting Bose gases in a unified way. In- occupation n = ψ ρψ leads to the ground state
0 0 0
h | | i
deed, we emphasize that (1) interpolates smoothly be- number statistics P(n ) (inset in Fig. 2). The average
0
tween the high-temperature limit T T , when inter- number of particles in state ψ turns out to be 12573.
c 0
≫
actions are negligible and thus our description is exact Our findings are nicely compatible with the “stochastic
anyway. At the opposite end, when T T , the SMFE Gross-Pitaevskii” results of [14], without, however over-
c
≪
3
3000 L=10µm L=24µm L=37µm L=51µm
00.0.00000235 densityprofile 3 31nK 31nK 31nK 31nK
groundstate 2 2 2
)0.0002 2.5
2500 (n00.00015 thermalpart ) 2 1.5 1.5 1.5
P0.0001 α
(1.5
5e-05 W 1 1 1
2000 0 1
lz× 0 50001n000001500020000 0.5 0.5 0.5 0.5
sity1500 00 0.5 1 1.5 2 2.500 0.5 1 1.5 2 2.500 0.5 1 1.5 2 2.500 0.5 1 1.5 2 2.5
en 2 60nK 1 60nK 1 60nK 1 60nK
d
1000 1.5 0.8 0.8 0.8
α) 0.6 0.6 0.6
( 1
500 W 0.4 0.4 0.4
0.5
0.2 0.2 0.2
0-30 -20 -10 z/0lz 10 20 30 00 0.5 1α1.5 2 2.500 0.5 1α1.5 2 2.5000.511α.522.53000.511α.522.53
FIG.2: Densityprofile(greensolid line)ofan1Dinteracting FIG.4: Distributionfunctionsoftheinterferencecontrastfor
Bosegasof87Rbatomswithω =2π×9Hz,ω =2π×36Hz different lengths L and different temperatures. The length-
z ⊥
at a temperature of 185nK simulated with our SMFE. The dependentnormalizedinterferencecontrastα= |A|2 issam-
groundstatecontribution(bluedashedline) andthethermal h|A|2i
pled over 10000 realizations of the SMFE. The calculation
part (red dashed-dotted line) are obtained with the Penrose-
is done for different temperatures and different integration
Oρ(nzs,azg′)e.r cIrnitethrieonup[2p6e]rblyeftcaclocurnlaetrintghethgerofuunllddestnastiteynmumatbreixr lengths L (n1D =59µm−1, ω⊥ =2π×3.0kHz).
statistics P(n0) from oursimulation is shown.
independent condensates are prepared in quasi-1D; af-
2e+06
ter expansion they interfere; the observed interference
SMFE 33nK pattern is integrated over a length L which determines
LLT 33nK
L/2
1.5e+06 LSMLTFE474n7KnK the contrast |A(L)|2, where A(L) = dzψˆ1†(z)ψˆ2(z).
SMFE 68nK −LR/2
Both,meanvalue A(L)2 (Fig. 3)andthefulldistribu-
LLT 68nK
2i tion W(α) of thehn|orma|liized moments defined through
Ah||1e+06 ∞W(α)αmdα = αm = h|A|2mi are determined (Fig.
h i h|A|2im
R0
4). In [1] it is shown that experimental results are well
5e+05
described by Luttinger-liquid theory (LLT) to which we
will compare our SMFE results.
In Fig. 3 we show the average contrast A(L)2 as a
00 1e-05 2e-05 3e-05 4e-05 5e-05 function of length L for different temperathu|res a|ndi find
L(m)
very good agreement with LLT (and thus with experi-
ment). Deviations for large L arise from density varia-
FIG.3: Lengthdependenceoftheaveragecontrast h|A(L)|2i tions along the gas: LLT results are based on a uniform
of an interference pattern oftwo uncoupled1D Bose gases in density. The gas contains some 4400 87Rb atoms with
a harmonic trap for temperatures of 33nK, 47nK and 68nK
(n1D ≈ 50µm−1, ω⊥ = 2π ×3.0kHz). The data from the astrceenngttrhalfodrentshiitsy1oDf nca1sDe i≈s a5g0aµinmg−1. =Th2e~ωintaera,cwtiiothn
SMFE (black plus signs, green crosses, brown stars) is com- 1D ⊥ s
ω = 2π 3.0kHz. In Fig. 4 we show the distribution
pared to Luttinger-liquid theory (LLT) (red solid line, blue ⊥
×
dashedline,yellowdashed-dottedline)whichagreeswellwith functionoftheinterferencecontrastW(α)asahistogram
the experimental measurements [1]. Small deviations arise with α(L) = A(L)2/ A(L)2 obtained from our sim-
from the variation of the density in theharmonic trap,while ulations. We|use a| cehn|tral d|einsity n = 59µm−1 in
1D
theLuttinger-liquid theory applies to a uniform density. line with the experimental setup (see [1]). LLT predicts
a change of the shape of the distribution function with
decreasing parameter F = ~2πn1D which is excellently
mkTL
estimating lowly occupied regions. reproduced by our simulations.
The SMFE (1) is ideally suited to study coherence
Finally, as in [4, 27], we investigate two-point den-
properties of interacting matter waves through the de-
sity correlation functions after expansion g (x;t) =
2
termination of spatial correlation functions. As an ap-
hψˆ†(x;t)ψˆ†(0;t)ψˆ(0;t)ψˆ(x;t)i . We chose t=27ms; note that
plication we show results of our SMFE applied to re- hψˆ†(x,t)ψˆ(x;t)ihψˆ†(0;t)ψˆ(0;t)i
cent experiments in the Schmiedmayer group [1]: two free (non-interacting) expansion can be assumed. A gas
4
8 used. As in [27], we Fourier transform to obtain den-
∞
7 SMFE 12nK sity ripples ρ(q)2 :=n2 exp(iqx)[g (x;t) 1]. In
SMFE 27nK h| |i 1D 2 −
)h6 [S2M7]F1E2n4K0nK Fig. 5ourresultsarecompa−reR∞dtocalculationsfrom[27].
2nξD15 [[2277]] 2470nnKK Again, we see very good agreement.
( Letussummarizeourachievement: Wepresentanovel
/
2i4 quantum stochastic matter field equation (SMFE) for a
t)| gas of N particles trapped in an arbitrary potential at
q,(3 any temperature T (canonical ensemble). The equation
ρ
h|2 is strictly valid for a non-interacting gas; we include in-
teractions in a stochastic mean-field sense and obtain
1
promising results over the entire relevant temperature
regime. The SMFE is capable of tackling problems in
0
0 0.1 0.2 0.3 0.4 1D to 3D with arbitrarytrapping potentials. Results for
qξh
groundstateoccupationdistributionanddensityprofiles
are shown. Of particular interest is the determination of
FIG. 5: Normalized spectrum of density ripples
spatialcorrelationfunctionsofarbitraryorder. Weapply
hw|ρit(hq)|a2ic/e(nnt21rDalξhd)enfosritya owfeank1lDy i≈nte4r0aµctmin−g1,Bionseagatrsap(87wRitbh) otiuornsapapnrdoadcehnstiotycarlicpupllaetseaisntreercfeenretlnycemceoansturraesdt.dOisturribrue--
transversal frequency w = 2π×2kHz; the expansion time
⊥
sults are in goodagreementwith Luttinger liquid theory
ist=27ms, healinglength isξ = ~ . Thesimulations of
h qgm whichhasproventodescribetheexperimentsadequately.
the SMFE are done for 12nK (black plus signs), 27nK(red
We are grateful for inspiring discussions with Anto-
crosses) and 40nK (blue stars) and compared to theory of
[27](12nK bluesolidline,27nK blackdashedlineand40nK nio Negretti, Carsten Henkel, J¨org Schmiedmayer, Igor
red dashed dotted line). Mazets, and Patrick Navez. S. H. acknowledges support
by the International Max Planck Research School, Dres-
den. Computing resources have been provided by the
of about 10000 atoms in a harmonic trap with ω = Zentrum fu¨r Informationsdienste und Hochleistungsrech-
⊥
2π 2.0kHz and a central density of n 40µm−1 is nen (ZIH) at the TU Dresden.
1D
× ≈
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