Table Of ContentVisibility of a Bose-condensed gas released from an optical lattice at finite
temperatures
Fabrice Gerbier
Laboratoire Kastler Brossel, D´epartement de Physique de l’ENS, 24, rue Lhomond, 75 005 Paris, France
7
0 Simon F¨olling, Artur Widera, and Immanuel Bloch
0
Institut fu¨r Physik, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany.
2
(Dated: 6 f´evrier 2008)
n
In response to a recent manuscript [1] on the analysis of interference patterns produced by ul-
a
J tracold atoms released from an optical lattice, wepoint out that in thepresence of aBose-Einstein
condensatetheinterferencepatterncanbestronglymodifiedbyinteractioneffectsandthepresence
7
of a harmonic trap superimposed on the lattice potential. Our results show that the visibility of
1
theinterferencepatternissignificant onlyifasizeable condensatefraction ispresentinthetrap,in
] strong contrast to thefindingsof ref. [1].
h
c PACSnumbers:03.75.Lm,03.75.Hh,03.75.Gg
e
m
- Introduction- Experiments with ultracold quantum – (ii) The mean-field broadening of the momentum
t
a gases in optical lattices rely heavily on time-of-flight peaks during time-of-flight expansion,
t expansion to probe the spatial coherence properties of – (iii) The finite time-of-flight in the experiments.
s
. the trapped gas [2, 3, 4, 5]. When the phase coherence Practically, these effects tend to broaden the peaks
t
a length is largecomparedto the lattice spacing,the post- presentinthe interferencepattern,hence diminishing its
m
expansion density distribution reveals a striking interfe- peakvalue.Inthepresenceofasignificantnon-condensed
- rence pattern with the same symmetry as the reciprocal fraction, this will also lower the visibility. In cases when
d
lattice. As the phase coherence length decreases, e.g. on the initial momentum width is very large, for example
n
o approaching the Mott insulator transition, the visibility for a cloud close to or in the Mott insulator regime, or a
c ofthisinterferencepatterndecreasesaccordingly,andfor thermalgaswellabovethecriticaltemperature,theycan
[ very short coherence length the underlying lattice order usually be neglected. However,the momentum width for
1 can only be revealed through higher-order correlations acondensateis converselyverynarrow.Evenwithoutan
v [6].Arecentmanuscript[1]hasraisedthequestion,whe- optical lattice, effects (i)-(iii) have been known to play
0 ther even a minute condensate fraction on the order of an important role in the interpretation of time-of-flight
2 1/(ǫN2/3) (N being the total particle number) should images (see, e.g. [7, 8, 9]). We point out that a number
4
give rise to an interference pattern with visibility very of other effects might be relevant in a given experimen-
1 V
0 close to unity, 1 ǫ. According to the calculations tal situation, for instance a finite imaging resolution, or
V ≈ −
7 in [1], a system of, e.g., N = 105 atoms should show an collisions during the initial expansion phase that would
0 interference pattern with a visibility of > 99% for scatter the atoms out of the condensate. Here we do not
/ V ≈
t minute condensate fractions 4% ( 5000 atoms). As consider these aspects further, and consider an idealized
a ≈ ≈
currentexperimentswithultracoldquantumgasesdonot experimentlimitedonlybythethreepointsabove.Below
m
observethis(the bestvisibilitiesthathavebeenachieved weoutlineourargumentsandpointoutthe mostcritical
-
d in our group being close to 90%), the authors conclude steps to arrive at the conclusions given in [1]. We pro-
n that the current experiments must be carried out in a pose a simple model to explain why the visibility of the
o thermalregimewithnocondensateorsuperfluidfraction interferencepatternfollowsthecondensatefractioninthe
c
present in the trap. systematleastinaqualitativeway.Wealsofindthatfor
:
v In this paper we would like to point out that this re- the parameters that correspond to current experiments,
Xi verseconclusionis an artifactof the model used in [1] to the visibility of an expanding, non-condensed thermal
describe the time-of-flight expansion. A central assump- cloud is always significantly lower than 1, a conclusion
r
a tion in Ref. [1] is to consider that the condensate mo- consistent with experiments [2, 3, 4, 5].
mentum distribution is Heisenberg-limited by the size of Setup and Optical Lattice Potentials - The optical
the systempriorto expansion,andthat this distribution lattice potential can be expressed as
is preservedintime-of-flight.Below,wewouldliketo ex-
V (r)=V sin2(k x)+sin2(k y)+sin2(k z) . (1)
plain that several effects spoil this assumption, as under OL 0 L L L
realistic experimental conditions, one needs to take into Here V is the(cid:0)lattice depth expressed in units(cid:1)of the
0
account : single-photon recoil energy E = h2/2mλ2. Here k =
R L L
– (i) The inhomogeneous trapping potential for the 2π/λ is the laserwavevector,λ is the laserwavelength
L L
quantum gas in the lattice, and m is the atomic mass. In addition to the lattice po-
2
tential,an“external”potentialispresent,duetothepre- Orders of magnitude - When condensed and non-
sence of a magnetic trap and also to the optical confi- condensed components coexist, the planar momen-
nement provided by the Gaussian-shaped lattice beams. tum distribution n splits naturally into a condensate
The external potential is nearly harmonic with trapping contribution n (r ⊥) and a non-condensed contribution
0
frequency n (r ). Let us fi⊥rst consider the relative scaling of the
nc
⊥
condensate and non-condensate contributions. We write
4(2V √V ) thecondensatemomentumdistribution,characterizedby
ω ω2 + 0− 0 , (2)
T ≈s m mw2 the condensedatomnumber N0 anda momentum width
∆k , as
0
whereω isthefrequencyofthemagnetictrap,assumed
isotropicm, and where w is the waist (1/e2 radius) of the n (k )=A N0 f |k| , (6)
lattice beams, assumed identical for all axes. This for- 0 ⊥ (∆k0·d)2 0(cid:18)∆k0(cid:19)
mula, slightly different from the one given in [5], takes
where f is a periodic, dimensionless function. Similarly,
0
into account the modification of the vibrational ground
introducing the number of of non-condensed atoms N ,
nc
state energy in each well due to the spatial variations
we write the non-condensed contribution as
of the laser intensities on the scale w (see [10]), which
amounts to a few percents change in ωT. n (k )=A Nnc f |k| . (7)
After releasing the cloud from the trap, the density nc ⊥ (∆knc·d)2 nc(cid:18)∆knc(cid:19)
distribution of the expanding cloud after a time of flight
We assume the condensate contribution to produce a
t is usually taken to be proportional to the momentum
sharpinterferencepattern(f (K )=1andf(K )
distribution, 0 max min ≈
0), whereas the non-condensed part extends over the
n(k)= w˜(ki)2 (k), (3) whole Brillouin zone (∆knc ≈ 1/d). The visibility then
| | S reads
i=x,y,z
Y
N +(∆k d)2N f( )
with a scaling factor rt.o.f. = (~t/m)k. Eq. (3) is va- = 0 0 nc − (8)
lid provided that two conditions are met : interactions V N0+(∆k0d)2Nnc f(+)
haveanegligibleinfluenceontheexpansion,andthetime
where we introduced
of flight is long enough to treat the initial particle dis-
tribution as point-like (“far-field” approximation). The f( ) =(f (K ) f (K ))/f (K ) (9)
± nc max nc min 0 max
envelope w˜ 2 is the Fourier transform of the Wannier ±
| | which take values of order unity. The “suppression fac-
function in the lowest Bloch band, which we approxi-
mate by a Gaussian, w˜(k )2 w0 exp k2w2 , with tor” (∆k0d)2 sets the importance of the non-condensed
w the size of the on-|siteiW|an≈nieπr1/f2unctio−n. iS(k0) is gi- fraction; if this ratio is very small, then even a tiny
0
venbyS(k)= i,jeik·(ri−rj)haˆ†iaˆji.Expe(cid:0)rimenta(cid:1)lly,one conUdneinfosremd nsuymstbemer w- illTyhieeldsyastvemisibciolintysidcelorseedtionu[1n]itiys. a
records the momentum distribution integrated over the
probe directionPz, which we call n . We assume for sim- gas in a box with Ns lattice sites in each direction (L
plicity that the width of (k) is l⊥ower or comparable to in the notation of Ref. [1]). The momentum width for
1/w0 (this should holdinSthe situations where the popu- tEhsesecnotniadlleynstahtiseaims oduiffnrtascttoiotnheliamssituemdpttoion∆tkh0at∼th1e/pNossdt-.
lationinthefirstexcitedBlochbandisnegligible).Then,
expansion momentum distribution ofthe condensate is a
substituting w˜(k )2 with w˜(0)2 leads to
| z | | | deltafunction.In[1],anystructureinthenon-condensed
component is neglected, so that f is uniform. Then,
n (k ) Ad dk (k), (4)
z
⊥ ⊥ ≈ S 1
Z (∆k d)2 10 4 (10)
whereweintroducedthelatticespacingd,aglobalfactor 0 ∼ Ns2 ∼ −
A = w0 w˜(k )2 w˜(k )2. The visibility as defined in
√πd| x | | y | foratypicalNs 100.ThisiswheretheresultinEq.(4)
[4] follows from of Ref. [1] come∼s from, up to a factor w /d which de-
0
pendsonhowthek integralishandled.InFig.1(upper
n n z
max min
= − , (5) panel) the momentum distribution for sucha case is dis-
V n +n
max min played.Essentially,the momentumpeaksapproachdelta
where n = n (0) + n (0) is measured at the first functions, explaining why tiny condensate fractions can
max 0 th
lateral peaks of the interference pattern, e.g. at K = lead to visibilities very close to unity. We note that a
max
(1/2,0) 2π/d, and where the minimum density n = better way of analyzing the visibility, is to measure the
min
n (K ×) is measured along a diagonal with the same weight, i.e. the integral over a peak over a finite area,
th min
distance from the central peak, i.e. at the point defined whichaccountsforfinite resolutioneffectsthatareinevi-
the vector K =(1/√2,1/√2) 2π/d. tably present in experiments.
min
×
3
actionenergyisreleasedaskineticexpansionenergy,and
second, for a finite time-of-flight t the initial size of the
4000
distributionis notnecessarilynegligible.Dueto theseef-
A 3000 fects, the actual momentum distribution will differ from
)/ the one calculated using Eq. (3).
0
,x 2000 Here, we handle them in a heuristic way, by approxi-
k
( matingthemaximaintheplanarmomentumdistribution
⊥
n 1000 for the condensate as Gaussians,
0 N π2A
−1 −0.5 0 0.5 1 n (k K ) 0 exp k2/∆k2 . (11)
k [2π/d] 0 ⊥ ≈ max ≈ (∆keffd)2 − ⊥ eff
x (cid:0) (cid:1)
Weassumeaneffectivespatialwidthgivenbythequadra-
50 ticsumoftheinitialwidthσ0 andoftheexpansionwidth
~∆k /mtcorrespondingtothereleaseenergy.Inmomen-
0
40
A tum space, this corresponds to ∆k2 = ∆k2 + mσ0 2.
/ eff ~t
) 30 IntheThomas-Fermilimit,boththepost-expansionmo-
0
(cid:0) (cid:1)
k,x 20 mentum width ∆k and the initialsizeσ0 aredetermined
( by the interaction energy, which itself is proportional to
⊥
n 10 the chemical potential µ. Accordingly, one has
0 ~2∆k2
−2 −1 0 1 2 =αµ, mω2σ2 =βµ, (12)
k [2π/d] m T 0
x
where the coefficients α and β are of order unity. The
final result is
Fig.1:Planarmomentumdistributionforky =0.Theupper
apnadneolniesaptloomttepderfosritae,uannidfofromraidceoanldgeanssewditfrhacNtison=f1c0∼6 s5it%es. ∆ke2ff ≈ m~2µ α+ (ωβt)2 , (13)
The lower panel is plotted for ourharmonic trap model with (cid:18) T (cid:19)
5int%eraacntdioansto(tuaslinnguαmb=er0)N,a=lso10fo6ruasicnogndthenesterdapfrapcatriaomneftcer∼s corresponding to a suppression factor (∆keffd)2 propor-
given in [5]. Note the vastly different scales on the vertical tional to m~d22µ ∼ µ/ER . 1 much larger than expected
axes. fromEq.(10)(weestimatebelowtheproportionalityfac-
tor). Hence, one expects the observed visibility to follow
theevolutionofthecondensedfractionatleastqualitati-
Ideal gas in a harmonic trap - As mentioned above, vely. The argument also holds for the quantum-depleted
current experiments are carried out in the presence of a part of the non-condensed cloud.
harmonic trap superimposed on the lattice. In this case, Visibility for a trapped gas model Tomakethe prece-
one expects the ground state wavefunction to be locali- ding arguments more precise, we introduce a semiclassi-
zed on a scale given by the harmonic potential ground caldescriptionofthethermalcloud,forwhichinteraction
state extension σ = ~/mω , and not by the “num- effects can usually be neglected to a first approximation
T T
ber of lattice sites” (roughly equivalent to the size w [9]. The thermal component is characterized by a distri-
p 0
of the laser beams). In fact, usually one has w σ . bution function in phase space given by
0 T
≫
The main point here is that the suppression factor is no
longer hopelessly small, especially when most atoms are ρ(r,k) = exp[(ǫk+V(r)1−µ)/kBT]−1, (14)
non-condensed. However, as well known, this ideal gas
model for the condensate is a poor description of realis- with the tight-binding dispersion relation ǫk =
2J (1 cos(k d)), with the trap potential
tic experimental situations [9]. i=x,y,z − i
V(r) = 1mω2r2. We can obtain the thermal fraction
Interacting gas in a harmonic trap - Toobtainabet- P 2 T
by integration over phase space,
ter description, it is necessary to include interatomic in-
twehraicchtiounsusailnlytahpeplpieiscttuoree.xpIneritmheentTahlosmituaas-tFioenrms,ithliemiint-, Nth =(2π)3/2 Rdth 3 ∞n=1 en(µ−n63J/)2/kBTI0(2nJ/kBT)(315.)
trapspatialcondensatewavefunctionwillbesignificantly (cid:0) (cid:1) P
broader than in the ideal gas case, implying a narrower Here I0 is a Bessel function and Rth = mkBωT2 the ther-
T
momentum distribution. However, as argued above, the malcloudradius.The criticaltemperatuqre for condensa-
post-expansion momentum profile can be very different tionfollowsfromEq.(15)bysetting the chemicalpoten-
from the sharply peaked in-trap profile. First, the inter- tial µ = 0 [9]. Integrating the phase space distribution
4
over position and k gives the planar momentum distri-
z
bution, n 1
o
n (k ) = A(2π)3/2 Rth 3 ∞ en(µ−2J)/kBT racti
⊥ ⊥ (cid:18) d (cid:19) nX=1 n3/2 ed f 0.5
I0(2nJ/kBT)e−nǫk⊥/kBT. (16) ens
d
This takes the form announced in Eq. (8). n
o
To estimate α and β in Eq. (13), we impose that our C 0
0 0.5 1 1.5 2 2.5
Gaussian model should reproduce the same interaction
and potential energy as a condensate in the Thomas-
1
Fermi limit, with chemical potential given by [10]
µ= 161√52πN0U 2/5 mωT2d2 2/5. (17) bility 0.5
(cid:18) (cid:19) si
The potential energy per part(cid:0)icle [9](cid:1)E = 3µ = Vi
pot 7
3mω2σ2 gives a coefficient β 12/21. The coefficient
4 T 0 ≈ 0
α depends on how the interaction energy is redistribu- 0 0.5 1 1.5 2 2.5
ted during the expansion. We consider the two limits, T/T
C
where the interaction energy is entirely concentrated ei-
ther into the single diffraction peak under consideration
(E =2µ/7=3~2∆k2/4m,orα=8/21),orinthe cen- Fig. 2: Upper plot : Calculated condensed fraction for an
int 0 idealBosegasinaperiodicpotential,withandwithoutaddi-
tralpeak avoidedby the visibility measurement(α=0).
tionalharmonictrap.Lowerplot:visibilityoftheinterference
This provides upper and lower bounds for the actual vi- observedafterreleasefromthetrap.Thedash-dottedlinecor-
sibility. The results of the calculations is plotted in Fig. responds to an uniform ideal gas, as in the model of [1], the
2 for various assumptions : no trap, ideal gas in a trap, solid line to a harmonically trapped ideal gas. The long and
interacting gas in a trap with α = 0 or α = 8/21. Com- short-dashedlinescorrespondtotwomodelsofaninteracting
condensate,whichshouldbeseen asupperandlower bounds
paring the two latter cases, one sees that the final result
for the visibility (α=0 and α=8/21, see text). The curves
depends sensitively on the value of α. A more complete areplottedforN =106 atoms,V0 =10ER andanexpansion
calculationwouldprobablyyieldacurvelyinginbetween
time t=20 ms.
the two bounds plotted. Nevertheless,irrespective of the
particularmodel,thecurvesdemonstratethatthevisibi-
lity falls well below unity in a broad temperature range,
where the condensate fraction stays non-zero.
In conclusion, we have calculated the visibility of the
interferencepatternobservedwhenreleasingafinitetem- [1] R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, cond-
perature, Bose-condensed gas from an optical lattice. mat/0609685 (2006).
[2] M.Greiner,O.Mandel,T.Esslinger,T.W.H¨ansch,and
Using a realistic model for the condensate momentum
I. Bloch, Nature415, 39 (2002).
distribution, we find that the evolution of the visibility
[3] T. St¨oferle, H. Moritz, C. Schori, M. K¨ohl, and T. Ess-
with temperature or lattice depth follows the evolution linger, Phys. Rev.Lett. 92, 130403 (2004).
ofthecondensedfractionatleastqualitatively.Wethere- [4] F.Gerbier,A.Widera,S.F¨olling,O.Mandel,T.Gericke,
foredisagreewiththe conclusionmade inRef. [1], thata and I.Bloch, Phys. Rev.Lett. 95, 050404 (2005).
measured visibility deviating from unity necessarily im- [5] F.Gerbier,A.Widera,S.F¨olling,O.Mandel,T.Gericke,
plies the absence of a Bose condensate in the system. and I.Bloch, Phys. Rev.A 72, 053606 (2005).
[6] S.F¨olling,F.Gerbier,A.Widera,O.Mandel,T.Gericke,
This claim originates essentially from neglecting interac-
and I.Bloch, Nature 434, 481 (2005).
tion effects on the gas expansion, which is a valid ap-
[7] Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys.
proximation close or in the Mott insulator regime, but Rev. A 54, R1753 (1996).
a poor one for a system including a sizeable condensate, [8] Y.CastinandJ.Dalibard,Phys.Rev.A55,4330(1997).
with or without [9] an optical lattice. We do however [9] F.Dalfovo,S.Giorgini,L.P.Pitaevskii,andS.Stringari,
agree with the authors of [1] on the importance of finite Rev. Mod. Phys. 71, 463 (1999).
temperature effects in current experiments, as implied [10] M. Greiner, Phd thesis, Ludwig-Maximilian University
Munich (2003).
by recent experimental [3, 11, 12] and theoretical work
[11] F. Gerbier, S. F¨olling, A. Widera, O. Mandel, and
[13, 14, 15, 16, 17], and certainly consider this topic to
I. Bloch, Phys.Rev. Lett. 96, 090401 (2006).
warrantfurther investigations.
[12] S.F¨olling,A.Widera,T.Mu¨ller,F.Gerbier,andI.Bloch,
We would like to thank Martin Zwierlein for useful Phys. Rev.Lett. 97, 060403 (2006).
comments on the manuscript. [13] B. DeMarco, C. Lannert, S. Vishveshwara, and T.-C.
5
Wei, Phys.Rev.A 71, 063601 (2005). [16] B. Capogrosso-Sansone, E. Kozik, N. Prokof’ev, and
[14] A.M.Rey,G.Pupillo,andJ.V.Porto,Phys.Rev.A73, B. Svistunov,cond-mat/0609600 (2006).
023608 (2006). [17] F. Gerbier, in preparation (2007).
[15] X.Lu and Y.Yu,cond-mat/0609230 (2006).
