Phase transition to Bose-Einstein condensation for a Bosonic gas confined in a combined trap

Abstract

We present a study of phase transition to macroscopic superfluidity for an ultracold bosonic gas confined in a combined trap formed by a one-dimensional optical lattice and a harmonic potential, focusing on the critical temperature of this system and the interference patterns of the Bose gas released from the combined trap. Based on a semiclassical energy spectrum, we develop an analytic approximation for the critical temperature $T_{c}$, and compare the analytic results with that obtained by numerical computations. For finite temperatures below $T_{c}$, we calculate the interference patterns for both the normal gas and the superfluid gas. The total interference pattern shows a feature of ``peak-on-a-peak". As a comparison, we also present the experimentally observed interference patterns of $^{87}$Rb atoms released from a one-dimensional optical lattice system in accord with our theoretical model. Our observations are consistent with the theoretical results. Comment: Author's address has been corrected; Typos corrected; Grammar errors corrected

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arXiv:1011.3140v2 [cond-mat.quant-gas] 19 Nov 2010
Phase transition to Bose-Einstein condensation for a Bosonic gas confined in a
combined trap
Baolong L¨ u,1Xinzhou Tan,1,2Bing Wang,1,2Lijuan Cao,1,2and Hongwei Xiong1, ∗
1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
2Graduate School of the Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
(Dated: November 22, 2010)
We present a study of phase transition to macroscopic superfluidity for an ultracold bosonic gas
confined in a combined trap formed by a one-dimensional optical lattice and a harmonic potential,
focusing on the critical temperature of this system and the interference patterns of the Bose gas
released from the combined trap. Based on a semiclassical energy spectrum, we develop an analytic
approximation for the critical temperature Tc, and compare the analytic results with that obtained
by numerical computations. For finite temperatures below Tc, we calculate the interference patterns
for both the normal gas and the superfluid gas. The total interference pattern shows a feature
of “peak-on-a-peak.” As a comparison, we also present the experimentally observed interference
patterns of87Rb atoms released from a one-dimensional optical lattice system in accord with our
theoretical model. Our observations are consistent with the theoretical results.
PACS numbers: 03.75.Lm, 67.25.dj, 37.10.Jk, 67.10.Ba
I.INTRODUCTION
Bosonic atoms confined in optical lattices have proved
to be a unique laboratory for investigating quantum
phase transitions from superfluids to Mott insulators [1–
3]. The momentum distribution of a lattice system can
be mapped out directly by the interference pattern of
the atomic cloud after a ballistic expansion over a time
of flight (TOF). The emergence of macroscopic bosonic
superfluid is usually identified by the appearance of in-
terference peaks. However, recent theoretical works [4, 5]
for homogeneous gases in a three-dimensional (3D) lat-
tice showed that this criterion of macroscopic superflu-
idity is not reliable since even a normal gas can have
sharp interference peaks. The underlying physical pic-
ture is that a lattice system at finite temperatures pos-
sesses a “V-shaped” phase diagram [4–6] which includes
a normal gas region between the Mott Insulator and the
superfluid. The true signature of macroscopic superflu-
idity is the δ-function momentum peaks with nearly unit
visibility [4]. Below critical temperature, the coexistence
of superfluid and normal gas in the homogeneous lattice
system should give rise to an interference pattern hav-
ing a feature of “peak on a peak” [5]. The new criterion
of macroscopic superfluidity makes it necessary to fur-
ther investigate the phase transition of bosonic atoms in
an optical lattice, particularly for the characteristics as-
sociated with the critical temperature and interference
patterns. Experimental investigations are also required
for comparison with relevant theoretical models.
There have been a few theoretical works [7–9] consider-
ing the translationally invariant (uniform) lattices. How-
ever, in a realistic experiment, an optical lattice is always
∗Electronic address: xionghongwei@wipm.ac.cn
accompanied by harmonic confinement in all dimensions,
arising from the focused Gaussian laser beams and/or an
external magnetic trap. A bosonic gas is, therefore, never
spatially uniform over the lattice range. Wild et al. [10]
have examined the critical temperature of the interacting
bosons in a one-dimensional (1D) lattice with additional
harmonic confinement. Ramakumar et al. [11] have in-
vestigated the condensate fraction and specific heat of
non-interacting bosons in 1D, two-dimensional (2D), and
3D lattices in the presence of harmonic potentials. Based
on a piecewise analytic density of states extended to ex-
cited bands, Blakie et al. [12] developed an analytical ex-
pression of the critical temperature for an ideal bosonic
gas in the combined harmonic lattice potential, and com-
pared the analytic result with their numerical compu-
tations. However, these studies on combined traps did
not mention interference patterns of the released bosonic
gases. A more recent theoretical paper [13] has investi-
gated the Bose-Einstein condensation (BEC) in a 3D in-
homogeneous optical lattice system, and predicted that a
bimodal structure in the momentum-space density profile
is a universal indicator of BEC transition.
The experiment of Spielman et al. [14] has examined
the superfluid to normal transition for a finite-sized 2D
optical lattice system. Their measurements confirm that
bimodal momentum distributions are associated with the
superfluid phase. For such a system with a typical density
of 1 atom per lattice site, the phase transition behavior
can be interpreted in terms of the commonly used Bose-
Hubbard model.
Unlike the 2D and 3D cases, an inhomogeneous 1D
optical lattice system is usually much more heavily pop-
ulated, with an atomic number up to several hundreds
in a single lattice site. In the superfluid phase, the on-
site interaction energy U varies from site to site because
of its dependence on the local population in single lat-
tice sites. This increases the complexity in searching for

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an analytical description of the phase transition. In this
paper we present a study of the critical temperature and
interference patterns of an ultracold bosonic gas confined
by a 1D optical lattice and an additional magnetic po-
tential. The interference patterns of the normal gas and
the condensed atoms are treated separately. The super-
position of the two parts gives rise to a feature of “peak
on a peak.” Different from a homogeneous lattice system,
however, the normal gas can-not produce sharp interfer-
ence peaks. Furthermore, the theoretical results are com-
pared with our preliminary experiment for a 1D lattice
system of87Rb Bose-Einstein condensates.
Our theoretical model relevant to the phase transition
is for ideal bosonic gases. In fact, interatomic interaction
may affect the shape of the interference pattern, espe-
cially for the condensed part which has a higher atomic
density. In order to obtain a better match with the exper-
iment result, we take the interaction energy into consid-
eration for the condensed atoms during the time of flight.
The computed result shows that interference peaks aris-
ing from the condensed atoms can be significantly broad-
ened due to the interaction effect.
This paper is organized as follows. In Sec. II, we be-
gin with a semiclassical energy spectrum for a combined
harmonic lattice trap. Under the tight-binding approxi-
mation and in the low energy limit, we derive an analyt-
ical expression of the critical temperature for the atoms
condensed to a superfluid state. The accuracy of the an-
alytical results are checked with respect to the numerical
calculations. Section III gives a description on how the
interference patterns are calculated for the normal gas,
as well as the Bose-condensed gas. In Sec. IV, we briefly
introduce the experiment, and present the observed in-
terference patterns for a comparison with our theoretical
results. Finally in Sec. V, we summarize the obtained
results.
II.CRITICAL TEMPERATURE
We now consider an ideal Bose gas confined in a 3D
harmonic potential with axial symmetry around the z
direction.The axial and transverse trapping frequen-
cies are ωz and ωx= ωy= ω⊥, respectively. Moreover,
we assume that the axial confinement is much weaker
than the radial confinement (ωz≪ ω⊥), so that the Bose
gas is made cigar-shaped. A 1D optical lattice potential,
V0sin2(kz), is applied along the z axis, where k = π/d is
the wave vector of the lattice light, d denotes the lattice
period, and V0denotes the potential depth of the lattice.
V0can be written in terms of the recoil energy Er, say,
V0 = sEr, where Er = ?2k2/2m, and m is the atomic
mass. The harmonic potential, together with the optical
lattice, forms a combined trap written as
V (x,y,z) =1
2mω2
⊥(x2+y2)+1
2mω2
zz2+V0sin2(kz). (1)
In practice, an optical lattice is usually produced by a
retro-reflected Gaussian laser beam which also produces
a transverse confining potential, that can be simply ab-
sorbed into ω⊥if it is non-negligible.
To obtain the eigenenergies of the combined trap sys-
tem, one needs to derive the single-particle Hamiltonian
of the system and then numerically diagonalize it [12].
Despite its accuracy for ideal Bose gases, this numer-
ical method can not provide an analytic expression of
the energy levels. The energy spectrum corresponding
to the transverse confinement is described by equally
spaced harmonic-oscillator states, whereas the oscillator
treatment is not applicable to the axial dimension due
to the presence of the optical lattice.
hereafter is based on the tight-binding approximation
that only the ground band is accessible to the system.
This approximation is valid when the thermal energy
of the atoms is much less than the first band gap of a
deep lattice. For a 1D uniform lattice, the eigen energy
can be written as a function of quasimomentum q [15],
ǫ(q) =1
2?? ω − 2J cos(qd/?). Here, ? ω is the frequency of
tunneling energy due to the hopping to a nearest neigh-
boring well, and it depends upon the lattice depth s in
the following form [15]
√πErs3/4exp?−2√s?.
It should be noted that Eq. (2) is valid only for deep
lattices. At s = 11, for example, it overestimates J by
approximately 18%. For the combined trap, it is a rea-
sonable assumption that Eq. (2) remains valid as long
as the trapping frequency ωz of the weak axial confine-
ment is much smaller than the tunneling rate J/?. We
are thus able to use a constant J over the entire lattice
system at a given lattice depth. For simplicity the energy
spectrum corresponding to the combined confinement in
the axial direction is approximated by the semiclassical
energy, ǫ(pz) +1
pz is the quasimomentum in the ground band. Now we
are able to write the total energy spectrum in an explicit
form
εnxny(z,pz) =?ω⊥(nx+ ny+ 1) +1
Our discussion
the local oscillation at each lattice well, while J is the
J =
4
(2)
2mω2
zz2, in the z-pzphase space, where
2mω2
zz2
+1
2?? ωz− 2J cos(pzd/?),
(3)
where {nx,ny} are non-negative integers.
For a semiclassical description of this system, we treat
the harmonic trap semiclassically while treating the op-
tical lattice quantum mechanically. Such a picture cor-
responds to a density distribution of the thermal cloud:
?
where
n(z) =
nx,ny
?
dpz
2π?F(pz,z)Md|Φpz(z)|2, (4)
F(pz,z) =
1
exp[β(εnxny− µ)] − 1,

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and
Φpz(z) =
1
√M
M/2
?
l=−M/2
w(z − ld)exp(ipzz/?).
Here, Φpzis the normalized wave function of a uniform
optical lattice system with an extension of M lattice sites,
and w(z − ld) is the Wannier wave function. The total
number of thermal atoms is then written as
?
=
2π?F(pz,z)Md|Φpz(z)|2.
Nth=n(z)dz
?
?
nx,ny
dpzdz
(5)
In the tight-binding limit, w(z) is well localized within a
single lattice site. In contrast, F(pz,z) is a slowly vary-
ing function of z. Therefore, Φpz(z) in Eq. (5) can be
integrated out. This results in a new integrand expressed
as a summation of discrete F(pz,z − ld)d, which in turn
can be approximated as an integral over z. By doing so,
one gets
?
Nth=
?
nx,ny
dpzdz
2π?
F(pz,z). (6)
Below a critical temperature Tcthe chemical potential
µ of the Bose gas reaches the bottom of the ground band
µ → µc= ?ω⊥+1
2?? ωz− 2J,
while the lowest state with pz= 0 becomes macroscopi-
cally populated which corresponds to the onset of Bose-
Einstein condensation. The condensed atoms exhibit
macroscopic superfluidity, whereas all other atoms be-
yond the lowest state form a so-called normal gas. Since
the condensate is actually a quantum fluid, we use “su-
perfluid” just as a synonym of BEC. The atomic number
of the normal gas is given by the sum,
?
Nnc= N − Nc=
?
nx,ny
1
exp[β(εnxny− µc)] − 1
dpzdz
2π?,
(7)
where N is the total number of the atoms, Ncthe atomic
number of the condensed part, β = 1/kBT, and kB the
Boltzmann constant. The integrand can be expanded in
powers of the exponential term using the formula 1/(ex−
1) =?∞
Performing the integration over nx, ny as well as that
over the coordinate z, one gets
n=1e−nx. Moreover, the sum over nx, nycan be
replaced by an integral if the atomic number N is large.
N − Nc=
1
(β?ω⊥)2
?
∞
?
n=1
1
n2
?
2π
nβmω2
z
?1/2
×
dpz
2π?exp[−nβ(2J − 2J cos(pzd/?))],
(8)
and the right side is a function of temperature T. Ap-
parently, Eq. (8) is suitable for numerical calculation of
the atomic number in the normal gas since the integra-
tion can be simply replaced by a summation over the pz
region.
By imposing that Nc = 0 at the transition, Eq. (8)
determines a critical temperature Tcfor a given N. Ap-
parently, to obtain the value of Tc, one needs to carry out
numerical computations based on Eq. (8). Nevertheless,
we can derive an analytic expression of Tcin a limiting
case. When the temperature of the Bose gas is so low
that most atoms occupy the states in the vicinity of the
bottom of the ground band, the relation pz≪ ?/d holds,
and the cosine function in Eq. (8) can be expanded to the
order of p2
z. With the pz-dependent function integrated
out, one has
N = (kBTc/?ω)3(m∗/m)1/2ζ(3),
where ω = (ω2
trapping frequencies, m∗= ?2/2Jd2the effective mass
of the atom, and ζ(α) =?∞
kBTc= 0.94?ωN1/3(m/m∗)1/6,
⊥ωz)1/3is the geometric average of the
n=11/nαthe Riemann zeta
function. Finally, one gets
(9)
which can be used as an analytic estimation of the critical
temperature.
FIG. 1: Critical temperature Tc versus the total number
of
obtained from the numerical calculation of Eq.
the analytical approximation of Tc (Eq. (9)), respectively.
The dotted line gives the full numerical result by diagonal-
izing the single-particle Hamiltonian. The lattice parameters
are d = 400nm and s = 11.2Er.
cies of the harmonic potential are ω⊥ = 2π × 83.7Hz and
ωz = 2π × 7.63Hz, respectively.
87Rb atoms. The solid curve and the dashed curve are
(8) and
The trapping frequen-
We recall that an ideal Bose gas trapped in a 3D har-
monic potential undergoes the phase transition to Bose-
Einstein condensation at a temperature [16] kBTc =
0.94?ωN1/3. Comparing this expression with Eq. (9),
one can see that Tcis changed by a factor of (m/m∗)1/6
due to the presence of the 1D lattice. Since m∗is always
larger than m [4] under tight-binding approximation, the

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combined trap Tc is actually reduced compared to the
case without lattice. Note also that a homogeneous 3D
lattice system has a reduced Tc as well [4], but with a
reducing factor
We have calculated the critical temperature Tc for a
87Rb gas in the combined trap (see Fig. 1). The trap
parameters are intentionally chosen to match our ex-
periment which will be described in the later section.
The numerically calculated Tc is displayed by the solid
curve, while the dashed curve is the analytic Tc calcu-
lated according to Eq.(9). The discrepancy between the
two curves becomes larger as the atom number N is in-
creased, showing that the accuracy of the analytic esti-
mation becomes worse for larger N. We thus use only the
numerically calculated Tcin the following computations.
?m/m∗instead.
FIG. 2: Condensate fraction as a function of T/Tc. Solid
line, numerical results based on Eq. (8). Dotted line is the
full numerical result based on diagonalizing the single-particle
Hamiltonian. The dash dot line is 1 − (T/Tc)3for T ≤ Tc.
The parameters of the combined trap are the same as in Fig.
1, and the atomic number is N = 5 × 104, corresponding to
Tc = 47.9nK.
It is well known that an ideal Bose gas in a 3D har-
monic potential shows a T dependence of the condensate
fraction as Nc∼ 1 − (T/Tc)3for T < Tc. We have also
calculated the condensate fraction for our combined trap
system with 5×104atoms, as shown by the solid line in
Fig. 2. It displays a noticeable deviation from the curve
of 1 − (T/Tc)3, but fits well to the characteristic shape,
1 − (T/Tc)α, with α = 2.679.
To justify our analytical approximation, we also cal-
culate the critical temperature and condensate fraction
based on the diagonalization of the single-particle Hamil-
tonian. The energy spectrum of the system is written as
εnxnynz= ?ω⊥(nx+ ny+ 1) + εnz. (10)
εnz can be obtained numerically from the following
single-particle Hamiltonian along the z direction,
?
Here εidescribes an energy offset at each lattice site due
to the presence of the harmonic trap along the z direc-
? Hz= −J
2
?i,j?
?
? a†
i? aj+ ? ai? a†
j
?
+
?
i
εi? a†
i? ai.(11)
tion. By diagonalizing the matrix
directly the energy spectrum εnz. Furthermore, with the
following formula,
?
i
???? Hz
???j
?
, one can get
N =
?
nx,ny,nz
1
e(εnxnynz−µ)/kBT− 1
,(12)
we give the full numerical results of the critical tempera-
ture and condensate fraction by the dotted lines in Figs.
1 and 2, respectively. Clearly, our semiclassical treat-
ment agrees with the full numerical method, and proves
to be reliable. Furthermore, it offers a convenient way
to analyze the spatial distribution of a confined atomic
cloud, which in turn simplifies the calculation of interfer-
ence patterns.
III. INTERFERENCE PEAKS
When the combined trap is suddenly shut off at the
moment t = 0, the Bose gas starts to expand freely. Af-
ter a time of flight τ, the expanded wavepackets initially
localized in single lattice wells overlap with each other,
forming a 3D density distribution. In the following cal-
culation, the x and y dependence of the atomic density
will be integrated out so as to obtain a density profile
along the z direction only. This is convenient for making
a comparison with the experimental results. Usually, an
absorption image is used to record the column density
profile of a released atomic cloud. Supposing that the
probe laser beam is applied along the direction of the x
axis, the density profile along z can be easily obtained by
integrating the column density over the y dimension.
A. Normal gas
In the combined trap, normal gas atoms are distributed
over the transverse harmonic modes labeled by a positive
quantum number q = nx+ ny. For a given q, there are
q + 1 degenerate states, and we hereafter call them sub-
states. The summation over nxand nyin the previously
mentioned equations is thus equivalent to?
the same q number have identical populations and spa-
tial distribution along the z direction. Due to optical
lattice potential, atoms in a single substate are further
distributed over the Bloch states with different quasimo-
mentum pz with pz/? ∈ (−π/d,π/d). Each pz compo-
nent can be treated semiclassically where the influence of
the optical lattice is given by a quantum wave packet de-
scription, while the influence of the harmonic trap along
the z direction is treated semiclassically. In such a pic-
ture, the single-particle wave function of a pzcomponent
at t = 0 takes the following form:
?
q(q +1)···.
From Eq. (7), one sees that the substates belonging to
Ψq
pz(t = 0) =
l
αq
lw(z − ld)exp(ipzz/?),(13)

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where (αq
located in the lth lattice site for the transverse harmonic
mode q.
Equation (7) shows that the atomic density of a sub-
state with pzhas an envelope as
l)2denotes the probability for a particle roughly
n(z) =∆pz
2π?
1
eβ[q?ω⊥+2J(1−cos(pzd/?))+1
2mω2
zz2]− 1, (14)
The atom where ∆pz denotes a small interval of pz.
number in the lth lattice site is then
nl=d∆pz
2π?
1
eβ[q?ω⊥+2J(1−cos(pzd/?))+1
Therefore, αq
Nq=?
be determined by solving the Schrodinger equation of Hz.
However, as shown lately, the thermal average of in-trap
density written in terms of |αp
pression obtained by semiclassical approximation, hence
within semiclassical approximation |αq
to (αq
In the tight-bindinglimit
approximated bya
(πσ2)−1/4exp(−z2/2σ2),
the oscillator length. After the free expansion over a
time of τ, the single-particle wave function of the atoms
with pzis written as
2mω2
zd2l2]− 1. (15)
l)2= nl/Nq, with
lis simply given by (αq
lnlbeing the total atom number of the pzcom-
ponent in the substate of interest. In principle, αq
lshould
l|2is matched to the ex-
l|2can be identified
l)2= nl/Nq.
w(z) can
wave
??/m? ωz
bewell
Gaussian
where σ
packet
=
is
Ψq
pz(t = τ)
?
Here, K(z,z′,τ) = ?z|exp(−iHτ/?)|z′? is the propaga-
tor, with H the Hamiltonian governing the expansion
process. If the interatomic interaction is neglected, H
contains only the kinetic energy, say, H = P2
this case, it is straightforward to get
?
For simplicity of expression and calculation, we will
use dimensionless units for the length in z, the quasimo-
mentum pz and the time t by the replacement z → zd
(and hence σ → σd), pz→ pz?/d, and t → t(2md2/?).
Inserting Eq. (17) into Eq. (16), and working out the
integration over z′, one gets
?
with A and B given by
=
l
αq
l
?
K(z,z′,τ)w(z′− ld)exp(ipzz′/?)dz′.
(16)
z/2m. In
K(z,z′,τ) =
m
i2π?τexp
?im
2?τ(z − z′)2
?
.(17)
Ψq
pz(t = τ) = A
l
αq
lB(pz,l,z),(18)
A = π−1/4(σ + i2τ/σ)−1/2,
B(pz,l,z) =
exp
?−2τσ2p2
z+ 2σ2zpz+ i[4lpzτ + (z − l)2]
2(σ4+ 4τ2)/(2τ + iσ2)
?
.
FIG. 3: (a) The solid lines are the calculated atomic distribu-
tion of the normal gas (87Rb) after 30ms of time of flight. The
total atom number N = 5.9 × 104, and the trap parameters
are the same as in Fig. 1, corresponding to Tc = 51.1nK. For
the top two curves, the temperature T/Tc is 0.98 and 0.95,
respectively. The other nine curves, from top to bottom, are
for T/Tc ranging from 0.9 to 0.1 with a step of 0.1. Vertical
scales of the curves have been adjusted so that the central
peaks have roughly the same height. (b) Solid circles repre-
sent the visibility of the side peaks in (a). The solid curve
connecting the points is added to guide the eye. Open circles
show the normal gas fraction for the given total atom number.
Taking into consideration all transverse modes and all
pzcomponents, one gets the atom density after the time
of flight,
nnc(z) =
∞
?
q=1
?
∞
?
pz
(q + 1)Nq|Ψq
pz(t = τ)|2
?????
= |A|2
q=1
?
pz
(q + 1)
?
l
√nlB(pz,l,z)
?????
2
.
(19)
In the numerical calculations of Eq. (19), the summa-
tion over transverse modes is cutoff at q = 200, while
n is cutoff at 30 and lattice number l at ±350. These
cutoff numbers are chosen to assure a high accuracy bet-
ter than 0.2% in the calculations of atom numbers. By
setting ∆pzto 0.05π, the whole pzrange is divided into
40 intervals. This step size of pzhas the order of h/dM,
where M ≃ 100 is the typical spatial extent in the z di-
rection. We have also checked that the calculated results
have almost no change when further reducing ∆pz. The
step size in z is set to be 18d (7.2µm), comparable to the