Exploring the thermodynamics of a two-dimensional Bose gas.

Abstract

Using in situ measurements on a quasi-two-dimensional, harmonically trapped (87)Rb gas, we infer various equations of state for the equivalent homogeneous fluid. From the dependence of the total atom number and the central density of our clouds with chemical potential and temperature, we obtain the equations of state for the pressure and the phase-space density. Then, using the approximate scale invariance of this 2D system, we determine the entropy per particle and find very low values (below 0.1k(B)) in the strongly degenerate regime. This shows that this gas can constitute an efficient coolant for other quantum fluids. We also explain how to disentangle the various contributions (kinetic, potential, interaction) to the energy of the trapped gas using a time-of-flight method, from which we infer the reduction of density fluctuations in a nonfully coherent cloud.

Full text

Page 1

Exploring the thermodynamics of a two-dimensional Bose gas
Tarik Yefsah, R´ emi Desbuquois, Lauriane Chomaz, Kenneth J. G¨ unter and Jean Dalibard
Laboratoire Kastler Brossel, CNRS, UPMC, Ecole Normale Sup´ erieure, 24 rue Lhomond, F-75005 Paris, France
(Dated: June 2, 2011)
Using in situ measurements on a quasi two-dimensional, harmonically trapped87Rb gas, we infer various
equations of state for the equivalent homogeneous fluid. From the dependence of the total atom number and the
central density of our clouds with the chemical potential and temperature, we obtain the equations of state for
the pressure and the phase-space density. Then using the approximate scale invariance of this two-dimensional
system, we determine the entropy per particle. We measure values as low as 0.06kBin the strongly degenerate
regime, which shows that a 2D Bose gas can constitute an efficient coolant for other quantum fluids. We also
explain how to disentangle the various contributions (kinetic, potential, interaction) to the energy of the trapped
gas using a time-of-flight method, from which we infer the reduction of density fluctuations in a non fully
coherent cloud.
PACS numbers: 03.75.-b, 05.10.Ln, 42.25.Dd
Physical properties of homogeneous matter at thermal equi-
librium are characterized by an equation of state (EOS), i.e. a
relationship between some relevant state variables. For a fluid
of particles, possible EOS’s consist in expressions of pressure,
density or entropy as functions of temperature T and chemical
potentialµ. Foranidealgasthe EOS canbecalculatedexactly
in any dimension for a Bose or Fermi gas. In the presence of
interactions, one has to resort to approximations or numerical
calculations, and a comparison with experiments is crucial to
test their validity. Trapped atomic gases at thermal equilib-
rium provide a powerful tool for this purpose [1]. Within the
local density approximation, any intensive state variable takes
at a point r in the trap the same value as in a homogenous
system with the same temperature and the shifted chemical
potential µ − V (r), where V (r) is the confining potential.
The case of an interacting two-dimensional (2D) Bose fluid
is particularly interesting in this context. Firstly, at non-zero
temperature the Mermin–Wagner theorem precludes Bose–
Einstein condensation [2, 3]. Therefore the EOS is expected
to be continuous at any point, in spite of the existence of
a superfluid, infinite-order phase transition, which is of the
Berezinskii–Kosterlitz–Thouless (BKT) type [4, 5]. Secondly
the EOS of a 2D Bose fluid is scale invariant [6] in the so-
called quasi–2D regime. The later refers to the experimen-
tally relevant situation where single-particle motion is frozen
along one axis, making it thermodynamically 2D, but where
collisions still keep their 3D character [7]. The (approximate)
scale invariance originates from the fact that in this regime,
the coupling strength ˜ g is an energy-independent dimension-
less coefficient, and thus provides no energy, nor length scale,
in contrast with the 1D or 3D cases. This implies in particular
that dimensionless thermodynamic variables such as the phase
space density D or the entropy per particle S are functions of
the ratio µ/kBT only, with ˜ g as a parameter.
Recent experiments with trapped 2D Bose atomic gases
have demonstrated the existence of a BKT-type transition at
a threshold phase space density. One line of investigation ex-
ploited matter-wave interference to monitor the appearance of
an extended coherence in the sample [8, 9], and another ap-
proach used a time-of-flight (ToF) technique to measure the
momentum distribution of the gas [10]. The steady-state scale
invariance was verified in [11]. In this paper we present a
detailed experimental investigation of several thermodynamic
properties of a 2D Bose gas. We describe measurements of
the EOS for the pressure from a count of the total atom num-
ber of a trapped 2D Bose gas, with a wide range of thermo-
dynamic parameters. From the same set of data we use the
central spatial density to access the EOS for phase space den-
sity. Combining these two measured EOS with the scale in-
variance we obtain the EOS for the entropy per particle. We
show that this quantity rapidly decreases around the super-
fluid transition point and then approaches zero in the highly
degenerate regime. We also present an original method to ex-
tract from a ToF in only one direction of space, the various
contributions (kinetic, potential, interaction) to the total en-
ergy of the trapped gas. This method is applicable to any low-
dimensional fluid. Here it shows that density fluctuations of
our 2D Bose gas are essentially frozen even when its thermal,
non coherent fraction is significant.
Our 2D Bose gases are prepared along the lines detailed in
[12]. We start with a 3D Bose–Einstein condensate of87Rb
atoms confined in a magnetic trap in their F = mF = 2 in-
ternal ground state with an adjustable temperature. We slice a
horizontal sheet of atoms with an off-resonant, blue-detuned
laser beam with an intensity node in the plane z = 0. It pro-
vides a strong confinement along the direction perpendicular
tothisplane, withoscillationfrequencyωz/2π = 1.9(2)kHz,
which correspond to the interaction strength ˜ g =√8πa/?z≈
0.1, where a is the 3D scattering length and ?z=
The energy ¯ hωzis similar to or larger than the thermal energy
kBT andtheinteractionenergyperparticle, sothatmostofthe
atoms occupy the ground state of the vibrational motion along
z. The magnetic trap provides a quasi-isotropic confinement
in the xy plane with frequency ω /2π = 20.6(1)Hz [13].
?¯ h/mωz.
After an equilibration time of 3 seconds in the combined
magnetic+laser trap, we measure the in situ density distri-
bution of the gas by performing absorption imaging with a
probe beam propagating along the vertical axis. The conven-
arXiv:1106.0188v1 [cond-mat.quant-gas] 1 Jun 2011

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2
a)
20µm
b)
0
20
40
60
80
0 25 50
Density in atoms/µm2
r in µm
c)
1
10
01000
r2in µm2
2000
Density in atoms/µm2
d)
FIG. 1: (Color on line) Absorption imaging of quasi-2D clouds of
87Rb atoms. (a) Image obtained with a short pulse (∼ 2µs) of an
intense probe beam (I/Isat = 40). (b) Image obtained with a longer
pulse (50 µs) of a weak probe beam (I/Isat = 0.5). (c) and (d)
Radial density profiles for image (a) (hollow circles ◦) and image (b)
(filled circles •) in linear (c) and logarithmic (d) scales.
tional procedure where one uses a weak probe beam with an
intensity I well below the saturation intensity Isatis prob-
lematic in this context [12]. Indeed for the relevant range
of temperatures (40–150 nK), the atomic thermal wavelength
λTis comparable to the optical wavelength used for probing,
λopt = 780 nm. Consequently in the highly degenerate re-
gion of the gas (D ≡ n(2D)λ2
between neighboring atoms is much smaller than λoptand the
absorption of a weak probe is strongly perturbed by collective
effects. To circumvent this problem we probe the gas with a
short pulse (duration ∼ 2µs) of an intense probe beam (typi-
cally I/Isat= 40 to 100) [14]. The interaction of any given
atom with light is then nearly independent of its neighbors.
High-intensity imaging, which was also used in [11], pro-
vides a faithful measurement of the atomic distribution in the
central region of the trap, where the density is large. How-
ever the quality of the images suffers from a large photon shot
noise, which spoils the detection of the low-density regions
of the cloud (Fig. 1a). In order to probe reliably these re-
gions on which we base our determination of temperature and
chemical potential, we complement the high-intensity imag-
ing procedure by the conventional low-intensity one (Fig. 1b).
In practice for any set of parameters to be studied, we perform
one run of the experiment with high-intensity imaging, and
one with low-intensity imaging immediately after. The repro-
ductibility of the experiment is checked by acquiring several
pairs of images for a given set of experimental parameters.
The procedure for image processing is detailed in the Aux-
iliary Material. In short, for each pair of images it provides
T? 1), the average distance
the temperature T, the chemical potential at center µ and the
density n(r) at any pixel of the image. We assume the atoms
in the excited states of the z motion to be described by the
Hartree–Fock mean-field (HFMF) theory [10, 15–17]; there-
fore, we can self-consistently calculate the populations of the
excited states, and subtract it from n(r) in order to obtain the
density distribution n0(r) in the ground state. The validity of
this procedure was checked by analyzing the results of a quan-
tum Monte Carlo calculation for a range of parameters similar
to ours [18].
We start our thermodynamic analysis by inferring the pres-
sure P(µ,T) of the homogeneous gas from our measure-
ments. Here we adapt to the two-dimensional case the tech-
nique presented in [1] and that has successfully been used
in 3D for Fermi gases [19]. We show that P(µ,T) is di-
rectly related to the atom number N0 =
harmonic trap. Indeed, the local density approximation re-
lates the density n0(r) to the density of the homogenous
gas n(2D)
V (r) = mω2r2/2 the total atom number is
?µ
and using the thermodynamic relation n(2D)
we find N0 = (2π/mω2)P(µ,T). Introducing the dimen-
sionless quantity P = Pλ2
reduced pressure, we then obtain
?¯ hω
where ω is to be replaced by the geometrical mean of ωxand
ωyfor an non-isotropic potential. Our results for the pressure
are summarized in Fig. 2a, where we plot P deduced from
Eq. (2) as function of µ/kBT. The temperatures of the data
entering in this plot range from 40 nK to 150 nK. The fact
that all data points collapse on the same line show that P is a
function of the ratio µ/kBT only, as expected from the scale
invariance of the system. The HFMF theory is represented by
a continuous line in the normal region and by a dotted line in
the superfluid region. The dashed line is the Thomas–Fermi
prediction at zero temperature P = π(µ/kBT)2/˜ g. The grey
area is the parameter subspace accessible for an ideal Bose
gas. Interestingly, although the phase space density D can
take arbitrarily large values in an ideal 2D Bose gas, one can
show that the reduced pressure is ≤ π2/6, where the equality
provides a local criterion for Bose–Einstein condensation in a
trapped ideal gas.
We show in Fig. 2b our measurements for the phase space
density D, obtained from the central density of each cloud. In
wide gray line we plot the prediction of [6], which is in good
agreement with our results. A measurement of D was also
reported in [11] for a quasi-2D Cesium gas, showing a simi-
lar agreement with [6]. To be quantitative we have fitted the
prediction of [6] to our data by multiplying it by a global fac-
tor ζ, and obtained ζ = 0.93 as optimal parameter. This 7%
?n0(r)d2r in our
hom[µ − V (r),T]. For an isotropic harmonic potential
N0=
2π
mω2
−∞
n(2D)
hom(µ?,T)dµ?,
(1)
hom= (∂P/∂µ)T,
T/kBT, which we refer to as the
P(µ,T) =
kBT
?2
N0,
(2)

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3
µ/kBT
Reduced pressure P
a)
125 nK – 150 nK
100 nK – 125 nK
75 nK – 100 nK
40 nK – 75 nK
0
5
10
15
20
-1 -0.500.51
µ/kBT
Phase space density D
b)
0
10
20
30
40
50
-1-0.50 0.51
µ/kBT
Entropy per particle S
c)
-0.5
0
0.5
1
1.5
2
2.5
-1-0.50 0.51
FIG. 2: (Color on line) Equations of state for (a) the reduced pressure P, (b) the phase space density D and (c) the entropy per particle S. The
Hartree-Fock mean field prediction is plotted in full line and extended in dotted line beyond the expected superfluid transition. The dashed line
indicates the Thomas-Fermi prediction. In (a) the grey area indicates the region of parameter space accessible to an ideal gas. In (b) the thick
grey line indicates the prediction from [6].
discrepancy may be due to residual loss of detectivity in high
density regions. Note that the measurement of the pressure
EOS is much less sensitive to this possible bias since it relies
on the count of the atom number N0over the whole cloud
rather than on the highest value of the spatial density.
From our measurements of P and D we also obtain the
equation of state for the entropy per particle S(µ,T):
S
kB
= 2P
D−
µ
kBT,
(3)
which can be derived starting from the entropy per unit area
s = (∂P/∂T)µ, assuming the EOS for P to be scale invari-
ant [20]. The corresponding result is shown in Fig. 2c. As
expected, S is large in the non-degenerate regime and rapidly
decreases around µ/kBT ≈ 0.15, where the superfluid tran-
sition is expected for our value of ˜ g [21]. Finally S tends
to zero in the Thomas–Fermi regime. Our data points with
the largest phase-space density (µ/kBT > 0.5) correspond to
S = 0.06(1)kBonly. Note that since the BKT transition is of
infinite order, one does not expect any discontinuous change
for P, D or S at the superfluid transition for an infinite homo-
geneous fluid, although the superfluid density jumps suddenly
from 0 to 4/λ2
We now turn to the last part of our study, where we il-
lustrate how to measure the various contributions to the en-
ergy of our trapped 2D gases: potential energy Epin the ex-
ternal trapping potential, kinetic energy of the particles Ek,
and interaction energy between atoms Ei. We first point out
the simple relation Ep = Ek+ Ei, obtained from virial
theorem assuming 2D contact interaction. We can measure
Ep =
need to disentangle the contributions of Ekand Eito the total
energy. This can be done by abruptly switching off interac-
tions at time t = 0. Each particle then undergoes a free har-
monic motion r(t) = cos(ωt)r(0) + sin(ωt)v(0)/ω. The
potential energy after a time t following the switching off of
T[22].
?n0(r)V (r)d2r, from an in situ image, but we still
the trapping laser is given by
Ep(t) = Ep(0)cos2(ωt) + Ek(0)sin2(ωt),
(4)
where we used the fact that the correlation ?r(0) · v(0)? is
zero at thermal equilibrium. Thus we can extract Ek(0) from
the time evolution of Ep, which we obtain from the density
profiles at different times t.
In order to implement this procedure, we perform a “one-
dimensional” ToF by switching off abruptly the laser provid-
ing the confinement along z while keeping the magnetic con-
finement in the xy plane. The gas then expands very fast along
the initially strongly confined direction z, as shown in figures
3a to 3d, and interactions between particles drop to a negli-
gible value after a time of a few ω−1
The subsequent evolution in the xy plane occurs on a longer
time scale given by ω−1∼ 8ms. From Eq. (4) and Ek(0) <
Ep(0), we expect the size of the gas to decrease for t<
which can be understood in simple physical terms. The equi-
librium state of the 2D gas results from a balance between
the trapping potential, which tends to compress the gas, and
the kinetic and interaction energies, which tend to increase its
area. When interaction energy drops to zero the equilibrium
is broken and the gas implodes in the xy plane. A similar 1D
ToF technique was used recently in Boulder with the value
of t fixed at π/2ω [10]. For this particular choice the initial
momentum distribution is converted into position distribution
and can thus be measured accurately [23].
We show in figure 3e an example of measurement of Ep(t)
for a gas with N0= 6.1104, T = 72nK, and µ/kBT = 0.59.
From the contraction of the gas, we infer Ek/Ep= 0.56(3),
from which we deduce Ei/Ep = 0.44(3) using virial theo-
rem. This configuration is thus neither completely in the very
dilute regime (Ei ? Ek ∼ Ep) nor in the Thomas–Fermi
regime (Ek ? Ei ∼ Ep) and contains comparable thermal
and quasi-coherent fractions.
The measurement of Ei is of particular interest in this
case since it gives access to the density fluctuations in the
z, where ω−1
z
∼ 100µs.
∼ω−1,

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4
a)
z
b)
c)
d)
40µm
18
18.5
19
19.5
20
20.5
01
Time of flight in ms
234
Ep/N in nK
e)
FIG. 3: (Color on line) (a) to (d) Side view of a cloud initially in
the 2D regime and expanding along z once the laser providing the
confinement in this direction has been switched off. (a) t = 1ms;
(b) t = 2ms; (c) t = 3ms; (d) t = 4ms. (e) Time evolution of the
potential energy Ep. The different lines represent a fit to the data of
a parabola (solid black line), the time evolution assuming completely
frozen fluctuations (dashed red line) and the one expected for a dilute
non-condensed gas (dash-dotted green line).
gas. Indeed, by definition Ei= (¯ h2˜ g/2m)??n2
parameter F which characterizes the degree to which density
fluctuation are reduced. In the limiting case of a very dilute,
non-condensed gas, one expects F = 2, since ?n2
while in the opposite limit of suppressed density fluctuations
F=1. Since our measurement provides us with Ei, we can
infer the value of F, from the comparison with the quantity
(¯ h2˜ g/2m)??n0(r)?2d2r, calculated using the in situ density
find F = 1.1(1), very close to 1 that would correspond to
completely frozen density fluctuations. Note that this is ob-
tained for a gas still far from the Thomas–Fermi limit since
Ek ∼ Ei. This “early” freezing of density fluctuations is
an important ingredient for the proper operation of the BKT
mechanism. This presuperfluid phase, whose existence was
also inferred by different methods in [10] and [11], consti-
tutes a medium that can support vortices, which pair at the
superfluid threshold.
In conclusion we have presented in this Letter various as-
pects of the thermodynamics of a 2D Bose gas, investigating
first the EOS’s for the pressure, the phase space density and
the entropy. Our results confirm the scale invariance that was
discussed theoretically in [6] and observed in [11] for D. We
point out that the entropy per particle drops notably below
0.1kBbeyond the transition point. With such a low entropy
these 2D Bose gases can constitue excellent coolants for other
quantum fluids such as 2D Fermi gases [24]. We have also
presented a method that allows one to extract the various con-
tributions to the total energy of the system. By applying it to a
degenerate but not fully coherent 2D cloud, we find that den-
0(r)? d2r =
(¯ h2˜ g/2m)F??n0(r)?2d2r, where we have introduced the
0? = 2?n0?2,
profile n0. For the experimental conditions of figure 3e, we
sity fluctuation are nearly frozen, marking the presuperfluid
phase.
We thank Gordon Baym, Yvan Castin, Markus Holzmann,
Werner Krauth, Christophe Salomon, Sandro Stringari and
Martin Zwierlein for helpful discussions. We are grateful to
Benno Rem for his help at an early stage of this project. We
are indebted to Aviv Keshet for letting us use the computer
code that he wrote for experimental control. This work is sup-
ported by IFRAF and ANR (project BOFL).
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5
Auxiliary material
Imaging a dense 2D atomic cloud.
sorption imaging consists in relating the number of missing
photons on a pixel to the number of atoms on this pixel. The
interaction between a probe beam and a single atom is char-
acterized by the absorption cross section σ defined by the re-
lation γ = σI/(¯ hωL), where γ is the photon scattering rate,
I the intensity of the beam on the atoms and ωL/2π its fre-
quency. In the case of a monochromatic resonant beam prob-
ing a two-level atom
The calibration of ab-
γ =Γ
2
I
I + Isat.
(5)
In the limit where I ? Isatthe absorption cross section is
σ0≡ Γ¯ hωL/2Isat.
In practice one must take into account stray magnetic fields,
non-zero linewidth of the probe laser, optical pumping effects,
etc. To model this complex situation, we heuristically replace
Isatby an effective saturation intensity αIsatand Γ by an ef-
fective linewidth βΓ. We then write the number of photons
Npscattered during an imaging pulse of given duration τ
Np≡ γτ =βΓ
2
I
I + αIsatτ,
(6)
or equivalently
σ = σ0
β
α + I/Isat.
(7)
At low intensity Npis proportional to I as in the two-level
case, but with a multiplicative coefficient β/α due (for exam-
ple) to the broadening of the resonance line. At large intensity
the number of scattered photons saturates at βΓτ/2 instead of
Γτ/2, which models a reduction that can be caused by optical
pumping effects, for instance.
We now turn to the description of absorption imaging of a
2D atomic cloud. The imaging process consists in shining a
resonant laser beam on an atomic sample, and in imaging the
transmission of the sample on a camera. In order to relate the
missing photon number to the atomic density n(x,y), we cal-
culate the probability for a photon of the probe beam to reach
a pixel of the camera. We introduce the area A associated to
this pixel in the atomic plane. In the limit where σ ? A a
photon has a probability σ/A to be absorbed by a given atom,
hence a probability Pt= (1−σ/A)Nto be transmitted, where
N is the number of atoms in the area A. Thus we find:
Pt≈ e−σn,
(8)
where we have used n = N/A, assuming that the atomic den-
sity varies smoothly over the pixel size. The intensity of the
beam at the output of the cloud is If= PtIiand we obtain:
?If(x,y)
− ln
Ii
?
= σ n(x,y),
(9)
where σ depends on the effective intensity I on the atoms [Eq.
(7)]. If the optical thickness of the cloud is large, i.e. if the
intensity Ifjust after the plane of atoms is significantly lower
than the intensity Iijust before this plane, the effective in-
tensity I must be determined in a self-consistent manner by
imposing:
If= Ii− nσ(I)I.
(10)
The elimination of the effective intensity I from Eqs. (7)-(10)
yields:
?If
It is interesting to note that even though the derivation in a 2D
system differs from the 3D case, the result is similar to the one
given in [14]. The first member of the right-hand side of Eq.
(11) is dominant in the weak intensity limit, and corresponds
to the 2D analog of the 3D Beer–Lambert law. In the high in-
tensity limit, the second member of the right-hand side dom-
inates. We calibrated α = 2.6(3) using the same method as
in [14]: we performed absorption imaging of clouds obtained
in similar experimental conditions with various intensities Ii
ranging from 0.1Isatto 6Isat, and imposed that these mea-
surements provide the same result for the left-hand side of Eq.
(11). Here we restricted ourselves to low atomic density re-
gions, to ensure that collective effects in the optical response
of the gas were negligible. The calibration of β = 0.40(2)
was performed as in [12], using the HFMF prediction as a fit
to the low-density parts of our atomic distributions, and using
µ, T and β as optimization parameters. In [12] where only
low intensity imaging was used, this calibration provided the
detectivity factor ξ, which is related to the present parameters
α and β by ξ = (15/7)β/α.
Obtaining a density profile from an absorption image.
The confinement potential in the xy plane is essentially pro-
vided by our magnetic trap, but it may also be affected by
some imperfections in the intensity profile of the beam that
freezes the z degree of freedom. These imperfections are re-
vealed by looking at the center of mass oscillations xcm(t)
and ycm(t), shown in Fig. 4a. Whereas the oscillation along
the direction of propagation of the “freezing laser” (x) shows
no deviation with respect to harmonic motion, the oscilla-
tion along y is damped. This is likely caused by irregular-
ities of the transverse intensity profile of the freezing laser.
In order to cope with these defects we have abandoned the
standard technique consisting in making angular average of
the images to produce radial density profiles.
take advantage of the separability of the potential in the xy
plane: V (x,y) = mω2
for the magnetic trapping potential and the irregularities of
the freezing laser. We consider cuts of the measured density
profile along the x direction, measured for various yi’s with
i = 1,...,q. In practice, we consider the q = 31 central
lines of our images. We expect that two cuts corresponding
to y1and y2coincide, provided we shift the second one by
nσ0β = −αln
Ii
?
+Ii− If
Isat
.
(11)
Instead we
xx2/2 + U(y), where U(y) accounts

Page 6

6
-5
0
5
00.1
holdtime in s
0.20.3
ycmin µm
b)
-5
0
5
xcmin µm
a)
-160 160
10
20
30
yi
U in nK
c)
FIG.4: (Coloronline)(a)and(b)Centerofmassoscillations(hollow
circles ◦) along x (a) and y (b). The red lines correspond to a fit with
a sine (a) and a damped sine (b). (c) Reconstructed potential along
the y axis (filled circles •) and a harmonic fit (red line).
making the substitution mω2
In practice we perform a least-square fit to optimize the su-
perposition of the various cuts, taking the numbers U(yi) as
parameters. We use a single set of U(yj) to fit a whole series
of images taken at a given temperature. The robustness of the
procedure is excellent, as shown in Fig. 4b, where we give the
reconstructed potential U(y), with bars corresponding to the
statistical errors of the U(yj)’s for various series of images
acquired at different temperatures.
Analyzing a density profile.
2D cloud, we first take a low-intensity image and then a high-
intensity image in the following run. From the density profile
of the low-intensity image, we determine the temperature T
and the chemical potential µ by fitting the low density region
with the HFMF prediction. Our fitting function takes into ac-
count the residual excitation of the z degree of freedom. The
high-intensity image provides the density profile n(r).
Once T and µ are known, we self-consistently determine
thepopulationoftheexcitedstatesusingthemethoddescribed
in [10, 15], assuming the atoms in the excited states j ≥ 1 of
the z motion to be in the HFMF regime. In practice we restrict
the analysis to the first ten levels. In order to give an estima-
tion of the contribution of the various levels j ≥ 1 to the
total density, we show in Fig. 5a numerical results obtained
by applying this procedure to a numerically generated pro-
file, produced using the prediction [6] with T = 100nK and
µ/kBT = 0.45. This temperature is on the high side of our
experimental range, where the influence of the atoms in the
excited states along z is expected to be the most important.
We plot in Fig. 5a the phase space density of the excited states
D(exc), distinguishing the contribution of the state(s) j = 1,
j = (1,2), j = (1,2,3), etc. For comparison we also plot the
profile D(0)obtained from [6], associated to the atoms in the
ground state. Note that the contribution of the states j > 4
xx2→ mω2
xx2+U(y2)−U(y1).
For each configuration of a
is already negligible. The phase space density associated to
each excited state is lower than 0.5, which justifies to treat the
atoms in these states within the HFMF approximation. The
flattened shape of the density distributions in the central re-
gion is due to the repulsive interaction with the atoms in the
ground state of the z motion.
This procedure also allows us to calculate the effective po-
tential felt by the atoms in j = 0, when the repulsive po-
tential W(r) created by the atoms in j ≥ 1 is taken into
account. Plotting together W(r) and the trapping potential
V (r) (Fig. 5b) we see that W(r) is essentially negligible
(<
∼1nK) and one can thus consider the density n0(r) to be
insensitive to the presence of the atoms in j ≥ 1.
Effect of the finite interaction energy.
√8πa/?zfor the interaction strength assumes that the atoms
are confined in the gaussian single-particle ground state of the
harmonic motion along the z direction. However, because the
interaction energy is not completely negligible compared to
¯ hωz, the state of the z motion at T = 0 is modified by these
interactions, which in turn modifies the coupling strength ˜ g.
To estimate the corresponding effect we used first-order per-
turbation theory to determine the modified ground state ϕ0(z)
of the z motion. We then calculated the coupling strength ˜ g,
which is proportional to?|ϕ0|4dz. We find that ˜ g is reduced
of our densest clouds, comparable to the noise level on our
data.
The expression ˜ g =
by the factor ≈ 1−1.4n(2D)a?z, which is ≈ 0.9 at the center
0.1
1
10
051015202530
Potential in nK
r in µm
b)
V (r)
W(r)
0
0.4
0.8
1.2
D(exc)and D(0)/25
a)
···
D(0)/25
j = 1 only
j = 1 and 2
j = 1,2 and 3
FIG. 5: (Color on line) (a) Phase space density of the ground (solid
red line) and excited state(s) of the z-motion. The n-th line from
the bottom corresponds to the contributions of excited levels 1 to
n. (b) Comparison of the trapping potential (red solid line) and the
repulsive potential created by the excited atoms on the population in
the ground state (blue dotted line).