Enhanced heat flow in the hydrodynamic collisionless regime.

Abstract

We study the heat conduction of a cold, thermal cloud in a highly asymmetric trap. The cloud is axially hydrodynamic, but due to the asymmetric trap radially collisionless. By locally heating the cloud we excite a thermal dipole mode and measure its oscillation frequency and damping rate. We find an unexpectedly large heat conduction compared to the homogeneous case. The enhanced heat conduction in this regime is partially caused by atoms with a high angular momentum spiraling in trajectories around the core of the cloud. Since atoms in these trajectories are almost collisionless they strongly contribute to the heat transfer. We observe a second, oscillating hydrodynamic mode, which we identify as a standing wave sound mode.

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Enhanced heat flow in the hydrodynamic-collisionless regime
R. Meppelink, R. van Rooij, J. M. Vogels and P. van der Straten1
1Atom Optics and Ultrafast Dynamics, Utrecht University,
P.O. Box 80,000, 3508 TA Utrecht, The Netherlands
(Dated: August 12, 2009)
We study the heat conduction of a cold, thermal cloud in a highly asymmetric trap. The cloud is
axially hydrodynamic, but due to the asymmetric trap radially collisionless. By locally heating the
cloud we excite a thermal dipole mode and measure its oscillation frequency and damping rate. We
find an unexpectedly large heat conduction compared to the homogeneous case. The enhanced heat
conduction in this regime is partially caused by atoms with a high angular momentum spiraling in
trajectories around the core of the cloud. Since atoms in these trajectories are almost collisionless
they strongly contribute to the heat transfer. We observe a second, oscillating hydrodynamic mode,
which we identify as a standing wave sound mode.
PACS numbers:
The field of Bose-Einstein condensation in dilute
atomic gases provides a fruitful playground to test well-
developed theories of quantum fluids.
ing Bose-Einstein condensates (BECs) can address open
questions relating to the many-body aspects of two-
component quantum liquids, namely the interaction be-
tween the hydrodynamic normal and the superfluid com-
ponent at finite temperatures [1]. After the first realiza-
tion of BEC some pilot experiments have been carried
out, but detailed experiments are missing [2, 3]. This
has to be compared to the case of liquid helium below
the λ point, where many experiments since the 1950s
have added to our understanding of novel phenomena in
quantum liquids, like collective excitations, first and sec-
ond sound, and others. One of the drawbacks of liquid
helium is that the interactions are so strong that a clear
distinction between the two components is difficult.
The reason for the lack of detailed experiments in
BECs to study quantum liquids and in particular the
hydrodynamical aspects of it, is the limited number of
atoms (typically 1–10 million) in the experiments leav-
ing the thermal atoms virtually collisionless. Efforts to
decrease the mean free path by increasing the confine-
ment limits the lifetime of the sample, since the density
is limited by three-body decay. This makes the observa-
tion of sound propagation in a BEC a challenge.
As to theory, hydrodynamical damping of trapped
Bose gases has been described above and below the tran-
sition temperature Tc [1, 4]. These theories yield the
oscillation frequencies and damping rates of several low-
lying modes, where it is assumed that the sample is fully
hydrodynamic in all directions. Experiments on dilute
clouds of cold atoms are generally conducted in highly
asymmetric traps. In these elongated, cigar-shaped ge-
ometries the mean free path of the atoms can become
much shorter than the size of the cloud in the long, axial
direction, but at the same time exceeds the size in the
other, radial directions. In this so-called hydrodynamic-
collisionless regime the system is axially hydrodynamic
Research us-
and radially collisionless. In our setup, described in de-
tail in Ref. [5], we have created BECs containing up to 3
× 108sodium atoms by evaporation of atoms in an ax-
ially strongly decompressed trap with an aspect ratio of
1:65. Hot atoms created in three-body collisions are able
to leave the sample in this highly asymmetric trap, before
they can heat other atoms in an avalanche [6]. The sam-
ple is axially hydrodynamic, but due to the large aspect
ratio collisionless in the radial direction. Such samples
seem ideal for the observation of sound propagation in
the axial direction, since the axial length of the conden-
sates exceeds a few mm. However, neither experiments
nor theoretical descriptions exist to determine if the colli-
sions in the radial direction will affect the damping rates
to a degree that the observation of sound remains elusive.
From a practical point of view, the hydrodynamic-
collisionless regime is of relevance for the realization of
a continuous atom laser by evaporatively cooling a mag-
netically guided atom beam [7]. Here the efficiency of
the cooling process is expected to be limited by the
heat transfer between the hot, upstream and cold, down-
stream parts of the beam.
In this Letter we report the experimental determina-
tion of the heat conduction in a cold, thermal gas above
the transition temperature Tc, which is hydrodynamic
in the axial direction, but collisionless in the radial di-
rections. The heat conduction is determined by locally
heating the cloud and subsequently observing the equili-
bration of the temperature distribution. Two previously
unobserved hydrodynamic modes are reported; a thermal
dipole mode and a standing wave sound mode. Further-
more, by reducing the number of atoms the transition
in the axial direction from the hydrodynamic regime to
the collisionless regime is observed. We find that the heat
conduction is five times stronger than calculations for the
homogeneous case predict.
We measure the heat flow by locally heating the ther-
mal cloud, after which the equilibration is studied. The
heat is induced by exciting the thermal cloud using Bragg
arXiv:0908.1659v1 [cond-mat.quant-gas] 12 Aug 2009

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Hold time (s)
Temp. gradient (a.u.)
a
b
c
FIG. 1: The temperature gradient as a function of the hold time τ for three values of the hydrodynamicity parameter ¯ γ: ¯ γ ≈ 1
(a), ¯ γ ≈ 3 (b), ¯ γ ≈ 7.5 (c). In figure (a) the behavior is collisionless, where the heat propagates through the MT almost
undamped. In Fig. (b) the behavior is neither collisionless nor hydrodynamic and the temperature gradient is nearly critically
damped. In Fig. (c) the behavior is hydrodynamic and the damping rate is therefore lower compared to (b), but a distinctive
oscillatory behavior can be identified. Due to the destructive imaging scheme used each point represents the temperature
gradient of a newly prepared cloud.
scattering with a laser beam aligned perpendicular to the
axial axis of the cloud, which is aimed at its tail and
retro-reflected [8]. The (1/e)-width of the intensity of
this beam is 0.8 mm, which is close to half of the axial
(1/e)-size of the cloud. The asymmetric excitation is cho-
sen since it yields the maximum separation between the
cold, unperturbed part and the heated part of the cloud,
resulting in a long observation time. The laser beam is
detuned 2 nm below of the23NaD2 transition in order
to reduce resonant scattering and prevent superradiant
scattering. The excited particles will locally redistribute
their momentum and energy through collisions with the
other particles, resulting after a few collisions in a local
thermal equilibrium.
The experiments are conducted on a cloud contain-
ing up to 1.3 × 109atoms confined in a clover leaf type
magnetic trap (MT) characterized by the radial trap fre-
quencies ωrad= 2π×95.6 Hz and the axial trap frequency
ωax = 2π × 1.46 Hz at a temperature between 1.2 and
2 µK, which is above Tc ≈ 1 µK. Once a cloud is ex-
cited, it is allowed to rethermalize in the MT during an
adjustable hold time τ, after which the confinement is
turned off and an absorption image is taken after time-
of-flight (TOF). The TOF duration is chosen in such a
way that the optical density will not exceed 3.5, result-
ing in a time-of-flight of 40 ms for the highest number of
atoms.
We introduce a measure for the hydrodynamicity in
the axial direction ¯ γ ≡ γcol/ωax, where the collision
rate γcol= neffσ vrelis the average number of collisions.
Here, the relative velocity vrel =
?8kBT/mπ is the thermal velocity at temperature T and
tion of two bosons with s-wave scattering length a. Fur-
thermore, neff=?n2(? r)dV/?n(? r)dV = n0/√8 for an
√2¯ vth, where ¯ vth =
m is the mass and σ = 8πa2is the isotropic cross sec-
equilibrium distribution in a harmonic potential, where
n0is the peak density. Written in terms of the number
of atoms N = n0
ric mean of the angular trap frequencies ¯ ω3≡ ω2
this results in γcol= Nmσ¯ ω3/(2π2kBT) ≈ 90 s−1for the
highest number of atoms and corresponds to a hydrody-
namicity of ¯ γ<
∼10 in the axial direction. Even at the
highest hydrodynamicity the lifetime of the cloud, lim-
ited by 3-body decay, is more than 60 s, which is more
than sufficient for the experiments described below. Note
that the hydrodynamicity parameter in the radial direc-
tion is due to the anisotropic trap potential ¯ γrad<
and the cloud is in the radially collisionless regime. By
reducing the number of atoms the heat flow through the
thermal cloud can also be measured in the axially colli-
sionless regime.
?2πkBT/?m¯ ω2??3/2and the geomet-
radωax,
∼10/65
The images are analyzed using a least square fit to a
2D Gaussian distribution. In this distribution the radial
size as a function of the axial position is modeled by a
hyperbolic tangent function which adds a gradient to the
width that resembles the asymmetric distribution. The
fit to the 2D distribution yields the temperature gradient,
axial and radial cloud sizes, and the optical density. In
the following we will focus on the temperature gradient,
which is a measure of the imbalance of the temperature
in the cloud. In this analysis we assume that a local
temperature equilibrium is established at all times. Since
this is not a valid assumption in the first few tenths of
milliseconds after excitation especially for lower collision
rates, we cannot accurately describe the data at these
times.
Heating the thermal cloud will also cause a quadrupole
motion of the atoms, since the cloud is excited non-
adiabatically to a higher temperature. Since the resulting
compression and decompression is homogeneous over the
cloud it does not influence the temperature gradient. The

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quadrupole mode, induced by perturbing the magnetic
confinement, has been studied experimentally by Buggle
and coworkers [9]. They found the damping rate and
oscillation frequency of this mode to be in good agree-
ment with a theoretical model [10]. Since the quadrupole
mode damps slower than the thermal dipole mode con-
sidered in this paper, the maximum hold time for most of
our measurement series turns out to be insufficient to ac-
curately determine the damping rate of the quadrupole
mode. Although our results are less accurate, we have
confirmed that the frequency and damping rate of the
quadrupole mode are in agreement with the results re-
ported in Refs. [9, 10].
A series of measurements consists of about 100 shots at
various hold time τ from which the temperature gradient
is determined. The number of atoms and temperature is
determined from an average over all shots, which yields
the hydrodynamicity ¯ γ. We plot the temperature gradi-
ent as a function of τ for three values of ¯ γ in Fig. 1.
Fig. 1(a) is the result of a measurement at small ¯ γ and
shows a slowly damped oscillation, where the temper-
ature gradient after half a trap oscillation has changed
sign. The heated atoms are then at the opposite side of
the cloud with respect to the excitation side, oscillating
a frequency ωd/ωax= 1. We refer to this mode as the
thermal dipole mode, which has not been observed previ-
ously. The oscillation frequency ωdand damping rate Γd
are determined by fitting the data to a damped sinusoidal
and are shown in Fig. 2 as a function of ¯ γ. For small col-
lision rates (¯ γ ≈ 1) the damping rate is proportional to
the collision rate. As a consequence the frequency of the
mode will decrease for increasing ¯ γ until it reaches zero,
when the system is critically damped. In the experiment
we observe oscillatory behavior for ¯ γ<
∼2.5.
For higher values of ¯ γ the temperature gradient as a
function of τ becomes critically damped, as can be seen
in Fig. 1(b). For even higher values of ¯ γ the system be-
comes hydrodynamic and atoms cannot move through
the cloud without colliding frequently. The heat trans-
port will become diffusive, which is a slower process than
the harmonic oscillation. As a consequence we expect the
damping of the temperature gradient to decrease, but re-
main non-oscillatory. A measurement for high ¯ γ is shown
in Fig. 1(c). For τ < 0.1 s, a double exponential decay
can be seen, where the fast decay due to higher order
modes and the slow decay of the lowest thermal dipole
mode can be discriminated from each other due to the
strong inequality of the damping rates. The reduced chi-
squared for all fits are of the order of unity, as can also be
seen from Fig. 1, since the curves go smoothly through
the data points.
The measurements in the hydrodynamic regime can be
analyzed by numerically solving the heat diffusion equa-

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Norm. frequency
Norm. damping rate
Hydrodynamicity
a
b
FIG. 2: The measured normalized damping rate Γd/ωax (a)
and normalized frequency ωd/ωax (b) of the thermal dipole
mode as a function of the hydrodynamicity ¯ γ. The solid line
in (a) is a fit of the data points with ¯ γ > 5 to the solution of
Eq. (1) with κ0 = 6.4. The dashed lines are a guide to the
eye. The vertical error bars show only statistical errors; the
main contribution to the uncertainty in ¯ γ is the uncertainty
in the number of atoms.
tion in the axial direction
cpn(z)∂
∂tT(z,t) =
∂
∂z
?
κ∂
∂zT(z,t)
?
.
(1)
Here the specific heat capacity cp= 7/2, the heat con-
ductivity κ ≡ κ0πvthΣ/(√8σ) with κ0the dimensionless
heat conductivity coefficient and Σ = 2πkBT/(mω2
is the effective cloud surface. The damping rate of the
lowest order solution in this regime is found to be Γd=
0.542κ0/¯ γ. Fitting the measurements for Γdwith ¯ γ > 5
yields κ0=6.4±0.4. This value is a factor of five larger
than the Chapman-Enskog value κ0= 75/64 ≈ 1.17 for
a homogeneous hydrodynamic system [11, 12].
The measurements in the hydrodynamic regime also
show a damped oscillation of the decay of the temper-
ature gradient for τ > 0.1 s (Fig. 1(c)). As this oscil-
lation is only seen in the hydrodynamic regime, where
the collisionless oscillation is completely damped out, we
conclude that it is the result of another hydrodynamic
mode, which we identify as a standing wave sound mode.
In our experiments this sound mode can only be seen for
values of ¯ γ exceeding 5. Using a least-square fit we de-
termine both the oscillation frequency ωs and damping
rate Γs of the sound mode for all values of ¯ γ > 5 as a
function of ¯ γ (see Fig. 4). The measured normalized fre-
quency ωs/ωax≈ 2.1 confirms that the mode differs from
a center-of-mass motion ωax and the quadrupole mode
ωq/ωax≈?12/5 [10].
the hydrodynamic equations [4] in the limit of no damp-
ing and we find ω =
resembles the quadrupole mode, although the standing
wave sound mode is even in the axial velocity vz and
rad)
This sound mode can be found theoretically by solving
?19/5ωax≈ 1.95ωax. This mode

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FIG. 3: Schematic representation of the oscillating, higher
order hydrodynamic mode, a standing wave sound mode. The
vertical lines indicate the velocity nodes, the arrows indicate
the motion of the atoms.
has two nodes in the velocity profile instead of one. A
schematic representation of this mode is given in Fig. 3.
As a consequence, this sound mode contributes to the
temperature gradient and can be observed in Fig. 1(c).
This is the first direct experimental observation of a ther-
mal sound mode in a cold gas.
A rigorous theoretical model to calculate the oscilla-
tion frequency and damping rate of the modes for arbi-
trary hydrodynamicity ¯ γ will be presented in Ref. [13].
The analysis yields κ0 = 5.98 in the hydrodynamic-
collisionless regime, which confirms the experiment. Fur-
thermore, for a linear confinement as is used for magnet-
ically guided atomic beams the enhancement of the heat
flow is even stronger; up to two orders of magnitude. This
result implies that the efficiency of evaporatively cooling
of a linearly guided atomic beam is strongly diminished
due to the large heat flow and questions the feasibility of
realizing a continuous atom laser. The increase of κ0is
found to be caused by two effects. First, the shape of the
density of states is altered, which results in a relatively
lower collision rate and causes the presence of more atoms
with a high transverse energy. The second effect lies in
the presence of atoms with a high angular momentum.
These atoms are in trajectories spiraling around the core
of the cloud. These trajectories are almost collisionless
and cause a very strong contribution to the heat flow.
In conclusion, we have successfully excited and mea-
sured two previously unobserved modes; a thermal dipole
mode and a standing wave sound mode. Observation of
the latter demonstrates both the hydrodynamic behavior
of the cloud and the presence of sound propagation in a
dilute thermal gas. In the hydrodynamic regime we have
measured the heat conduction coefficient κ0 =6.4±0.4,
which is five times higher than calculations for the ho-
mogeneous case predict. This effect is expected to be
even stronger for a linear confined cloud in the axially
hydrodynamic-radially collisionless regime. This result
implies that the efficiency of evaporative cooling in a con-
tinuous atom laser, where the confinement is linear, is
strongly reduced by the large heat transfer between the
hot and cold parts of the beam.
This work is supported by the Stichting voor Fun-
damenteel Onderzoek der Materie “FOM” and by the
Nederlandse Organisatie voor Wetenschaplijk Onderzoek

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Norm. frequency
Norm. damping rate
Hydrodynamicity
a
b
FIG. 4: The measured normalized damping rate Γs/ωax (a)
and normalized frequency ωs/ωax (b) of the hydrodynamic
sound mode as a function of the hydrodynamicity ¯ γ. The
dashed line shows the frequency of this mode in the limit of
no damping, ω =
p19/5.
“NWO”. We are grateful to W. C. Germs for contribut-
ing in the early stages to the theoretical description and
to D. Gu´ ery-Odelin and J. Dalibard for stimulating dis-
cussions.
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