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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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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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