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The superfluid fountain effect in a Bose-Einstein condensate
Tomasz Karpiuk,1Benoˆıt Gr´emaud,1,2,3Christian Miniatura,1,2,4and Mariusz Gajda 5,6
1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
3Laboratoire Kastler Brossel, Ecole Normale Sup´erieure,
CNRS, UPMC; 4 Place Jussieu, 75005 Paris, France
4Institut Non Lin´eaire de Nice, UMR 6618, UNS,
CNRS; 1361 route des Lucioles, 06560 Valbonne, France
5Instytut Fizyki PAN, Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland
6Faculty of Mathemathics and Sciences, Cardinal Stefan Wyszy´nski University, Warsaw, Poland
(Dated: December 13, 2010)
We consider a simple experimental setup, based on a harmonic confinement, where a Bose-Einstein
condensate and a thermal cloud of weakly interacting alkali atoms are trapped in two different vessels
connected by a narrow channel. Using the classical field approximation, as described in J. Phys.
B40, R1 (2007) and optimized in Phys. Rev. A 81, 013629 (2010) for an arbitrary trapping
potential, we theoretically investigate the analog of the celebrated superfluid helium fountain effect.
We show that this thermo-mechanical effect might indeed be observed in this system. By analyzing
the dynamics of the system, we are able to identify the superfluid and normal components of the
flow as well as to distinguish the condensate fraction from the superfluid component. We show that
the superfluid component can easily flow from the colder vessel to the hotter one while the normal
component is practically blocked in the latter.
I. INTRODUCTION
The experimental discovery of superfluidity in helium
II by Kapitsa [1] and Allen and Misener [2] in 1938 has
triggered a great theoretical interest in this phenomenon.
One of the most spectacular effects related to superflu-
idity of helium II is its ability to flow through narrow
channels with apparently zero viscosity. Extensive stud-
ies of this system were very important for the founda-
tion of the theory of Bose and Fermi quantum liquids.
In this system however, even at the lowest temperatures,
the strong interactions between the helium atoms deplete
the population of the Bose-Einstein condensate to about
10% of the total mass whereas the superfluid fraction is
almost 100%.
The situation is substantially different with dilute ul-
tracold atomic gases. The first implementation of a
Bose-Einstein condensation [3, 4] in alkali atoms has
opened new possibilities to explore Bose quantum liq-
uids at much higher level of control. Indeed, contrary
to liquid helium, large condensate fractions are routinely
obtained with dilute atomic gases as the atoms are very
weakly interacting. To date, many phenomena previ-
ously observed in liquid helium below the lambda point
have found their experimental counterpart with ultracold
alkali gases even if the experimental evidence of superflu-
idity in atomic condensates has been a very challenging
task. One of the main signatures of superfluid flow is the
generation of quantized vortices when the system is set
into rotation. After many efforts such quantized vortices,
and also arrays of vortices, were observed in atomic con-
densates [5–7]. Observation of the first sound [8, 9], of
scissor modes [10] or of the critical velocity [11] beyond
which the superfluid flow breaks down, are other exam-
ples of the manifestation of this spectacular macroscopic
quantum phenomenon in trapped ultracold atomic sys-
tems.
In addition to the above-mentioned properties, helium
II exhibits also a very unusual feature related to the flow
of heat. Variations of temperature propagate in this sys-
tem in a form of waves known as the second sound. Both
these extraordinary features, i.e. non viscous flow and
unusual heat transport, manifest themselves in full glory
in the helium fountain effect, called also the thermo-
mechanical effect. Its first observation was reported by
Allen and Jones [12]. In their original setup, the lower
part of a U-tube packed with fine emery powder was im-
mersed into a vessel containing liquid helium II. A tem-
perature gradient was created by shining a light beam
on the powder which got heated due to light absorption.
As a result of the temperature gradient, a superfluid flow
is generated from the cold liquid helium reservoir to the
hotter region. This flow can be so strong that a jet of he-
lium is forced up through the vertical part of the U-tube
to a height of several centimeters, hence the fountain ef-
fect name.
Up to now, there exists many different experimental
implementations of this spectacular effect and one of
them is shown in Fig.1. A small vessel, connected to a
bulb filled with emery powder forming a very fine capil-
lary net, is immersed in the container of liquid helium II.
When the electric heater is off the superfluid liquid flows
freely through the capillary net in the bulb and fills in the
small vessel as shown in panel (A) of Fig.1. If now the su-
perfluid helium inside the small vessel is heated then the
level of the liquid in the vessel increases above the level
of the liquid in the big container, see panel (B). A con-
tinuous heating sustains the flow from the colder part of
the system to the hotter one, an observation at variance
arXiv:1012.2225v1 [cond-mat.quant-gas] 10 Dec 2010
2
with our ordinary everyday life experience. Eventually
liquid helium reaches the top of the small vessel where it
forms the helium fountain, see panel (C).
The explanation for this counter-intuitive thermo-
mechanical effect is closely related to the notion of the
second sound and to the two-fluid model developed by
Tisza and Landau [14, 15]. This approach assumes the
existence of two co-existing components of the liquid he-
lium: the superfluid and the normal one. The normal
component is viscous and can transport heat. On con-
trary, the superfluid component has no viscosity and can-
not transport heat. Because it is viscous, the normal
component cannot flow through the capillary net but the
superfluid can. Heat transport is thus forbidden because
it can only be carried by the normal component. As a
consequence, the system cannot reach thermal equilib-
rium and the temperature in the reservoir keeps smaller
than the temperature in the small vessel. Only mechani-
cal equilibrium is allowed, i.e. the chemical potentials in
both vessels have to equilibrate. Heating of the superfluid
component inside the small vessel leads to a reduction of
the chemical potential per unit mass in this vessel and
this reduction is compensated by the flow of the super-
fluid component from the reservoir.
The two-fluid model for helium II assumes a local ther-
mal equilibrium which signifies a hydrodynamic regime
where the collision time is the shortest time scale. If
this is indeed the case for superfluid helium II, which
is a strongly interacting system, it is generally not for
trapped ultracold dilute atomic gases where reaching this
regime proves extremely difficult. For example, second
sound has only been observed recently [16]. As a con-
sequence, the usual two-fluid model fails to apply and,
up to our knowledge, there are no theoretical predictions
about heat transport in weakly-interacting atomic con-
densates. It is not even obvious if an effect similar to the
helium fountain can be observed in these dilute atomic
systems. This is probably why the analog of this classic
textbook phenomenon has not yet been studied in the
context of ultracold trapped atomic gases.
Nevertheless the question of the nature of heat trans-
port in these weakly-interacting atomic condensates
seems to be well posed. There are not many experi-
ments where a non-equilibrium transfer of atoms related
to temperature differences have been studied. We should
recall here the experiment of the MIT group, where dis-
tillation of a condensate was observed [17]. The authors
studied how the superfluid system ‘discovers’ the exis-
tence of a dynamically-created global minimum of the
trapping potential and how the system gets to this mini-
mum. Theoretical studies of the corresponding 1D situa-
tion suggested different dynamical behaviors of the ther-
mal fraction and of the superfluid component which, in
some sense, resemble the fountain effect [18].
In the present work we theoretically study the non-
equilibrium dynamics of a Bose-Einstein condensate
which is driven by a temperature gradient. We will show
that an effect qualitatively very similar to the helium
fountain may be observed in experiments with trapped
ultracold dilute atomic gases.
FIG. 1. A cartoon picture showing the idea of the superfluid
fountain experiment.
The paper is organized as follows: In Sec. II, we briefly
introduce the classical field approximation. Sec. III de-
scribes the system under consideration and our numerical
procedure. Then in Sec. IV we present and analyze our
numerical data. We show that the thermo-mechanical ef-
fect is indeed present in our system and we highlight the
importance of distinguishing between the superfluid, nor-
mal, condensate and thermal components of the system.
Finally, we give in Sec. V some concluding remarks.
II. CLASSICAL FIELDS APPROXIMATION
There exist different effective methods to describe and
study dynamical effects in condensates at nonzero tem-
perature. For example, the Zaremba-Nikuni-Griffin for-
malism assumes a splitting of the system into a conden-
sate and a thermal cloud [19] whereas different versions of
the classical fields method describe both the condensate
and the thermal cloud by a single Gross-Pitaevskii equa-
tion [20–24]. Here, we will use the classical fields method
as described in [25] and optimized in [26] for arbitrary
trapping potentials. Since this latter paper describes
quite extensively how to prepare the classical fields in a
thermodynamic equilibrium state characterized by tem-
perature T, total number of atoms N, condensate frac-
tion N0/N (N0being the number of condensed atoms),
and scattering length a, we will just briefly describe here
the main ingredients of this approach.
A. Formalism
We start with the usual bosonic field operator ˆ
Ψ(r, t)
which annihilates an atom at point rand time t. It obeys
the standard bosonic commutation relations:
hˆ
Ψ(r, t),ˆ
Ψ+(r0, t)i=δ(r−r0)
[ˆ
Ψ+(r, t),ˆ
Ψ+(r0, t)] = 0
[ˆ
Ψ(r, t),ˆ
Ψ(r0, t)] = 0,(1)
3
and evolves according to the Heisenberg equation of mo-
tion:
i~∂
∂t ˆ
Ψ(r, t) = −~2
2m∇2+Vtr(r, t)ˆ
Ψ(r, t)
+gˆ
Ψ+(r, t)ˆ
Ψ(r, t)ˆ
Ψ(r, t),(2)
where Vtr(r, t) is the (possibly) time-dependent trapping
potential and g= 4π~2a/m is the coupling constant ex-
pressed in terms of the s-wave scattering length a.
The field operator itself can be expanded in a basis of
one-particle wave functions φα(r), where αdenotes the
set of all necessary one-particle quantum numbers:
ˆ
Ψ(r, t) = X
α
φα(r)ˆaα(t).(3)
In the presence of a trap, a natural choice for the
one-particle modes φαwould be the harmonic oscillator
modes, otherwise one generally uses plane wave states.
The classical fields method is an extension of the Bogoli-
ubov idea to finite temperatures and gives some micro-
scopic basis to the two-fluid model. The main idea is to
assume that modes φαin expansion (3) having an energy
Eαless than a certain cut-off energy Ecare macroscop-
ically occupied and, consequently, to replace all corre-
sponding annihilation operators by c-number amplitudes:
ˆ
Ψ(r, t)'X
Eα≤Ec
φα(r)aα(t) + X
Eα>Ec
φα(r)ˆaα(t).(4)
Assuming further that the second sum in (4) is small and
can be neglected, the field operator ˆ
Ψ(r, t) is turned into
a classical complex wave function:
ˆ
Ψ(r, t)→Ψ(r, t) = X
Eα≤Ec
φα(r)aα(t).(5)
In this way, both the condensate and a thermal cloud of
atoms, interacting with each other, will be described by
a single classical field Ψ(r, t). Injecting (5) into (2), we
obtain the equation of motion for the classical field:
i~∂
∂t Ψ(r, t) = −~2
2m∇2+Vtr(r, t)Ψ(r, t)
+gΨ∗(r, t)Ψ(r, t)Ψ(r, t).(6)
In numerical implementations, one controls a total en-
ergy, a number of macroscopically occupied modes φα
and a value of gN. The energy-truncation constraint
Eα≤Ecis usually implemented by solving Eq. (6) on a
rectangular grid using the Fast Fourier Transform tech-
nique. The spatial grid step determines the maximal mo-
mentum per particle, and hence the energy, in the system
whereas the use of the Fourier transform implies projec-
tion in momentum space.
Equation (6) looks identical to the usual Gross-
Pitaevskii equation describing a Bose-Einstein conden-
sate at zero temperature. However, the interpretation of
the complex wave function Ψ(r, t) is here different. It de-
scribes all the atoms in the system. Therefore, the ques-
tion arises on how to extract all these modes out of the
time-evolving classical field Ψ(r, t). For this purpose, we
follow the definition of Bose-Einstein condensation orig-
inally proposed by Penrose and Onsager [28] where the
condensate is assigned to be described by the eigenvec-
tor corresponding to the dominant eigenvalue of the one-
particle density matrix. This one-particle density matrix
reads:
%(1)(r,r0;t) = 1
NΨ(r, t) Ψ∗(r0, t),(7)
and obviously corresponds to a pure state with all atoms
in the condensate mode. This is because Eq. (7) is the
spectral decomposition of the one-particle density ma-
trix. To extract the modes out of the classical field some
kind of averaging is needed. In a typical experiment, one
generally measures the column density integrated along
some direction. We implement here the same type of
procedure and define the coarse-grained density matrix:
¯%(x, y, x0, y0;t) = 1
NZdz Ψ(x, y, z, t) Ψ∗(x0, y0, z, t),
(8)
from which we extract the corresponding eigenvalues in
order to apply the Penrose-Onsager criterion. We have
tested the ability of this averaging procedure for a clas-
sical field at thermal equilibrium in a harmonic trap by
comparing it to the results obtained when the original
density matrix is averaged long enough over time. With
a 1% accuracy, both methods give the same results.
Solving the eigenvalue problem for the coarse-grained
density matrix (8) leads to the decomposition:
¯%(x, y, x0, y0;t) =
K
X
k=0
nk(t)ϕk(x, y, t)ϕ∗
k(x0, y 0, t),(9)
where the relative occupation numbers nk(t) = Nk(t)/N
of the orthonormal macroscopically occupied modes ϕk
are ordered according to n0(t)≥n1(t)≥(. . .)≥nK(t).
For future convenience, we define the eigenmodes of
the coarse-grained one-particle density matrix which are
normalized to the relative occupation numbers of these
modes and the corresponding one-particle density matrix
¯ρk:
ψk(x, y, t) = rNk
Nϕk(x, y, t),
¯%k(x, y, x0, y0;t) = ψk(x, y , t)ψ∗
k(x0, y0, t),(10)
such that ¯%=PK
k=0 ¯%kand %T=PK
k=1 ¯%k, the conden-
sate being described by ¯%0.
According to the standard definition, the condensate
wave function corresponds to k= 0 and the thermal den-
sity is simply:
ρT(x, y, t) = ¯%(x, y, x, y;t)− |ψ0(x, y , t)|2.(11)
4
In an equilibrium situation, the relative occupation num-
bers nkdo not depend on time. In this case, the to-
tal number of atoms is determined from the smallest
eigenvalue of the one-particle density matrix through
nKN=ncut, where ncut ≈0.46 for the 3D harmonic
oscillator [29], from which one can infer the value of the
interaction strength g. The temperature Tof the sys-
tem is then given by the energy of this highest occupied
mode.
Let us note that any initial state evolving with the
Gross-Pitaevskii equation reaches a state of thermal equi-
librium characterized by a temperature, a total number
of particles and an interaction strength g. However to
obtain an equilibrium classical field for a given set of pa-
rameters, one has to properly choose the energy of the
initial state and the cut-off parameter Ec. This task is
time consuming because the temperature and the num-
ber of particles can only be assigned to the field after
the equilibrium is reached. To speed up the prepara-
tion of the initial equilibrium state, we first solve the
self-consistent Hartree-Fock model (SCHFM) [35]. This
allows us to estimate quite accurately the energy of the
state for a given temperature and particle number. The
detailed description of this procedure can be found in
[26]. Here we only recall the SCHFM equations:
ρ0(r) = 1
g[µ−Vtr(r)−2g ρT(r)] (12)
f(r,p) = e[p2/2m+Ve(r)−µ]/kBT−1−1(13)
ρT(r) = 1
λ3
T
g3/2e[µ−Ve(r)]/kBT(14)
Ve(r) = Vtr(r)+2g ρ0(r)+2g ρT(r) (15)
µ=g ρ0(0) + 2 g ρT(0) + Vtr(0) ,(16)
where
λT=h
√2πmkBT(17)
is the thermal de Broglie wavelength. The g3/2(z) func-
tion is given by the expansion:
g3/2(z) =
∞
X
n=1
zn
n3/2.(18)
The main variables in this approach are the condensate
density ρ0(r) and the phase space distribution function
f(r,p) of thermal component. The thermal density ρT(r)
can be obtained from f(r,p) by integrating over mo-
menta. The effective potential Ve(r) and the chemical
potential µare functions of the condensate density and
of the thermal density. The condensate and thermal den-
sities can be found iteratively for a given number of atoms
and condensate fraction by taking into account that the
total number of atoms is N=Rdr(ρ0(r) + ρT(r)).
The SCHFM is known to work well for the homoge-
neous harmonic trap [36] and for inhomogeneous traps
with small aspect ratio [37]. In the present work we need
the SCHFM not only to speed up the preparation of an
initial state but also because we will use the chemical
potential and the thermal atoms distribution function in
the section IV to explain some aspects of the studied
phenomena.
III. EXPERIMENTAL SYSTEM
Following [30], we consider here a cloud of Na atoms
prepared in the |3S1/2, F = 1, mF=−1istate and
confined in a harmonic trap with trapping frequencies
ωx=ωy= 2π×51Hz and ωz= 2π×25Hz. The scat-
tering length for this system is a= 2.75nm. In subse-
quent calculations, we use the harmonic oscillator length
`osc =p~/mωz= 4.195µm, time τosc = 1/ωz= 6.366ms
and energy osc =~ωzas space, time and energy units
(oscillatory units).
A. Preparation of initial states
The preparation of an initial state in the harmonic
trap Vtr(r) = 1
2m(ω2
xx2+ω2
yy2+ω2
zz2) follows the steps
described in the previous section. An example of such
state is shown in the first row of Fig.3. In this particular
case the temperature of the system is 100nK and the
condensate fraction is about 20%. We also prepared two
more initial states corresponding to different condensate
fractions, temperatures and numbers of atoms, see Table
I. When T= 0, the initial state is simply the ground
N N0/N T [nK]kBT[osc]µ[osc ]
250000 1.0 0. 0. 22.7
250000 0.5 84. 69.1 16.2
250000 0.2 100. 83.7 12.
TABLE I. Numerical values of the condensate fraction N0/N,
temperature T, thermal energy kBT, and chemical potential
µused in our simulations.
state of the Gross-Pitaevskii equation.
In the next step, we split the cloud of atoms into two
approximately equal parts by rising a Gaussian potential
barrier Vb(r, t) = Vb(r)f(t) at the center of the harmonic
trap by means of a linear time-ramp f(t), see Fig.2. Such
a barrier, with height Vband width wb
Vb(r) = Vbe−x2/w2
b,(19)
can be created by optical means using a blue-detuned
laser light sheet perpendicular to the x-direction. The
barrier is ramped at a time t0, chosen at the end of the
initial equilibration phase, and we have fixed the barrier
rising time at τ= 78.54τosc in our simulations. After
this perturbation, we let the system reach again equi-
librium in the double-well trap by evolving the state for
an additional time τ. Finally the system is split into
5
FIG. 2. Sketch of the experimental time-sequence. Solid and
dashed lines correspond to linear time-ramps f(t) and h(t)
respectively.
two separate clouds containing each about 125000 atoms.
In all our simulations, the barrier parameters are fixed
at Vb= 432osc and wb= 2.529`osc (≈10.6µm). The
full width at half-maximum (FWHM) of the barrier is
Wb= 2√ln 2wb= 4.21`osc (≈17.7µm). The height of
the barrier has been chosen much larger than kBTand
the chemical potential µso that both thermal and con-
densed atoms cannot flow through the barrier, see Table
I.
As we can create the equilibrium state in a double-well
potential corresponding to different initial temperatures,
we can also easily prepare our system in a state where
temperatures in both wells are different. This can be
done by replacing the zero temperature component in
the right well by a nonzero temperature cloud as shown
in the third row of Fig.3. The numbers of atoms in each
well are approximately equal. We have designed all steps
of the preparation stage of the initial state of two sub-
systems with different temperatures having in mind a
possible and realistic experimental realization. Only the
last step, i.e. replacing the zero temperature component
in one subsystem by a finite temperature state has to
be done differently in the experiment. Heating only one
subsystem localized in a given well could be done by a
temporal modulation of the well, followed by a thermal-
ization.
B. Opening the channel between the two vessels
Having prepared two subsystems at different temper-
atures separated by the potential barrier, we can now
study their dynamics when a thin channel is rapidly
opened between the two wells. This is done by switch-
ing on the channel potential Vc(r, t) = Vc(r)h(t), where
the linear time-ramp h(t) starts after the equilibration of
the two subsystems created by the barrier, i.e. at time
t0+ 2τ, see Fig.2. Its duration has been fixed to τ /10 in
all our numerical simulations.
From an experimental point of view, there are various
FIG. 3. Three-dimensional surface plots of the trapping po-
tential and averaged atomic column density at the different
stages of the simulations. The upper row shows the ini-
tial harmonic trap (left) during the preparation of the initial
state. The corresponding atomic density at thermal equilib-
rium is shown on the right side. The second row shows the
double-well trap obtained by rising the barrier at the center
of the harmonic trap (left) and the corresponding equilib-
rium atomic density (right). The third row shows the den-
sity of atoms in the double-well trap at zero temperature
(left) and when the temperatures in each well are different
(right). In this example, the left well contains a pure con-
densate (T= 0) whereas the condensate fraction in the right
well is 20% (T= 100nK). The last row shows the two wells
connected through a thin channel (left). The corresponding
atomic density at some stage of the evolution is shown on the
right.
ways to create the channel potential Vc(r). For example,
starting from an harmonic trap, one could use two or-
thogonal sheets of blue-detuned laser light propagating in
the (Oy, O z) plane. These two sheets build together the
barrier described earlier in this section and by putting
two obstacles along their direction of propagation, one
would create two shadows. Their intersection would open
the desired channel between the two wells but the min-
6
imum channel width would then be constrained by the
diffraction effects induced by the two obstacles. However
widths of the order of few µm should be feasible. Alter-
native methods would be to use TE0,1Hermite-Gaussian
modes, or properly designed separate traps [31–34] and
then focus a red-detuned Gaussian beam. The corre-
sponding channel potential would be:
Vg(r) = −Vb
w2
c
w2
c(x)e−(y2+z2)
w2
c(x),(20)
wc(x) = wcq1 + x2/w2
b,
where the Rayleigh length xR=kLw2
cof the chan-
nel laser beam (kLis the laser wavenumber) has been
matched to the barrier parameter wb. For wc= 3.5µm,
one would have wb= 133µm. The sum of the barrier
potential Vb(r) and of the new channel potential Vg(r)
is shown in the left frame of Fig.4. In this case, the
opened channel would have two ”potholes” separated by
a relatively small barrier and these spurious wells would
trap atoms. In order to observe a superfluid flow and the
fountain effect, one would then have to make sure that
the chemical potential µis larger than this small bar-
rier height ≈Vb/5. We have run numerical simulations
(not shown here) and checked that the fountain effect is
indeed present in this case.
As this spurious trapping would unnecessarily compli-
cate (but not kill) our proof-of-principle analysis of the
fountain effect, we have chosen to work with the following
channel potential in all our numerical simulations:
Vc(r) = −Vbe−(y2+z2)/w2
ce−x2/w2
b.(21)
It has the opposite barrier strength Vb, a Gaussian profile
with width wcin the (Oy, O z) plane and same width wb
as the barrier potential along Ox. The sum of Vb(r) and
Vc(r) creates a smooth channel between the two vessels
as it is shown in the right frame of Fig.4.
FIG. 4. Comparison between the combined barrier and chan-
nel potential obtained by using a focused Gaussian laser beam
(left frame) and the one used in our simulations (right frame).
In the first, atoms get trapped in the ”potholes” and the num-
ber of atoms in the vessels has to be increased in order to
observe the fountain effect.
The FWHM-width of the channel is Wc= 2√ln 2wc.
The final shape of the total potential (harmonic trap in-
cluded) is shown in the last row of Fig.3 on the left, while
a typical example of the column density of the evolving
atomic cloud is shown on the right. In our subsequent nu-
merical simulations we will use different channel widths
Wcto compare the behavior of the thermal flow to the
superfluid one.
IV. NUMERICAL RESULTS
The main observations of this paper concern the time
evolution of two dilute atomic clouds at two different tem-
peratures and initially prepared in two different potential
wells (vessels). At a certain time, a ”trench” is dug in
the potential barrier separating the two vessels and the
atoms can flow from one vessel to the other through the
channel which has been opened. For classical systems
one would expect a heat transport from the hotter cloud
to the colder one, followed by a fast thermalization pro-
cess. The hot vessel is the potential well on the right and
it contains only 20% of condensed atoms (T= 100nK).
The left well is the cold vessel and it initially contains a
pure condensate (T= 0). In our simulations, we clearly
see that, shortly after the two vessels are connected, the
condensate is flowing very fast from the left cold vessel
to the right hot vessel as shown in Fig.5. In the six initial
frames we clearly see that the atomic density in the right
hot vessel is increasing significantly while it is decreasing
in the left cold vessel. During the same time there is no
visible transfer of thermal atoms from the hot vessel to
the cold one. Atoms from the cold vessel are rapidly in-
jected into the hot vessel. This scenario clearly has the
flavor of the helium fountain experiment where the su-
perfluid helium is flowing from the colder big vessel to
the smaller hot vessel through a thin net of capillaries
and finally streams through the small hole in the lid to
form a jet. In our case we do not see a true fountain
effect but instead some increase of the atomic density
in the hot vessel. In fact this physical effect could be
easily observed in an experiment using standard imaging
techniques.
One has to note that, in the original helium fountain
experiment, there is always a very big reservoir of super-
fluid atoms. Therefore the fountain effect can persist as
long as the small vessel is heated. In our case the initial
number of atoms in each wells is the same. The reservoir
of cold atoms is thus almost emptied very fast. Then the
situation gets reversed: the right vessel contains more
cold atoms than the left one and the atomic cloud starts
to oscillate back and forth between the two vessels. This
is clearly seen in Fig.5, where frames 6 −11 clearly show
temporal oscillations of the total atomic density between
the two vessels (left column).
7
FIG. 5. Snapshots of the time evolution of the column
atomic densities when the cold left vessel (pure condensate,
T= 0) and the hot right vessel (condensate fraction 20%,
T= 100nK) are connected by a channel. The initial num-
ber of atoms in each vessel is about 125000. The left column
of the different frames shows the total atomic density, the
middle column shows the condensate density and the right
column shows the density of thermal atoms. The channel
width is Wc= 2.4`osc (10µm). The time interval between the
different frames is about 2.5τosc (≈15.9ms).
A. Condensate and thermal component
The above qualitative findings can be quantified. To
this end we first have to split the classical field into con-
densed and thermal components as described in Sec. II.
The evolution of these components is shown in the middle
and the right panels of Fig.5. The flow starts when the
channel between the two vessels is fully opened, which
approximately corresponds to the third frame in Fig.5.
Analyzing the condensate part, we see that its initial
flow is quite turbulent and a series of shock waves ap-
pears (frames 3 – 5). This is because the velocity of
the superfluid component reaches and exceeds the criti-
cal velocity. As a result thermal atoms are produced in
the right well (frame 5, left column) and the condensate
gets fragmented (frame 5, middle column). After this ini-
tial turbulent evolution, the flow becomes laminar. We
have checked that these initial effects are significantly re-
duced when the temperature difference between the two
subsystems is smaller.
A quantitative analysis of the dynamics requires an es-
timation of temperature of both subsystems. In this dy-
namical nonequilibrium situation, the notion of tempera-
ture is questionable. However we can use the condensate
fraction in the left and the right well as an estimate of
the ’temperature’ of both subsystems. To this end, us-
ing Eq.(10), we split the relative occupation numbers of
the one-particle density matrix modes into left and right
components:
nL
k(t) = R0
−∞ dx R∞
−∞ dy ¯%k(x, y, x, y;t),
nR
k(t) = R∞
0dx R∞
−∞ dy ¯%k(x, y, x, y;t).(22)
This gives, for each vessel, the condensate, the thermal
cloud and the total relative occupation numbers: nX
0(t),
nX
T(t) = PK
k=1 nX
k(t) and nX(t) = nX
0(t) + nX
T(t) (X=
L, R).
We have drawn the above quantities in Fig.6 for two
different initial condensate fractions in the right well,
20% (T=100nK) for the top frame and 50% for the bot-
tom frame (T= 84nK). The thin and thick lines corre-
spond to the condensate and thermal fractions respec-
tively. The main observations are the following: (i) The
initial injection of the left condensate at T= 0 into the
right well lasts about 47τosc (300ms) in the upper frame,
and about 31τosc (200ms) in the lower frame; (ii) Af-
ter the initial injection, the condensate fractions in both
wells oscillate with a small amplitude around a mean
value – some condensed atoms flow from one well to the
other; (iii) The thermal components stay almost constant
in both wells.
However, a more detailed analysis shows some ini-
tial increase of the thermal component during the first
8−16τosc (50 −100ms) in the left well which is followed
by a very slow flow of the thermal cloud from the hot to
the cold part of the system. The initial increase of the
thermal component can be easily explained. First of all,
the opening of the channel between the two wells is not
adiabatic and a thermal fraction is excited in the process
– see the first three panels in Fig.5. Secondly, the initial
flow of the condensed component is very fast and turbu-
lent so it is another source of thermal excitations. Finally
a small thermal fraction of atoms is initially present in
the region of the barrier.
To prove that the thermo-mechanical effect is indeed
present in our system, we have to show that mechanical
equilibrium is reached at once whereas thermal equilib-
8
FIG. 6. Time evolution of the condensate and thermal rela-
tive occupation numbers in the left and right vessels for two
different initial condensate fractions in the right well. The
time unit is τosc = 6.366ms. Top frame: initial right con-
densate fraction of 20% (T= 100nK), final channel width
Wc= 0.96`osc (≈4µm). Bottom frame: initial right con-
densate fraction of 50% (T= 83nK), final channel width
Wc=`osc (≈4.2µm). Condensate relative occupation num-
bers: nL
0(t) (thin dotted line) and nR
0(t) (thin dashed line).
Thermal relative occupation numbers: nL
T(t) (thick dotted
line) and nR
T(t) (thick dashed line). As one can see, after
some time, the left and right condensate relative occupation
numbers oscillate around half the total condensate fraction
n0(t)/2 (thin solid line) whereas the thermal fractions stay
roughly constant.
rium is never reached during the considerably long com-
putation time of our simulations. To this end we first
compute and compare the relative condensate and ther-
mal fractions fX
0(t) and fX
T(t) in the left (X=L) and
the right (X=R) vessels:
fX
0(t) = NX
0(t)
NX(t)=nX
0(t)
nX(t),(23)
fX
T(t) = NX
T(t)
NX(t)= 1 −fX
0(t).(24)
The upper frame of Fig.7 shows these quantities for an
initial right condensate fraction of 50% (T= 83nK)
and the thinest channel width considered here, i.e Wc=
`osc ≈4.2µm. It is clearly visible that after 157τosc (1s),
FIG. 7. The upper frame shows the time evolution of the con-
densate (solid line) and of the thermal (dashed line) fractions
in the left vessel (thin lines) and in the right vessel (thick
lines). The time unit is τosc = 6.366ms. The initial conden-
sate fraction in the right vessel is about 50% (T= 83nK) and
the channel width is Wc=`osc ≈4.2µm . As one can see,
the condensate and thermal components in each vessel never
equilibrate meaning that the system does not reach thermal
equilibrium. The lower frame shows the local chemical poten-
tials calculated in the left (thin line) and in the right (thick
line) vessels. As one can see, the system is able to reach
rapidly, in about 31τosc (200ms), a state very close to me-
chanical equilibrium (µL∼µR). The two distinctive features
of the helium fountain effect are thus recovered: mechanical,
but not thermal, equilibrium.
the condensate fraction in the left well is much larger
than in the right well. This situation will hold obvi-
ously even longer. Similarly the thermal components in
both vessels are very different. This signifies that both
subsystems are not in thermal equilibrium. During the
evolution, the initial hot cloud in the right vessel always
remains much hotter then in the left part.
To show that the system (almost) reaches mechanical
equilibrium after a short period of time, we have to con-
sider the chemical potential defined according to (16):
µ(r) = g ρ0(r)+2g ρT(r) + Vtr(r).(25)
At mechanical equilibrium the chemical potential should
be position-independent. For comparison we choose two
positions on opposite sides of the barrier located near the
9
maximum of the initial atomic densities in each wells,
rR= (x, y, z) and rL= (−x, y, z), and we calculate the
corresponding local chemical potentials µL=µ(rL) and
µR=µ(rR). There is however, one technical difficulty.
The condensate and thermal densities are obtained from
the diagonalization of the column-averaged one-particle
density matrix. Therefore, in fact we only know the 2D
densities in the (Ox, Oy ) plane for all eigenmodes. To
estimate the 3D densities, we need to calculate the Oz-
width of each eigenmode along the channel. For this we
average the one-particle density matrix Eq.(7) along Oy
ending up with column densities in the (Ox, Oz ) plane.
We extract the Oz-width wz
k(x) for each mode along
the channel as the FWHM of the corresponding column
densities. The 3D density is then estimated through
ρk(x, 0,0) = ρxy
k(x, 0)/wz
k(x), where ρxy
kis the column
density of the k-th mode in the (Ox, Oy ) plane.
FIG. 8. Time evolution of the relative occupation numbers of
the condensate nL,R
0(t) (thin lines) and of the thermal cloud
nL,R
T(t) (thick lines) in the left (dotted lines) and in the right
(dashed lines) vessels. The time unit is τosc = 6.366ms. The
initial condensate fraction in the right well is 50% (T= 83nK)
and the final width of the channel is Wc= 4.0`osc (16.8µm).
As one can see, after a short initial stage, the right and left
condensate fractions oscillate around a mean value which is
half the total condensate fraction n0(t)/2 (thin solid line).
The thermal part, after a while, stays roughly constant but,
as clearly seen, some part of the thermal cloud flows in phase
with the condensed atoms.
Having the 3D densities, we can calculate the chemical
potentials µLand µRas the average over a few points
located around x=−4.6 and x= 4.6`osc (19.3µm) re-
spectively. The time evolution of these chemical poten-
tials is shown in the lower frame of Fig.7. Although the
curves look a bit ragged, we nevertheless see that the
system rapidly reaches a state very close to mechanical
equilibrium, µL∼µR, in about 31τosc (200ms). In fact
we observe small out-of-phase oscillations of the chemi-
cal potentials caused by the back-and-forth oscillations
of the condensed atoms.
As a main conclusion of the above discussions, we see
that our system does present all the three distinctive
features of the helium fountain experiment: (i) the sys-
tem cannot achieve thermal equilibrium, (ii) it can only
achieve mechanical equilibrium, and (iii) the component
which flows through the very narrow channel connecting
the two vessels at different temperatures does not trans-
port heat.
B. Superfluid and normal component
To show that our system was not reaching thermal
equilibrium, we had to divide the classical field into a
condensate and a thermal component. As the condensate
component corresponds to the dominant eigenvalue of a
coarse-grained one-particle density matrix, the thermal
cloud consists of many modes with relatively small occu-
pation numbers. This coarse-graining procedure splits
the system into many different modes. On the other
hand the standard two-fluid model of the helium fountain
is solely based on the distinction between a superfluid
and a normal component. For liquid helium, which is a
strongly interacting system, there is an essential differ-
ence between the condensate and the superfluid compo-
nent. This difference is much less pronounced in the case
of weakly-interacting trapped atomic condensates, but is
nevertheless noticeable as pointed out in [38] where the
macroscopic excitation of a nonzero momentum mode has
been studied within the classical fields formalism. The
authors showed that not only the condensate but also
phonon-like excitations do participate in the frictionless
flow. Both the condensate part and these phonon modes
thus contribute to the superfluid.
As will be shown in this section, this is also the case
in the system studied here. A careful reader might have
already noticed that in Fig.5 some part of the thermal
component oscillates together with the condensate. This
effect is very small for very narrow channels but is becom-
ing quite pronounced for wider channels. Fig.8 shows the
dynamics of the relative occupation numbers of the con-
densate and of the thermal components when the channel
width is Wc= 4.0`osc (16.8µm), the initial condensate
fraction in the right vessel being 50%. It is clearly visible
that a certain amount of excited atoms is flowing in phase
with the condensate, back and forth from one vessel to
the other.
To explain this behavior, we show in Fig.9 the time
evolution of the relative occupation numbers of the first
seven dominant eigenmodes (in the right well) of the one-
particle density matrix, the largest occupation number
corresponding to the condensate. Apart from the con-
densate, the next two modes in the hierarchy exhibit
very similar, fast and in-phase oscillations and their oc-
cupation numbers are significantly larger than those of
the remaining other modes. These two modes, together
with the condensate, constitute the three largest coherent
‘pieces’ of the system. They might be viewed as three in-
dependent ‘condensates’ each of them characterized by a
particular mode structure, occupation number and heal-
ing length.
10
FIG. 9. Time evolution of the relative occupation numbers of
the seven dominant modes in the right vessel of the system
(top frame). The time unit is τosc. The solid circles, solid
squares, and solid diamonds correspond respectively to the
condensate and to the next two highest occupied modes. The
remaining four thin lines correspond to the next four modes
of smaller occupation numbers. The bottom frame shows the
local healing lengths of these modes. The thin horizontal line
corresponds to the half the width of the channel, Wc/2. The
initial condensate fraction in the right vessel is about 50%
(T= 83nK) and the channel width is Wc= 4.0`osc (16.8µm).
To explain why these three modes can flow freely
from one vessel to the ether, we calculate the local
healing length for each of these modes, ξk(x, 0,0) =
1/p8πaρk(x, 0,0) where xis a distance along the chan-
nel direction, ais the scattering length and ρkis the 3D
density of the mode estimated through the procedure de-
scribed previously. The healing lengths are shown in the
bottom frame of Fig.9. The thin horizontal line corre-
sponds to half the width of the channel, Wc/2. We im-
mediately see that the modes flowing together with the
condensate fulfill the condition:
ξk.Wc
2,(26)
where ξkis the ”typical” healing length of mode k(for
example, taken at the middle of the channel). As a gen-
eral rule, we infer that only modes with a typical heal-
ing length smaller than half the channel width can flow
freely. These modes, condensate included, form the su-
perfluid component. Higher modes, having a typical heal-
ing length larger than Wc/2, cannot fit into the channel
and cannot flow: they form the normal component.
The superfluid fraction and superfluid density are re-
spectively defined as
nS(t) = PkS
k=0 nk(t),
ρS(x, y, t) = PkS
k=0 |ψk(x, y, t)|2,(27)
where kSis the index of the highest occupied one-particle
density matrix eigenmode fulfilling ξk< Wc/2. Anal-
ogously one can define corresponding quantities for the
normal component, i.e. nN(t) and ρN(x, y, t). It is more-
over convenient to split the superfluid and normal frac-
tions into their left and right components nL,R
S,N (t).
FIG. 10. Time evolution of the relative occupation num-
bers of the superfluid (thin lines) and normal (thick lines)
components in the left (dotted lines) and right (dashed lines)
vessels. The time unit is τosc. The system contains initially
50% (T= 83nK) of condensed atoms in the right poten-
tial well and the final width of the channel is Wc= 4.0`osc
(16.8µm). The thin and thick solid lines show half the to-
tal superfluid and normal fractions respectively. The normal
fraction flows smoothly and slowly from the hotter vessel to
the colder one as expected while the superfluid fraction oscil-
lates back and forth between the two vessels around a mean
value being half the total superfluid fraction nS(t)/2 (thin
solid line). The thick solid line represents half the total nor-
mal fraction (1 −nS(t))/2 which is never reached by the left
and right normal components during the time scale of the
simulation.
These quantities are shown in Fig.10. It can be seen
that the normal component flows only very slowly from
the hotter to the colder well as it is expected for the
superfluid fountain effect. Comparison with Fig.8 fur-
ther shows that the normal component, contrary to
the thermal one, does not exhibit any temporal oscil-
lations. In Fig.11, we plot the superfluid column density
ρS(x, y, t) (middle column), the normal column density
ρN(x, y, t) (right column), and the total atomic density
¯ρ(x, y, x, y;t) (left column) for the same parameters as in
Fig.5. The left column is identical as in fig. 5 and is put
here as a reference. One can clearly see that essentially
11
only the superfluid component travels back and forth be-
tween the two vessels. The normal component remains
mainly located in the right hotter vessel and its flow to
the colder left vessel is almost invisible.
To estimate the rate of flow of the superfluid fraction,
we wait for the system to reach its oscillatory regime
and then fit the (damped) oscillations of the superfluid
fraction in the left vessel by:
F(t) = Asin (2πνt +φ)e−γt +Bt +C, (28)
and extract the oscillation frequency νand the oscilla-
tion amplitude Aof the superfluid flow. The maximal
superfluid flux through the channel is FS= 2πAN ν . We
also fit the slow decrease of the normal fraction in the
left vessel by the linear function G(t) = αt +β. The
maximal flux of the normal atoms is then FN=αN. All
these quantities are collected in Tables II and III.
Wc[osc.u.] A ν[Hz] α[s−1]FS[at
ms ]FN[at
ms ]
0.96 0.0254 3.82 0.0201 152 5.03
1.2 0.036 5.27 0.0277 298 6.93
2.4 0.06 12.6 0.0502 1188 12.55
TABLE II. The relevant coefficients obtained from our fitting
procedure and the calculated superfluid and normal rates of
flow. The initial occupation number of the condensate in the
right well is 20% (T= 100nK).
Wc[o.u.] A ν[Hz] α FS[at
ms ]FN[at
ms ]
1.0 0.036 3.8 0.013 215 3.25
2.0 0.078 9.1 0.027 1150 6.75
4.0 0.085 20.1 - 2684 -
TABLE III. The relevant coefficients obtained from our fitting
procedure and the calculated superfluid and normal rates of
flow. The initial occupation number of the condensate in the
right well is 50% (T= 83nK).
Note, that the last row of Table III does not contain
any value for the αcoefficient nor for the corresponding
normal flux FN. This is because, for wider channels,
the rate of flow of the normal component is changing
significantly in time and fitting the decrease by a linear
function is no longer reasonable. In this case, the flow is
fastest at the beginning as it is visible in Fig.10.
We did not include the value of the coefficients B,C,
and βin the Tables, even if they increase the precision
of our fitting procedure, as they are essentially irrelevant
four our considerations. For channel widths Wc≤5`osc ,
the coefficients γturns out to be smaller than the statis-
tical error (γ∼0) and are also not included in Tables II
and III. This observation is in agreement with the fact
that the dynamics takes place in the collisionless regime
as mentioned in the Introduction.
We see that both superfluid and normal flow rates in-
crease with the channel width. Moreover, the superfluid
FIG. 11. Snapshots of the time evolution of the total (left),
superfluid (middle) and normal (right) column densities. The
initial condensate fractions are 100% (T= 0) in the left ves-
sel and about 20% (T= 100nK) in the right vessel. The
final channel width is Wc= 2.4`osc (10µm). The time inter-
val between the frames is about 2.5τosc (15.9ms). As clearly
seen, the superfluid component oscillates back and forth be-
tween the two vessels while the normal component is essen-
tially trapped in the hotter right vessel.
flow rate is in all cases larger by two or three orders of
magnitude then the normal one. We expect that the nor-
mal component behaves like a classical fluid. Therefore,
its flow rate should correspond to the flux of atoms dis-
tributed initially according to the classical phase space
distribution as obtained from the SCHFM equations de-
scribed in the second section. Our SCHFM calculations
indeed give a value very close to the one obtained from
the classical fields dynamics. For example the flux of
thermal atoms for a system initially prepared with 50%
12
of condensed atoms in the hotter vessel and for a final
channel width Wc= 6.0`osc (25.2µm) is found to be
FN≈115.4 atoms/ms. The classical field approxima-
tion gives a similar result FN≈182.5 atoms/ms. In-
deed, the very slow transfer of the normal component is
a phase space distribution effect – a very small fraction of
thermal atoms have velocities aligned along the channel.
On the contrary, the superfluid component is built from
coherent modes. The coherence of these modes extends
over the entire two vessels and is established on a short
time scale of about 16τosc (100ms).
V. CONCLUSION
In conclusion, we have shown that the analog of the
thermo-mechanical effect, observed in the celebrated su-
perfluid helium II fountain, could be also observed with
present-day experiments using weakly-interacting degen-
erate trapped alkali gases. We have proposed a realis-
tic experimental setup based on a standard harmonic
confinement potential and analyzed it with the help of
the classical fields aproximation method. The trapped
ultracold gas is first split in two subsytems thanks to
a potential barrier. Each of the two independent sub-
systems achieve their own thermal equilibrium, the final
temperature in the two vessels being different. At a later
time, a communication channel is opened between the
two vessels, and the atoms are allowed to flow from one
vessel to the other. We have shown that the transport
of atoms between the two subsystems prepared at two
different temperatures exhibits the two main features of
the superfluid fountain effect: the mechanical equilibrium
is obtained almost instantly while the thermal equilib-
rium is never reached. We have further shown that the
superfluid component of this system is composed of all
eigenmodes of the one-particle density matrix having a
sufficiently small healing length that can fit into the com-
munication channel. The superfluid flow is at least two
orders of magnitude faster than the flow of the normal
component. The slow flow of the normal component can
be understood as a phase space effect.
ACKNOWLEDGMENTS
The Authors wish to thank Miros law Brewczyk, Bj¨orn
Hessmo, Cord M¨uller and David Wilkowski for discus-
sions and valuable comments. TK and MG acknowledge
support from the Polish Goverment research funds for
the period 2009-2011 under the grant N N202 104136.
Some of the present results have been obtained using
computers at the Department of Physics of University
of Bia lystok (Poland). ChM and BG acknowledge sup-
port from the CNRS PICS Grant No. 4159 and from
the France-Singapore Merlion program, FermiCold grant
No. 2.01.09. The Centre for Quantum Technologies is
a Research Centre of Excellence funded by the Ministry
of Education and the National Research Foundation of
Singapore.
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