Superfluid fountain effect in a Bose-Einstein condensate

We consider a simple experimental setup, based on a harmonic confinement, where a Bose-Einstein condensate and a thermal cloud of weakly interacting alkali-metal atoms are trapped in two different vessels connected by a narrow channel. Using the classical field approximation, we theoretically investigate the analog of the celebrated superfluid-helium fountain effect. We show that this thermomechanical 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. In the long-time limit, the superfluid component exhibits periodic oscillations reminiscent of the ac Josephson effect obtained in superfluid weak-link experiments.

Figure 1

A cartoon picture showing the idea of the superfluid fountain experiment. A small vessel, connected to a bulb filled with emery powder forming a very fine capillary net, is immersed in a larger vessel containing 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 so that the liquid levels in both vessels are the same (a). When the superfluid helium inside the small vessel is heated, then the liquid level increases above the liquid level in the larger vessel (b). A continuous heating sustains the flow from the colder part of the system to the hotter one, an observation at variance with our ordinary everyday life experience. Eventually liquid helium reaches the top of the small vessel where it forms the helium fountain (c).

Figure 2

Sketch of the experimental time sequence. Solid and dashed lines correspond to linear time ramps $f\left(t\right)$ and $h\left(t\right)$, respectively.

Figure 3

Three-dimensional surface plots of the trapping potential and averaged atomic column density at the different stages of the simulations. The top row shows the initial harmonic trap (left) during the preparation of the initial state. The corresponding atomic density at thermal equilibrium is shown on the right side. The second row shows the double-well trap obtained by raising the barrier at the center of the harmonic trap (left) and the corresponding equilibrium atomic density (right). The third row shows the density 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 condensate ($T=0$) whereas the condensate fraction in the right well is 20% ($T=100$ nK). 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.

Figure 4

Comparison between the combined barrier and channel potentials 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 number of atoms in the vessels has to be increased in order to observe the fountain effect.

Figure 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%, $T=100$ nK) are connected by a channel. The initial number 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 $W_c=2.4{\ell }_{\mathrm{osc}}$ ($10\mu$m). The time interval between the different frames is about $2.5{\tau }_{\mathrm{osc}}$ ($\approx$$15.9$ ms).

Figure 6

Time evolution of the condensate and thermal relative occupation numbers, as given by Eq. (5), in the left and right vessels for two different initial condensate fractions in the right well. The time unit is ${\tau }_{\mathrm{osc}}=6.366$ ms. Top frame: initial right condensate fraction of 20% ($T=100$ nK), final channel width $W_c=0.96{\ell }_{\mathrm{osc}}$ ($\approx$$4\mu$m). Bottom frame: initial right condensate fraction of 50% ($T=83$ nK), final channel width $W_c={\ell }_{\mathrm{osc}}$ ($\approx$$4.2\mu$m). Condensate relative occupation numbers: $n_0^L\left(t\right)$ (thin dotted line) and $n_0^R\left(t\right)$ (thin dashed line). Thermal relative occupation numbers: $n_T^L\left(t\right)$ (thick dotted line) and $n_T^R\left(t\right)$ (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 $n_0\left(t\right)/2$ (thin solid line), whereas the thermal fractions stay roughly constant.

Figure 7

The upper frame shows the time evolution of the condensate (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 ${\tau }_{\mathrm{osc}}=6.366$ ms. The initial condensate fraction in the right vessel is about 50% ($T=83$ nK) and the channel width is $W_c={\ell }_{\mathrm{osc}}\approx 4.2\mu$m. As one can see, the condensate and thermal fractions in each vessel never reach the same level, meaning that the system does not reach thermal equilibrium. The lower frame shows the local chemical potentials 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 $31{\tau }_{\mathrm{osc}}$ (200 ms), a state very close to thermodynamical equilibrium (${\mu }_L\sim {\mu }_R$). The two distinctive features of the helium fountain effect are thus recovered: because the system achieves local thermodynamical equilibrium, a temperature gradient is immediately compensated by a pressure difference, which generates a particle flow.

Figure 8

Time evolution of the relative occupation numbers of the condensate $n_0^{L,R}\left(t\right)$ (thin lines) and of the thermal cloud $n_T^{L,R}\left(t\right)$ (thick lines) in the left (dotted lines) and in the right (dashed lines) vessels. The time unit is ${\tau }_{\mathrm{osc}}=6.366$ ms. The initial condensate fraction in the right well is 50% ($T=83$ nK) and the final width of the channel is $W_c=4.0{\ell }_{\mathrm{osc}}$ ($16.8\mu$m). 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 $n_0\left(t\right)/2$ (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.

Figure 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 ${\tau }_{\mathrm{osc}}$. 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 ${\xi }_k(x,0,0)$ for these modes. The thin horizontal line corresponds to half the width of the channel, $W_c/2$. The initial condensate fraction in the right vessel is about 50% ($T=83$ nK) and the channel width is $W_c=4.0{\ell }_{\mathrm{osc}}$ ($16.8\mu$m).

Figure 10

Time evolution of the relative occupation numbers of the superfluid (thin lines) and normal (thick lines) components in the left (dotted lines) and right (dashed lines) vessels. The time unit is ${\tau }_{\mathrm{osc}}$. The system contains initially 50% ($T=83$ nK) of condensed atoms in the right potential well and the final width of the channel is $W_c=4.0{\ell }_{\mathrm{osc}}$ ($16.8\mu$m). The thin and thick solid lines show half the total 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 oscillates back and forth between the two vessels around a mean value that is half the total superfluid fraction $n_S\left(t\right)/2$ (thin solid line). The thick solid line represents half the total normal fraction $[1-n_S\left(t\right)]/2$, which is never reached by the left and right normal components during the time scale of the simulation.

Figure 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 vessel and about 20% ($T=100$ nK) in the right vessel. The final channel width is $W_c=2.4{\ell }_{\mathrm{osc}}$ ($10\mu$m). The time interval between the frames is about $2.5{\tau }_{\mathrm{osc}}$ ($15.9$ ms). As clearly seen, the superfluid component oscillates back and forth between the two vessels while the normal component is essentially trapped in the hotter right vessel.