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arXiv:physics/0508082 · Physical Review A

Dynamics of a relative superflow between a Bose-Einstein condensate and the thermal cloud

Łukasz Zawitkowski, Mariusz Gajda, Kazimierz Rza̧żewski

Published Oct 2, 2006• 10 citations
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Abstract

We use the classical fields approximation to study a translational flow of the condensate with respect to the thermal cloud in a weakly interacting Bose gas confined in a three-dimensional box. We study both subcritical and supercritical relative velocity cases and analyze in detail a state of stationary flow which is reached in the dynamics. This state corresponds to the thermal equilibrium, which is characterized by the relative velocity of the condensate and the thermal cloud. We observe two processes---re-thermalization and drag, both of which lead to a reduction of a relative velocity of the superflow. Yet only the drag process, which is observed above the critical velocity ${v}_{cr}$, results in transferring a Bose-Einstein condensate to a slower moving mode. In this case the relative velocity of the flow suddenly drops to a value close to zero. Finally, we report the critical velocity to be for our parameters ${v}_{cr}=0.21c$ for the initial condition and ${v}_{cr}^{eq}=0.12c$ for re-thermalized superflow ($c$ being the Landau speed of sound), which is strikingly lower than Landau critical velocity, yet consistent with experiments.

DOI
10.1103/physreva.74.043601

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The flow of a uniform Bose gas at speeds greater than the Landau critical velocity vc{v}_{c}vc​ does not necessarily destroy superfluidity, but rather needs only to lead to a decrease of the superfluid mass density ${\ensuremath{\rho}}_{s}$. Analyzing a weakly interacting Bose gas with a finite-range interparticle interaction that leads to a Landau critical velocity at nonzero quasiparticle momentum, we explicitly construct the (nonuniform) condensate for fluid flow faster than vc{v}_{c}vc​ and calculate the accompanying decrease in ${\ensuremath{\rho}}_{s}$. We briefly comment on the relation of the physics to other problems in superfluids, e.g., solitons and vortices in Bose-Einstein condensates, and critical currents in superconductors.
We present measurements of the critical velocity for vortex shedding in a highly oblate Bose-Einstein condensate with a moving repulsive Gaussian laser beam. As a function of the barrier height V0{V}_{0}V0​, the critical velocity vc{v}_{c}vc​ shows a dip structure having a minimum at ${V}_{0}\ensuremath{\approx}\ensuremath{\mu}$, where $\ensuremath{\mu}$ is the chemical potential of the condensate. At fixed ${V}_{0}\ensuremath{\approx}7\ensuremath{\mu}$, we observe that the ratio of vc{v}_{c}vc​ to the speed of sound cs{c}_{s}cs​ monotonically increases for decreasing $\ensuremath{\sigma}/\ensuremath{\xi}$, where $\ensuremath{\sigma}$ is the beam width and $\ensuremath{\xi}$ is the condensate healing length. We explain our results with the density reduction effect of the soft boundary of the Gaussian obstacle, based on the local Landau criterion for superfluidity. The measured value of vc/cs{v}_{c}/{c}_{s}vc​/cs​ with our stiffest obstacle is about 0.4, which is in good agreement with theoretical predictions for a two-dimensional superflow past a circular cylinder.
We consider the setup employed in a recent experiment (Ramanathan et al 2011 Phys. Rev. Lett. 106 130401) devoted to the study of the instability of the superfluid flow of a toroidal Bose–Einstein condensate in the presence of a repulsive optical barrier. Using the Gross–Pitaevskii mean-field equation, we observe, consistently with what we found in Piazza et al (2009 Phys. Rev. A 80 021601), that the superflow with one unit of angular momentum becomes unstable at a critical strength of the barrier and decays through the mechanism of phase slippage performed by pairs of vortex–antivortex lines annihilating. While this picture qualitatively agrees with the experimental findings, the measured critical barrier height is not very well reproduced by the Gross–Pitaevskii equation, indicating that thermal fluctuations can play an important role (Mathey et al 2012 arXiv:1207.0501). As an alternative explanation of the discrepancy, we consider the effect of the finite resolution of the imaging system. At the critical point, the superfluid velocity in the vicinity of the obstacle is always of the order of the sound speed in that region, v barr = c l . In particular, in the hydrodynamic regime (not reached in the above experiment), the critical point is determined by applying the Landau criterion inside the barrier region. On the other hand, the Feynman critical velocity v f is much lower than the observed critical velocity. We argue that this is a general feature of the Gross–Pitaevskii equation, where we have v f = c l with being a small parameter of the model. Given these observations, the question still remains open about the nature of the superfluid instability.
Based on the dielectric formalism in the generalized random phase approximation, we generalize the description of a Bose-Einstein condensed gas to allow for a relative velocity between the superfluid and normal fluid. In this model, we determine the critical velocity dynamically as the transition point between stable and unstable dynamics. Unlike the zero temperature case, at finite temperature the relative critical velocity of a dilute Bose-Einstein gas is lower than the sound velocity. The requirement of Galilean invariance essential to obtain an invariant result necessitates the use of a gapless and conserving approximation.