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arXiv:1208.0734 · Journal of Physics B Atomic Molecular and Optical Physics

Critical velocity for a toroidal Bose–Einstein condensate flowing through a barrier

F Piazza, L A Collins, A Smerzi

Published Apr 26, 2013• 42 citations

Abstract

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.

DOI: 10.1088/0953-4075/46/9/095302

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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.

We study superfluid dissipation due to phase slips for a Bose-Einstein condensate flowing through a repulsive barrier inside a torus. The barrier is adiabatically raised across the annulus, while the condensate flows with a finite quantized angular momentum. At a critical height, a vortex moves from the inner region and reaches the barrier to eventually circulate around the annulus. At a higher critical height, an antivortex also enters into the torus from the outer region. Both vortex and antivortex decrease the total angular momentum by leaving behind a $2\ensuremath{\pi}$ phase slip. When they collide and annihilate or orbit along the same loop, the condensate suffers a global $2\ensuremath{\pi}$ phase slip, and the total angular momentum decreases by one quantum. In hydrodynamic regime, the instability sets in when the local superfluid velocity equals the sound speed inside the barrier region.

Using a numerical implementation of the truncated Wigner approximation, we simulate the experiment reported by Ramanathan et al. in Phys. Rev. Lett. 106, 130401 (2011), in which a Bose-Einstein condensate is created in a toroidal trap and set into rotation via a phase imprinting technique. A potential barrier is then placed in the trap to study the decay of the superflow. We find that the current decays via thermally activated phase slips, which can also be visualized as vortices crossing the barrier region in the radial direction. Adopting the notion of critical velocity used in the experiment, we determine it to be lower than the local speed of sound at the barrier, in contradiction to the predictions of the zero-temperature Gross-Pitaevskii equation. We map out the superfluid decay rate and critical velocity as a function of temperature and observe a strong dependence. Thermal fluctuations offer a partial explanation of the experimentally observed reduction of the critical velocity from the phonon velocity.

We study the instability of a superfluid flow through a constriction in three spatial dimensions. We consider a Bose–Einstein condensate at zero temperature in two different geometries: a straight waveguide and a torus. The constriction consists of a broad, repulsive penetrable barrier. In the hydrodynamic regime, we find that the flow becomes unstable as soon as the velocity at the classical (Thomas–Fermi) surface equals the sound speed inside the constriction. At this critical point, vortex rings enter the bulk region of the cloud. The nucleation and dynamics scenario is strongly affected by the presence of asymmetries in the velocity and density of the background condensate flow.

We study the phase slips superfluid dissipation mechanism with a BEC flowing through a repulsive barrier inside a torus. The barrier is adiabatically raised across the annulus while the condensate is flowing with a finite quantized angular momentum. We found that, at a critical height, a vortex reaches the barrier moving radially from the inner region to eventually circulate along the annulus. At a slightly higher barrier, an anti-vortex also enters into the annulus from the outward region. The vortex and anti-vortex decrease the total angular momentum by leaving behind their respective paths a 2{pi} phase slip. When they collide or orbit along the same loop, the condensate suffers a global 2{pi} phase slip and the total angular momentum decreases by one quantum. The analysis is based on numerical simulations of the dynamical Gross-Pitaevskii equation both in two- and three-dimensions, the latter with the experimental parameters of the torus trap recently created at the NIST institute.

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We numerically model experiments on the superfluid critical velocity of an elongated, harmonically trapped Bose-Einstein condensate as reported by [P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)]. These experiments swept an obstacle formed by an optical dipole potential through the long axis of the condensate at constant velocity. Their results found an increase in the resulting density fluctuations of the condensate above an obstacle velocity of

v\approx 0.3

mm/s, suggestive of a superfluid critical velocity substantially less than the average speed of sound. However, our analysis shows that the that the experimental observations of Engels and Atherton are in fact consistent with a superfluid critical velocity equal to the local speed of sound. We construct a model of energy transfer to the system based on the local density approximation to explain the experimental observations, and propose and simulate experiments that sweep potentials through harmonically trapped condensates at a constant fraction of the local speed of sound. We find that this leads to a sudden onset of excitations above a critical fraction, in agreement with the Landau criterion for superfluidity.