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arXiv:1011.5041 · New Journal of Physics

Instability and vortex ring dynamics in a three-dimensional superfluid flow through a constriction

F Piazza, L A Collins, Augusto Smerzi

Published Apr 8, 2011• 28 citations
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Abstract

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.

DOI
10.1088/1367-2630/13/4/043008

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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 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 use classical field simulations of the homogeneous Bose gas to study the breakdown of superflow due to vortex nucleation past a cylindrical obstacle at finite temperature. Thermal fluctuations modify the vortex nucleation from the obstacle, turning antiparallel vortex lines (which would be nucleated at zero temperature) into wiggly lines, vortex rings, and even vortex tangles. We find that the critical velocity for vortex nucleation decreases with increasing temperature and scales with the speed of sound of the condensate, becoming zero at the critical temperature for condensation.
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.
We propose and numerically validate an experimentally feasible on-demand protocol for the nucleation and manipulation of stable quantum vortex rings in trapped Bose-Einstein condensates. The method relies on sweeping a laser-sheet barrier that locally constricts the superflow and triggers vortex-ring formation. By tuning the barrier height and width, and by scanning the barrier velocity, we identify the onset of periodic generation of vortex rings above the critical velocity and achieve direct, deterministic control over the ring nucleation position, radius, and hence propagation speed. After its formation, optical potentials are applied to reshape the vortex ring, creating clean Kelvin-wave excitations. Our results provide a foundation for systematic studies of three-dimensional vortices in atomic superfluids and open the door to tailored vortex dynamics and interactions, enabling controlled access to quantum turbulence.
Using a velocity formula derived by matched asymptotic expansion, we study the dynamics of a vortex ring in an axisymmetric Bose-Einstein condensate in the Thomas-Fermi limit. The trajectory for an axisymmetrically placed and oriented vortex ring shows that it generally precesses in a condensate. The linear instability due to bending waves is investigated both numerically and analytically. General stability boundaries for various perturbed wave numbers are computed. Our analysis suggests that a slightly oblate trap is needed to prevent the vortex ring from becoming unstable.
In the present work, we explore the existence, stability, and dynamics of single- and multiple-vortex-ring states that can arise in Bose-Einstein condensates. Earlier works have illustrated the bifurcation of such states in the vicinity of the linear limit for isotropic or anisotropic three-dimensional harmonic traps. Here, we extend these states to the regime of large chemical potentials, the so-called Thomas-Fermi limit, and explore their properties such as equilibrium radii and inter-ring distance for multi-ring states, as well as their vibrational spectra and possible instabilities. In this limit, both the existence and stability characteristics can be partially traced to a particle picture that considers the rings as individual particles oscillating within the trap and interacting pairwise with one another. Finally, we examine some representative instability scenarios of the multi-ring dynamics, including breakup and reconnections, as well as the transient formation of vortex lines.