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Physical Review A

Superfluid vortex dynamics on a spherical film

Sálvio Jacob Bereta, M. A. Caracanhas, Alexander L. Fetter

Published May 6, 2021• 48 citations
Publisher

Abstract

Motivated by ongoing experimental efforts to make closed Bose-Einstein condensate shells in microgravity environments, this work studies the energy and dynamics of singly quantized vortices on a thin spherical superfluid shell, where the overall vortex charge must vanish (as on any compact surface). For each vortex, stereographic projection yields the corresponding complex potential on the tangent plane. The resulting stream function then provides both the total energy and the dynamics of a system of overall neutral vortices on a spherical film. Although a single-vortex dipole follows a simple dynamical orbit, four vortices can present a variety of situations. We study a few symmetric initial configurations and then focus on the special case of two small vortex dipoles.

DOI
10.1103/physreva.103.053306

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Motivated by ongoing experimental efforts to make closed Bose-Einstein condensate shells in microgravity environments, this work studies the energy and dynamics of singly quantized vortices on a thin spherical superfluid shell, where the overall vortex charge must vanish (as on any compact surface). For each vortex, stereographic projection yields the corresponding complex potential on the tangent plane. The resulting stream function then provides both the total energy and the dynamics of a system of overall neutral vortices on a spherical film. Although a single-vortex dipole follows a simple dynamical orbit, four vortices can present a variety of situations. We study a few symmetric initial configurations and then focus on the special case of two small vortex dipoles.
Understanding the behavior of quantized vortices is essential to gaining insight into diverse superfluid phenomena, from critical-current densities in superconductors to quantum turbulence in superfluids. We observe the real-time dynamics of quantized vortices in trapped dilute-gas Bose-Einstein condensates by repeatedly imaging the vortex cores. The precession frequency of a single vortex is measured by explicitly observing its time dependence and is found to be in good agreement with theory. We further characterize the dynamics of vortex dipoles in two distinct configurations: (i) an asymmetric configuration, in which the vortex trajectories are dynamic and nontrivial, and (ii) a stable, symmetric configuration, in which the dipole is stationary.
Shell-shaped hollow Bose-Einstein condensates (BECs) exhibit behavior distinct from their filled counterparts and have recently attracted attention due to their potential realization in microgravity settings. Here we study distinct features of these hollow structures stemming from vortex physics and the presence of rotation. We focus on a vortex-antivortex pair as the simplest configuration allowed by the constraints on superfluid flow imposed by the closed-surface topology. In the two-dimensional limit of an infinitesimally thin shell BEC, we characterize the long-range attraction between the vortex-antivortex pair and find the critical rotation speed that stabilizes the pair against energetically relaxing towards self-annihilation. In the three-dimensional case, we contrast the bounds on vortex stability with those in the two-dimensional limit and the filled sphere BEC, and evaluate the critical rotation speed as a function of shell thickness. We thus demonstrate that analyzing vortex stabilization provides a nondestructive means of characterizing a hollow sphere BEC and distinguishing it from its filled counterpart.
The static and dynamic properties of vortices in dipolar Bose-Einstein condensates (dBECs) can be considerably modified relative to their nondipolar counterparts by the anisotropic and long-ranged nature of the dipole-dipole interaction. Working in a uniform dBEC, we analyze the structure of single vortices and the dynamics of vortex pairs, investigating the deviations from the nondipolar paradigm. For a straight vortex line, we find that the induced dipolar interaction potential is axially anisotropic when the dipole moments have a nonzero projection orthogonal to the vortex line. This results in a corresponding elongation of the vortex core along this projection as well as an anisotropic superfluid phase and enhanced compressibility in the vicinity of the vortex core. Consequently, the trajectories of like-signed vortex pairs are described by a family of elliptical and oval-like curves rather than the familiar circular orbits. Similarly for opposite-signed vortex pairs their translation speeds along the binormal are found to be dipole-interaction dependent. We expect that these findings will shed light on the underlying mechanisms of many-vortex phenomena in dBECs such as quantum turbulence, vortex reconnections, and vortex lattices.
The static and dynamic properties of vortices in dipolar Bose-Einstein condensates (dBECs) can be considerably modified relative to their nondipolar counterparts by the anisotropic and long-ranged nature of the dipole-dipole interaction. Working in a uniform dBEC, we analyze the structure of single vortices and the dynamics of vortex pairs, investigating the deviations from the nondipolar paradigm. For a straight vortex line, we find that the induced dipolar interaction potential is axially anisotropic when the dipole moments have a nonzero projection orthogonal to the vortex line. This results in a corresponding elongation of the vortex core along this projection as well as an anisotropic superfluid phase and enhanced compressibility in the vicinity of the vortex core. Consequently, the trajectories of like-signed vortex pairs are described by a family of elliptical and oval-like curves rather than the familiar circular orbits. Similarly for opposite-signed vortex pairs their translation speeds along the binormal are found to be dipole-interaction dependent. We expect that these findings will shed light on the underlying mechanisms of many-vortex phenomena in dBECs such as quantum turbulence, vortex reconnections, and vortex lattices. Published by the American Physical Society 2024
Superfluid vortex dynamics on an infinite cylinder differs significantly from that on a plane. The requirement that a condensate wave function be single valued upon once encircling the cylinder means that such a single vortex cannot remain stationary. Instead, it acquires one of a series of quantized translational velocities around the circumference, the simplest being $\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}/(2MR)$, with MMM the mass of the superfluid particles and RRR the radius of the cylinder. A generalization to a finite cylinder automatically includes these quantum-mechanical effects through the pairing of the single vortex and its image in either the top or bottom end of the surface. The dynamics of a single vortex on this surface provides a hydrodynamic analog of Laughlin pumping. The interaction energy for two vortices on an infinite cylinder is proportional to the classical stream function $\ensuremath{\chi}({\mathbit{r}}_{12})$, and it crosses over from logarithmic to linear when the intervortex separation ${\mathbit{r}}_{12}$ becomes larger than the cylinder radius. An appendix summarizes the connection to an earlier study of Ho and Huang for one or more vortices on an infinite cylinder. A second appendix reviews the topologically equivalent planar annulus, where such quantized vortex motion has no offset, but Laughlin pumping may be more accessible to experimental observation.
We study the dynamics of vortices formed in a superfluid film adsorbed on the curved two-dimensional surface of a cone. To this aim, we observe that a cone can be unrolled to a sector on a plane with periodic boundary conditions on the straight sides. The sector can then be mapped conformally to the whole plane, leading to the relevant stream function. In this way, we show that a superfluid vortex on the cone precesses uniformly at fixed distance from the apex. The stream function also yields directly the interaction energy of two vortices on the cone. We then study the vortex dynamics on unbounded and bounded cones. In suitable limits, we recover the known results for dynamics on cylinders and planar annuli.