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arXiv:2101.06672 · Physical Review A

Superfluid vortex dynamics on an ellipsoid and other surfaces of revolution

M. A. Caracanhas, Pietro Massignan, Alexander L. Fetter

Published Feb 7, 2022• 33 citations
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Abstract

We present a study of superfluid vortex dynamics on general axisymmetric compact surfaces with no holes. Specifically, we develop a general method to transform conformally from the axisymmetric surface to a plane, where we use familiar methods based on the hydrodynamic stream function. Our approach shows that vortices constitute a Hamiltonian dynamical system, with their angular positions on the surface as canonical variables. The dynamical motion conserves both the total energy and the angular momentum, and it features a marked anticorrelation between the vortex speed and the local curvature of the surface.

DOI
10.1103/physreva.105.023307

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We present a study of superfluid vortex dynamics on general axisymmetric compact surfaces with no holes. Specifically, we develop a general method to transform conformally from the axisymmetric surface to a plane, where we use familiar methods based on the hydrodynamic stream function. Our approach shows that vortices constitute a Hamiltonian dynamical system, with their angular positions on the surface as canonical variables. The dynamical motion conserves both the total energy and the angular momentum, and it features a marked anticorrelation between the vortex speed and the local curvature of the surface.
The superfluid flow velocity is proportional to the gradient of the phase of the superfluid order parameter, leading to the quantization of circulation around a vortex core. In this work, we study the dynamics of a superfluid film on the surface of a torus. Such a compact surface allows only configurations of vortices with zero net vorticity. We derive analytic expressions for the flow field, the total energy, and the time-dependent dynamics of the vortex cores. The local curvature of the torus and the presence of noncontractable loops on this multiply connected surface alter both the superfluid flow and the vortex dynamics. Finally, we consider more general surfaces of revolution, called toroids.
Recent advances in cold-atom platforms have made real-time dynamics accessible, renewing interest in the motion of superfluid vortices in two-dimensional domains. Here we show that the energy and the trajectories of arbitrary vortex configurations may be computed on a complicated (curved or bounded) surface, provided that one knows a conformal map that links the latter to a simpler domain (like the full plane, or a circular boundary). We also prove that Hamilton’s equations based on the vortex energy agree with the complex dynamical equations for the vortex dynamics, demonstrating that the vortex trajectories are constant-energy curves. We use these ideas to study the dynamics of vortices in a two-dimensional incompressible superfluid with an elliptical boundary, and we derive an analytical expression for the complex potential describing the hydrodynamic flow throughout the fluid. For a vortex inside an elliptical boundary, the orbits are nearly self-similar ellipses.
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.
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.
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.
We develop a neutral vortex fluid theory on closed surfaces with zero genus. The theory describes collective dynamics of many well-separated quantum vortices in a superfluid confined on a closed surface. Comparing to the case on a plane, the covariant vortex fluid equation on a curved surface contains an additional term proportional to Gaussian curvature multiplying the circulation quantum. This term describes the coupling between topological defects and curvature in the macroscopic level. For a sphere, the simplest nontrivial stationary vortex flow is obtained analytically and this flow is analogous to the celebrated zonal Rossby-Haurwitz wave in classical fluids on a nonrotating sphere. For this flow the difference between the coarse-grained vortex velocity field and the fluid velocity field generated by vortices is solely driven by curvature and vanishes in the corresponding vortex flow on a plane when the radius of the sphere goes to infinity. Published by the American Physical Society 2024