Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli

TL;DR

This paper explores the unique dynamics of quantized superfluid vortices on cylindrical and annular surfaces, revealing quantum effects on vortex motion, interactions, and potential for experimental observation.

Contribution

It introduces a hydrodynamic model of vortex behavior on cylinders, including quantized velocities and interaction energy crossover, extending understanding of superfluid vortex dynamics to curved geometries.

Findings

Vortices on cylinders have quantized translational velocities.

Interaction energy transitions from logarithmic to linear with separation.

Planar annuli exhibit similar vortex quantization without offset.

Abstract

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 $\pm \hbar/(2MR)$, with $M$ the mass of the superfluid particles and $R$ 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 $\chi({\bf r}_{12})$, and it crosses over from logarithmic to linear when the intervortex separation ${\bf r}_{12}$ 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.