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arXiv:1708.08903 · Physical Review A

Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli

Nils-Eric Guenther, Pietro Massignan, Alexander L. Fetter

Published Dec 6, 2017• 39 citations
arXivHTMLPDF

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 $\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}/(2MR)$, 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 $\ensuremath{\chi}({\mathbit{r}}_{12})$, 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.

DOI
10.1103/physreva.96.063608

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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±\else\textpm\fi\ensuremathℏ/(2MR)\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}/(2MR)\ifmmode±\else\textpm\fi\ensuremathℏ/(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 \ensuremathχ(\mathbitr12)\ensuremath{\chi}({\mathbit{r}}_{12})\ensuremathχ(\mathbitr12​), and it crosses over from logarithmic to linear when the intervortex separation \mathbitr12{\mathbit{r}}_{12}\mathbitr12​ 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.
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Here, we investigate the motion of a thin circular quantum vortex filament near the infinite planar surface. The fluid surrounding this surface moves with a non-zero velocity v\bf{v}v, which is parallel to the surface. We study the specific features of this quantum system and show that they are quite suitable for the boundary layer theory. The developed model allows us to calculate the vortex energy spectrum, E=E(p)E = E({\bf p})E=E(p), where p{\bf p}p is the total momentum of a vortex ring. We have demonstrated that this function has complex non-trivial dependence on the velocity v\bf{v}v. It is stated that the inverse effective mass of a vortex under consideration is of a tensorial nature. In certain quantum states, the system shows the possibility of both negative and positive effective mass existing. This study employs a novel quantization method for classical closed vortex filaments, developed by the author earlier.
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In this thesis, we analytically and numerically study two different scenarios of vortex dynamics in a two-dimensional superfluid. The first system we explore is an expanding cluster of like-sign vortices, which recently suggested it could explain observations in a superfluid optomechanics experiment. We employ a recent theoretical model that approximates large collections of vortices as a fluid and present an analytic solution for its dynamics. Analysis suggests that as a cluster expands due to thermal friction, it forms a Rankine vortex (i.e., a top-hat distribution) that expands self-similarly in time regardless of initial vorticity distribution. Numerical simulations of point vortices confirm this universal expanding regime and further demonstrates vortex lattice formation that goes beyond the vortex fluid theory. We analyse correlations that emerge in the resulting lattices and observe decaying disorder as the cluster expands. Furthermore, we analyse experimental data from a quasi-two-dimensional 87Rb Bose-Einstein condensate, which supports the vortex fluid theory’s predictions.The second scenario considers the dynamics of a scattering event where a single quantum vortex interacts with a uniform, circular pinning potential in a steady two-dimensional superfluid flow. We first develop an approximate hydrodynamical model to understand the interaction between the vortex and the pinning potential in the conservative regime and find it is in good agreement with the mean-field Gross-Pitaevskii equation in the low-velocity limit. We further solve the Gross-Pitaevskii equation for higher background velocities to study how the vortex moves around pinning sites, interacts with it, and ultimately becomes pinned. We uncover four different regimes of vortex pinning, each marked with a characteristic emission of sound energy. Finally, we study how vortices become unpinned from a pinning site, revealing large vortex dipoles form to stabilise the system.Our work in this thesis takes significant steps towards advancing knowledge of both the collective dynamics of vortices and their individual interaction with defects upon surface substrates. Our numerical and analytical results opens up new avenues of exploration be it further understanding dissipation mechanisms in two-dimensional superfluids, or elucidating the nature of vortex pinning in systems such as neutron stars.