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Physical review. B, Condensed matter

Interactions of vortices with boundaries in superfluid films

Joseph L. McCauley, Clinton W. Allen

Published May 1, 1982• 6 citations
Publisher

Abstract

We present a general method for calculating vortex creation and annihilation rates at arbitrary boundaries in two-dimensional superfluids and give explicit results for the cases of parallel film edges and cylinders. We explain why cylindrical boundaries may provide a reasonable description of vortex shedding and pinning in regions of a superfluid film where the superfluid density is reduced locally to zero, for example, by substrate imperfections. The relation of our work to other recent contributions is discussed, and the current status of the comparison between theoretical predictions and experimental results for the decay of persistent currents in superfluid films is discussed and clarified.

DOI
10.1103/physrevb.25.5680

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We present a general method for calculating vortex creation and annihilation rates at arbitrary boundaries in two-dimensional superfluids and give explicit results for the cases of parallel film edges and cylinders. We explain why cylindrical boundaries may provide a reasonable description of vortex shedding and pinning in regions of a superfluid film where the superfluid density is reduced locally to zero, for example, by substrate imperfections. The relation of our work to other recent contributions is discussed, and the current status of the comparison between theoretical predictions and experimental results for the decay of persistent currents in superfluid films is discussed and clarified.
An exact analytic solution for the dynamics of vortex pairs is obtained for rapid temperature quenches of a superfluid film starting from the line of critical points below the critical temperature T(KT). An approximate solution for quenches at and above above T(KT) provides insights into the origin of logarithmic transients in the vortex decay, and is in general agreement with recent simulations of the quenched XY model. These results confirm that there is no "creation" of vortices whose density increases with the quench rate as predicted by the Kibble-Zurek theory but only monotonic decay of the thermal vortices already present at the initial temperature.
We characterize the mechanisms of vortex pinning in a superfluid thin film described by the two-dimensional Gross-Pitaevskii equation. We consider a vortex "scattering experiment" whereby a single vortex in a superfluid flow interacts with a circular, uniform pinning potential. By an analogy with linear dielectrics, we develop an analytical hydrodynamic approximation that predicts vortex trajectories, the vortex fixed point and the unpinning velocity. We then solve the Gross-Pitaevskii equation to validate this model, and build a phase portrait of vortex pinning. We identify two different dynamical pinning mechanisms marked by distinctive phonon emission signatures: one enabled by acoustic radiation and another mediated by vortex dipoles nucleated within the pin. Relative to obstacle size, we find that pinning potentials on the order of the healing length are more effective for vortex capture. Our results could be useful in mitigating the negative effects of drag due to vortices in superfluid channels, in analogy to maximizing supercurrents in type-II superconductors.
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.
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.
The coupling of superfluid film to a moving vortex is a gauge coupling entirely dictated by topology. From the definition of a linking number, one can define a gauge field ${\mathrm{A}}^{\mathrm{\ensuremath{\mu}}}$ , whose (2+1)-dimensional curl is the vortex three-current ${\mathrm{J}}^{\mathrm{\ensuremath{\mu}}}$ , and to which the superfluid is minimally coupled. We compute the superfluid density and current response to a moving vortex. Exploiting the analogy to (2+1)-dimensional electrodynamics, we compute the effective vortex mass M(\ensuremath{\omega}) and find that it is logarithmically divergent in the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, with a constant imaginary part, yielding a super-Ohmic dissipation in the presence of an oscillating superflow. Numerical integration of the nonlinear Schr\"odinger equation supports these conclusions. The interaction of vortices with impurities coupling to the density also is discussed.