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Journal of Physics A Mathematical and Theoretical

Accurate numerical determination of a self-preserving quantum vortex ring

Simone Zuccher, Marco Caliari A

Published Nov 12, 2020• 3 citations

Abstract

Abstract We compute simultaneously the translational speed, the magnitude and the phase of a quantum vortex ring for a wide range of radii, within the Gross–Pitaevskii model, by imposing its self preservation in a co-moving reference frame. By providing such a solution as the initial condition for the time-dependent Gross–Pitaevskii equation, we verify a posteriori that the ring’s radius and speed are well maintained in the reference frame moving at the computed speed. Convergence to the numerical solution is fast for large values of the radius, as the wavefunction tends to that of a straight vortex, whereas a continuation technique and interpolation of rough solutions are needed to reach convergence as the ring tends to a disk. Comparison with other strategies for generating a quantum ring reveals that all of them seem to capture quite well the translational speed, whereas none of them seems to preserve the radius with the accuracy reached in the present work.

DOI: 10.1088/1751-8121/abc9d9

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Abstract We compute simultaneously the translational speed, the magnitude and the phase of a quantum vortex ring for a wide range of radii, within the Gross–Pitaevskii model, by imposing its self preservation in a co-moving reference frame. By providing such a solution as the initial condition for the time-dependent Gross–Pitaevskii equation, we verify a posteriori that the ring’s radius and speed are well maintained in the reference frame moving at the computed speed. Convergence to the numerical solution is fast for large values of the radius, as the wavefunction tends to that of a straight vortex, whereas a continuation technique and interpolation of rough solutions are needed to reach convergence as the ring tends to a disk. Comparison with other strategies for generating a quantum ring reveals that all of them seem to capture quite well the translational speed, whereas none of them seems to preserve the radius with the accuracy reached in the present work.

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We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how vortex knots break up into vortex rings.

We study vortex ring solutions of the Gross-Pitaevskii equation for Bose-Einstein condensates confined in toroidal traps. Analogously to the Feynman-Onsager ansatz for a single vortex line, we assume a condensate wavefunction that describes a quantized vortex ring. This leads to a modified Gross-Pitaevskii equation that incorporates vortex ring solutions. By numerically solving this equation, we have obtained the density profiles and the vortex core structure. We have also calculated the nucleation energy of a vortex ring, and have analyzed its dependence on the ring position. Finally, a qualitative description of the vortex ring dynamics is presented.

The time evolution and scattering interactions of superfluid vortex rings are studied using the Gross-Pitaevskii model, in order to consider the maintenance and generation of vorticity in superfluid flows, and to compare ring dynamics in Navier-Stokes and superfluids. For a single ring, we verify previous analytic results on stability and the value of the translation velocity. When two vortex rings scatter, we generically observe merger and subsequent breakup in cases of close approach. Depending on the initial conditions, the final state can contain zero, two, or more final rings.

Vortex structures in dilute quantum fluids are studied using the time-independent Gross-Pitaevskii equation. The velocity and momentum of vortex rings with multiple circulation are determined and their core structures analysed. For flow around a spherical object, we study the encircling and pinned ring solutions, and determine their excitation energies as a function of velocity for both penetrable and impenetrable objects. The ring and laminar flow solutions converge at a critical velocity, which decreases with increasing object size. We also study the vortex solutions associated with flow past a surface bump which indicate that surface roughness also reduces the critical velocity. This effect may have important implications for the threshold of dissipation in superfluids and superconductors.