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Journal of Physics A Mathematical and General

Motion in a Bose condensate: VIII. The electron bubble

Natalia G. Berloff, Paul Harry Roberts

Published Dec 19, 2000• 53 citations
Publisher

Abstract

The Bose condensate model is used to elucidate the motion of the electron bubble in superfluids. An asymptotic expansion is developed for steady subcritical flow. Numerical integration of the coupled nonlinear Schrödinger equations, that describe the evolution of the wavefunctions of the Bose condensate and the impurity, is used to study the nucleation and capture of vortex rings. Because an electron bubble is made oblate by its motion relative to the condensate, the critical velocity for the vortex nucleation is reduced by about 20%, in agreement with experiments.

DOI
10.1088/0305-4470/34/1/306

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The Bose condensate model is used to elucidate the motion of the electron bubble in superfluids. An asymptotic expansion is developed for steady subcritical flow. Numerical integration of the coupled nonlinear Schrödinger equations, that describe the evolution of the wavefunctions of the Bose condensate and the impurity, is used to study the nucleation and capture of vortex rings. Because an electron bubble is made oblate by its motion relative to the condensate, the critical velocity for the vortex nucleation is reduced by about 20%, in agreement with experiments.
The Bose condensate model is used to analyse the superfluid flow around an ion (modelled as a solid sphere) and to elucidate the mechanism of vortex ring emission from the sphere that occurs if its velocity exceeds a critical value. An asymptotic expansion is developed for the steady subcritical flow, using the ratio of the healing length to the radius of the sphere as a small parameter. This expansion allows for the compressibility of the condensate, and converges well enough for the critical ion velocity to be calculated accurately. The flow for supercritical ion velocities is computed numerically. Particular attention is paid to the question of why the vortex rings are emitted at a preferred location on the sphere's surface.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.
Nucleation of vortex rings accompanies the collapse of ultrasound bubbles in superfluids. Using the Gross-Pitaevskii equation for a uniform condensate we elucidate the various stages of the collapse of a stationary spherically symmetric bubble and establish conditions necessary for vortex nucleation. The minimum radius of the stationary bubble, whose collapse leads to vortex nucleation, was found to be 28+/-1 healing lengths. The time after which the nucleation becomes possible is determined as a function of the bubble's radius. We show that vortex nucleation takes place in moving bubbles of even smaller radius if the motion makes them sufficiently oblate.
Nucleation of vortex rings accompanies the collapse of ultrasound bubbles in superfluids. Using the Gross-Pitaevskii equation for a uniform condensate we elucidate the various stages of the collapse of a stationary spherically symmetric bubble and establish conditions necessary for vortex nucleation. The minimum radius of the stationary bubble, whose collapse leads to vortex nucleation, was found to be 28+/-1 healing lengths. The time after which the nucleation becomes possible is determined as a function of the bubble's radius. We show that vortex nucleation takes place in moving bubbles of even smaller radius if the motion makes them sufficiently oblate.
We study the motion of an electron bubble in the zero-temperature limit where neither phonons nor rotons provide a significant contribution to the drag exerted on an ion moving within the superfluid. By using the Gross-Clark model, in which a Gross-Pitaevskii equation for the superfluid wave function is coupled to a Schr\"odinger equation for the electron wave function, we study how vortex nucleation affects the measured drift velocity of the ion. We use parameters that give realistic values of the ratio of the radius of the bubble with respect to the healing length in superfluid 4He^{4}\mathrm{He}4He at a pressure of one bar. By performing fully three-dimensional spatiotemporal simulations of the superfluid coupled to an electron, that is modeled within an adiabatic approximation and moving under the influence of an applied electric field, we are able to recover the key dynamics of the ion-vortex interactions that arise and the subsequent ion-vortex complexes that can form. Using the numerically computed drift velocity of the ion as a function of the applied electric field, we determine the vortex nucleation limited mobility of the ion to recover values in good agreement with measured data.