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arXiv:cond-mat/0104404

Ginzburg-Landau Description of Vortex Nucleation in Rotating Superfluid

Igor S. Aranson, Victor Steinberg

Published Apr 21, 2001• 0 citations
arXivHTMLPDF

Abstract

Nucleation of vortices in rotating superfluid by spin-up and rapid thermal quench is discussed in the framework of the time-dependent Ginzburg--Landau equation (TDGLE). An analysis of the instability in inhomogeneous rotationally-invariant system results in the expression for the critical rotational velocity. A stability analysis of multicharged vortices is presented. It is shown that they are very long-living objects with lifetime inversely proportional to the dissipation rate. It was found by numerical and analytical solution of the TDGLE that vortex nucleation by rapid thermal quench in the presence of superflow is dominated by a transverse instability of the moving normal-superfluid interface.

DOI
10.48550/arxiv.cond-mat/0104404

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Nucleation of vortices in rotating superfluid by spin-up and rapid thermal quench is discussed in the framework of the time-dependent Ginzburg--Landau equation (TDGLE). An analysis of the instability in inhomogeneous rotationally-invariant system results in the expression for the critical rotational velocity. A stability analysis of multicharged vortices is presented. It is shown that they are very long-living objects with lifetime inversely proportional to the dissipation rate. It was found by numerical and analytical solution of the TDGLE that vortex nucleation by rapid thermal quench in the presence of superflow is dominated by a transverse instability of the moving normal-superfluid interface.
The flow of a superfluid past a solid boundary can generate vortices, with a key criterion being that the local flow speed reaches the speed of sound (Landau). However, the physical mechanism underlying this vortex generation remains unresolved. We consider the canonical problem of uniform flow past an impenetrable planar wall, to examine the vortex nucleation process generated by a solid obstruction moving through a superfluid. A linear stability analysis is conducted of this uniform two-dimensional flow governed by the Gross–Pitaevskii equation (GPE). This shows that the flow is marginally stable to two-dimensional, normal-mode perturbations. Direct numerical solutions of the GPE confirm this finding for small perturbations of finite amplitude. They also reveal the existence of a critical amplitude above which instabilities, in the form of vortices, are nucleated at the solid wall in finite time. Once nucleated, these vortices convect away from the solid wall with infinite initial speed. The physical mechanism underlying this nonlinear phenomenon is discussed, and a proposal is advanced to deepen its understanding.
In this thesis we demonstrate the power of the Gross-Pitaevskii and the time-dependent Ginzburg-Landau equations by numerically solving them for various fundamental problems related to superfluidity and superconductivity. We start by studying the motion of a massive object through a quantum fluid modelled by the Gross-Pitaevskii equation. Below a critical velocity, the object does not exchange momentum or energy with the fluid. This is a manifestation of its superfluid nature. We discuss the effect of applying a constant force to the object and show that for small forces a vortex ring is created to which the object becomes attached. For a larger force the object detaches from the vortex ring and we observe periodic shedding of rings. All energy transfered to the system is contained within the vortex rings and the drag force on the object is due to the recoil of the vortex emission. If we exceed the speed of sound, there is an additional contribution to the drag from sound emission. To make a link to superconductivity, we then discuss vortex states in a rotating system. In the ground state, regular arrays of vortices are observed which, for systems containing many vortices, mimic solid-body rotation. In the second part of the thesis, we initially review solutions to the Ginzburg-Landau equations in an applied magnetic field. For superconducting disks we observe vortex arrays similar to those in rotating superfluids. Finally, we study an electrical current flow along a superconducting wire subject to an external magnetic field. We observe the motion of flux lines, and hence dissipation, due to the Lorentz force. We measure the V - I curve which is analogous to the drag force in a superfluid. With the introduction of impurities, flux lines become pinned which gives rise to an increased critical current. (author)
We study numerically the stability of axially symmetric vortex lines in trapped dilute gases subject to rotation. For this purpose, we solve numerically both the Gross-Pitaevskii and the Bogoliubov equations for a three-dimensional condensate in spherically and cylindrically symmetric traps, from small to very large nonlinearities. In the stationary case we find that the vortex states with m=1m=1m=1 and m=2m=2m=2 are energetically unstable. In the rotating trap it is found that this energetic instability may only be suppressed for the m=1m=1m=1 vortex line, and that the multicharged vortices are never a local minimum of the energy functional. This result implies that the absolute minimum of the energy is not an eigenstate of the Lz{L}_{z}Lz​ operator, when the angular speed is above a certain value.
We study the response of a trapped Bose-Einstein condensate to a sudden turn on of a rotating drive by numerically solving the two-dimensional Gross-Pitaevskii equation. A weakly anisotropic rotating potential excites a quadrupole shape oscillation and its time evolution is analyzed by a quasiparticle projection method. In a quadrupolar resonant regime, which depends on the trap anisotropy, simple periodic oscillations in surface-mode populations disappear and the system exhibits stochastic dynamics. In the presence of the phenomenological dissipation, an initially irrotational condensate is found to undergo damped elliptic deformation followed by unstable surface ripple excitations, some of which develop into quantized vortices that eventually form a lattice. Recent experimental results on the vortex nucleation should be explained not only by the dynamical instability but also by the Landau instability; the latter is necessary for the vortices to penetrate into the condensate.
We have measured the critical velocity ${\ensuremath{\upsilon}}_{\mathrm{c}}$ at which ${}^{3}\mathrm{He}\ensuremath{-}\mathrm{A}$ in a rotating cylinder becomes unstable against the formation of quantized vortex lines with continuous (singularity-free) core structure. We find that ${\ensuremath{\upsilon}}_{\mathrm{c}}$ is distributed between a maximum and a minimum limit, which we ascribe to a dependence on the texture of the orbital angular momentun axis $\stackrel{^}{\mathbf{l}}(\mathbf{r})$ in the cylinder. Slow cooldown through Tc{T}_{\mathrm{c}}Tc​ in rotation yields $\stackrel{^}{\mathbf{l}}(\mathbf{r})$ textures for which the measured ${\ensuremath{\upsilon}}_{\mathrm{c}}$'s are in good agreement with the calculated instability of the expected $\stackrel{^}{\mathbf{l}}$ texture.