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arXiv:1706.07137 · Physical Review A

Vortex precession dynamics in general radially symmetric potential traps in two-dimensional atomic Bose-Einstein condensates

P. G. Kevrekidis, Wenlong Wang, R. Carretero-González, D. J. Frantzeskakis, Shuangquan Xie

Published Oct 13, 2017• 11 citations
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Abstract

We consider the motion of individual two-dimensional vortices in general radially symmetric potentials in Bose-Einstein condensates. We find that although in the special case of the parabolic trap there is a logarithmic correction in the dependence of the precession frequency $\ensuremath{\omega}$ on the chemical potential $\ensuremath{\mu}$, this is no longer true for a general potential $V(r)\ensuremath{\propto}{r}^{p}$. Our calculations suggest that for $p>2$, the precession frequency scales with $\ensuremath{\mu}$ as $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\mu}}^{\ensuremath{-}2/p}$. This theoretical prediction is corroborated by numerical computations, not only at the level of spectral (Bogolyubov--de Gennes) stability analysis by identifying the relevant precession mode dependence on $\ensuremath{\mu}$ but also through direct numerical computations of the vortex evolution in the large-$\ensuremath{\mu}$, so-called Thomas-Fermi, limit. Additionally, the dependence of the precession frequency on the distance to the trap center of an initially displaced vortex is examined, and the corresponding predictions are tested against numerical results.

DOI
10.1103/physreva.96.043612

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We consider the motion of individual two-dimensional vortices in general radially symmetric potentials in Bose-Einstein condensates. We find that although in the special case of the parabolic trap there is a logarithmic correction in the dependence of the precession frequency \ensuremathω\ensuremath{\omega}\ensuremathω on the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ, this is no longer true for a general potential V(r)\ensuremath∝rpV(r)\ensuremath{\propto}{r}^{p}V(r)\ensuremath∝rp. Our calculations suggest that for p>2p>2p>2, the precession frequency scales with \ensuremathμ\ensuremath{\mu}\ensuremathμ as \ensuremathω\ensuremath∼\ensuremathμ\ensuremath−2/p\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\mu}}^{\ensuremath{-}2/p}\ensuremathω\ensuremath∼\ensuremathμ\ensuremath−2/p. This theoretical prediction is corroborated by numerical computations, not only at the level of spectral (Bogolyubov--de Gennes) stability analysis by identifying the relevant precession mode dependence on \ensuremathμ\ensuremath{\mu}\ensuremathμ but also through direct numerical computations of the vortex evolution in the large-\ensuremathμ\ensuremath{\mu}\ensuremathμ, so-called Thomas-Fermi, limit. Additionally, the dependence of the precession frequency on the distance to the trap center of an initially displaced vortex is examined, and the corresponding predictions are tested against numerical results.
Based on the method of matched asymptotic expansions and on a time-dependent variational analysis, we study the dynamics of a vortex in a three-dimensional disk-shaped nonaxisymmetric condensate in the Thomas-Fermi limit. Both methods show that a vortex in a trapped Bose-Einstein condensate has formally unstable normal mode(s) with positive normalization and negative frequency, corresponding to a precession of the vortex line around the center of the trap. In a rotating trap, the solution becomes stable above an angular velocity Omega(m), characterizing the onset of metastability with respect to small transverse displacements of the vortex from the central axis.
We numerically study properties of the dynamics of vortices in nonrotating Bose-Einstein condensates in the Thomas-Fermi regime. On the one hand, we compute the vortex energy as a function of its position and we predict, using the expression of the Magnus force, the vortex precession velocity. On the other hand, we calculate the temporal evolution of the vortex-state and test the accuracy of the previous prediction. We also investigate the validity of analytical formulae of this velocity involving the healing length. In addition, we analyze the velocity field and the angular momentum and we compare them to available analytical expressions.
We study vortices in a radially inhomogeneous superfluid, as realized by a trapped degenerate Bose gas in a uniaxially symmetric potential. We show that, in contrast to a homogeneous superfluid, an off-axis vortex corresponds to an anisotropic superflow whose profile strongly depends on the distance to the trap axis. One consequence of this superflow anisotropy is vortex precession about the trap axis in the absence of an imposed rotation. In the complementary regime of a finite prescribed rotation, we compute the minimum-energy vortex density, showing that in the rapid-rotation limit it is extremely uniform, despite a strongly inhomogeneous (nearly) Thomas-Fermi condensate density ${\ensuremath{\rho}}_{s}(r)$. The weak radially dependent contribution $[\ensuremath{\propto}{\ensuremath{\nabla}}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\rho}}_{s}(r)]$ to the vortex distribution, that vanishes with the number of vortices Nv{N}_{v}Nv​ as 1∕Nv1∕{N}_{v}1∕Nv​, arises from the interplay between vortex quantum discreteness (namely their inability to faithfully support the imposed rigid-body rotation) and the inhomogeneous superfluid density. This leads to an enhancement of the vortex density at the center of a typical concave trap, a prediction that is in quantitative agreement with recent experiments. One striking consequence of the inhomogeneous vortex distribution is an azimuthally directed, radially shearing superflow.
Based on the two-dimensional Gross–Pitaevskii equation model, we investigated the asymmetric vortex evolution of two-dimensional Bose–Einstein condensates in a harmonic potential trap with polar direction perturbation. This corresponds to the generation of an asymmetric vortex evolution mode under initial vortex light manipulation together with polar direction perturbation. Unlike most of the prior work on this topic that uses a pure numerical method for the system under study, we use an exact analytical method rather than numerical simulation to investigate the key features of the system evolution dynamics in this study. Based on the variational method, and for different system parameter settings including the strength of the harmonic oscillator potential and the nonlinear interaction, we derived two evolution modes, namely, the periodic evolution oscillation mode and the monotonic decay mode, and pictorially demonstrated the evolution patterns of the system. In addition, we investigated the scenario when the system is in the rotating state, which corresponds to the action of the quadratic centrifugal potential. Notably, we also identified the damping effects for the two modes with increasing angular velocity, whose upper limit value corresponds to the constant vortex pattern of the system. Our theoretical results can be used to guide the experimental investigation of asymmetric vortex evolution in two-dimensional Bose–Einstein condensates.
The dynamics of vortices in trapped Bose-Einstein condensates are investigated both analytically and numerically. In axially symmetric traps, the critical rotation frequency for metastability of an isolated vortex coincides with the largest vortex precession frequency (or anomalous mode) in the Bogoliubov excitation spectrum. The number of anomalous modes increases for an elongated condensate. The largest mode frequency exceeds the thermodynamic critical frequency and the nucleation frequency at which vortices are created dynamically. Thus, anomalous modes describe both vortex precession and the critical rotation frequency for creation of the first vortex in an elongated condensate.
We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.