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arXiv:cond-mat/0103489 · Physical Review A

Vortex stabilization in a small rotating asymmetric Bose-Einstein condensate

Marion Linn, Matthias Niemeyer, Alexander L. Fetter

Published Jul 3, 2001• 40 citations
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Abstract

We use a variational method to investigate the ground-state phase diagram of a small, asymmetric Bose-Einstein condensate with respect to the dimensionless interparticle interaction strength $\ensuremath{\gamma}$ and the applied external rotation speed $\ensuremath{\Omega}.$ For a given $\ensuremath{\gamma},$ the transition lines between no-vortex and vortex states are shifted toward higher $\ensuremath{\Omega}$ relative to those for the symmetric case. We also find a re-entrant behavior, where the number of vortex cores can decrease for large $\ensuremath{\Omega}.$ In addition, stabilizing a vortex in a rotating asymmetric trap requires a minimum interaction strength. For a given asymmetry, the evolution of the variational parameters with increasing $\ensuremath{\Omega}$ shows two different types of transitions (sharp or continuous), depending on the strength of the interaction. We also investigate transitions to states with higher vorticity; the corresponding angular momentum increases continuously as a function of $\ensuremath{\Omega}.$

DOI
10.1103/physreva.64.023602

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We use a variational method to investigate the ground-state phase diagram of a small, asymmetric Bose-Einstein condensate with respect to the dimensionless interparticle interaction strength \ensuremathγ\ensuremath{\gamma}\ensuremathγ and the applied external rotation speed \ensuremathΩ.\ensuremath{\Omega}.\ensuremathΩ. For a given \ensuremathγ,\ensuremath{\gamma},\ensuremathγ, the transition lines between no-vortex and vortex states are shifted toward higher \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ relative to those for the symmetric case. We also find a re-entrant behavior, where the number of vortex cores can decrease for large \ensuremathΩ.\ensuremath{\Omega}.\ensuremathΩ. In addition, stabilizing a vortex in a rotating asymmetric trap requires a minimum interaction strength. For a given asymmetry, the evolution of the variational parameters with increasing \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ shows two different types of transitions (sharp or continuous), depending on the strength of the interaction. We also investigate transitions to states with higher vorticity; the corresponding angular momentum increases continuously as a function of \ensuremathΩ.\ensuremath{\Omega}.\ensuremathΩ.
Abstract Vortices in a Bose-Einstein condensate are modelled as spontaneously symmetry breaking minimum energy solutions of the time dependent Gross-Pitaevskii equation, using the method of constrained optimization. In a non-rotating axially symmetric trap, the core of a single vortex precesses around the trap center and, at the same time, the phase of its wave function shifts at a constant rate. The precession velocity, the speed of phase shift, and the distance between the vortex core and the trap center, depend continuously on the value of the conserved angular momentum that is carried by the entire condensate. In the case of a symmetric pair of identical vortices, the precession engages an emergent gauge field in their relative coordinate, with a flux that is equal to the ratio between the precession and shift velocities.
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.
It has been demonstrated experimentally that non-axisymmetric vortices precess around the center of a Bose-Einstein condensate. Two types of single vortex states have been observed, usually referred to as the S vortex and the U vortex. We study theoretically the single vortex excitations in spherical and elongated condensates as a function of the interaction strength. We solve numerically the Gross-Pitaevskii equation and calculate the angular momentum as a function of precession frequency. The existence of two types of vortices means that we have two different precession frequencies for each angular momentum value. As the interaction strength increases the vortex lines bend and the precession frequencies shift to lower values. We establish that for given angular momentum the S vortex has higher energy than the U vortex in a rotating elongated condensate. We show that the S vortex is related to the solitonic vortex, which is a nonlinear excitation in the nonrotating system. For small interaction strengths the S vortex is related to the dark soliton. In the dilute limit a lowest Landau level calculation provides an analytic description of these vortex modes in terms of the harmonic oscillator states.
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 dynamics of small vortex clusters with a few (2-4) corotating vortices in Bose-Einstein condensates by means of experiments, numerical computations, and theoretical analysis. All of these approaches corroborate the counterintuitive presence of a dynamical instability of symmetric vortex configurations. The instability arises as a pitchfork bifurcation at sufficiently large values of the vortex system angular momentum that induces the emergence and stabilization of asymmetric rotating vortex configurations. The latter are quantified in the theoretical model and observed in the experiments. The dynamics is explored both for the integrable two-vortex particle system, where a reduction of the phase space of the system provides valuable insight, as well as for the nonintegrable three- (or more) vortex case, which additionally admits the possibility of chaotic trajectories.
We study the vortex lattices in a Bose-Einstein condensate in a rotating anisotropic harmonic trap. We first investigate the single-particle wave functions obtained by the exact solution of the problem and give simple expressions for these wave functions in the small anisotropy limit. Depending on the strength of the interactions, a few or a large number of vortices can be formed. In the limit of many vortices, we calculate the density profile of the cloud and show that the vortex lattice stays triangular. We also find that the vortex lattice planes align themselves with the weak axis of the external potential. For a small number of vortices, we numerically solve the Gross-Pitaevskii equation and find vortex configurations that are very different from the vortex configurations in an axisymmetric rotating trap.