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arXiv:0902.4205 · Physical Review A

Small numbers of vortices in anisotropic traps

Suzanne McEndoo, Thomas Busch

Published May 12, 2009• 29 citations
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Abstract

We investigate the appearance of vortices and vortex lattices in two-dimensional, anisotropic, and rotating Bose-Einstein condensates. Once the anisotropy reaches a critical value, the positions of the vortex cores in the ground state are no longer given by an Abrikosov lattice geometry but by a linear arrangement. Using a variational approach, we determine the critical stirring frequency for a single vortex as well as the equilibrium positions of a small number of vortices.

DOI
10.1103/physreva.79.053616

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We investigate the appearance of vortices and vortex lattices in two-dimensional, anisotropic, and rotating Bose-Einstein condensates. Once the anisotropy reaches a critical value, the positions of the vortex cores in the ground state are no longer given by an Abrikosov lattice geometry but by a linear arrangement. Using a variational approach, we determine the critical stirring frequency for a single vortex as well as the equilibrium positions of a small number of vortices.
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.
Vortex lattices have become one of the most interesting areas of research in the area of cold atomic physics in recent years. When inducing a large amount of angular momentum into the ground state of a superfluid Bose-Einstein condensate, the sample responds by forming a so-called Abrikosov lattice. Within this lattice, vortices with winding number one are arranged in a triangular pattern that extends over the whole size of the sample. While the Abrikosov lattice is the exact solution for homogeneous condensates, it is also a very good solution for systems in isotropic harmonic traps. Due to the inhomogeneous potential however, the vortex lattice as a whole rotates within the trap.
The nucleation of vortices and the resulting structures of vortex arrays in dilute, trapped, zero-temperature Bose-Einstein condensates are investigated numerically. Vortices are generated by rotating a three-dimensional, anisotropic harmonic atom trap. The condensate ground state is obtained by propagating the Gross-Pitaevskii equation in imaginary time. Vortices first appear at a rotation frequency significantly larger than the critical frequency for vortex stabilization, consistent with a critical velocity mechanism for vortex nucleation. At higher frequencies, the structures of the vortex arrays are strongly influenced by trap geometry.
Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.
For Bose–Einstein condensation of neutral atoms in anisotropic traps at zero temperature, we present simple analytical methods for computing the properties of ground state and single vortex of Bose–Einstein condensates, and compare those results to extensive numerical simulations. The critical angular velocity for production of vortices is calculated for both positive and negative scattering lengths a, and find an analytical expression for the large- N limit of the vortex critical angular velocity for a >0, and the critical number for condensate population approaches the point of collapse for a <0, by using approximate variational method.
We study a harmonically confined Bose-Einstein condensate under rotation. Vortex lattice configurations are investigated through a variational approach. Vortices with more than a unit of angular momentum are not stable. We explicitly show that the critical rotational frequency is quite small, as was reported in a recent experiment, with an asymmetric harmonic confining potential. The width of the condensate along the z direction (the anisotropy direction) and in the xy plane and the vortex density are determined as functions of the rotational frequency. The aspect ratio becomes small for large rotational frequencies, making the condensate an effectively two-dimensional system.