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Journal of Physics B Atomic Molecular and Optical Physics

Metastable off-axis vortices in nonrotating Bose–Einstein condensates

Dora M. Jezek, P. Capuzzi, Horacio M. Cataldo

Published Feb 11, 2008• 7 citations
Publisher

Abstract

We propose a scenario for studying vortices confined by nonrotating traps. For this purpose we model a polynomial trapping potential that can sustain locally stable off-axis vortices. We also describe a simple numerical method for generating vortices without relying on rotating frames or centrifugal potential techniques. This method offers other advantages related to the control in the number of vortices to be generated and in their location. Finally, we compute the vortex energy as a function of its position using three approaches and discuss their validity. We show that a well-defined energy barrier between the vortex and the ground state is obtained.

DOI
10.1088/0953-4075/41/4/045304

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We propose a scenario for studying vortices confined by nonrotating traps. For this purpose we model a polynomial trapping potential that can sustain locally stable off-axis vortices. We also describe a simple numerical method for generating vortices without relying on rotating frames or centrifugal potential techniques. This method offers other advantages related to the control in the number of vortices to be generated and in their location. Finally, we compute the vortex energy as a function of its position using three approaches and discuss their validity. We show that a well-defined energy barrier between the vortex and the ground state is obtained.
We numerically obtain stationary rings of vortices in pancake-shaped Bose-Einstein condensates confined in a three-dimensional nonrotating trap. For this purpose we use a static axisymmetric trapping potential that can sustain locally stable off-axis vortices. We analyze the maximum number of vortices the system can host as a function of the number of particles. We also show that this system provides a very suitable scenario for predicting vortex dynamics in inhomogeneous media. Specifically, on the one hand, we derive an exact and simple analytical expression for the velocity field in a particular vortex position due to the presence of the other vortices of the array. On the other hand, using the fact that in stationary conditions this field should balance all other contributions to the velocity, we investigate the applicability of approximated expressions for the precession velocity obtained in previous works for condensates with a single vortex.
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.
We present a numerical method for generating vortex rings in Bose-Einstein condensates confined in axially symmetric traps. The vortex ring is generated using the line-source approximation for the vorticity, i.e., the curl of the superfluid velocity field is different from zero only on a circumference of a given radius located on a plane perpendicular to the symmetry axis and coaxial with it. The particle density is obtained by solving a modified Gross-Pitaevskii equation that incorporates the effect of the velocity field. We discuss the appearance of density profiles, the vortex core structure, and the vortex nucleation energy, i.e., the energy difference between vortical and ground-state configurations. This is used to present a qualitative description of the vortex dynamics.
For a Bose-Einstein condensate placed in a rotating trap and strongly confined along the z axis, we set a framework of study for the Gross-Pitaevskii energy in the Thomas-Fermi regime for an effective two-dimensional (2D) situation in the $x\ensuremath{-}y$ plane. We investigate an asymptotic expansion of the energy, the critical angular velocities of nucleation of vortices with respect to a small parameter $\ensuremath{\varepsilon},$ and the location of vortices. The limit $\ensuremath{\varepsilon}$ going to zero corresponds to the Thomas-Fermi regime. The nondimensionalized energy is similar to the Ginzburg-Landau energy for superconductors in the high-\ensuremath{\kappa} high-field limit and our estimates rely on techniques developed for this latter problem. We also take advantage of this similarity to develop a numerical algorithm for computing the Bose-Einstein vortices. Numerical results and energy diagrams are presented.
We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross–Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.
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.