Multi-vortex crystal lattices in Bose-Einstein Condensates with a rotating trap

TL;DR

This paper develops a reduced ODE model for vortex configurations in rotating Bose-Einstein Condensates, accurately predicting stable vortex arrangements, densities, and maximum vortex numbers, validated by numerical simulations.

Contribution

The authors derive a novel reduced ODE system for vortex dynamics in BECs, providing analytical predictions for vortex crystal structures and stability, extending to anisotropic traps.

Findings

Reduced ODE system accurately predicts vortex configurations.

Vortex crystal density and maximum vortex number derived.

Anisotropic traps cause vortices to align along the long axis.

Abstract

We consider vortex dynamics in the context of Bose-Einstein Condensates (BEC) 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.