# Vortex lattice in a uniform Bose-Einstein condensate in a box trap

## TL;DR

This paper numerically investigates vortex lattice formation in rapidly rotating uniform Bose-Einstein condensates confined in square and circular box traps, revealing lattice structures and stability characteristics consistent with theoretical predictions.

## Contribution

It provides a detailed numerical analysis of vortex lattice configurations in different box geometries using the Gross-Pitaevskii equation, highlighting geometric effects on vortex arrangements.

## Key findings

- Square box traps form perfect square vortex lattices at specific vortex counts.
- Circular box traps produce concentric vortex orbits with polygonal and circular shapes.
- Vortex lattices are dynamically stable under small perturbations.

## Abstract

We study numerically the vortex-lattice formation in a rapidly rotating uniform quasi-two-dimensional Bose-Einstein condensate (BEC) in a box trap. We consider two types of boxes: square and circle. In a square-shaped 2D box trap, when the number of generated vortices is the square of an integer, the vortices are found to be arranged in a perfect square lattice, although deviations near the center are found when the number of generated vortices is arbitrary. In case of a circular box trap, the generated vortices in the rapidly rotating BEC lie on concentric closed orbits. Near the center, these orbits have the shape of polygons, whereas near the periphery the orbits are circles. The circular box trap is equivalent to the rotating cylindrical bucket used in early experiment(s) with liquid He II. The number of generated vortices in both cases is in qualitative agreement with Feynman's universal estimate. The numerical simulation for this study is performed by a solution of the underlying mean-field Gross-Pitaevskii (GP) equation in the rotating frame, where the wave function for the generated vortex lattice is a stationary state. Consequently, the imaginary-time propagation method can be used for a solution of the GP equation, known to lead to an accurate numerical solution. We also demonstrated the dynamical stability of the vortex lattices in real-time propagation upon a small change of the angular frequency of rotation, using the converged imaginary-time wave function as the initial state.