Poincar\'e index formula and analogy with the Kosterlitz-Thouless transition in a non-rotated cold atom Bose-Einstein condensate

TL;DR

This paper models a Bose-Einstein condensate in a harmonic trap using the Gross-Pitaevskii equation, analyzing vortices and stagnation points, and draws an analogy with the Kosterlitz-Thouless transition based on angular momentum.

Contribution

It introduces the analysis of stagnation points in the condensate and relates vortex-stagnation point dynamics to the Kosterlitz-Thouless transition analogy.

Findings

Vortices increase with angular momentum and form Abrikosov lattices.

Stagnation points tend to pair with vortex cores at high numbers.

The difference between vortices and stagnation points remains constant, as per Poincaré index formula.

Abstract

A dilute gas of Bose-Einstein condensed atoms in a non-rotated and axially symmetric harmonic trap is modelled by the time dependent Gross-Pitaevskii equation. When the angular momentum carried by the condensate does not vanish, the minimum energy state describes vortices (or antivortices) that propagate around the trap center. The number of (anti)vortices increases with the angular momentum, and they repel each other to form Abrikosov lattices. Besides vortices and antivortices there are also stagnation points where the superflow vanishes; to our knowledge the stagnation points have not been analyzed previously, in the context of the Gross-Pitaevskii equation. The Poincar\'e index formula states that the difference in the number of vortices and stagnation points can never change. When the number of stagnation points is small, they tend to aggregate into degenerate propagating structures. But when the number becomes sufficiently large, the stagnation points tend to pair up with the vortex cores, to propagate around the trap center in regular lattice arrangements. There is an analogy with the geometry of the Kosterlitz-Thouless transition, with the angular momentum of the condensate as the external control parameter instead of the temperature.