Stability of a vortex in a small trapped Bose-Einstein condensate

TL;DR

This paper analyzes the stability of vortices in small trapped Bose-Einstein condensates using second-order expansions of the Gross-Pitaevskii and Bogoliubov equations, revealing detailed conditions for vortex creation and stability.

Contribution

It provides a second-order theoretical framework for vortex stability in small condensates, extending previous first-order analyses and including effects of trap anisotropy.

Findings

Critical angular velocity for vortex creation is determined.

Vortex stability depends on external rotation and trap anisotropy.

Metastable vortex states appear at specific angular velocities.

Abstract

A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity Omega_c for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency omega_a of the anomalous mode. Although Omega_c = -omega_a through first order, the second-order contributions ensure that the absolute value |omega_a| is always smaller than the critical angular velocity Omega_c. With increasing external rotation Omega, the dynamical instability of the condensate with a vortex disappears at Omega*=|omega_a|, whereas the vortex state becomes energetically stable at the larger value Omega_c. Both second-order contributions depend explicitly on the axial anisotropy of the trap. The appearance of a local minimum of the free energy for a vortex at the center determines the metastable angular velocity Omega_m. A variational calculation yields Omega_m=|\omega_a| to first order (hence Omega_m also coincides with the critical angular velocity Omega_c to this order). Qualitatively, the scenario for the onset of stability in the weak-coupling limit is the same as that found in the strong-coupling (Thomas-Fermi) limit.