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

A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity Ωc for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency ωa of the anomalous mode. Although Ωc=-ωa through first order, the second-order contributions ensure that the absolute value |ωa| is always smaller than the critical angular velocity Ωc. With increasing external rotation Ω, the dynamical instability of the condensate with a vortex disappears at Ω*=|ωa|, whereas the vortex state becomes energetically stable at the larger value Ωc. 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 Ωm. A variational calculation yields Ωm=|ωa| to first order (hence Ωm also coincides with the critical angular velocity Ωc 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.