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 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. [S1050-2947(99)08512-1].