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.