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

Abstract

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