The basic physics of dilute trapped atomic gases reflects both the interparticle interactions and the quantum degeneracy (which occurs when the thermal De Broglie wavelength becomes comparable with the interparticle spacing). In the simplest case that most of the particles are in the condensate, the time-dependent Gross-Pitaevskii equation describes well the dynamical evolution (this is formally equivalent to a nonlinear Schrodinger equation). The stability of a vortex in a rotating condensate can be studied in at least three distinct ways. (i) The first examines how the energy changes as the vortex is displaced from the central position and predicts the onset of metastability at a critical angular velocity Omega_m (below this value, the central position is a local maximum of the energy, whereas above this value, the central position becomes a local minimum). (ii) A more direct dynamical approach considers the small-amplitude perturbations with the Bogoliubov equations and finds a negative frequency if the angular velocity is smaller than that for onset of metastability (this behavior indicates an instability of the Landau type). (iii) The most physical approach derives the local velocity of each element of the vortex, confirming the previous analyses. In addition, this latter method predicts large-amplitude periodic tipping orbits in a nearly spherical condensate. Experiments confirm these latter motions and also the predictions about the precession of a nearly straight vortex in an axisymmetric condensate.
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