Vortices in a rotating Bose-Einstein condensate: Critical angular velocities and energy diagrams in the Thomas-Fermi regime

For a Bose-Einstein condensate placed in a rotating trap and strongly confined along the z axis, we set a framework of study for the Gross-Pitaevskii energy in the Thomas-Fermi regime for an effective two-dimensional (2D) situation in the $x-y$ plane. We investigate an asymptotic expansion of the energy, the critical angular velocities of nucleation of vortices with respect to a small parameter $\varepsilon ,$ and the location of vortices. The limit $\varepsilon$ going to zero corresponds to the Thomas-Fermi regime. The nondimensionalized energy is similar to the Ginzburg-Landau energy for superconductors in the high-κ high-field limit and our estimates rely on techniques developed for this latter problem. We also take advantage of this similarity to develop a numerical algorithm for computing the Bose-Einstein vortices. Numerical results and energy diagrams are presented.