Ground, Symmetric and Central Vortex States in Rotating Bose-einstein Condensates

We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically. We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics. We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum. We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization. Ground, symmetric and central vortex states, as well as their energy configurations, are reported in 2D and 3D for a rotating BEC. Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime. Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.