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 con¯nement and present its semiclassical scaling and geometrical optics. We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and ¯nd 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 °ow (CNGF) method with a backward Euler ¯nite di®erence (BEFD) discretization. Ground, symmetric and central vortex states, as well as their energy con¯gurations, are reported in 2D and 3D for a rotating BEC. Through our numerical study, we ¯nd various con¯gurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of di®erent states in a strong replusive interaction regime. Finally we report the critical angular velocity at which the ground state loses symmetry, numerical veri¯cation 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.