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

Abstract

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 numer- ically. 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 conflnement and present its semiclassical scaling and geometrical optics. We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and flnd that symmetric and central vortex states are independent of the angular rotational momentum. We perform numerical experiments computing these states using a continuous normal- ized gradient ∞ow (CNGF) method with a backward Euler flnite difierence (BEFD) discretization. Ground, symmetric and central vortex states, as well as their energy conflgurations, are reported in 2D and 3D for a rotating BEC. Through our numerical study, we flnd various conflgurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of difierent states in a strong replusive interaction regime. Finally we report the critical angular velocity at which the ground state loses symmetry, numerical veriflcation 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.