Unified semiclassical approximation for Bose-Einstein condensates: Application to a BEC in an optical potential

We present semiclassical descriptions of Bose-Einstein condensates for configurations with spatial symmetry, e.g., cylindrical symmetry, and without any symmetry. The description of the cylindrical case is quasi-one-dimensional (Q1D), in the sense that one only needs to solve an effective 1D nonlinear Schrodinger equation, but the solution incorporates 3D aspects of the problem, as a result of which the 1D equation is supplemented by a noncanonical (quartic) normalization condition. The solution in classically allowed regions is matched onto that in classically forbidden regions by a connection formula that properly accounts for the nonlinear mean-field interaction. Special cases for vortex solutions are treated too. Comparisons of the Q1D solution with the full 3D and Thomas-Fermi ones are presented, and conditions for the applicability of the effective low-dimensional equations are obtained.