Mean-field equations for cigar- and disc-shaped Bose and Fermi superfluids

Abstract

Starting from the three-dimensional (3D) time-dependent nonlinear Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) and the density-functional (DF) equation for a Fermi superfluid at the unitarity and Bardeen-Cooper-Schrieffer (BCS) limits, we derive effective one-(1D) and two-dimensional (2D) mean-field equations, respectively, for the dynamics of a trapped cigar-and disc-shaped BEC and Fermi superfluid by using the adiabatic approximation. The reduced 1D and 2D equations for a cigar-and disc-shaped Fermi superfluid have simple analytic non-linear terms and at unitarity produce results for stationary properties and non-stationary breathing oscillation and free expansion in excellent agreement with the solution of the full 3D equation.