Numerical Method for Evolving the Dipolar Projected Gross-Pitaevskii Equation


We describe a method for evolving the projected Gross-Pitaevskii equation 􏰔PGPE􏰕 for an interacting Bose gas in a harmonic-oscillator potential, with the inclusion of a long-range dipolar interaction. The central difficulty in solving this equation is the requirement that the field is restricted to a small set of prescribed modes that constitute the low-energy c-field region of the system. We present a scheme, using a Hermite- polynomial-based spectral representation, which precisely implements this mode restriction and allows an efficient and accurate solution of the dipolar PGPE. We introduce a set of auxiliary oscillator states to perform a Fourier transform necessary to evaluate the dipolar interaction in reciprocal space. We extensively charac- terize the accuracy of our approach and derive Ehrenfest equations for the evolution of the angular momentum.

Physical Review E