Fourier-Splitting methods for the dynamics of rotating Bose-Einstein condensates

TL;DR

This paper introduces a novel Fourier-splitting method for simulating the dynamics of rotating Bose-Einstein condensates with time-dependent potentials, combining algebraic techniques for efficient and accurate propagation.

Contribution

It develops a new high-order splitting scheme using Magnus expansions and Fourier transforms, tailored for nonautonomous quadratic Hamiltonians in Bose-Einstein condensate simulations.

Findings

Efficient for small nonlinearities and perturbed potentials

Highly accurate for large nonlinearities with higher order methods

Adaptable to include dissipation effects

Abstract

We present a new method to propagate rotating Bose-Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose-Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.