Recently, there has been notable interest within the quantum gas community in the study of ultracold bosonic clouds subject to dipolar interparticle forces. Using numerical simulations, we attempt to predict how such anisotropic long-range forces affect the collective modes of the system (i.e. excitation spectrum).
Solution of the excitation spectrum efficiently requires a medley of iterative methods: Firstly, we must solve the Gross-Pitaevskii equation (nonlinear Schrödinger equation) to obtain a stationary state which determines, e.g. the particle density and the mass currents in the cloud. This is carried out using a relaxation method. Secondly, we solve the excitations for the stationary state from the Bogoliubov-de Gennes equation (eigenvalue equation) by using the Arnoldi iteration. To this end, we utilize the shift and invert mode of ARPACK library routines, which requires the implementation of an efficient method for solving a large linear system. This is achieved with a biconjugate gradient method.