A Method for the Dynamics of Vortices in a Bose-Einstein Condensate: Analytical Equations of the Trajectories of Phase Singularities

We present a method to study the dynamics of a quasi-two dimensional Bose-Einstein condensate which initially contains several vortices at arbitrary locations. The method allows one to find the analytical solution for the dynamics of the Bose-Einstein condensate in a homogeneous medium and in a parabolic trap, for the ideal non-interacting case. Secondly, the method allows one to obtain algebraic equations for the trajectories of the position of phase singularities present in the initial condensate along with time (the vortex lines). With these equations, one can predict quantities of interest, such as the time at which a vortex and an antivortex contained in the initial condensate will merge. For the homogeneous case, this method was introduced in the context of photonics. Here, we adapt it to the context of Bose-Einstein condensates, and we extend it to the trapped case for the first time. Also, we offer numerical simulations in the non-linear case, for repulsive and attractive interactions. We use a numerical split-step simulation of the non-linear Gross-Pitaevskii equation to determine how these trajectories and quantities of interest are changed by the interactions. We illustrate the method with several simple cases of interest, both in the homogeneous and parabolically trapped systems.

Automated summary

We propose a method for studying the dynamics of a quasi-two dimensional Bose-Einstein condensate with vortices at arbitrary locations. The method provides an analytical solution for the condensate's dynamics in a homogeneous medium and in a parabolic trap, assuming ideal non-interacting conditions. It also allows for predicting the merging time of vortices in the condensate by obtaining algebraic equations for the trajectories of phase singularities. Additionally, we adapt the method from photonics to Bose-Einstein condensates and extend it to trapped systems for the first time, and present numerical simulations considering nonlinear cases.