Poincar\'e index formula, vortices and superflow saddle points in a non-rotating cold atom Bose-Einstein condensate

Abstract

A dilute gas of Bose-Einstein condensed atoms, in a non-rotating and axially symmetric harmonic trap, can be modeled by the time dependent Gross-Pitaevskii equation. The minimum energy solutions describe vortices that propagate around the trap center. The number of vortices increases with increasing angular momentum, and the vortices repel each other to form Abrikosov lattices. Besides vortices there are also saddle points, where the velocities of the superflow of distinct vortices cancel each other. The Poincar\'e index formula states that the difference in the number vortices and saddles points can never change. When the number of saddle points is small, they aggregate and form degenerate propagating structures. However, when their number becomes sufficiently large there is a transition and the saddle points start dispersing. They pair up with vortices and propagate around the trap center in regular arrangements akin Abrikosov lattices.