Nucleation and kinematics of vortices in stirred Bose Einstein condensates

We apply the Halperin-Mazenco formalism within the Gross-Pitaevskii theory to characterise the kinematics and nucleation of quantum vortices in a two-dimensional stirred Bose Einstein condensate. We introduce a smooth defect density field measuring the superfluid vorticity and is a topologically conserved quantity. We use this defect density field and its associated current density to study the precursory pattern formations that occur inside the repulsive potential of an obstacle and determine the onset of vortex nucleation and shedding. We demonstrate that phase slips form inside hard potentials even in the absence of vortex nucleation, whereas for soft potentials they occur only above a critical stirring velocity leading to vortex nucleation. The Halperin-Mazenco formalism provides an elegant and accurate method of deriving the point vortex dynamic directly from the Gross-Pitaevskii equation.