Modelling of vorticity, sound and their interaction in two-dimensional superfluids

Vorticity in two-dimensional superfluids is subject to intense research efforts since it is responsible for the onset of superfluidity in these systems as described by the BKT transition. Interaction of sound and vortices is of broad importance in Bose-Einstein condensates and superfluid helium. However, both the modelling of the vortex flow field and of its interaction with sound are complicated hydrodynamic problems, with analytic solutions only available in special cases. In this work, we develop a method to compute the vortex flow field in an arbitrary two-dimensional domain. Further, we analyse the dispersive interaction of vortices with sound modes in a two-dimensional superfluid and develop a model that quantifies this interaction for any vortex distribution on any two-dimensional bounded domain, exploiting analogies with fluid dynamics of an ideal gas and electrostatics. We show how this technique can be used for experimental vortex tracking in a superfluid helium thin film and propose an experiment that should be able to unambiguously detect single circulation quanta in a helium thin film