Snell’s Law and Necklace States for Vortex Dipoles in a Quantum Gas

Abstract

A quantum vortex is an excitation of a superfluid that carries angular momentum, in which the superfluid is expelled from and circulates around a region known as the vortex core. A quantum vortex dipole consists of two bound quantum vortices with opposite circulation, which together carry linear momentum. The topic of this thesis is two-fold, with a common theme of quantum vortex dipoles. In the first part of the thesis, the motion of a quantum vortex dipole incident upon a step-change in the background superfluid density of an otherwise uniform two-dimensional Bose-Einstein condensate is investigated, both analytically and numerically. Due to the conservation of fluid momentum and energy, the incident and refracted angles of the dipole satisfy a relation analogous to Snell’s law, when crossing the interface between regions of different density. The predictions of the analogue Snell’s law relation are confirmed for a wide range of incident angles by systematic numerical simulations of the Gross-Pitaevskii equation (GPE). Near the critical angle for total internal reflection, we identify a regime of anomalous Snell’s law behaviour where the finite size of the dipole causes transient capture by the interface. Remarkably, despite the extra complexity of the surface interaction, the incoming and outgoing dipole paths obey Snell’s law. In the second part of the thesis, the point-vortex model is used to find the stationary states of an arbitrary neutral number of vortices of alternating sign arranged on a ring in the simply bounded domain. These stationary states are collectively referred to as necklace states, due to the symmetry of their appearance. Curiously, the necklace states are found to have a simple relation to the metallic means, which are generalisations of the golden ratio. The necklace state is numerically simulated with different initial perturbations in both the point-vortex model and the GPE with a comparison of the results. Perturbations from the necklace solution evolve differently in the point-vortex model and GPE due to the healing length scale in the GPE enforcing a departure from the point-vortex model predictions. We introduce a simple numerical scheme to compensate for the healing length and confirm the necklace as a steady of the GPE. We find that the agreement between the point-vortex model and GPE for evolution of significant perturbations persists for longer as the healing length reduces compared to system size.