Superfluid vortex dynamics on a torus and other toroidal surfaces of revolution

Abstract

The superfluid flow velocity is proportional to the gradient of the phase of the superfluid order parameter, leading to the quantization of circulation around a vortex core. In this work, we study the dynamics of a superfluid film on the surface of a torus. Such a compact surface allows only configurations of vortices with zero net vorticity. We derive analytic expressions for the flow field, the total energy, and the time-dependent dynamics of the vortex cores. The local curvature of the torus and the presence of non-contractable loops on this multiply connected surface alter both the superfluid flow and the vortex dynamics. Finally we consider more general surfaces of revolution, called toroids.

Superfluid vortex multipoles and soliton stripes on a torus

“…In tandem with the procurement of steady states and their characterization (stability), we will study the corresponding elements in the dynamically reduced model where the vortices are considered as point-particles, as per the fundamental work of Ref. [34]. Note that, in comparison to the latter, we are using here adimensional variables which correspond to the models in these works with = M = 1 where M is the mass of the particles forming the atomic BEC.…”

“…Following Ref. [34], one defines the isothermal coordinates (u, v) related to the toroidal (φ, θ) ones through [38]…”

“…II we introduce the original, full, spatio-temporal NLS model on the torus and briefly review (for completeness) the main aspects of its reduction to an effective point-vortex model, as obtained in Ref. [34]. Section III presents the bulk of our results by studying the existence of vortex-and stripe-bearing solutions, their stability and dynamics.…”

“…In the present work, our platform of choice will be the surface a torus, i.e., the simplest compact and multiply connected surface. This is motivated by the above developments, the torus' nontrivial topological structure, by the proposal by experimental groups of the realization of an optical lattice on its surface [33] and by the recent formulation of the effective vortex particle dynamics on its surface [34]. More concretely, we extend our earlier considerations in the realm of bright solitary waves [35] to the self-repulsive condensate setting and the primarily vortical (but also dark soliton stripe) set of structures that can arise therein, while also respecting the periodicity of the torus in both of its angular directions.…”

“…The fundamental work of Ref. [34] has set the stage by providing a description at the level of ordinary differential equations (ODEs) for the vortex particles. Yet, the latter work still left many questions unanswered.…”