Vortices vacate vales and other singular tales

Quantized vortices in superfluids are a microscopic analogue of the vortices found in a sink as the water rushes down the drain. Unlike ordinary vortices, vortices in superfluid helium or artificial Bose condensates are particle-like because they are very long-lived. My thesis discusses forces experienced by these vortices which have a geometric origin. The first part focuses on vortices in a layer of liquid helium on a curved substrate . Such vortices experience a force that has Gaussian curvature as its source; vortices are attracted to negative and repelled from positive Gaussian curvature. Ideas about contexts where this force might be observed experimentally and the conformal mapping techniques for calculating the strength of the force are presented. The second part focuses on predictions about vortices in a dilute optically trapped Bose condensate of spinor atoms and derives their properties from first principles. Now the atoms' internal spin space provides the nontrivial geometry; spinor states can be classified by introducing a set of points on the sphere called "spin roots". The spin two phase diagram is determined as a function of the interaction parameters. Homotopy theory shows that multiple types of vortices can be created in a condensate of spin two atoms. The interactions between these vortices can be calculated using the spin roots' symmetry group. Metastable vortex states can arise in both contexts. A vortex on a curved surface be trapped near a saddle point. If thermal fluctuations push the vortex away from the saddle, it can move to the edge of the film and disappear, releasing the kinetic energy of the superfluid currents that circle the vortex. Sets of vortices in a spinor condensate can trap each other in a molecule (in the presence of a magnetic field). After a long time, thermal fluctuations can bring the component molecules together, leading to a "chemical reaction" in which new vortices that are not trapped are produced. Then the molecule breaks apart and releases kinetic energy as in the previous scenario.