Bose-Einstein Condensate Vortices with Hypercylindrical Symmetry using Zeroth-order Dimensional Perturbation Theory

We investigate D-dimensional atomic Bose-Einstein condensates in a hypercylindrical trap with a vortex core along the z-axis and quantized circulation. We use D dimensions so that we can understand the squeezing effect of anisotropic traps on lower effective dimensionality of the system. In addition, we explore higher-dimensional vortex properties (D>=3), which have potential applications in the emerging field of synthetic dimensions, which are created by experimentally manipulating internal degrees of freedom so that they mathematically behave like extra external degrees of freedom such as extra spatial dimensions. We analytically approximate the hypercylindrical Gross-Pitaevskii equation using dimensional perturbation theory, where the perturbation parameter is δ = 1/(D + 2|m| − 3). Making use of the zeroth-order δ approximation, which can be thought of as a large-D or large-m approximation, we keep contributions from the kinetic energy as well as keeping the full nonlinear interaction term from the Schrodinger equation. We derive zeroth-order (δ → 0) semiclassical approximations for the condensate density, energy, chemical potential, and critical vortex rotation speed in arbitrary dimensions.