Critical Rotational Speeds for Superfluids in Homogeneous Traps

We present an asymptotic analysis of the effects of rapid rotation on the ground state properties of a superfluid confined in a two-dimensional trap. The trapping potential is assumed to be radial and homogeneous of degree larger than two in addition to a quadratic term. Three critical rotational velocities are identified, marking respectively the first appearance of vortices, the creation of a `hole' of low density within a vortex lattice, and the emergence of a giant vortex state free of vortices in the bulk. These phenomena have previously been established rigorously for a `flat' trap with fixed boundary but the `soft' traps considered in the present paper exhibit some significant differences, in particular in the giant vortex regime, that necessitate a new approach. These differences concern both the shape of the bulk profile and the size of vortices relative to the size of the bulk close to the giant vortex transition. For `flat' traps the profile is of Thomas-Fermi type whereas it is gaussian for soft traps, and the `last' vortices to survive in the bulk before the giant vortex transition are small (relatively to the bulk of the fluid) in the former case but large in the latter.