Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps

Abstract

We study a rotating Bose-Einstein Condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of 2D Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as $1/ε^2$ and we are interested in the limit $ε\to 0$ (TF limit) with the angular velocity $Ω$ depending on $ε$. We derive rigorously the leading asymptotics of the ground state energy and the density profile when $Ω$ tends to infinity as a power of $1/ε$. If $Ω(ε)=Ω_0/ε$ a ``hole'' (i.e., a region where the density becomes exponentially small as $1/ε\to\infty$) develops for $Ω_0$ above a certain critical value. If $Ω(ε)\gg 1/ε$ the hole essentially exhausts the container and a ``giant vortex'' develops with the density concentrated in a thin layer at the boundary. While we do not analyse the detailed vortex structure we prove that rotational symmetry is broken in the ground state for ${\rm const.}|\logε|<Ω(ε)\lesssim \mathrm{const.}/ε$.