Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps

TL;DR

This paper rigorously analyzes the asymptotic behavior of a rapidly rotating Bose-Einstein condensate in a strongly anharmonic trap, revealing the formation of holes and giant vortices as the rotation speed increases.

Contribution

It provides the first rigorous derivation of ground state energy asymptotics and density profiles for rotating BECs in flat traps with high angular velocities.

Findings

Development of a density hole at critical rotation speeds.

Formation of a giant vortex at very high rotation speeds.

Rotational symmetry is broken in the ground state for certain rotation regimes.

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 $1/\epsilon^2$ and we are interested in the limit $\epsilon\to 0$ (TF limit) with the angular velocity $\Omega$ depending on $\epsilon$. We derive rigorously the leading asymptotics of the ground state energy and the density profile when $\Omega$ tends to infinity as a power of $1/\epsilon$. If $\Omega(\epsilon)=\Omega_0/\epsilon$ a ``hole'' (i.e., a region where the density becomes exponentially small as $1/\epsilon\to\infty$) develops for $\Omega_0$ above a certain critical value. If $\Omega(\epsilon)\gg 1/\epsilon$ 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 ${\rm const.}|\log\epsilon|<\Omega(\epsilon)\lesssim \mathrm{const.}/\epsilon$.