Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

TL;DR

This paper rigorously proves that the Gross-Pitaevskii equation accurately describes the ground state properties of rapidly rotating dilute Bose gases, including phenomena like Bose-Einstein condensation and vortex formation.

Contribution

It extends the validity of the Gross-Pitaevskii equation to rapidly rotating Bose gases and simplifies previous proofs, also analyzing symmetry breaking due to vortices.

Findings

Gross-Pitaevskii equation describes ground state energy and density matrix

Complete Bose-Einstein condensation occurs in the system

Quantized vortices cause spontaneous symmetry breaking

Abstract

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.