Critical Rotational Speeds in the Gross-Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions

Abstract

We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas in a disc-shaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range | log ε| ≪ Ω ε −2 | log ε| −1 where Ω is the rotational velocity and the coupling parameter is written as ε −2 with ε ≪ 1. Three critical speeds can be identified. At Ω = Ωc 1 ∼ | log ε| vortices start to appear and for | log ε| ≪ Ω < Ωc 2 ∼ ε −1 the vorticity is uniformly distributed over the disc. For Ω ≥ Ωc 2 the centrifugal forces create a hole around the center with strongly depleted density. For Ω ≪ ε −2 | log ε| −1 vorticity is still uniformly distributed in an annulus containing the bulk of the density, but at Ω = Ωc 3 ∼ ε −2 | log ε| −1 there is a transition to a giant vortex state where the vorticity disappears from the bulk. The energy is then well approximated by a trial function that is an eigenfunction of angular momentum but one of our results is that the true minimizers break rotational symmetry in the whole parameter range, including the giant vortex phase.MSC: 35Q55,47J30,76M23. PACS: 03.75.Hh, 47.37.+q.

Vortex patterns and sheets in segregated two component Bose–Einstein condensates

“…We have made the choice to include a complete square in the first term of the energy without subtracting the centrifugal term Ω 2 x 2 u 2 , which for Ω 2 ε 2 ≪ 1 leads to the same energy expansion and vortex patterns. At Ω = 1/ε, as explained in [17], the energy without the centrifugal term displays a change of behaviour: the bulk of the condensate becomes annular. The two energy yield the same structures for rotationnal velocities much lower than 1/ε; in the case when δ tends to 1, velocities up to 1/ε can be less than 1/ε if ε ≪ε 2 .…”

Surface superconductivity in presence of corners

“…We are thus left with the energy contributions of the smooth pieces of the boundary layer. There we can pass to boundary coordinates and use the same trick, i.e., a suitable integration by parts, involved in the proofs of our earlier results [CR2,CR3] and inspired by other works on rotating condensates (see, e.g., [CR1,CRY,CPRY1,CPRY2,CPRY3]).…”

On the third critical speed for rotating Bose-Einstein condensates

“…Sections 3 contains some preliminary estimates and a detailed analysis of the effective functionals that will play a significant role throughout the proofs. In Section 3.4 we prove the main properties of the cost function and in particular its positivity, which is the main mathematical tool used in the proof of the giant vortex transition as in several other works [CPRY1,CPRY2,CPRY3,CR1,CRY].…”