The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

Abstract

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as $ 1 / \eps^2 $ we consider the asymptotic regime $ \eps \to 0 $ with the angular velocity $\Omega$ proportional to $ (\eps^2|\log\eps|)^{-1} $. We prove that if $ Ω= Ω_0 (\eps^2|\log\eps|)^{-1} $ and ${\Omega }_0>2(3\pi {)}^{-1}$ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.

52 pages, PDFLaTex. Minor corrections, sign convention modified. To be published in Commun. Math. Phys

References in corpus (7)

• Rapidly Rotating Atomic Gases
• Energy and Vorticity in Fast Rotating Bose-Einstein Condensates
• Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps
• Rapidly Rotating Bose-Einstein Condensates in Homogeneous Traps
• Vortex Rings in Fast Rotating Bose-Einstein Condensates
• The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
• Non existence of vortices in the small density region of a condensate

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• Giant vortex phase transition in rapidly rotating trapped Bose-Einstein condensates
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• On the Third Critical Speed for Rotating Bose-Einstein Condensates
• Surface Effects in Superconductors with Corners
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• Existence and Asymptotic Behavior of Ground States for Rotating Bose-Einstein Condensates
• Differential Equations of Quantum Mechanics
• Almost Flat Angles in Surface Superconductivity
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