The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate

Ignat, Radu; Millot, Vincent

doi:10.1016/j.jfa.2005.06.020

Cited by 85 publications

(220 citation statements)

References 23 publications

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“…Also there is an analogy between our (somewhat informal) terminology about critical speeds and that of critical fields in GL theory. In particular, the analogy between the first critical speed and the field H c1 is well-known and of great use in the papers [AAB,IM1,IM2]. We want to emphasize however that the Gross-Pitaevskii theory in the regime we consider largely deviates from the Ginzburg-Landau theory.…”

Section: Theorem 12 (Asymptotics For the Explicit Vorticity)mentioning

confidence: 92%

Vortex Rings in Fast Rotating Bose–Einstein Condensates

Rougerie

2011

Arch Rational Mech Anal

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When Bose-Einstein condensates are rotated sufficiently fast, a giant vortex phase appears, that is the condensate becomes annular with no vortices in the bulk but a macroscopic phase circulation around the central hole. In a former paper [M. Correggi, N. Rougerie, J. Yngvason, Communications in Mathematical Physics 303, 451-308 (2011)] we have studied this phenomenon by minimizing the two dimensional Gross-Pitaevskii energy on the unit disc. In particular we computed an upper bound to the critical speed for the transition to the giant vortex phase. In this paper we confirm that this upper bound is optimal by proving that if the rotation speed is taken slightly below the threshold there are vortices in the condensate. We prove that they gather along a particular circle on which they are uniformly distributed. This is done by providing new upper and lower bounds to the GP energy.MSC: 35Q55,47J30,76M23. PACS: 03.75.Hh, 47.32.-y, 47.37.+q.

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Section: Theorem 12 (Asymptotics For the Explicit Vorticity)mentioning

confidence: 92%

Vortex Rings in Fast Rotating Bose–Einstein Condensates

Rougerie

2011

Arch Rational Mech Anal

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“…These have been studied in various contexts (see for instance [8,10,65,125,163]). As we will see in this paper, some are actually far from optimal.…”

Section: Known Resultsmentioning

confidence: 99%

“…The point being that (4) ensures that minimizing sequences of G ε in H cannot have their mass escaping at infinity (see also [178]); in fact the imbedding H → L 2 (R 2 , C) is compact (see [125,208], or more generally [27,Lemma 3.1]). (One can also ignore the mass constraint, and instead minimize the functional…”

Section: The Problemmentioning

confidence: 99%

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The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

Karali

Sourdis

2015

Arch Rational Mech Anal

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We study the ground state which minimizes a Gross-Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the ThomasFermi limit where a small parameter ε tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Bose-Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrödinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as ε → 0, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, to understand the superfluid flow around an obstacle, and it also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the clas-

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“…In these situations, the limit algebraic equation (the analog of (1.4)) typically undergoes a pitchfork or saddle-node bifurcation as the parameter y crosses a curve Γ. The case of pitchfork bifurcation has received a lot of attention recently, as it occurs when minimizing a Gross-Pitaevskii functional under the unit mass constraint (see [1], [2], [3], [19], and [25]). Due to the irregular nature of the singular limit (it does not belong in the Sobolev space H 1 ), standard weak convergence arguments are not applicable.…”

Section: 2mentioning

confidence: 99%

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

Karali

Sourdis

2012

Communications in Partial Differential Equations

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Resonance phenomena in a singular perturbation problem in the case of exchange of stabilities. GEORGIA KARALI AND CHRISTOS SOURDISAbstract. We consider the following singularly perturbed elliptic problem:where Ω is a bounded domain in R 2 with smooth boundary, ε > 0 is a small parameter, n denotes the outward normal of ∂Ω, and a, b are smooth functions. We assume that the zero set of a − b is a simple closed curve Γ, contained in Ω, and ∇(a − b) = 0 on Γ. We will construct solutions uε that converge in the Hölder sense to max{a, b} in Ω, and their Morse index tends to infinity, as ε → 0, provided that ε stays away from certain critical numbers. Even in the case of stable solutions, whose existence is well established for all small ε > 0, our estimates improve previous results.

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The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate

Cited by 85 publications

References 23 publications

Vortex Rings in Fast Rotating Bose–Einstein Condensates

Vortex Rings in Fast Rotating Bose–Einstein Condensates

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

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