Energy Expansion and Vortex Location for a Two-Dimensional Rotating Bose–einstein Condensate

Ignat, Radu; Millot, Vincent

doi:10.1142/s0129055x06002607

Cited by 52 publications

(104 citation statements)

References 21 publications

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“…A strictly negative value of this energy will indicate the presence of vortices. We note that this energy functional is very similar to a functional appearing in [AAB] where an annular condensate at slow rotation speeds is considered (see also [IM1,IM2]). The major difference is that the domain A depends on ε in a crucial way : its width tends to zero proportionally to ε| log ε| when ε → 0.…”

Section: Formal Derivationsmentioning

confidence: 55%

“…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%

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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: Formal Derivationsmentioning

confidence: 55%

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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“…In the case of uniform rotation, that is V (x) = x ⊥ = (−x 2 , x 1 ) and with D a disk, Serfaty [17] studied minimizers of a closely related functional (see Remark 1.5) to determine the critical value Ω 1 = Ω 1 (ε) of the angular speed Ω at which vortices first appear (see also [10,11] for BECs). She finds that minimizers acquire vorticity at Ω 1 = k(D)| ln ε|+O(ln | ln ε|) for an explicitly determined constant k(D).…”

Section: Introductionmentioning

confidence: 99%

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

Alama

Bronsard

Millot

2011

JAMA

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Abstract. We study the variational convergence of a family of two-dimensional GinzburgLandau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that, suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

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“…Since u ε − u ap = ϕ * ∈ Y , we see that (126), (127) (105), (106), (126), and (129), we find that ϕ * satisfies…”

Section: Propositionmentioning

confidence: 85%

“…Estimates (38)- (42) for η ε follow readily from the corresponding estimates (137)- (141) for u ε . Relation (44) follows easily from (106) and (126). Next, we will derive estimate (45) by building on estimates (25), (137)- (141), and using that η ε L 2 (R 2 ) = 1.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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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Energy Expansion and Vortex Location for a Two-Dimensional Rotating Bose–einstein Condensate

Cited by 52 publications

References 21 publications

Vortex Rings in Fast Rotating Bose–Einstein Condensates

Vortex Rings in Fast Rotating Bose–Einstein Condensates

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

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

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