Giant Vortex and the Breakdown of Strong Pinning in a Rotating Bose-Einstein Condensate

Abstract

We consider a two-dimensional model for a rotating Bose-Einstein condensate (BEC) in an anharmonic trap. The special shape of the trapping potential, negative in a central hole and positive in an annulus, favors an annular shape for the support of the wave function u. We study the minimizers of the energy in the Thomas-Fermi limit, where a small parameter tends to 0, for two different regimes of the rotational speed Ω. When Ω is independent of , we observe that the energy minimizers acquire vorticity beyond a critical Ω, but the vortices are strongly pinned in the central hole where the potential is negative. In this regime, minimizers exhibit no vortices in the annular bulk of the condensate. There is a critical rotational speed Ω = O(| ln |) for which this strong pinning effect breaks down and vortices begin to appear in the annular bulk. We derive an asymptotic formula for the critical Ω, and determine precisely the location of nucleation of the vortices at the critical value. These results are related to very recent experimental and numerical observations on BEC.

Vortex Rings in Fast Rotating Bose–Einstein Condensates

“…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.…”

“…We note that several authors (see e.g. [AAB,ASS,AB1,AB2,ABM,SS]) have already successfully used the representation (2.45) for the computation of similar quantities. However, in our case, the particular geometry of A (fixed radius but shrinking width) makes it difficult to obtain the properties of G required in the computation.…”

“…We thus construct a test function displaying the optimal number of vortices evenly distributed on the circle C R * . To evaluate the energy of such a test function we adapt the well-established technique used e.g in [AAB,ASS,AB1,AB2,ABM,SS]. It involves in particular a Green representation of the electrostatic energy (1.30).…”

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

“…for some constant C > 0, provided ε is small (see [8,10,130], and Remark 18 herein). (We remark that the logarithmic term appears because (6), (7) imply that ∇ √ A + is not square-integrable near ∂D 0 .)…”

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

“…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.…”