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A numerical study of vortex nucleation in 2D rotating Bose-Einstein condensates

Guillaume Dujardin, Ingrid Lacroix-Violet, Anthony Nahas

TL;DR

This work develops a FFT-based discretization and an explicit projected gradient (EPG) method to minimize a constrained Gross–Pitaevskii energy for rotating 2D Bose–Einstein condensates, including single- and two-component regimes with segregation or coexistence. It introduces post-processing tools to detect and index vortices and vortex sheets, and validates the approach through extensive numerical experiments across rotational regimes, comparing performance against GPELab. The results reproduce known theoretical regimes (e.g., vortex lattices, annular structures, and vortex sheets) and demonstrate that the EPG method can outperform linearly implicit schemes in speed while maintaining accuracy in capturing complex vortex structures. The methodology provides a versatile, scalable framework for exploring vortex phenomena in multi-component BECs and offers a practical tool for SEO-friendly scientific summaries and retrieval.

Abstract

This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.

A numerical study of vortex nucleation in 2D rotating Bose-Einstein condensates

TL;DR

This work develops a FFT-based discretization and an explicit projected gradient (EPG) method to minimize a constrained Gross–Pitaevskii energy for rotating 2D Bose–Einstein condensates, including single- and two-component regimes with segregation or coexistence. It introduces post-processing tools to detect and index vortices and vortex sheets, and validates the approach through extensive numerical experiments across rotational regimes, comparing performance against GPELab. The results reproduce known theoretical regimes (e.g., vortex lattices, annular structures, and vortex sheets) and demonstrate that the EPG method can outperform linearly implicit schemes in speed while maintaining accuracy in capturing complex vortex structures. The methodology provides a versatile, scalable framework for exploring vortex phenomena in multi-component BECs and offers a practical tool for SEO-friendly scientific summaries and retrieval.

Abstract

This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.
Paper Structure (51 sections, 1 theorem, 38 equations, 28 figures)

This paper contains 51 sections, 1 theorem, 38 equations, 28 figures.

Table of Contents

  1. Introduction
  2. The model and the regimes
  3. The model
  4. The different regimes
  5. One component condensates
  6. Two components condensates in the segregation regime ($\delta>1$)
  7. Two components condensates in the coexistence regime ($\delta < 1$)
  8. The discretisation of the problem and the minimization method
  9. Discretization of the energy
  10. Computation of the gradient of the discrete energy \ref{['energie_discr']}
  11. A criterion for the minimization of $E^\Delta_{\varepsilon,\delta}$ under constraints
  12. Gradient Method
  13. Post-processing algorithms
  14. Indices of the Vortices
  15. Indices of the vortex sheets
  16. ...and 36 more sections

Key Result

Proposition 3

If $(\psi^{1*}, \psi^{2*}) \in \mathbb{C}^{(N+2)^2}$ are minimizers of $E_{\varepsilon,\delta}^\Delta$ under the constraints eq:contraintesdiscretes satisfying the homogeneous Dirichlet boundary conditions, then the four $N\times N$ real-valued matrices $K^\Delta_{1,P}, K^\Delta_{2,P}, K^\Delta_{1,Q and for $\ell\in\{1,2\}$, vanish.

Figures (28)

  • Figure 1: $\mathbb{S}_\lambda(n,k)$ with angles $\theta_0$ to $\theta_{8\lambda}$.
  • Figure 2: An example of the contours detected after the first step of the contour detection algorithm ($m=0.4$, $M=0.6$, and $tol_3=0.3$). We display just a part of the minimizer obtained in Fig \ref{['fig:2_espece_om_15_seg_1']}.
  • Figure 3: An example of the contours detected and orientated by the contour detection algorithm after the third and fourth steps.
  • Figure 4: $\Omega$ as a function of $\varepsilon$ and the associated zones.
  • Figure 5: Convergence test for the gradient method with $N=128$ for one component without rotation.
  • ...and 23 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Remark 4
  • Remark 5: Computation of the index of a giant hole
  • Remark 6
  • Remark 7