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Numerical study of the Gross-Pitaevskii equation on a two-dimensional ring and vortex nucleation

Quentin Chauleur, Radu Chicireanu, Guillaume Dujardin, Jean-Claude Garreau, Adam Rançon

Abstract

We consider the Gross-Pitaevskii equation with a confining ring potential with a Gaussian profile. By introducing a rotating sinusoidal perturbation, we numerically highlight the nucleation of quantum vortices in a particular regime throughout the dynamics. Numerical computations are made via a Strang splitting time integration and a two-point flux approximation Finite Volume scheme based on a particular admissible triangulation. We also develop numerical algorithms for vortex tracking adapted to our finite volume framework.

Numerical study of the Gross-Pitaevskii equation on a two-dimensional ring and vortex nucleation

Abstract

We consider the Gross-Pitaevskii equation with a confining ring potential with a Gaussian profile. By introducing a rotating sinusoidal perturbation, we numerically highlight the nucleation of quantum vortices in a particular regime throughout the dynamics. Numerical computations are made via a Strang splitting time integration and a two-point flux approximation Finite Volume scheme based on a particular admissible triangulation. We also develop numerical algorithms for vortex tracking adapted to our finite volume framework.
Paper Structure (28 sections, 2 theorems, 94 equations, 11 figures)

This paper contains 28 sections, 2 theorems, 94 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The Gross-Pitaevskii model
  3. Physical motivation
  4. Stationary 2D trapping ring potential
  5. Rotating potential modulation
  6. The dimensionless model
  7. Assumption on the geometry
  8. Discretization of the problem
  9. Time discretization by Strang splitting
  10. Space discretization by Finite Volumes
  11. Triangulation of the geometry
  12. Building the symmetric mesh
  13. Concentrated symmetric mesh
  14. Computation of the ground state
  15. Post-processing algorithms
  16. ...and 13 more sections

Key Result

Proposition 1

We fix a stepsize $h>0$. Let $N_c=\left\lceil\frac{r_{\max}-r_{\min}}{h} \right\rceil$ and Then the triangulation $\mathcal{T}$ satisfy the strict Delaunay condition.

Figures (11)

  • Figure 1: Atomic BEC 2D ring trap configuration. a) Schematic representation of the setup. The LG laser beams (not shown) are focused down to the BEC (red) through high numeric-aperture optics. In the vertical direction, the BEC is trapped in a single node of an additional optical lattice, which 'freezes' the motion of atoms along $z$, effectively reducing the system's dimensionality. b) Radial profile of the ring trapping potential, proportional to the intensity of the LG mode $L_1^0$ (inset). c) Rotating potential modulation used to stir the BEC into motion.
  • Figure 2: GMSH triangulation of 15222 triangles with $r_{\min}=0.4$, $r_{\max}=1.6$ and element size factor$h=0.08$.
  • Figure 3: Concentrated symmetric triangulation of $N=14850$ triangles with $r_{\min}=0.4$, $r_{\max}=1.6$, step size$h=0.05$, a number of circles $N_c=25$ and a number of points per circle $N_p=297$.
  • Figure 4: The set $\mathbb{S}_{1}(n)$ with angles $\theta_0$ to $\theta_8$.
  • Figure 5: Convergence of $A_{\mathcal{T}}$ on the triangulation $\mathcal{T}$ towards $\Delta$.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof