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arXiv:0812.4049 · Physical Review A

Dynamics of quantum vortices in a toroidal trap

Peter Mason, Natalia G. Berloff

Published Apr 21, 2009• 44 citations

Abstract

The dynamics of quantum vortices in a two-dimensional annular condensate held under a toroidal trapping potential composed of a harmonic term plus Gaussian term are considered by numerically simulating the Gross-Pitaevskii equation. Families of solitary wave sequences are reported, both without and with a persistent flow, for various values of interaction strength. It is shown that in the toroidal geometry the dispersion curve of solutions is much richer than in the cases of a semi-infinite channel or uniform condensate studied previously. In particular, the toroidal condensate is found to have states of single vortices at the same position and circulation that move with different velocities. The stability of the solitary wave sequences for the annular condensate without a persistent flow is also investigated by numerically evolving the solitary wave solutions in time. In addition, the interaction of vortex-vortex pairs and vortex-antivortex pairs is considered and it is demonstrated that the collisions are either elastic or inelastic depending on the magnitude of the angular velocity. The similarities and differences between numerically simulating the Gross-Pitaevskii equation and using a point vortex model for these collisions are elucidated.

DOI: 10.1103/physreva.79.043620

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The dynamics of quantum vortices in a two-dimensional annular condensate held under a toroidal trapping potential composed of a harmonic term plus Gaussian term are considered by numerically simulating the Gross-Pitaevskii equation. Families of solitary wave sequences are reported, both without and with a persistent flow, for various values of interaction strength. It is shown that in the toroidal geometry the dispersion curve of solutions is much richer than in the cases of a semi-infinite channel or uniform condensate studied previously. In particular, the toroidal condensate is found to have states of single vortices at the same position and circulation that move with different velocities. The stability of the solitary wave sequences for the annular condensate without a persistent flow is also investigated by numerically evolving the solitary wave solutions in time. In addition, the interaction of vortex-vortex pairs and vortex-antivortex pairs is considered and it is demonstrated that the collisions are either elastic or inelastic depending on the magnitude of the angular velocity. The similarities and differences between numerically simulating the Gross-Pitaevskii equation and using a point vortex model for these collisions are elucidated.

We study the dynamics of vortices in ideal and weakly interacting Bose-Einstein condensates using a Ritz minimization method to solve the two-dimensional Gross-Pitaevskii equation. For different initial vortex configurations we calculate the trajectories of the vortices. We find conditions under which a vortex-antivortex pair annihilates and is created again. For the case of three vortices we show that at certain times two additional vortices may be created, which move through the condensate and annihilate each other again. For a noninteracting condensate this process is periodic, whereas for small interactions the essential features persist, but the periodicity is lost. The results are compared to exact numerical solutions of the Gross-Pitaevskii equation confirming our analytical findings.

Abstract We investigate solitary vortex evolution in two-dimensional Bose-Einstein condensates based on the Gross-Pitaevskii equation model. Through the variational method, together with the novel Gaussian ansatz incorporating asymmetric perturbation effects, we arrive at the analytical solitary vortex solution with two typical forms: a symmetric quasi-stable solution under certain parametric settings and a diverging propagation case arising from an initial asymmetric perturbation. The derived pictorial evolutionary patterns of the solitary vortices are compared with those from a pure numerical analysis, and by identifying the key qualitative features, we show the applicability of the theoretical treatment presented here.

We studied the evolution of an asymmetric solitary vortex in a two-dimensional Bose-Einstein condensate starting from a quasi-one dimensional state with external harmonic trapping potential modulation. We derived the analytical solution of the asymmetric solitary vortex based on the two-dimensional Gross-Pitaevskii equation model and variational method. Furthermore, we identified the oscillation mode between the extreme asymmetric state and circular symmetric state for the derived solitary vortex. The obtained theoretical results can be used as guides for relevant experimental studies of asymmetric solitary vortices under similar physical conditions.

Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (

p

) -- substitution energy (

\hat{\cal E}

) plane with

p\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/c

as

U \to c

, where

ρ

is the density,

c

is the speed of sound,

κ

is the quantum of circulation and

U

is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particularly, it is shown that during collapse the vortex core becomes significantly compressed and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.

We consider a rotating Bose-Einstein condensate confined in combined harmonic and quartic traps, following recent experiments [V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004)]. We investigate numerically the behavior of the wave function which solves the three-dimensional Gross Pitaevskii equation and analyze in detail the structure of vortices. For a quartic-plus-harmonic potential, as the angular velocity increases, the vortex lattice evolves into a vortex array with hole. The merging of vortices into the hole is highly three dimensional, starting from the top and bottom of the condensate to reach the center. We also investigate the case of a quartic-minus-harmonic potential, not covered by experiments or previous numerical works. For intermediate repulsive potentials, we show that the transition to a vortex array with hole takes place for lower angular velocities, when the lattice is made up of a small number of vortices. For the strong repulsive case, a transition from a giant vortex to a hole with a circle of vortices around is observed.

Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (

p

)-substitution energy ($\stackrel{^}{\mathcal{E}}$) plane with $p\ensuremath{\rightarrow}0.09\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/{c}^{2}$, $\stackrel{^}{\mathcal{E}}\ensuremath{\rightarrow}0.091\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/c$ as $U\ensuremath{\rightarrow}c$, where $\ensuremath{\rho}$ is the density,

c

is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and

U

is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particular, it is shown that during collapse the vortex core becomes significantly compressed, and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.