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arXiv:1512.02809 · Physical Review A

Hamiltonian dynamics of two same-sign point vortices

Anderson V. Murray, Andrew J. Groszek, Pekko Kuopanportti, Tapio Simula

Published Mar 29, 2016• 13 citations
arXivHTMLPDF

Abstract

We have studied numerically the Hamiltonian dynamics of two same-sign point vortices in an effectively two-dimensional, harmonically trapped Bose-Einstein condensate. We have found in the phase space of the system an impenetrable wall that divides the dynamics into two distinct and exhaustive types. In the two-dimensional position-coordinate space, the first type corresponds to intersecting single-vortex orbits and the second type to orbits that have no points in common. The two types are also easily distinguished in the two-dimensional space spanned by the radial and angular velocities of the vortices: In the first type, both single-vortex orbits are the same simple loop in this two-dimensional space, whereas in the second type the two orbits constitute two nonintersecting loops. The phase-space-dividing wall is distinct from the bifurcation curve of rigidly rotating states found by Navarro et al. [Phys. Rev. Lett. 110, 225301 (2013)].

DOI
10.1103/physreva.93.033649

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We have studied numerically the Hamiltonian dynamics of two same-sign point vortices in an effectively two-dimensional, harmonically trapped Bose-Einstein condensate. We have found in the phase space of the system an impenetrable wall that divides the dynamics into two distinct and exhaustive types. In the two-dimensional position-coordinate space, the first type corresponds to intersecting single-vortex orbits and the second type to orbits that have no points in common. The two types are also easily distinguished in the two-dimensional space spanned by the radial and angular velocities of the vortices: In the first type, both single-vortex orbits are the same simple loop in this two-dimensional space, whereas in the second type the two orbits constitute two nonintersecting loops. The phase-space-dividing wall is distinct from the bifurcation curve of rigidly rotating states found by Navarro et al. [Phys. Rev. Lett. 110, 225301 (2013)].
We consider a rotating Bose-Einstein condensate in a harmonic trap and investigate numerically the behavior of the wave function which solves the Gross-Pitaevskii equation. Following recent experiments [P. Rosenbuch, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 89, 200403 (2002)], we study in detail the line of a single quantized vortex, which has a U or S shape. We find that a single vortex can lie only in the $x\ensuremath{-}z$ or $y\ensuremath{-}z$ plane. S-type vortices exist for all values of the angular velocity $\ensuremath{\Omega}$ while U vortices exist for $\ensuremath{\Omega}$ sufficiently large. We compute the energy of the various configurations with several vortices and study the three-dimensional structure of vortices.
Dynamics of vortex clusters is essential for understanding diverse superfluid phenomena. In this paper, we examine the dynamics of vortex quadrupoles in a trapped two-dimensional (2D) Bose-Einstein condensate. We find that the movement of these vortex-clusters fall into three distinct regimes which are fully described by the radial positions of the vortices in a 2D isotropic harmonic trap, or by the major radius (minor radius) of the elliptical equipotential lines decided by the vortex positions in a 2D anisotropic harmonic trap. In the "recombination" and "exchange" regimes the quadrupole structure maintains, while the vortices annihilate each other permanently in the "annihilation" regime. We find that the mechanism of the charge flipping in the "exchange" regime and the disappearance of the quadrupole structure in the "annihilation" regime are both through an intermediate state where two vortex dipoles connected through a soliton ring. We give the parameter ranges for these three regimes in coordinate space for a specific initial configuration and phase diagram of the vortex positions with respect to the Thomas-Fermi radius of the condensate. We show that the results are also applicable to systems with quantum fluctuations for the short-time evolution.
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.
In this paper, we consider the dynamics of two interacting point vortex rings in a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose – Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented.
This study aims to research on dynamics of two quantized vortices in miscible two-component Bose--Einstein condensates, trapped by a circular box potential by using the Gross--Pitaevskii equation. We consider a situation in which two vortices belong to different components and they are initially close. Their dynamics are significantly different from those of a vortex pair in a single-component condensate. When two vortices are initially co-located, they are split by dynamical instability or remain co-located depending on their intercomponent interaction. If the two split vortices have the same sign of circulation, they rotate around each other. Then, the angular velocity can be given as a function of the distance between two vortices, which is understood through the equations of motion and their interaction. If the circulations of two close vortices have different signs, however, they move in the same direction initially, then overlap during the dynamics. This subsequent overlap can be categorized into two types based on the initial distance between them. The mechanism of this overlapping can be understood from the interaction between the vortex and its image vortex, which makes the two vortices close to each other even if the intrinsic interaction between them is repulsive.