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Effects of System Geometry on the Dynamics of Ideal Point Vortices in a Two-Dimensional Quantum Gas

Markus Lakaniemi

Published Aug 14, 2007• 0 citations
OpenAlex

Abstract

A Lagrangian describing a gas of ideal pointlike vortices interacting with a quantum gas of spin-0 2 particles confined in an infinite or finite subset of is derived. Single- and two-vortex dynamics is studied in the special case of a homogeneous gas in infinite and finite systems. Special attention is given to the case of the annulus (ring) for which new analytic and numerical solutions are found. We define the concept of stability radius for single- and two-vortex systems and see that the existence of these radii has great impact on the behaviour of the vortices in the annulus. It is shown that the vortex-vortex pair and the vortex-antivortex pair can display many similar types of solutions because of the existence of the stability radii. Especially an analytic corotating solution similar to the case of a vortex-vortex pair in 2 exist both for the vortex-vortex and vortex-antivortex pair. 1

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A Lagrangian describing a gas of ideal pointlike vortices interacting with a quantum gas of spin-0 2 particles confined in an infinite or finite subset of is derived. Single- and two-vortex dynamics is studied in the special case of a homogeneous gas in infinite and finite systems. Special attention is given to the case of the annulus (ring) for which new analytic and numerical solutions are found. We define the concept of stability radius for single- and two-vortex systems and see that the existence of these radii has great impact on the behaviour of the vortices in the annulus. It is shown that the vortex-vortex pair and the vortex-antivortex pair can display many similar types of solutions because of the existence of the stability radii. Especially an analytic corotating solution similar to the case of a vortex-vortex pair in 2 exist both for the vortex-vortex and vortex-antivortex pair. 1
The dynamics of one and two pointlike vortices in a planar quantum gas of spin-0 particles confined in an annular region is considered. New analytical and numerical solutions are found. The concept of stationarity radius, related to the doubly connected nature of the annulus, is defined. It is seen that the existence of these radii has great impact on the behaviour of the vortices. It is shown that, because of the existence of the stationarity radii, vortices exhibit similar behaviour regardless of the sign of their winding number. The energetically stable vortex solutions are studied qualitatively.
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.
A family of exact analytical solutions of vortices in quantum fluid governed by a two-dimensional time-dependent Schrödinger equation is presented, which describes different kinds of vortex structures. The dynamics of different vortex clusters, such as the single vortex, vortex pair, vortex dipole and vortex trimer in a two-dimensional quantum fluid are analytically studied based on these exact solutions. The time evolutions of the wave of such vortices are demonstrated, and the orbits of motion of singular points in the vortices are also explored. The interactions of vortices in many-vortex clusters are discussed. A repulsive interaction between vortices with the same topological charge, and inter-annihilation and inter-creation of vortices with opposite topological charge, are shown.
We study the dynamics of a single vortex and a pair of vortices in quasi two-dimensional Bose-Einstein condensates at finite temperatures. To this end, we use the stochastic Gross--Pitaevskii equation, which is the Langevin equation for the Bose--Einstein condensate. For a pair of vortices, we study the dynamics of both the vortex-vortex and vortex-antivortex pairs, which are generated by rotating the trap and moving the Gaussian obstacle potential, respectively. Due to thermal fluctuations, the constituent vortices are not symmetrically generated with respect to each other at finite temperatures. This initial asymmetry coupled with the presence of random thermal fluctuations in the system can lead to different decay rates for the component vortices of the pair, especially in the case of two corotating vortices.
In this paper, we investigate the dynamics of the vortices in the homogeneous two-dimensional Bose–Einstein condensate by considering the condensate density variation. It is shown that the condensate density variation plays an important role when two vortices are close together. By the time-dependent Lagrangian functional method and the numerical simulation, we study how the interaction strength of the vortices varies with the distance between two vortices with the same or different topological charges, and generalize the results for the vortex pairs to the multi-vortex case. The numerical simulations demonstrate the validity of our theory.
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.