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International Conference Nonlinearity, Information and Robotics (NIR)

Dynamics of interacting two vortex N-gons in BECs

Elizaveta Markovna Artemova, Alexander Alexandrovich Kilin

Published Aug 26, 2021• 0 citations
Publisher

Abstract

Equations governing the motion of vortex N-gons in a Bose-Einstein condensate enclosed in a harmonic trap are obtained. It is shown that these equations have two first integrals: the integral of energy and the integral of the moment. The reduction on the fixed level set of the integral of moment is carried out. The equilibrium positions of the system considered are found. Phase portraits of the system of two vortex N-gons, each of which consists of two vortices, are constructed.

DOI
10.1109/nir52917.2021.9666121

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Equations governing the motion of vortex N-gons in a Bose-Einstein condensate enclosed in a harmonic trap are obtained. It is shown that these equations have two first integrals: the integral of energy and the integral of the moment. The reduction on the fixed level set of the integral of moment is carried out. The equilibrium positions of the system considered are found. Phase portraits of the system of two vortex N-gons, each of which consists of two vortices, are constructed.
The problem of stability of rotating regular vortex N-gons (Thomson's configurations) in a Bose–Einstein condensate in a harmonic trap is considered. A reduction procedure on the level set of the momentum integral is proposed. The dependence of the velocity of rotation ω of vortex polygon about the center of the trap is obtained as a function of the number of vortices N and the radius of the configuration, R. The analysis of the orbital linear and nonlinear stability of the motion of such configurations is carried out. For N ⩽ 6, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for N>6 are unstable for any parameters of the system. In this paper, we study the stability of rotating regular vortex N-gons in a Bose–Einstein condensate in a harmonic trap. The analysis of the orbital linear and nonlinear stability of motion is carried out. The dependence of the stability of regular vortex N-gons on the number of vortices N and the parameters of the system is given.
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.
We consider the problem of the stability of rotating regular vortex N-gons (Thomson configurations) in a Bose-Einstein condensate in a harmonic trap. The dependence of the rotation velocity ω of the Thomson configuration around the center of the trap is obtained as a function of the number of vortices N and the radius of the configuration R. The analysis of the stability of motion of such configurations in the linear approximation is carried out. For N⩽6, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for N > 6 are unstable for any parameters of the system.
We study the stability of vortices in a binary system of Bose-Einstein condensat es, with their wave functions modeled by a set of coupled, time-dependent Gross-Pitaevskii equations. Beginning with an effective two-dimensional system, we identify miscible and immiscible regimes characterized by the inter- and intra-atomic interactions and the initial configuration of the system. We then consider a binary system of Bose-Einstein condensates placed in a rotating harmonic trap and study the single united vortex state in this system. We derive an approximate form for the energy of a single united vortex in the binary system and the critical angular velocity for the global stability of a vortex at the center of the trap. We also compute the metastability onset angular velocity for the local stability of a vortex at the center of the trap. Numerical solutions to the Gross-Pitaevskii equations support these expressions. These rotational results inform us of a novel bimodal formation within the miscible regime of the binary condensate system. We thus demonstrate the nontrivial aspects of vortex stability in interacting binary Bose-Einstein condensates as a result of their nonlinear interactions.
We present a method to study the dynamics of a quasi-two dimensional Bose-Einstein condensate which initially contains several vortices at arbitrary locations. The method allows one to find the analytical solution for the dynamics of the Bose-Einstein condensate in a homogeneous medium and in a parabolic trap, for the ideal non-interacting case. Secondly, the method allows one to obtain algebraic equations for the trajectories of the position of phase singularities present in the initial condensate along with time (the vortex lines). With these equations, one can predict quantities of interest, such as the time at which a vortex and an antivortex contained in the initial condensate will merge. For the homogeneous case, this method was introduced in the context of photonics. Here, we adapt it to the context of Bose-Einstein condensates, and we extend it to the trapped case for the first time. Also, we offer numerical simulations in the non-linear case, for repulsive and attractive interactions. We use a numerical split-step simulation of the non-linear Gross-Pitaevskii equation to determine how these trajectories and quantities of interest are changed by the interactions. We illustrate the method with several simple cases of interest, both in the homogeneous and parabolically trapped systems.