Dynamics of Vortex Dipoles in Anisotropic Bose--Einstein Condensates

Abstract

Abstract. We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ordinary differential equations describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system we are able to construct complex periodic orbits in the original, partial differential equation, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.

Bifurcations of Multi-Vortex Configurations in Rotating Bose–Einstein Condensates

“…Existence and stability of stationary states were analyzed in [9,10] with the amplitude equations for the Hermite function decompositions and their truncation at the continuous resonant equation. Vortex dipoles were studied with normal form equations and numerical approximations in [12]. Numerical evidences of existence, bifurcations, and stability of such vortex and dipole solutions can be found in a vast literature [22,23,25,28,31].…”

Multi-vortex crystal lattices in Bose–Einstein condensates with a rotating trap

“…This concludes the derivation of formula (2), which is the starting point for all the subsequent results of this paper. The fundamental element of novelty in our dynamical equations lies in the treatment of the interaction terms, as both the anisotropic and the dissipative cases have been recently considered in a similar vein from the viewpoint of effective particle dynamics; see, e.g., [29] and [40] for respective examples. In what follows, we will proceed to analyze the resulting systems for N = 2, as well as for general N number of vortices for both isotropic and anisotropic traps, comparing the conclusions to those stemming from direct numerical simulations.…”

“…The fundamental element of novelty in our dynamical equations lies in the treatment of the interaction terms, as both the anisotropic and the dissipative cases have been recently considered in a similar vein from the viewpoint of effective particle dynamics; see, e.g. [31,42] for respective examples.…”

An inverse problem from condensed matter physics

“…dipoles in the presence of anisotropic backgrounds is extremely rich and has been studied, see for example [12].…”