Dynamics of three noncorotating vortices in Bose-Einstein condensates

Abstract

In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the energy (which is expressed through the Hamiltonian H) and the angular momentum L of the system. By using the integral of the angular momentum, we reduce the system to a 2-degrees-of-freedom one with L as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.

Bifurcation Diagram of One Generalized Integrable Model of Vortex Dynamics

“…A special attention is paid to vortex structures in a Bose-Einstein condensate obtained for ultracold atomic gases [1]. In this paper we consider a generalized mathematical model which covers such two problems as dynamics of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate [2], [3], [4] and dynamics of two point vortices bounded by a circular region in an ideal fluid [5], [6], [7]. This model leads to a completely Liouville integrable Hamiltonian system with two degrees of freedom, therefore topological methods used for such systems can be applied.…”

“…Physical parameter "c" expresses the extent of the vortices' interaction, ε is a parameter of deformation. These parameters determine two limiting cases, namely, the model of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate (ε = 0) [2], [3], [4] and the model of two point vortices bounded by a circular region in an ideal fluid (c = 0, ε = 1) [5], [6], [7]. The phase space P is defined as a direct product of two open disks of radius 1 with the exception of vortices' collision points P = {(z 1 , z 2 ) : |z 1 | < 1, |z 2 | < 1, z 1 = z 2 }.…”

Dynamics of trapped interacting vortices in Bose–Einstein condensates: a role of breathing degree of freedom

“…The effective dynamics with frozen width degree of freedom can reproduce neither rippled structures (: rapid radial oscillation superposed on the slower rotational motion) nor the global geometry of trajectories. Likewise, the effective dynamics proposed so far [3,4,15,16] does not include the coupling with w(t), and therefore cannot generate these rippled structures.…”

“…[3,4,15,16] which has also a nonlinear spring force and a renormalization of the strength of the inter-vortex interaction but has no coupling with the width degree of freedom. The minor discrepancy of the spring force constant and renormalized strength of inter-vortex interaction between the present and preceding works are due to the Thomas-Fermi approximation in the latter works which suppresses the kinetic energy term in GPE.…”

“…On the other hand, the effective dynamics of interacting vortices in trapped BECs with strong nonlinearity is recently proposed and numerically analyzed [3,4,15,16], but little theoretical attention has been paid to how to couple the core-coordinates of interacting vortices with the breathing width degree of freedom.…”