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Journal of Experimental and Theoretical Physics Letters

Propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate

V. A. Mironov, Lev A. Smirnov

Published Aug 1, 2012• 7 citations
Publisher

Abstract

An asymptotic theory has been developed to describe the propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate with the repulsive interaction between atoms. It has been shown that the trajectories of vortex pairs coincide with geometric-optical rays in an equivalent isotropic medium with the refractive index depending both on the density of the unperturbed inhomogeneous condensate and on the energy of structures under study. The transformation of vortex pairs to vortex-free quasisolitons and back has been described.

DOI
10.1134/s0021364012110057

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An asymptotic theory has been developed to describe the propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate with the repulsive interaction between atoms. It has been shown that the trajectories of vortex pairs coincide with geometric-optical rays in an equivalent isotropic medium with the refractive index depending both on the density of the unperturbed inhomogeneous condensate and on the energy of structures under study. The transformation of vortex pairs to vortex-free quasisolitons and back has been described.
We develop an asymptotic theory describing the dynamics of two-dimensional dark quasisolitons in a smoothly inhomogeneous Bose-Einstein condensate (BEC) with repulsive interaction between atoms. It is shown that the trajectories of such quasisolitons coincide with the geometric-optical rays in an equivalent isotropic medium characterized by the effective refractive index dependent on both the density of the undisturbed condensate and the energy of the structures. An expression for this effective refractive index is found and specific features of the refraction of quasisolitons with different initial energies by the condensate inhomogeneities are studied. The obtained results are confirmed by direct numerical simulation using the Gross-Pitaevskii (GP) equation.
We analyze the scattering of vortex pairs (the particular case of 2D dark solitons) by a single quantum vortex in a Bose–Einstein condensate with repulsive interaction between atoms. For this purpose, an asymptotic theory describing the dynamics of such 2D soliton-like formations in an arbitrary smoothly nonuniform flow of a ultracold Bose gas is developed. Disregarding the radiation loss associated with acoustic wave emission, we demonstrate that vortex–antivortex pairs can be put in correspondence with quasiparticles, and their behavior can be described by canonical Hamilton equations. For these equations, we determine the integrals of motion that can be used to classify various regimes of scattering of vortex pairs by a single quantum vortex. Theoretical constructions are confirmed by numerical calculations performed directly in terms of the Gross–Pitaevskii equation. We propose a method for estimating the radiation loss in a collision of a soliton-like formation with a phase singularity. It is shown by direct numerical simulation that under certain conditions, the interaction of vortex pairs with a core of a single quantum vortex is accompanied by quite intense acoustic wave emission; as a result, the conditions for applicability of the asymptotic theory developed here are violated. In particular, it is visually demonstrated by a specific example how radiation losses lead to a transformation of a vortex–antivortex pair into a vortex-free 2D dark soliton (i.e., to the annihilation of phase singularities).
We study the process of scattering of two-dimensional dark solitons, and their vortex-antivortex pairs as a specific case, by a single quantum vortex in a Bose-Einstein condensate with repulsive interaction between atoms. An asymptotic theory describing the dynamics of such solitonlike structures in a smoothly inhomogeneous flow of ultracold Bose gas is developed. An analytical expression for the angle of scattering of two-dimensional dark solitons (including vortex pairs) by a single-phase singularity is obtained in the limit of large impact parameters. All theoretical concepts are confirmed by numerical calculations performed directly within the Gross-Pitaevskii equation. It is shown that for small impact parameters, the considered solitonlike structures interact inelastically with the core of a single quantum vortex, scattering over large angles and radiating sound waves.
Abstract Understanding quantum dynamics in a two-dimensional Bose–Einstein condensate relies on understanding how vortices interact with each other microscopically and with local imperfections of the potential which confines the condensate. Within a system consisting of many vortices, the trajectory of a vortex–antivortex pair is often scattered by a third vortex, an effect previously characterised. However, the natural question remains as to how much of this effect is due to the velocity induced by this third vortex and how much is due to the density inhomogeneity which it introduces. In this work, we describe the various qualitative scenarios which occur when a vortex–antivortex pair interacts with a smooth density impurity whose profile is identical to that of a vortex but lacks the circulation around it.
The mechanisms behind the motion of vortices has recently been re-examined. The findings have given a more detailed and complete view as to how vortices move in Bose-Einstein condensation. Specifically, vortex propagation is not seen as purely the result of a density gradient which would arise because of the gradient of the trapping potential, but is also made up of a contribution from the boundary eects. I will consider the motion of vortices in an asymptotically inhomogeneous condensate under various trapping potentials. By first considering a linear potential trap and then a semi-infinite quadratic trap, the eects that govern the motion can be separated. It will be shown that for large density gradient backgrounds the motion can be analysed from the energy-momentum dispersion relationship, whereas for low density gradient backgrounds the motion is dominated by an eective shift in the image term caused by the eects of the boundary. Finally, a separate case of the motion of vortices in an infinitely long two-dimensional channel will be presented where the interaction strength plays a key role in the dynamics.
Vortex dynamics in inhomogeneous Bose-Einstein condensates are studied numerically in two and three dimensions. We simulate the precession of a single vortex around the center of a trapped condensate, and use the Magnus force to estimate the precession frequency. Vortex ring dynamics in a spherical trap are also simulated, and we discover that a ring undergoes oscillatory motion around a circle of maximum energy. The position of this locus is calculated as a function of the number of condensed atoms. In the presence of dissipation, the amplitude of the oscillation will increase, eventually resulting in self-annihilation of the ring.