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Journal of Mathematical Sciences

Equations of Motion of the Vortices in Bose–Einstein Condensates: Influence of Rotation and The Inhomogeneity of Density

T. I. Zueva

Published Nov 14, 2016• 1 citation
Publisher
DOI
10.1007/s10958-016-3169-3

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The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.
A dissipative model of vortex motion within a two-dimensional Gross-Pitaevskii equation is studied. With the asymptotic coordination method of solutions, a system of ordinary first-order differential equations describing vortex motion in rotating Bose-Einstein condensate was derived. The model takes into account the effect of various external factors: magnetic trap frequencies, number of particles in the condensate, angular rotational velocity of the condensate, dissipation parameter, etc. In special cases of dissipation-free motion, the results are coordinated with the known results of other authors. The addition of dissipation generalizes the known equations and makes it possible to see the vortex motion to the equilibrium points and to define the equilibrium configurations of any number of vortices. The model is illustrated by a large number of examples.
In two spatial dimensions, vortex-vortex interactions approximately vary with the logarithm of the inter-vortex distance, making it possible to describe an ensemble of vortices as a Coulomb gas. We introduce a duality between vortices in a quasi-two-dimensional (quasi-2D) scalar Bose-Einstein condensates (BEC) and effective Maxwell's electrodynamics. Specifically, we address the general scenario of inhomogeneous, time-dependent BEC number density with dissipation or rotation. Starting from the Gross-Pitaevskii equation (GPE), which describes the mean-field dynamics of a quasi-2D scalar BEC without dissipation, we show how to map vortices in a quasi-2D scalar BEC to 2D electrodynamics beyond the point-vortex approximation, even when dissipation is present or in a rotating system. The physical meaning of this duality is discussed.