Vortex dynamics near the surface of a Bose-Einstein condensate

U. Al Khawaja

Phys. Rev. A 71, 063611 – Published 22 June 2005

Abstract

The center-of-mass dynamics of a vortex in the surface region of a Bose-Einstein condensate is investigated both analytically using a variational calculation and numerically by solving the time-dependent Gross-Pitaevskii equation. We find, in agreement with previous works, that away from the Thomas-Fermi surface, the vortex moves parallel to the surface of the condensate with a constant velocity. We obtain an expression for this velocity in terms of the distance of the vortex core from the Thomas-Fermi surface that fits accurately with the numerical results. We find also that, coupled to its motion parallel to the surface, the vortex oscillates along the direction normal to the surface around a minimum point of an effective potential.

Received 20 June 2004

DOI:https://doi.org/10.1103/PhysRevA.71.063611

Authors & Affiliations

U. Al Khawaja

Physics Department, United Arab Emirates University, P.O. Box 17551, Al-Ain, United Arab Emirates

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Issue

Vol. 71, Iss. 6 — June 2005

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Images

Figure 1
The planar geometry and coordinate system used in this paper. The small circle represents the core of the vortex which is of the order of the coherence length $ξ$ . The coordinates of the core of the vortex are $x_{0}$ and $y_{0}$ . The radius of the condensate is $R$ . The bulk of the condensate is located in the region $x < 0$ . The point $P$ has Cartesian coordinates $x^{'}$ and $y^{'}$ related to the polar coordinates $ρ$ and $ϕ$ by $x^{'} = ρ \sin ϕ$ and $y^{'} = ρ \cos ϕ$ .Reuse & Permissions
Figure 2
The vortex velocity in the $y$ direction as a function of the magnitude of the distance between the vortex core and the Thomas-Fermi surface. The velocity is in units of $δ ∕ τ$ . The surface depth $δ$ is defined by $ℏ^{2} ∕ m δ^{2} = F δ$ , and the time $τ$ is defined by $τ = ℏ ∕ F δ$ . The dots on the upper curve are the results of the numerical solution of a vortex moving near the surface of the condensate. The dots on the lower curve correspond to a vortex moving parallel to a hard wall in a uniform density background. The solid curve is the result of the variational calculation of Sec. 2, given by Eq. (20). The long-dashed curve corresponds to the expression derived by Anglin [20]. The short-dashed curve is given by $v_{y} = ℏ ∕ 2 m x_{0}$ .Reuse & Permissions
Figure 3
The $x$ coordinate of the vortex core measured from the Thomas-Fermi surface. The points are the results of the numerical solution of the time-dependent Gross-Pitaevskii equation, Eq. (4) for different initial distances. The initial velocity is taken to be zero. The solid and dashed curves are the results of the variational calculation of Sec. 2C. The solid curve is the solution of Eq. (31), and the dashed curve is an analytic approximate solution given by Eq. (35). The value of $b$ used here is $5 δ$ .Reuse & Permissions
Figure 4
The coherence length associated with the $x$ and $y$ cross sections of the vortex density profile. It is calculated by fitting the numerical density profile of the vortex to Eq. (8). Circles represent the coherence length associated with the $y$ cross section of the vortex density profile, and diamonds represent the coherence length for the $x$ cross section.Reuse & Permissions
Figure 5
Density profile of the vortex. The upper row of pictures corresponds to the initial state with the vortex core located at $x_{0} (0) = - 3.1 δ$ . The left column of pictures is a front view in which the $x$ axis is normal to the picture towards the viewer. The right column is a side view in which the $y$ axis is normal to the picture towards the viewer. The vertical line in the right column of pictures is to reference the core location with respect to the initial position, and shows clearly that the vortex core oscillates in the $x$ direction.Reuse & Permissions
Figure 6
The vortex $y$ coordinate. Starting from the upper curve, the values of the initial position of the vortex core $x_{0}$ are $- 3 δ, - 4 δ, - 5 δ$ , and $- 7 δ$ . The constant slope indicates that the velocity component parallel to the surface is constant.Reuse & Permissions

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