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arXiv:cond-mat/0406454 · Physical Review A

Vortex dynamics near the surface of a Bose-Einstein condensate

Usama Al Khawaja

Published Jun 22, 2005• 7 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.71.063611

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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.
The subject of the present thesis is the study of the interaction between a Bose-Einstein condensate (BEC) and a superconductor.The BEC performs center-of-mass oscillations perpendicular to the superconducting surface.Each atom in the BEC carries a magnetic dipole moment, which induces eddy currents in the surface.These eddy currents are a source of a magnetic field which modifies the potential in which the BEC is trapped.This leads to a change of the dynamical behavior of the BEC.In particular the BEC center-of-mass oscillation frequency is shifted compared to the case without the surface interaction.Additionally, the potential generated by the superconductor excites shape oscillations of the BEC.To investigate these effects, the Gross-Pitaevskii equation for a dipolar BEC close to a superconducting surface is solved numerically.Also analytical approximations are presented and compared to the numerically obtained results.It is shown that the interaction with the superconducting surface generates a frequency shift, that is large enough to be detected in an experiment.Furthermore, different methods to identify the eddy current effect are presented.It is also discussed how the effects on the BEC can be enhanced by choosing the proper dipole orientation and by adjusting the geometry of the setup.The BEC shape oscillations can be additionally enhanced by making use of a resonance phenomenon.4 Center-of-mass frequency shift 4.1 Frequency shift within TF approximation . . . . . . . . . .4.1.1Frequency shift based on a Thomas-Fermi ellipsoid 4.1.2Frequency shift based on the column density model 4.2 Numerical results for the frequency shift . . . . . . . . . . .4.3 Different polarizations of the BEC . . . . . . . . . . . . . .4.3.1 Polarization in the y-direction . . . . . . . . . . . .4.3.2Polarization in the x-direction (perpendicular to the surface) . . . . . . . . . . .
The theory of vortex motion in a dilute superfluid of inhomogeneous density demands a boundary layer approach, in which different approximation schemes are employed close to and far from the vortex, and their results matched smoothly together. The most difficult part of this procedure is the hydrodynamic problem of the velocity field, many healing lengths away from the vortex core. This paper derives and exploits an exact solution of this problem in the two-dimensional case of a linear trapping potential, which is an idealization of the surface region of a large condensate. It thereby shows that vortices in inhomogeneous clouds are effectively ``dressed'' by a nontrivial distortion of their flow fields, that image vortices are not relevant to Thomas-Fermi surfaces, and that for condensates large compared to their surface depths, the energetic barrier to vortex penetration disappears at the Landau critical velocity for surface modes.
We numerically study properties of the dynamics of vortices in nonrotating Bose-Einstein condensates in the Thomas-Fermi regime. On the one hand, we compute the vortex energy as a function of its position and we predict, using the expression of the Magnus force, the vortex precession velocity. On the other hand, we calculate the temporal evolution of the vortex-state and test the accuracy of the previous prediction. We also investigate the validity of analytical formulae of this velocity involving the healing length. In addition, we analyze the velocity field and the angular momentum and we compare them to available analytical expressions.
We present a numerical calculation of the velocity field of an off-axis vortex within a harmonic trapping potential. We consider a condensate with a large number of particles which, in spite of being in the Thomas-Fermi regime, possesses a non-totally-negligible vortex core size. Via a simple, yet realistic, modeling of the vortex-state density, we derive, by treating the equation of continuity outside and inside the vortex core, respective equations for the velocity field and the vortex precession velocity. We find that the vortex precession velocity is given by two contributions: the background velocity field evaluated around the border of the core and another term which depends on the core shape. Our findings concerning the velocity field outside the vortex core are in good agreement with previous theoretical predictions in a narrow region around it, while far away from the vortex we observe a field with a large asymmetry with respect to the vortex position. We show that a better approximation may be obtained by adding the velocity field produced by an antivortex located outside the condensate.
The dynamics and interaction of quantized vortices in Bose-Einstein condensates (BECs) are investigated by using the two-dimensional Gross-Pitaevskii equation (GPE) with/without an angular momentum rotation term. If all vortices have the same winding number, they would rotate around the trap center but never collide. In contrast, if the winding numbers are different, their interaction highly depends on the initial distance between vortex centers. The analytical results are presented to describe the dynamics of the vortex centers when β=0. While if β≠0, there is no analytical result but some conclusive numerical findings are provided for the further understanding of vortex interaction in BECs. Finally, the dynamic laws describing the relation of vortex interaction in nonrotating and rotating BECs are presented.
One of the key issues related to superfluidity is the existence of quantized vortices. Following recent experiments on Bose–Einstein condensates exhibiting vortices, we investigate the behavior of the wave function which solves the Gross–Pitaevskii equation. For a rotating Bose–Einstein condensate in a harmonic trap, we give a simplified expression of the Gross–Pitaevskii energy in the Thomas–Fermi regime, which only depends on the number and shape of the vortex lines. This allows us to study in detail the structure of the line of a single quantized vortex, which has a U or S shape. S type vortices exist for all values of the angular velocity Ω but are not minimizers of the energy while U vortices are minimizers and exist only for Ω sufficiently large. Finally, we study the drag created by the movement of a detuned laser beam in a condensate and the nucleation of vortices in the low density region close to the boundary layer of the cloud.