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Existence, Stability and Nonlinear Dynamics of Vortices and Vortex Clusters in Anisotropic Bose-Einstein Condensates

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Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations

Part of the book series: Progress in Optical Science and Photonics ((POSP,volume 1))

Abstract

This chapter is devoted to the study of vortex excitations in one-component Bose-Einstein condensates, with a special emphasis on the impact of anisotropic confinement on the existence, stability and dynamical properties of vortices and particularly few-vortex clusters. Symmetry breaking features are pervasive within this system even in its isotropic installment, where cascades of symmetry breaking bifurcations give rise to the multi-vortex clusters, but also within the anisotropic realm which naturally breaks the rotational symmetry of the multi-vortex states. Our first main tool for analyzing the system consists of a weakly nonlinear (bifurcation) approach which starts from the linear states of the problem and examines their continuation and bifurcation into novel symmetry-broken configurations in the nonlinear case. This is first done in the isotropic limit and the modifications introduced by the anisotropy are subsequently presented. The second main tool concerns the highly nonlinear regime where the vortices can be considered as individual topologically charged "particles" which precess within the parabolic trap and interact with each other, similarly to fluid vortices. The conclusions stemming from both the bifurcation and the interacting particle picture are corroborated by numerical computations which are also used to bridge the gap between these two opposite-end regimes.

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Notes

  1. 1.

    The connection provided through the numerical results is often essential as some configurations may e.g. be stable in the above two limits but possess instabilities in finite intermediate ranges of parameter values that would not be observable by restricting our view to the analytically tractable limits. A notable example of this type is offered by the vortex quadrupole configuration (see e.g. Fig.  8 of [16] and equivalently the isotropic limit of both Fig. 10d and e below). Such a state is found to be linearly stable in both of the above quasi-analytical limits and its intermediate range of instability parameter values is only detected by the bridging numerical continuation.

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Authors and Affiliations

  1. Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany

    J. Stockhofe & P. Schmelcher

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, 01003-4515, MA, USA

    P. G. Kevrekidis

Authors
  1. J. Stockhofe
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  2. P. G. Kevrekidis
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  3. P. Schmelcher
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Correspondence to P. Schmelcher .

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Editors and Affiliations

  1. Faculty of Engineering The Iby and Aladar Fleischman, Tel Aviv University, Ramat Aviv, Israel

    Boris A. Malomed

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Stockhofe, J., Kevrekidis, P.G., Schmelcher, P. (2012). Existence, Stability and Nonlinear Dynamics of Vortices and Vortex Clusters in Anisotropic Bose-Einstein Condensates. In: Malomed, B. (eds) Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations. Progress in Optical Science and Photonics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10091_2012_10

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  • DOI: https://doi.org/10.1007/10091_2012_10

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