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Conference on Localized Excitations in Nonlinear Complex Systems Sevilla, July 9-12, 2012 Existence, stability and nonlinear dynamics of vortex clusters in anisotropic Bose-Einstein condensates J. Stockhofe, S. Middelkamp, P. Schmelcher

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Existence, stability and nonlinear dynamics

  • f vortex clusters in anisotropic

Bose-Einstein condensates

  • J. Stockhofe, S. Middelkamp, P. Schmelcher

Center for Optical Quantum Technologies, University of Hamburg

P.G. Kevrekidis

Department of Mathematics and Statistics, University of Massachusetts - Amherst

Conference on Localized Excitations in Nonlinear Complex Systems Sevilla, July 9-12, 2012

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Outline

Motivation and introduction to the theoretical model

Particle-like description of vortices and their dynamics

Numerical results

Near-linear (bifurcation) approach

Conclusions

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Motivation

Objective of our work: Study arrangements of few matter-wave quantum vortices in a non-rotating, repulsive, quasi-2D BEC in the presence of a harmonic trap with ωx ǂ ωy (in general) Why such ”vortex clusters”? Recent experiments, new methods for creating and imaging vortices Why anisotropic confinement? Experimentally controllable parameter, relevance for stability properties expected, theoretical results so far incomplete and partially incoherent

Freilich et al., Science 329, 1182 (2010) Neely et al., Phys. Rev. Lett. 104, 160401 (2010) Seman et al., Phys. Rev. A 82, 033616 (2010) Middelkamp et al., PRA 84, 011605(R) (2011) Crasovan et al., Phys. Rev. A 68, 063609 (2003) Möttönen et al., Phys. Rev. A 71, 033626 (2005) Pietilä et al., Phys. Rev. A 74, 023603 (2006) Middelkamp et al., Phys. Rev. A 82, 013646 (2010)

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Mean-field model

  • 2D Gross-Pitaevskii equation, z-direction ”frozen out”
  • Repulsive interaction (defocusing nonlinearity)
  • Parameter α controls anisotropy
  • Obtain stationary solutions by factorizing order parameter:
  • Stationary vortex solution:
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Vortex precession

Precessional motion of vortices modified due to anisotropic confinement: elliptical

  • rbits

Precession frequency (from matched asymptotics approach):

assume Thomas-Fermi background (highly nonlinear limit)

neglect off-center effects

Svidzinsky/Fetter, Phys. Rev. A 62, 063617 (2000) Fetter/Svidzinsky, J. Phys.: Condens. Matter 13, R135 (2001) Middelkamp et al., J. Phys. B 43, 155303 (2010)

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Vortex interaction

Each vortex moves with velocity field created by all the other vortices

Interaction contribution to dynamics:

Model implicitly assumes homogeneous background: inhomogeneous Thomas-Fermi profile partially accounted for by using an effective B

Middelkamp et al., PRA 82, 013646 (2010) Middelkamp et al., PRA 84, 011605(R) (2011)

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”Particle picture” equations of motion

Combine effects of precessional motion and interaction to describe dynamics of vortices by coupled ODEs: Fixed points of these ODEs ↔ Stationary vortex clusters of the GPE Linearization modes of the ODEs ↔ BdG modes of the vortex cluster

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Reminder: Bogoliubov-de Gennes analysis

Having identified a stationary solution u(x,y), study linearized dynamics around it:

Linearizing time-dependent GPE → BdG eigenvalue problem

Non-zero imaginary parts → stationary state dynamically unstable

Lowest-order contribution to GPE energy functional:

→ ”anomalous” mode, energetically unstable

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Numerics: Single vortex

 Predicted precession frequency ωpr (black) coincides with

anomalous mode (red) in the BdG spectrum:

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Vortex dipole

2 vortices of opposite charge

Equilibria symmetrically along trap's main axes

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Vortex tripole

3 vortices, charges alternating

Full stabilization of this ”tripole” for strong enough transversal confinement

Particle picture ODEs:

Numerically:

taken from: Seman et al., Phys. Rev. A 82, 033616 (2010)

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Destabilized dipole: dynamics

Elongating the cloud perpendicularly to the dipole axis leads to instability (remember imaginary BdG mode for α > 1)

Random perturbation induces onset of (periodic) dynamics → decays and revivals

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Stabilizing the vortex tripole

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(De)stabilizing effect of anisotropy

  • n aligned vortex clusters

 stabilizing  (further) destabilizing

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Near-linear bifurcation analysis

So far: highly nonlinear regime, vortices localized ”point-like” entities

Understanding (de)stabilization of vortex clusters due to anisotropy close to the linear (Schrödinger) limit?

Well known in isotropic setting: dipole, tripole,... bifurcate from dark soliton branch as chemical potential (or particle number) is increased:

Cascade of symmetry-breaking bifurcations inducing the ”snaking instability” of the dark soliton stripe in 2D

Crasovan et al., Phys. Rev. A 68, 063609 (2003) Li et al., Phys. Rev. A 77, 053610 (2008) Middelkamp et al., Phys. Rev. A 82, 013646 (2010)

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Anisotropy effectively shifts bifurcations from the dark soliton stripe

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Anisotropy effectively shifts bifurcations from the dark soliton stripe

First bifurcating vortex cluster branch inherits stability from the parental dark soliton stripe

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Quantitative understanding within Galerkin-type approach

Near linear limit of vanishing particle number: aligned n-vortex state well approximated by harmonic oscillator modes

Within the two-dimensional subspace of the full Hilbert space: prediction of bifurcation's μcr possible

μcr crucially depends on energy difference between linear modes, which in turn is controlled by the anisotropy parameter α

Result: n-vortex line inherits soliton's stability below critical value

Reproduces numerical results for dipole and tripole

Stabilization occurs even for large numbers of vortices n

Theocharis et al., Phys. Rev. E 74, 056608 (2006) Middelkamp et al., Phys. Rev. A 82, 013646 (2010)

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String-like oscillations in a stabilized 17-vortex cluster

 ”fundamental”  ”first overtone”

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Conclusions and references

Anisotropy of the trapping potential strongly affects the stability of aligned vortex configurations such as the dipole and tripole

Elongation of the cloud along the cluster's axis has a stabilizing effect, elongation in the perpendicular direction destabilizes the vortex configuration

These effects can be understood both from a particle-like ODE description (valid in the highly nonlinear regime) and a near-linear bifurcation analysis

Large aligned vortex clusters which are stabilized by anisotropy support string-like oscillations

Methods also apply to non-aligned vortex clusters, in particular quadrupoles Further information:

  • J. Stockhofe, S. Middelkamp, P.G. Kevrekidis, P. Schmelcher: Europhys. Lett. 93, 20008 (2011)

  • J. Stockhofe, P.G. Kevrekidis, P. Schmelcher: arXiv 1203.4762 (2012)

(to be published as a chapter in the forthcoming volume ”Spontaneous Symmetry Breaking, Self-Trapping and Josephson Oscillations”, ed. by B. Malomed, Springer)