Existence, stability and nonlinear dynamics of vortex clusters in - PowerPoint PPT Presentation

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 Center for Optical Quantum Technologies, University of Hamburg P.G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts - Amherst

Outline Motivation and introduction to the theoretical model  Particle-like description of vortices and their dynamics  Numerical results  Near-linear (bifurcation) approach  Conclusions  Sevilla, July 12, 2012 2

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 ω x ǂ ω y (in general) Why such ”vortex clusters”? Recent experiments, new methods for creating and imaging vortices 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) Why anisotropic confinement? Experimentally controllable parameter, relevance for stability properties expected, theoretical results so far incomplete and partially incoherent 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) Sevilla, July 12, 2012 3

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: Sevilla, July 12, 2012 4

Vortex precession Precessional motion of vortices modified  due to anisotropic confinement: elliptical orbits Precession frequency (from matched  asymptotics approach): 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) assume Thomas-Fermi background  (highly nonlinear limit) neglect off-center effects  Sevilla, July 12, 2012 5

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) Sevilla, July 12, 2012 6

”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 Sevilla, July 12, 2012 7

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  Sevilla, July 12, 2012 8

Numerics: Single vortex  Predicted precession frequency ω pr (black) coincides with anomalous mode (red) in the BdG spectrum: Sevilla, July 12, 2012 9

Vortex dipole 2 vortices of opposite charge  Equilibria symmetrically along  trap's main axes Sevilla, July 12, 2012 10

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) Sevilla, July 12, 2012 11

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 Sevilla, July 12, 2012 12

Stabilizing the vortex tripole Sevilla, July 12, 2012 13

(De)stabilizing effect of anisotropy on aligned vortex clusters  stabilizing  (further) destabilizing Sevilla, July 12, 2012 14

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: 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) Cascade of symmetry-breaking bifurcations inducing the ”snaking  instability” of the dark soliton stripe in 2D Sevilla, July 12, 2012 15

Anisotropy effectively shifts bifurcations from the dark soliton stripe Sevilla, July 12, 2012 16

Anisotropy effectively shifts bifurcations from the dark soliton stripe First bifurcating vortex cluster branch inherits stability from the parental dark soliton stripe Sevilla, July 12, 2012 17

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 μ cr possible Theocharis et al., Phys. Rev. E 74 , 056608 (2006) Middelkamp et al., Phys. Rev. A 82 , 013646 (2010) μ cr 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  Sevilla, July 12, 2012 18

String-like oscillations in a stabilized 17-vortex cluster  ”fundamental”  ”first overtone” Sevilla, July 12, 2012 19

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) Sevilla, July 12, 2012 20