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arXiv:0803.3251 · Physical Review A

Stability of quantized vortices in a Bose-Einstein condensate confined in an optical lattice

Kody J. H. Law, L. Qiao, P. G. Kevrekidis, Ioannis G. Kevrekidis

Published May 27, 2008• 16 citations
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Abstract

We investigate the existence and especially the linear stability of single- and multiple-charge quantized vortex states of nonlinear Schr\"odinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions. The study is motivated by an examination of pancake-shaped Bose-Einstein condensates in the presence of magnetic and optical confinement. A two-parameter space of the condensate's chemical potential versus the periodic potential's strength is scanned for both single- and double-quantized vortex states located at a local minimum or a local maximum of the lattice. Triple-charged vortices are also briefly discussed. Single-charged vortices are found to be stable for cosinusoidal potentials and unstable for sinusoidal ones above a critical strength. Higher-charge vortices are more unstable for both types of potentials, and their dynamical evolution leads to a breakup into single-charged vortices.

DOI
10.1103/physreva.77.053612

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•May 4, 2004•2 citations
We investigate the existence and especially the linear stability of single- and multiple-charge quantized vortex states of nonlinear Schr"odinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions. The study is motivated by an examination of pancake-shaped Bose-Einstein condensates in the presence of magnetic and optical confinement. A two-parameter space of the condensate's chemical potential versus the periodic potential's strength is scanned for both single- and double-quantized vortex states located at a local minimum or a local maximum of the lattice. Triple-charged vortices are also briefly discussed. Single-charged vortices are found to be stable for cosinusoidal potentials and unstable for sinusoidal ones above a critical strength. Higher-charge vortices are more unstable for both types of potentials, and their dynamical evolution leads to a breakup into single-charged vortices.
We study the existence and stability of solutions of the two-dimensional nonlinear Schrodinger equation in the combined presence of a parabolic and a periodic potential. The motivating physical example consists of Bose-Einstein condensates confined in an harmonic (e.g., magnetic) trap and an optical lattice. By connecting the nonlinear problem with the underlying linear spectrum, we examine the bifurcation of nonlinear modes out of the linear ones for both focusing and defocusing nonlinearities. In particular, we find real-valued solutions (such as multipoles) and complex-valued ones (such as vortices). A primary motivation of the present work is to develop "rules of thumb" about what waveforms to expect emerging in the nonlinear problem and about the stability of those modes. As a case example of the latter, we find that among the real-valued solutions, the one with larger norm for a fixed value of the chemical potential is expected to be unstable.
We investigate the dynamics of vortices in repulsive Bose–Einstein condensates in the presence of an optical lattice (OL) and a parabolic magnetic trap. The dynamics is sensitive to the phase of the OL potential relative to the magnetic trap, and depends less on the OL strength. For the cosinusoidal OL potential, a local minimum is generated at the trap’s centre, creating a stable equilibrium for the vortex, while in the case of the sinusoidal potential, the vortex is expelled from the centre, demonstrating spiral motion. Cases where the vortex is created far from the trap’s centre are also studied, revealing slow outward-spiralling drift. Numerical results are explained in an analytical form by means of a variational approximation. Finally, motivated by a discrete model (which is tantamount to the case of the strong OL lattice), we present a novel type of vortex consisting of two pairs of antiphase solitons.
We study the dynamics of a BEC with a singly quantized vortex, placed in the combined potential of a 1D (2D) optical lattice and an axi-symmetric harmonic trap. A time-dependent variational Lagrangian analysis shows that an optical lattice helps to stabilize the vortex which in the absence of the optical lattice is unstable. We find that the normal modes are stable only if the depth of the optical potential is more than a certain critical value. This critical value of the optical potential depends on the 2D interaction parameter. In general, the higher the interaction parameter, the lower the value of the optical potential required to stabilize the vortex. The BEC with the singly quantized vortex is found to be relatively more unstable in a 2D optical lattice compared to a 1D optical lattice.
In this work, we study pancake-shaped Bose-Einstein condensates confined by both a cylindrically symmetric harmonic potential and an optical lattice with equal periodicity in two orthogonal directions. We first identify the spectrum of the underlying two-dimensional linear problem through multiple-scale techniques. Then, we use the results obtained in the linear limit as a starting point for a nonlinear existence and stability analysis of the lowest energy states, emanating from the linear ones, in the nonlinear problem. Two-parameter continuations of these states are performed for increasing nonlinearity and optical lattice strengths, and their instabilities and temporal evolution are investigated. It is found that the ground state as well as one of the excited states are either stable or weakly unstable for both attractive and repulsive interatomic interactions.
We study the stability of solitary vortices in a two-dimensional trapped Bose-Einstein condensate (BEC) with a spatially localized region of self-attraction. Solving the respective Bogoliubov--de Gennes equations and running direct simulations of the underlying Gross-Pitaevskii equation reveals that vortices with a topological charge up to S=6S=6S=6 (at least) are stable above a critical value of the chemical potential (i.e., below a critical number of atoms, which sharply increases with SSS). The largest nonlinearity-localization radius admitting stabilization of higher-order vortices is estimated analytically and accurately identified in numerical form. To the best of our knowledge, this is the first example of a setting which gives rise to stable higher-order vortices, S>1S>1S>1, in a trapped self-attractive BEC. The same setting may be realized in nonlinear optics too.
The nonlinear Schrodinger equation models a wide variety of different physical phenomena ranging from nonlinear optics, water waves, magnetization of ferromagnets to Bose-Einstein condensates (BEC). The structure of the equation supports existence of topologically non-trivial solutions - vortices. Surprisingly, we demonstrate that the Landau-Lifshitz magnetization equation which is formally also a nonlinear Schrodinger equation does not admit such solutions. On the other hand, the contrary is true for the Gross-Pitaevskii equation which describes the mean-field approximation of BEC. We investigate stability of vortex solutions by means of a very reliable, sensitive and robust technique - the Evans function. This method, although limited to two dimensions, allows us to study rotating axisymmetric BEC for large particle numbers which can be unattainable by other means.We found a singly-quantized vortex linearly stable.The linear stability of multi-quantized vortices depends on the diluteness of a condensate, with alternating intervals of stability and instability. This work justifies previous results in the literature obtained by less reliable methods and opens up a few interesting questions.