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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Ground states and vortices of matter-wave condensates and optical guided waves

Tristram J. Alexander, Luc Bergé

Published Jan 25, 2002• 76 citations
Publisher

Abstract

We analyze the shape and stability of localized states in nonlinear cubic media with space-dependent potentials modeling an inhomogeneity. By means of a static variational approach, we describe the ground states and vortexlike stationary solutions, either in dilute atom gases or in optical cavities, with an emphasis on parabolic-type potentials. First, we determine the existence conditions for soliton and vortex structures for both focusing and defocusing nonlinearity. It is shown that, even for a defocusing medium, soliton modes can exist with a confining potential. Second, step potentials and boundedness effects in hollow capillaries are investigated, which both proceed from a similar analysis. Finally, we discuss applications of this procedure to charged vortices in dilute quantum gases and to Bose-Einstein condensates trapped in the presence of a light-induced Gaussian barrier.

DOI
10.1103/physreve.65.026611

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We overview our recent theoretical studies on nonlinear atom optics of the Bose-Einstein condensates (BECs) loaded into optical lattices. In particular, we describe the band-gap spectrum and nonlinear localization of BECs in one- and two-dimensional optical lattices. We discuss the structure and stability properties of spatially localized states (matter-wave solitons) in 1D lattices, as well as trivial and vortex-like bound states of 2D gap solitons. To highlight similarities between the behavior of coherent light and matter waves in periodic potentials, we draw useful parallels with the physics of coherent light waves in nonlinear photonic crystals and optically-induced photonic lattices.
We consider two-dimensional (2D) localized vortical modes in the three-wave system with the quadratic ( χ ( 2 ) ) nonlinearity, alias nondegenerate second-harmonic (SH)-generating system, guided by the isotropic harmonic-oscillator (alias parabolic) confining potential. In addition to the straightforward realization in optics, the system models mixed atomic-molecular Bose–Einstein condensates. The main issue is stability of the vortex modes, which is investigated through computation of instability growth rates for eigenmodes of small perturbations, and by means of direct simulations. The threshold of parametric instability for single-color beams, represented solely by the SH with zero vorticity, is found in an analytical form with the help of the variational approximation. Trapped states with vorticities ( + 1 , − 1 , 0 ) in the two fundamental-frequency components and the SH one (the so-called hidden-vorticity modes) are completely unstable. Also unstable are semi-vortices , with component vorticities ( 1 , 0 , 1 ) . However, full vortices, with charges ( 1 , 1 , 2 ) , have a well-defined stability region. Unstable full vortices feature regions of robust dynamical behavior, where they periodically split and recombine, keeping their vortical content.
The novel phenomena arising from Bose-Einstein condensates with spatially and spatiotemporally modulated nonlinearities in external potential is reviewed from a theoretical viewpoint. We first present theoretical analysis and numerical studies of the localized nonlinear matter waves in one-dimensional single and two-component BECs with spatially and spatiotemporally modulated nonlinearities, respectively. It is shown that the spatially or spatiotemporally modulated nonlinearity can support stable novel localized nonlinear matter waves such as the breathing solitons and moving solitons. Then the quasi-two-dimensional BEC with spatially modulated nonlinearity is investigated and we show that all of the BECs, similar to the linear harmonic oscillator, can have an arbitrary number of localized nonlinear matter waves with discrete energies. Their properties are determined by the principal quantum number and secondary quantum number. Moreover, we investigate the quantized vortices in a rotating BEC with spatiotemporally modulated interaction in harmonic and anharmonic potentials, respectively. The exact quantized vortex and giant vortex solutions are constructed explicitly by similarity transformation. Their stability behavior is examined by numerical simulation, which shows that a new series of stable vortex states which are defined by radial and angular quantum numbers, can be supported by the spatiotemporally modulated interaction in this system. We find that there exist stable quantized vortices with large topological charges in repulsive condensates with spatiotemporally modulated interaction.
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.