Stability of small amplitude normal modes of a Bose-Einstein condensate with a singly quantized vortex confined in an optical lattice

We consider two-dimensional (2D) localized vortical modes in the three-wave system with the quadratic ($\chi ^{(2)}$) nonlinearity, alias nondegenerate second-harmonic-generating system, guided by the isotropic harmonic-oscillator (HO) (alias parabolic) confining potential. In addition to the straightforward realization in optics, the system models mixed atomic-molecular Bose-Einstein condensates (BECs). The main issue is stability of the vortex modes, which is investigated t...

This chapter presents an overview of the properties of a Bose-Einstein condensate (BEC) trapped in a periodic potential. This system has attracted a wide interest in the last years, and a few excellent reviews of the field have already appeared in the literature (see, for instance, [1-3] and references therein). For this reason, and because of the huge amount of published results, we do not pretend here to be comprehensive, but we will be content to provide a flavor of the ri...

We consider the manipulation of Bose-Einstein condensate vortices by optical potentials generated by focused laser beams. It is shown that for appropriate choices of the laser strength and width it is possible to successfully transport vortices to various positions inside the trap confining the condensate atoms. Furthermore, the full bifurcation structure of possible stationary single-charge vortex solutions in a harmonic potential with this type of impurity is elucidated. Th...

We develop stability analysis for matter-wave solitons in a two-dimensional (2D) Bose-Einstein condensate loaded in an optical lattice (OL), to which periodic time modulation is applied, in different forms. The stability is studied by dint of the variational approximation and systematic simulations. For solitons in the semi-infinite gap, well-defined stability patterns are produced under the action of the attractive nonlinearity, clearly exhibiting the presence of resonance f...

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 f...

We present a theoretical study of vortices within a harmonically trapped Bose-Einstein condensate in a rotating optical lattice. Due to the competition between vortex-vortex interactions and pinning to the optical lattice we find a very complicated energy landscape, which leads to hysteretic evolution. The qualitative structure of the vortex configurations depends on the commensurability between the vortex density and the site density -- with regular lattices when these are c...

A vortex in a trapped Bose-Einstein condensate can experience at least two types of instabilities. (1). Macroscopic hydrodynamic motion of the vortex core relative to the center of mass of the condensate requires some process to dissipate energy. (2). Microscopic small-amplitude normal modes can also induce an instability. In one specific example, the vortex core again moves relative to the overall center of mass, suggesting that there may be only a single physical mechanism.

We study the stability of singly- and doubly-quantized vortex states of harmonically trapped dipolar Bose-Einstein Condensates (BECs) by calculating the low-lying excitations of these condensates. We map the dynamical stability of these vortices as functions of the dipole-dipole interaction strength and trap geometry by finding where their excitations have purely real energy eigenvalues. In contrast to BECs with purely contact interactions, we find that dipolar BECs in singly...

We consider two-dimensional (2D) localized modes in the second-harmonic-generating \chi ^{(2)} system with the harmonic-oscillator (HO) trapping potential. In addition to its realization in optics, the system describes the mean-field dynamics of mixed atomic-molecular Bose-Einstein condensates (BECs). The existence and stability of various modes is determined by their total power, N, topological charge, m/2 [m is the intrinsic vorticity of the second-harmonic (SH) field], and...

We show that the Kapitza stabilization can occur in the context of nonlinear quantum fields. Through this phenomenon, an amplitude-modulated lattice can stabilize a Bose-Einstein condensate with repulsive interactions and prevent the spreading for long times. We present a classical and quantum analysis in the framework of Gross-Pitaevskii equation, specifying the parameter region where stabilization occurs. Effects of nonlinearity lead to a significant increase of the stabili...