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arXiv:cond-mat/0208136 · Physical Review A

Stability of gap solitons in a Bose-Einstein condensate

Karen Marie Hilligsøe, Markus K. Oberthaler, Karl-Peter Marzlin

Published Dec 20, 2002• 59 citations
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Abstract

We analyze the dynamical stability of gap solitons formed in a quasi-one-dimensional Bose-Einstein condensate in an optical lattice. Using two different numerical methods we show that, under realistic assumptions for experimental parameters, a gap soliton is stable only in a truly one-dimensional situation. In two and three dimensions, resonant transverse excitations lead to dynamical instability. The time scale of the decay is numerically calculated and shown to be large compared to the characteristic time scale of solitons for realistic physical parameters.

DOI
10.1103/physreva.66.063605

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We analyze the dynamical stability of gap solitons formed in a quasi-one-dimensional Bose-Einstein condensate in an optical lattice. Using two different numerical methods we show that, under realistic assumptions for experimental parameters, a gap soliton is stable only in a truly one-dimensional situation. In two and three dimensions, resonant transverse excitations lead to dynamical instability. The time scale of the decay is numerically calculated and shown to be large compared to the characteristic time scale of solitons for realistic physical parameters.
Bose–Einstein condensate (BEC) exhibits a variety of fascinating and unexpected macroscopic phenomena, and has attracted sustained attention in recent years—particularly in the field of solitons and associated nonlinear phenomena. Meanwhile, optical lattices have emerged as a versatile toolbox for understanding the properties and controlling the dynamics of BEC, among which the realization of bright gap solitons is an iconic result. However, the dark gap solitons are still experimentally unproven, and their properties in more than one dimension remain unknown. In light of this, we describe, numerically and theoretically, the formation and stability properties of gap-type dark localized modes in the context of ultracold atoms trapped in optical lattices. Two kinds of stable dark localized modes—gap solitons and soliton clusters—are predicted in both the one- and two-dimensional geometries. The vortical counterparts of both modes are also constructed in two dimensions. A unique feature is the existence of a nonlinear Bloch-wave background on which all above gap modes are situated. By employing linear-stability analysis and direct simulations, stability regions of the predicted modes are obtained. Our results offer the possibility of observing dark gap localized structures with cutting-edge techniques in ultracold atoms experiments and beyond, including in optics with photonic crystals and lattices.
We investigated the stability properties of breather soliton trains in a three-dimensional Bose–Einstein condensate (BEC) with Feshbach-resonance management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is confined only by a one-dimensional optical lattice and we consider strong, moderate and weak confinement. By strong confinement we mean a situation in which a quasi two-dimensional soliton is created. Moderate confinement admits a fully three-dimensional soliton. Weak confinement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in effective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully confirmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully three-dimensional analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We find that in the three-dimensional case the stability region splits into two parts. However, as we tighten the confinement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for the quasi two-dimensional case. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust three-dimensional solitons in experiments in a BEC in a one-dimensional lattice.
Abstract Vector gap solitons in quasi-two-dimensional Bose–Einstein condensate loaded in a square optical lattice with spin-orbit and Rabi coupling are investigated theoretically. The solitons are obtained by the Newton-Conjugate-Gradient method for various physical parameters. The stability properties of gap solitons are theoretically analyzed by direct nonlinear dynamical evolution. It is found that the existence of gap solitons is sensitive to the spin-orbit and Rabi coupling strength. Smaller Rabi coupling strength is beneficial for the excitation of solitons in the semi-infinite gap. Conversely, larger Rabi coupling strength is beneficial for the soliton excitation in the first gap. The dynamical stability of these gap solitons depends on the spin-orbit and Rabi coupling strength, and the location of the soliton in the bandgap. These findings may contribute to understanding the topological excitations in condensed matter systems.
We consider stationary matter-wave gap solitons realized in Bose–Einstein condensates loaded in one-dimensional (1D) optical lattices and investigate whether the effective 1D equation proposed in Muñoz Mateo and Delgado (2008 Phys. Rev. A 77 013617) can be a reliable alternative to the three-dimensional treatment of this kind of system in terms of the Gross–Pitaevskii equation (GPE). Our results demonstrate that, unlike the standard 1D GPE (which is not applicable in most realistic situations), the above effective model is able to correctly predict the distinctive trajectories characterizing the different gap soliton families as well as the corresponding axial wavefunctions along the entire band gaps. It can also predict the stability properties of the different gap soliton families as follows from both a linear stability analysis and a representative set of numerical computations. In particular, by numerically solving the corresponding Bogoliubov–de Gennes equations we show that the effective 1D model gives the correct spectrum of the complex eigenfrequencies responsible for the dynamical stability of the system, thus providing us with a useful tool for the physical description of stationary matter-wave gap solitons in 1D optical lattices.
We suggest and study the stable disc- and cigar-shaped gap solitons of a dipolar Bose–Einstein condensate of 52 Cr atoms localized in the lowest band gap by three optical-lattice (OL) potentials along orthogonal directions. The one-dimensional version of these solitons of experimental interest confined by an OL along the dipole moment direction and harmonic traps in transverse directions is also considered. Important dynamics of (i) the breathing oscillation of a gap soliton upon perturbation and (ii) the dragging of a gap soliton by a moving lattice along the axial z direction demonstrates the stability of gap solitons. A movie clip (available at http://stacks.iop.org/0953-4075/44/121001/mmedia ) of the dragging of the three-dimensional gap soliton is included.