Gap solitons in elongated geometries: the one-dimensional Gross-Pitaevskii equation and beyond

TL;DR

This paper systematically analyzes matter-wave gap solitons in three-dimensional Bose-Einstein condensates within elongated traps and optical lattices, comparing the accuracy of 1D models and exploring stability across different confinement regimes.

Contribution

It demonstrates that the nonpolynomial 1D equation accurately describes gap solitons in various confinement conditions, extending beyond the usual 1D cubic GPE's applicability.

Findings

The nonpolynomial 1D equation provides accurate approximations for GSs in all considered cases.

Most fundamental GSs are stable according to systematic simulations.

Bound states of GSs can be stable if the optical lattice potential is sufficiently deep.

Abstract

We report results of a systematic analysis of matter-wave gap solitons (GSs) in three-dimensional self-repulsive Bose-Einstein condensates (BECs) loaded into a combination of a cigar-shaped trap and axial optical-lattice (OL) potential. Basic cases of the strong, intermediate, and weak radial (transverse) confinement are considered, as well as settings with shallow and deep OL potentials. Only in the case of the shallow lattice combined with tight radial confinement, which actually has little relevance to realistic experimental conditions, does the usual one-dimensional (1D) cubic Gross-Pitaevskii equation (GPE) furnish a sufficiently accurate description of GSs. However, the effective 1D equation with the nonpolynomial nonlinearity, derived in Ref. [Phys. Rev. A \textbf{77}, 013617 (2008)], provides for quite an accurate approximation for the GSs in all cases, including the situation with weak transverse confinement, when the soliton's shape includes a considerable contribution from higher-order transverse modes, in addition to the usual ground-state wave function of the respective harmonic oscillator. Both fundamental GSs and their multipeak bound states are considered. The stability is analyzed by means of systematic simulations. It is concluded that almost all the fundamental GSs are stable, while their bound states may be stable if the underlying OL potential is deep enough.