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

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 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.

Effective equations for matter-wave gap solitons in higher-order transversal states

“…(14) in terms of α and γ . Minimization of this latter equation with respect to the variational parameter then leads to the following z-dependent condensate width:…”

“…(14) and using that μ ⊥ (n 1 ) = ∂E/∂N, one finds the following transverse local chemical potential:…”

Stable symmetry-protected 3D embedded solitons in Bose–Einstein condensates

“…They are localized nonlinear stationary states of the GPE whose chemical potentials lie in the forbidden band gaps of the linear spectrum. In a repulsive condensate, gap soliton solutions bifurcate from the upper edge of the linear Bloch bands, forming distinct one-parameter families characterized by their chemical potential μ as a function of the number of atoms N 12 , 13 . These families describe continuous trajectories in the μ − N parameter plane and, in general, vanish as they approach the vicinity of an upper band, where the gap solitons become resonant with the extended Bloch waves residing therein.…”

“…Embedded solitons have received almost no attention in the field of Bose–Einstein condensation. This may be, in part, because most theoretical studies on gap solitons have focused on quasi-1D BECs 10 , 12 , 13 , 19 . Nonetheless, 3D gap waves 20 (a type of gap solitons that can be viewed as truncated nonlinear Bloch waves) and 3D gap solitons 21 , 22 have been obtained, respectively, in BECs loaded in 3D and in 1D optical lattices.…”

Effective Equations for Repulsive Quasi‐One Dimensional Bose–Einstein Condensates Trapped with Anharmonic Transverse Potentials

“…[20][21][22][23][24][25][26][27][28][29] and via standard adiabatic approximations in Refs. [30][31][32][33][34][35][36][37][38][39]. However, in all these papers the effective equations are obtained based on the assumption that the transverse potential is quadratic.…”