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Chinese Journal of Physics

An Effective-Mass Theory for Bright Gap Solitons in Optical Lattices

Szu-Cheng Cheng

Published Aug 1, 2005• 2 citations
OpenAlex

Abstract

A one-band effective-mass (EFM) theory was developed to study the properties of the localized states of a Bose-Einstein condensate (BEC) in a periodic optical potential with either attractive or repulsive atom interactions. The localized states of the BEC are solitons in the gaps or gap solitons. The analytic solutions of the gap solitons are a Bloch function from periodicity, modulated by a soliton envelope function of the nonlinear Schrödinger equation. The soliton width is determined by the detuning chemical potential from the band edge. The detuning chemical potential in the nonlinear Schrödinger equation indicates that not all the atoms in the condensate are used for forming gap solitons, and that some atoms form the background of nonlinear modes. The background atoms are determined by the chemical potential at the band edge. The gap solitons form on top of the background atoms. PACS numbers: 03.75.Lm, 03.75.Kk, 05.30.Jp I.

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A one-band effective-mass (EFM) theory was developed to study the properties of the localized states of a Bose-Einstein condensate (BEC) in a periodic optical potential with either attractive or repulsive atom interactions. The localized states of the BEC are solitons in the gaps or gap solitons. The analytic solutions of the gap solitons are a Bloch function from periodicity, modulated by a soliton envelope function of the nonlinear Schrödinger equation. The soliton width is determined by the detuning chemical potential from the band edge. The detuning chemical potential in the nonlinear Schrödinger equation indicates that not all the atoms in the condensate are used for forming gap solitons, and that some atoms form the background of nonlinear modes. The background atoms are determined by the chemical potential at the band edge. The gap solitons form on top of the background atoms. PACS numbers: 03.75.Lm, 03.75.Kk, 05.30.Jp I.
We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions.
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 systematically study the properties of lattice solitons in Bose-Einstein condensates with either attractive or repulsive atom interactions. This is done, by exactly solving the mean-field Gross-Pitaevskii equation in the presence of a periodic potential. We find new families of lattice soliton solutions that are characterized by the position of the energy eigenvalue within the associated band structure. These include lattice solitons in condensates with either attractive or repulsive atom interactions that exist in finite or semi-infinite gaps, as well as nonlinear modes that exhibit atomic population cutoffs.
We discuss localized ground states of Bose-Einstein condensates (BEC's) in optical lattices with attractive and repulsive three-body interactions in the framework of a quintic nonlinear Schr\"odinger equation which extends the Gross-Pitaevskii equation to the one-dimensional case. We use both a variational method and a self-consistent approach to show the existence of unstable localized excitations which are similar to Townes solitons of the cubic nonlinear Schr\"odinger equation in two dimensions. These solutions are shown to be located in the forbidden zones of the band structure, very close to the band edges, separating decaying states from stable localized ones (gap solitons) fully characterizing their delocalizing transition. In this context the usual gap solitons appear as a mechanism for arresting the collapse in low-dimensional BEC's in optical lattices with an attractive real three-body interaction. The influence of the imaginary part of the three-body interaction, leading to dissipative effects in gap solitons, and the effect of atoms feeding from the thermal cloud are also discussed. These results may be of interest for both BEC's in atomic chips and Tonks-Girardeau gas in optical lattices.
By developing a multiple-scale method, we study analytically the dynamics of the soliton inside the semi-infinite band gap (SIBG) of quasi-one-dimensional Bose–Einstein condensates trapped in an optical lattice. In the linear case, a stable condition of soliton formation is obtained. For a weak nonlinearity, whether there occurs a spatially propagating or localized gap soliton is determined by the lattice depth. Meanwhile, we predict the existence of the dark (bright) gap solitons for the repulsive (attractive) interactions in the SIBG, different from that of the gap solitons in other energy gaps. And the collision of two dark (or bright) solitons is nearly elastic under a safe range of atomic numbers. An experimental protocol is further designed for observing these phenomena.