A model of the Bose-Einstein condensate of dipolar atoms, confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction, is introduced, taking into regard the dipole-dipole (DD) and contact interactions. The model is based on the discrete nonlinear Schr\"odinger equation with an additional nonlocal term accounting for the DD interactions. The existence and stability of fundamental unstaggered solitons are studied for attractive and repulsive signs of both the local and nonlocal interactions. The DD forces strongly affect the shape and stability of on-site and intersite discrete solitons. The corresponding existence and stability regions in the parametric space are summarized in the form of diagrams, which feature a multiple stability exchange between the on-site and intersite families; in the limit of the dominating DD attraction, the on-site solitons are stable, while their intersite counterparts are not. We also demonstrate that the DD interactions reduce the Peierls-Nabarro barrier and enhance the mobility of the discrete solitons.
A model of the Bose-Einstein condensate of dipolar atoms, confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction, is introduced, taking into regard the dipole-dipole (DD) and contact interactions. The model is based on the discrete nonlinear Schr"odinger equation with an additional nonlocal term accounting for the DD interactions. The existence and stability of fundamental unstaggered solitons are studied for attractive and repulsive signs of both the local and nonlocal interactions. The DD forces strongly affect the shape and stability of on-site and intersite discrete solitons. The corresponding existence and stability regions in the parametric space are summarized in the form of diagrams, which feature a multiple stability exchange between the on-site and intersite families; in the limit of the dominating DD attraction, the on-site solitons are stable, while their intersite counterparts are not. We also demonstrate that the DD interactions reduce the Peierls-Nabarro barrier and enhance the mobility of the discrete solitons.
A model of the Bose–Einstein condensate (BEC) of dipolar atoms, confined in a combination of a cigar-shaped trap and optical lattice acting in the axial direction, is studied in the framework of the one-dimensional (1D) nonpolynomial Schrödinger equation (NPSE) with additional terms describing long-range dipole–dipole (DD) interactions. The NPSE makes it possible to describe the collapse of localized modes, which was experimentally observed in the self-attractive BEC confined in tight traps, in the framework of the 1D description. We study the influence of the DD interactions on the dynamics of bright solitons, especially concerning their collapse-induced instability. Both attractive and repulsive contact and DD interactions are considered. The results are summarized in the form of stability/collapse diagrams in a respective parametric space. In particular, it is shown that the attractive DD interactions may prevent the collapse instability in the condensate with attractive contact interactions.
We analyze the formation and dynamics of bright unstaggered solitons in the disk-shaped dipolar Bose-Einstein condensate, which features the interplay of contact (collisional) and long-range dipole-dipole (DD) interactions between atoms. The condensate is assumed to be trapped in a strong optical-lattice potential in the disk's plane, hence it may be approximated by a two-dimensional (2D) discrete model, which includes the on-site nonlinearity and cubic long-range (DD) interactions between sites of the lattice. We consider two such models, which differ by the form of the on-site nonlinearity, represented by the usual cubic term, or more accurate nonpolynomial one, derived from the underlying three-dimensional Gross-Pitaevskii equation. Similar results are obtained for both models. The analysis is focused on the effects of the DD interaction on fundamental localized modes in the lattice (2D discrete solitons). The repulsive isotropic DD nonlinearity extends the existence and stability regions of the fundamental solitons. New families of on-site, inter-site, and hybrid solitons, built on top of a finite background, are found as a result of the interplay of the isotropic repulsive DD interaction and attractive contact nonlinearity. By themselves, these solutions are unstable, but they evolve into robust breathers which exist on an oscillating background. In the presence of the repulsive contact interactions, fundamental localized modes exist if the DD interaction (attractive isotropic or anisotropic) is strong enough. They are stable in narrow regions close to the anticontinuum limit, while unstable solitons evolve into breathers. In the latter case, the presence of the background is immaterial.
Abstract. A model of the Bose-Einstein condensate (BEC) of dipolar atoms, confined in a combination of a cigar-shaped trap and optical lattice acting in the axial direction, is studied in the framework of the one-dimensional (1D) nonpolynomial Schrödinger equation (NPSE) with additional terms describing long-range dipoledipole (DD) interactions. The NPSE makes it possible to describe the collapse of localized modes, which was experimentally observed in the self-attractive BEC confined in tight traps, in the framework of the 1D description. We study the influence of the DD interactions on the dynamics of bright solitons, especially as concerns their collapseinduced instability. Both attractive and repulsive contact and DD interactions are considered. The results are summarized in the form of stability/collapse diagrams in a respective parametric space. In particular, it is shown that the attractive DD interactions may prevent the collapse instability in the condensate with attractive contact interactions. Collapse instability of solitons in the nonpolynomial Schrödinger equation with dipole-dipole interactions2 1
We analyse the existence, stability and dynamics of localized discrete modes with intrinsic vorticity S = 1 and S = 2 in the disc-shaped dipolar Bose-Einstein condensate loaded into a deep two-dimensional optical lattice. The condensate, which features the interplay of local contact and nonlocal dipole-dipole (DD) interactions between atoms, is modelled by the 2D discrete Gross-Pitaevskii equation which includes the long-range DD term. Various species of discrete vortex solitons, which are known in the model of the condensate with local interactions, are found to exist in the presence of the DD interaction too. In locally self-attractive condensates, the isotropic DD repulsion, which corresponds to the orientation of atomic dipoles perpendicular to the confinement plane, helps to extend the region of the vortex stability, while in the case of anisotropic DD interactions, corresponding to the in-plane orientation of the dipoles, vortices are unstable. In the former case, those vortices which are unstable may evolve into robust ring-shaped breathers. The attractive isotropic DD interaction can create localized vortices in the condensate with the local self-repulsion, but they all are unstable, evolving into single-peak asymmetric structures.
The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schrodinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce "model 1" (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. "Model 2," which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2-in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.
We consider intersite discrete bright solitons in one-dimensional Bose-Einstein condensate in parabolic trapped optical lattice. Analytical and numerical calculations are performed to determine the existence and stability of bright solitons. Analysis is based on continuous Gross-Pitaevskii equation and discrete nonlinear Schrodinger equation. It is observed that the strength of external magnetic trap can change the stability of bright solitons. Stability windows of bright solitons are presented and stability approximations are derived using perturbation theory, with their numerical results.