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Solitons in the Tonks–Girardeau gas with dipolar interactions

Bakhtiyor B. Baizakov, F. Kh. Abdullaev, Boris A. Malomed, Mario Salerno

Published Jan 1, 2009• 38 citations
OpenAlex

Abstract

The existence of bright solitons in the model of the Tonks–Girardeau (TG) gas with dipole–dipole (DD) interactions is reported. The governing equation is taken as the quintic nonlinear Schrödinger equation (NLSE) with the nonlocal cubic term accounting for the DD attraction. In different regions of the parameter space (the dipole moment and atom number), matter-wave solitons feature flat-top or compacton-like shapes. For the flat-top states, the NLSE with the local cubic-quintic (CQ) nonlinearity is shown to be a good approximation. Specific dynamical effects are studied assuming that the strength of the DD interactions is ramped up or drops to zero. Generation of dark-soliton pairs in the gas shrinking under the action of the intensifying DD attraction is observed. Dark solitons exhibit particle-like collision behaviour. Peculiarities of dipole solitons in the TG gas are highlighted by comparison with the NLSE including the local CQ terms. Collisions between the bright solitons are also studied. In many cases, the collisions result in the merger of the solitons into a breather, due to a strong attraction between them. (Some figures in this article are in colour only in the electronic version) 1.

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The existence of bright solitons in the model of the Tonks–Girardeau (TG) gas with dipole–dipole (DD) interactions is reported. The governing equation is taken as the quintic nonlinear Schrödinger equation (NLSE) with the nonlocal cubic term accounting for the DD attraction. In different regions of the parameter space (the dipole moment and atom number), matter-wave solitons feature flat-top or compacton-like shapes. For the flat-top states, the NLSE with the local cubic-quintic (CQ) nonlinearity is shown to be a good approximation. Specific dynamical effects are studied assuming that the strength of the DD interactions is ramped up or drops to zero. Generation of dark-soliton pairs in the gas shrinking under the action of the intensifying DD attraction is observed. Dark solitons exhibit particle-like collision behaviour. Peculiarities of dipole solitons in the TG gas are highlighted by comparison with the NLSE including the local CQ terms. Collisions between the bright solitons are also studied. In many cases, the collisions result in the merger of the solitons into a breather, due to a strong attraction between them. (Some figures in this article are in colour only in the electronic version) 1.
The existence of bright solitons in the model of the Tonks-Girardeau (TG) gas with dipole-dipole (DD) interactions is reported. The governing equation is taken as the quintic nonlinear Schrödinger equation (NLSE) with the nonlocal cubic term accounting for the DD attraction. In different regions of the parameter space (the dipole moment and atom number), matter-wave solitons feature flat-top or compacton-like shapes. For the flat-top states, the NLSE with the local cubic-quintic (CQ) nonlinearity is shown to be a good approximation. Specific dynamical effects are studied assuming that the strength of the DD interactions is ramped up or drops to zero. Generation of dark-soliton pairs in the gas shrinking under the action of the intensifying DD attraction is observed. Dark solitons exhibit the particle-like collision behavior. Peculiarities of dipole solitons in the TG gas are highlighted by comparison with the NLSE including the local CQ terms. Collisions between the solitons are studied too. In many cases, the collisions result in merger of the solitons into a breather, due to strong attraction between them.
We consider one-dimensional mixtures of an atomic Bose–Einstein condensate (BEC) and Tonks–Girardeau (TG) gas. The mixture is modeled by a coupled system of the Gross–Pitaevskii equation for the BEC and the quintic nonlinear Schrödinger equation for the TG component. An immiscibility condition for the binary system is derived in a general form. Under this condition, three types of BEC–TG interfaces are considered: domain walls (DWs) separating the two components; bubble-drops (BDs), in the form of a drop of one component immersed into the other (BDs may be considered as bound states of two DWs); and bound states of bright and dark solitons (BDSs). The same model applies to the copropagation of two optical waves in a colloidal medium. The results are obtained by means of systematic numerical analysis, in combination with analytical Thomas–Fermi approximations (TFAs). Using both methods, families of DW states are produced in a generic form. BD complexes exist solely in the form of a TG drop embedded into the BEC background. On the contrary, BDSs exist as bound states of TG bright and BEC dark components, and vice versa.
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
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.