Abstract Spinor Bose–Einstein condensates (BECs) are formed when atoms in the multi-component BECs possess single hyperfine spin states but retain internal spin degrees of freedom. This study concentrates on a (1+1)-dimensional three-couple Gross–Pitaevskii system to depict the macroscopic spinor BEC waves within the mean-field approximation. Regarding the distribution of the atoms corresponding to the three vertical spin projections, a known binary Darboux transformation is utilized to derive the N matter-wave soliton solutions and triple-pole matter-wave soliton solutions on the zero background, where N is a positive integer. For those multiple matter-wave solitons, the asymptotic analysis is performed to obtain the algebraic expressions of the soliton components in the N matter-wave solitons and triple-pole matter-wave solitons. The asymptotic results indicate that the matter-wave solitons in the spinor BECs possess the property of maintaining their energy content and coherence during the propagation and interactions. Particularly, in the N matter-wave solitons, each soliton component contributes to the phase shifts of the other soliton components; and in the triple-pole matter-wave solitons, stable attractive forces exist between the different matter-wave soliton components. Those multiple matter-wave solitons are graphically illustrated through three-dimensional plots, density plot and contour plot, which are consistent with the asymptotic analysis results. The present analysis may provide the explanations for the complex natural mechanisms of the matter waves in the spinor BECs, and may have potential applications in designs of atom lasers, atom interferometry and coherent atom transport.
Abstract Spinor Bose–Einstein condensates (BECs) are formed when atoms in the multi-component BECs possess single hyperfine spin states but retain internal spin degrees of freedom. This study concentrates on a (1+1)-dimensional three-couple Gross–Pitaevskii system to depict the macroscopic spinor BEC waves within the mean-field approximation. Regarding the distribution of the atoms corresponding to the three vertical spin projections, a known binary Darboux transformation is utilized to derive the N matter-wave soliton solutions and triple-pole matter-wave soliton solutions on the zero background, where N is a positive integer. For those multiple matter-wave solitons, the asymptotic analysis is performed to obtain the algebraic expressions of the soliton components in the N matter-wave solitons and triple-pole matter-wave solitons. The asymptotic results indicate that the matter-wave solitons in the spinor BECs possess the property of maintaining their energy content and coherence during the propagation and interactions. Particularly, in the N matter-wave solitons, each soliton component contributes to the phase shifts of the other soliton components; and in the triple-pole matter-wave solitons, stable attractive forces exist between the different matter-wave soliton components. Those multiple matter-wave solitons are graphically illustrated through three-dimensional plots, density plot and contour plot, which are consistent with the asymptotic analysis results. The present analysis may provide the explanations for the complex natural mechanisms of the matter waves in the spinor BECs, and may have potential applications in designs of atom lasers, atom interferometry and coherent atom transport.
We investigate the different types of matter-wave solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Using mean-field theory and adopting the multiscale perturbation method, the original five-component Gross-Pitaevskii spin-orbit-coupled spin-2 spinor Bose-Einstein condensate model can be reduced to a single effective nonlinear Schrödinger equation, which allows us to find analytical soliton solutions of this system. In this way, for different regimes of the spin-orbit coupling, Raman coupling, and interatomic interactions, we find approximate bright and dark soliton solutions. Particularly, the type of solitons depends on the dispersion properties of the system. When the lowest-energy band has a single-well structure, we find there only exist positive mass bright or dark solitons due to the dispersion coefficient of effective nonlinear Shrödinger equation always positive. However, when the lowest-energy band has a double-well structure, there will appear positive (negative) mass bright or dark solitons because the sign of the dispersion coefficient can be positive (negative) under different momentum. We employ direct numerical simulation of the original five-component Gross-Pitaevskii equations to confirm the analytical results.
Following our previous work [J. Ieda, T. Miyakawa, M. Wadati, cond-mat/0404569] on a novel integrable model describing soliton dynamics of an F=1 spinor Bose--Einstein condensate, we discuss in detail the properties of the multi-component system with spin-exchange interactions. The exact multiple bright soliton solutions are obtained for the system where the mean-field interaction is attractive (c_0 < 0) and the spin-exchange interaction is ferromagnetic (c_2 < 0). A complete classification of the one-soliton solution with respect to the spin states and an explicit formula of the two-soliton solution are presented. For solitons in polar state, there exists a variety of different shaped solutions including twin peaks. We show that a "singlet pair" density can be used to distinguish those energetically degenerate solitons. We also analyze collisional effects between solitons in the same or different spin state(s) by computing the asymptotic forms of their initial and final states. The result reveals that it is possible to manipulate the spin dynamics by controlling the parameters of colliding solitons.
Bose–Einstein Condensation (BEC) of atomic gases has attracted a renewed theoretical and experimental interest in quantum many-body systems at extremely low temperatures (Pethick & Smith; 2002). This excitement stems from two favorable features: (1) by applying magnetic fields and lasers, most of the system parameters, such as the shape, dimensionality, internal states of the condensates, and even the strength of the interatomic interactions, are controllable; (2) due to the diluteness, the mean-field theory explains experiments quite well. In particular, the Gross–Pitaevskii (GP) equation demonstrates its validity as a basic equation for the condensate dynamics. The GP equation is the counterpart of the nonlinear Schrodinger (NLS) equation in nonlinear optics. Thus, a study based on nonlinear analysis is possible and important. In nonlinear physics, a soliton is remarkable object not only for the fact that exact solutions can be obtained but also for its usefulness as a communications tool due to its robustness. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the former is attributed to the interatomic interactions, while the latter comes from the kinetic energy. Either dark or bright solitons are allowable depending on the positive or negative sign of the interatomic coupling constants g, respectively, and indeed have been observed in a quasione-dimensional (q1D) optically constructed waveguide (Strecker et al.; 2002) (Khaykovich et al.; 2002). Such matter-wave solitons are expected in atom optics for applications in atom laser, atom interferometry, and coherent atom transport (Meystre; 2002). In this chapter, we extend the analysis of the matter-wave solitons to a multicomponent case by considering the so-called spinor condensate (Stenger et al.; 1998) whose spin degrees of freedom are liberated under optical traps. Based on theoretical and experimental results, we introduce a new integrable model which describes the dynamical properties of the matter-wave soliton of spinor condensates (Ieda et al.; 2004a). We employ the inverse scattering method to solve this model exactly. As a result, we predict the occurrence of undiscovered physical phenomena such as macroscopic spin precession and spin switching. The chapter is organized as follows. In Sec. 2, the mean field theory of condensate is briefly reviewed. Section 3 introduces an effective interatomic coupling in a q1D condensate. Using these results, we consider a spinor condensate in q1D regime in Sec. 4. Then, in Sec. 5, we
We propose an integrable model of a multicomponent spinor Bose-Einstein condensate in one dimension, which allows an exact description of the dynamics of bright solitons with spin degrees of freedom. We consider specifically an atomic condensate in the F=1 hyperfine state confined by an optical dipole trap. When the mean-field interaction is attractive (c(0)<0) and the spin-exchange interaction of a spinor condensate is ferromagnetic (c(2)<0), we prove that the system possesses a completely integrable point leading to the existence of multiple bright solitons. By applying results from the inverse scattering method, we analyze a collision law for two-soliton solutions and find that the dynamics can be explained in terms of the spin precession.
We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (nonintegrable) model of a spinor (three-component) model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.
We investigate dynamical properties of bright solitons with a finite background in the F=1 spinor Bose-Einstein condensate (BEC), based on an integrable spinor model which is equivalent to the matrix nonlinear Schrödinger equation with a self-focusing nonlineality. We apply the inverse scattering method formulated for nonvanishing boundary conditions. The resulting soliton solutions can be regarded as a generalization of those under vanishing boundary conditions. One-soliton solutions are derived in an explicit manner. According to the behaviors at the infinity, they are classified into two kinds, domain-wall (DW) type and phase-shift (PS) type. The DW-type implies the ferromagnetic state with nonzero total spin and the PS-type implies the polar state, where the total spin amounts to zero. We also discuss two-soliton collisions. In particular, the spin-mixing phenomenon is confirmed in a collision involving the DW-type. The results are consistent with those of the previous studies for bright solitons under vanishing boundary conditions and dark solitons. As a result, we establish the robustness and the usefulness of the multiple matter-wave solitons in the spinor BECs.