We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.
We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.
Solitons are self-trapped modes existing in various nonlinear systems. Creating stable solitons in two- and three-dimensional settings is a challenging goal in various branches of physics. Several methods have been developed theoretically and experimentally to achieve this, but few of them can support stable multi-dimensional solitons in free space. Recently, a new scheme using spin-orbit-coupling (SOC) has been proposed to create stable 2D solitons in Bose–Einstein condensates (BECs). This paper reviews recent theoretical progress on creating stable 2D solitons in spinor dipolar BEC with SOC, combined with long-range dipole-dipole interaction (DDI), Zeeman splitting (ZS) and contact nonlinearity, in free space. The continuous family of stable symmetric vortex solitons (SVS), asymmetric vortex solitons (AVS), as well as gap solitons (GS) is found via different settings. Their existence and stability conditions are summarized and discussed in detail. The mobility properties of these types of solitons are also addressed. For SVS, a potential method to manipulate its shape and mobility is investigated. These results are supposed to enrich our understanding of 2D solitons and help create multi-dimensional solitons in experiments.
We analytically and numerically discuss the dynamics of two pseudospin components Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in deep optical lattices. Rich localized phenomena, such as breathers, solitons, self-trapping, and diffusion, are revealed and strongly depend on the strength of the atomic interaction, SOC, Raman detuning, and the spin polarization (i.e., the initial population difference of atoms between the two pseudospin components of BECs). The critical conditions for the transition of localized states are derived analytically. Based on the critical conditions, the detailed dynamical phase diagram describing the different dynamical regimes is derived. When the Raman detuning satisfies a critical condition, localized states with a fixed initial spin polarization can be observed. When the critical condition is not satisfied, we use two quenching methods, i.e., suddenly and linearly quenching Raman detuning from the soliton or breather state, to discuss the spin dynamics, phase transition, and wave packet dynamics by numerical simulation. The sudden quenching results in a damped oscillation of spin polarization and transforms the system to a new polarized state. Interestingly, the linear quenching of Raman detuning induces a controllable phase transition from an unpolarized phase to an expected polarized phase, while the soliton or breather dynamics is maintained.
We introduce one- and two-dimensional (1D and 2D) continuum and discrete models for the two-component BEC, with the spin-orbit (SO) coupling of the Rashba type between the components, and attractive cubic interactions, assuming that the condensate is fragmented into a quasidiscrete state by a deep optical-lattice potential. In 1D, it is demonstrated, in analytical and numerical forms, that the ground states of both the discrete system and its continuum counterpart switch from striped bright solitons, featuring deep short-wave modulations of its profile, to smooth solitons, as the strength ratio of the inter- and intracomponent attraction, γ, changes from γ<1 to γ>1. At the borderline, γ=1, there is a continuous branch of stable solitons, which share a common value of the energy and interpolate between the striped and smooth ones. Unlike the 2D system, the 1D solitons, which do not represent the ground state at given γ, are nevertheless stable against small perturbations, and they remain stable too in collisions with other solitons. In 2D, a transition between two different types of discrete solitons, which represent the ground state, viz., semivortices and mixed modes, also takes place at γ=1. A specific property of 2D discrete solitons of both types is their discontinuous transition into a delocalized state at a critical value of the SO-coupling strength. We also address the continuum 2D model in the borderline case of γ=1, which was not studied previously, and demonstrate the existence of an energy-degenerate branch of dynamically stable solitons connecting the semivortex and the mixed mode. Last, it is demonstrated that 1D and 2D discrete solitons are mobile, in a limited interval of velocities.
We study two-dimensional (2D) matter-wave solitons in spinor Bose–Einstein condensates under the action of the spin–orbit coupling and opposite signs of the self- and cross-interactions. Stable 2D two-component solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collapse threshold. Further, in the case of the repulsive self-interaction and inter-component attraction, stable 2D self-trapped modes, which may be considered as quantum droplets (QDs), are created if the beyond-mean-field Lee–Huang–Yang terms are added to the self-repulsion in the underlying system of coupled Gross–Pitaevskii equations. Stable QDs of the mixed-mode type, of a large size with an anisotropic density profile, exist with arbitrarily large values of the norm, as the Lee–Huang–Yang terms eliminate the collapse. The effect of the spin–orbit coupling term on characteristics of the QDs is systematically studied. We also address the existence and stability of QDs in the case of SOC with mixed Rashba and Dresselhaus terms, which makes the density profile of the QD more isotropic. Thus, QDs in the spin-orbit-coupled binary Bose–Einstein condensate are for the first time studied in the present work.
Abstract In ultracold atomic gases, optical lattices and spin-orbit coupling are two important and useful linear means to study and control the nonlinear and quantum dynamics of matter waves, and both have been well realized in experiments. Here, nonlinear localization of spin-orbit-coupled Bose-Einstein condensates in moiré optical lattices is revealed theoretically, stressing the existence and stability of localized modes of diverse types including fundamental, half-vortex matter-wave gap solitons, and full-vortex ones (of both components). Notably, to our knowledge, a new type of vector vortex gap solitons, which is composed of a gap vortex soliton (singular mode) and a rhombic-array gap vortex (quadruple mode), is predicted in our two-component theoretical framework. It is also found that the model exhibits unique linear Bloch band structures including band splitting and extremely-flat bands, because of which all the predicted localized modes tend to be robustly stable, backed by our direct perturbed simulations. Our results demonstrate that the moiré pattern (and its joint cooperation with spin-orbit coupling) provides great possibilities to explore flat-band and moiré physics in ultracold atoms, and to further investigate the nonlinear dynamics.
A Theory is presented to construct several types of self-trapping nonlinear modes in the two-dimensional model of binary Bose-Einstein condensate (BEC) with the repulsive dipole-dipole interaction (DDI) and the spatially confined spin-orbit coupling (SOC). The spatially confined SOC can be induced by illuminating spinor BEC with a Gaussian beam. It is found that the two-dimensional SOC of Rashba type within a confined area can help trap the composite matter-wave solitons. Under the combined effects of the repulsive DDI and the SOC, stable solitons of semi-vortex type and mixed-mode type can be found in ground state as well as in excited state. These solitons can still be stable when the topological number is up to 11. The properties and stability of the nonlinear modes is also studied in a moving frame of reference.