Abstract We introduce a discrete model for binary spin–orbit-coupled Bose–Einstein condensates trapped in a deep one-dimensional optical lattice. Two different types of the couplings are considered, with spatial derivatives acting inside each species, or between the species. The discrete system with inter-site couplings dominated by the spin–orbit coupling (SOC), while the usual hopping is negligible, emulates condensates composed of extremely heavy atoms, as well as those with opposite signs of the effective atomic masses in the two components. Stable localized composite states of miscible and immiscible types are constructed. The effect of the SOC on the immiscibility–miscibility transition in the localized complexes, which emulates the phase transition between insulating and conducting states in semiconductors, is studied.
Abstract We introduce a discrete model for binary spin–orbit-coupled Bose–Einstein condensates trapped in a deep one-dimensional optical lattice. Two different types of the couplings are considered, with spatial derivatives acting inside each species, or between the species. The discrete system with inter-site couplings dominated by the spin–orbit coupling (SOC), while the usual hopping is negligible, emulates condensates composed of extremely heavy atoms, as well as those with opposite signs of the effective atomic masses in the two components. Stable localized composite states of miscible and immiscible types are constructed. The effect of the SOC on the immiscibility–miscibility transition in the localized complexes, which emulates the phase transition between insulating and conducting states in semiconductors, is studied.
We study localized nonlinear excitations of a dilute Bose-Einstein condensate (BEC) with spin-orbit coupling in a deep optical lattice (OL). We use Wannier functions to derive a tight-binding model that includes the spin-orbit coupling (SOC) at the discrete level in the form of a generalized discrete nonlinear Schödinger equation. Spectral properties are investigated and the existence and stability of discrete solitons and breathers with different symmetry properties with respect to the OL is demonstrated. We show that the symmetry of the modes can be changed from on-site to inter-site and to asymmetric modes simply by changing the interspecies interaction. Asymmetric modes appear to be novel modes intrinsic of the SOC.
Binary mixtures of Bose-Einstein condensates (BECs) trapped in deep optical lattices and subjected to equal contributions of Rashba and Dresselhaus spin-orbit coupling (SOC) are investigated in the presence of a periodic time modulation of the Zeeman field. SOC tunability is explicitly demonstrated by adopting a mean-field tight-binding model for the BEC mixture and by performing an averaging approach in the strong modulation limit. In this case, the system can be reduced to an unmodulated vector discrete nonlinear Schr\"odinger equation with a rescaled SOC tuning parameter $\ensuremath{\alpha}$, which depends only on the ratio between amplitude and frequency of the applied Zeeman field. We consider the attractive interaction case and focus on the effect of the SOC tuning on the localized ground states. The dependence of the spectrum of the linear system on $\ensuremath{\alpha}$ has been analytically characterized. In particular, we show that extremal curves (ground and highest excited states) of the linear spectrum are continuous piecewise functions (together with their derivatives) of $\ensuremath{\alpha}$, which consist of a finite number of decreasing band lobes joined by constant lines. This structure also remains in the presence of inter- and intra-species interactions, the nonlinearity mainly introducing a number of localized states in the band gaps. The stability of ground states in the presence of the modulating field has been demonstrated by real-time evolutions of the original (unaveraged) system. Localization properties of the ground state induced by the SOC tuning, and a parameter design for possible experimental observation, have also been discussed.
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.
Abstract We present a systematic investigation of symmetries and ground state phase transitions in two-dimensional Bose–Einstein condensates with rotational and spin–orbit couplings confined within one-dimensional optical lattices. For lattice phases φ = π 2 , the vortex arrangement exhibits mirror symmetry about the y-axis, and the density distribution in momentum space maintains mirror symmetry about the k x -axis. For φ = 0, in contrast, the system exhibits spontaneous symmetry breaking, leading to ground state degeneracy. The optical lattice divides the condensate into distinct units. With a fixed spin–orbit coupling strength, the two-component condensates in different units successively undergo immiscible to miscible transitions as the rotational angular frequency increases. For φ = 0, in particular, the unit of condensate located in the center of the external potential consistently remains in the phase separation.
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.