We consider an effectively one-dimensional binary Bose-Einstein condensate (BEC) with nonlinear repulsive interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the presence of the trapping harmonic-oscillator (HO) potential, we report the existence of even, odd, and asymmetric spatial modes. They feature alternating domains with opposite directions of the pseudospin, i.e., antiferromagnetic structures, which is explained by the interplay of the linear couplings, HO confinement, and repulsive self-interaction. The number of the domains is determined by the strength of the SO coupling. The modes are constructed analytically in the weakly nonlinear system. The dynamical stability of the modes is investigated by means of the Bogoliubov--de Gennes equations and direct simulations. A notable result is that the multi-domain-wall (DW) structures are stable, alternating between odd and even shapes, while the simplest single-DW structure is unstable. Thus, the system features a transition to the complex ground states under the action of the SO coupling. The addition of the Zeeman splitting transforms the odd modes into asymmetric ones via spontaneous symmetry breaking. The results suggest possibilities for switching the binary system between states with opposite (pseudo)magnetization by external fields, and realization of similar stable states and dynamical effects in solid-state and nonlinear-optical settings emulated by the SO-coupled BECs.
We consider an effectively one-dimensional binary Bose-Einstein condensate (BEC) with nonlinear repulsive interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the presence of the trapping harmonic-oscillator (HO) potential, we report the existence of even, odd, and asymmetric spatial modes. They feature alternating domains with opposite directions of the pseudospin, i.e., antiferromagnetic structures, which is explained by the interplay of the linear couplings, HO confinement, and repulsive self-interaction. The number of the domains is determined by the strength of the SO coupling. The modes are constructed analytically in the weakly nonlinear system. The dynamical stability of the modes is investigated by means of the Bogoliubov--de Gennes equations and direct simulations. A notable result is that the multi-domain-wall (DW) structures are stable, alternating between odd and even shapes, while the simplest single-DW structure is unstable. Thus, the system features a transition to the complex ground states under the action of the SO coupling. The addition of the Zeeman splitting transforms the odd modes into asymmetric ones via spontaneous symmetry breaking. The results suggest possibilities for switching the binary system between states with opposite (pseudo)magnetization by external fields, and realization of similar stable states and dynamical effects in solid-state and nonlinear-optical settings emulated by the SO-coupled BECs.
We consider a binary Bose-Einstein condensate with linear and nonlinear interactions between its components, which emulate the spinor system with spin-orbit (SO) and Rabi couplings. For a relatively dense condensate, one-dimensional coupled equations with the nonpolynomial nonlinearity of both repulsive and attractive signs are derived from the three-dimensional Gross-Pitaevskii equations. Profiles of modes confined in an external potential under the action of the self-repulsion, and self-trapped solitons in the case of the self-attraction, are found in a numerical form and by means of analytical approximations. In the former case, the interplay of the SO and Rabi couplings with the repulsive nonlinearity strongly distorts shapes of the trapped modes, adding conspicuous sidelobes to them. In the case of the attractive nonlinearity, the most essential result is reduction of the collapse threshold under the action of the SO and Rabi couplings.
Abstract We investigate the ground-state properties of spin–orbit (SO)-coupled Bose–Einstein condensates confined in a spin-dependent asymmetric potential composed of harmonic and quartic contributions. The combined effect of the two potentials gives rise to an effective spatially inhomogeneous Zeeman field along the z -direction, and the complex competition among SO coupling, interatomic interactions, and this effective field leads to the emergence of nontrivial topological phases and complex spin textures in the system. For weak interatomic interactions, as the SO coupling strength increases, the system undergoes a series of transitions between ground-state phases: from a spin-polarized state, to an Anderson–Toulouse (AT) vortex state, a high-order radial dark soliton state, and finally to a domain structure. For strong interatomic interactions, the condensate evolves from a spin-polarized state to an AT vortex state, and then to a spatially coexisting state of AT and Mermin–Ho vortices, which corresponds to a novel topological excitation in pseudospin space: a skyrmion-dominated texture embedded with circular-hyperbolic antimeron pairs, realizing the coexistence and nesting of multiple topological defects. For sufficiently strong SO coupling, both components eventually evolve into similar multiring-shaped higher-order radial dark soliton states, with the number of soliton rings increasing as the SO coupling strength increases. Our results enrich the ground-state phases and topological excitations of cold atomic systems and provide new insights for exploring and manipulating nontrivial topological excitations in such systems.
Abstract We report results of systematic analysis of confined steadily rotating patterns in the two-component BEC including the spin–orbit coupling (SOC) of the Rashba type, which acts in the interplay with the attractive or repulsive intra-component and inter-component nonlinear interactions and confining potential. The analysis is based on the system of the Gross–Pitaevskii equations (GPEs) written in the rotating coordinates. The resulting GPE system includes effective Zeeman splitting. In the case of the attractive nonlinearity, the analysis, performed by means of the imaginary-time simulations, produces deformation of the known two-dimensional SOC solitons (semi-vortices and mixed-modes). Essentially novel findings are reported in the case of the repulsive nonlinearity. They demonstrate patterns arranged as chains of unitary vortices which, at smaller values of the rotation velocity Ω, assume the straight (single-string) form. At larger Ω, the straight chains become unstable, being spontaneously replaced by a trilete star-shaped array of vortices. At still larger values of Ω, the trilete pattern rebuilds itself into a star-shaped one formed of five and, then, seven strings. The transitions between the different patterns are accounted for by comparison of their energy. It is shown that the straight chains of vortices, which form the star-shaped structures, are aligned with boundaries between domains populated by plane waves with different wave vectors. A transition from an axisymmetric higher-order (multiple) vortex state to the trilete pattern is investigated too.
We study spontaneous symmetry breaking in a system of two parallel quasi-one-dimensional traps (cores), equipped with optical lattices (OLs) and filled with a Bose-Einstein condensate (BEC). The cores are linearly coupled by tunneling (the model may also be interpreted in terms of spatial solitons in parallel planar optical waveguides with a periodic modulation of the refractive index). Analysis of the corresponding system of linearly coupled Gross-Pitaevskii equations (GPEs) reveals that spectral band gaps of the single GPE split into subgaps. Symmetry breaking in two-component BEC solitons is studied in cases of the attractive (AA) and repulsive (RR) nonlinearity in both traps; the mixed situation, with repulsion in one trap and attraction in the other (RA), is considered too. In all the cases, stable asymmetric solitons are found, bifurcating from symmetric or antisymmetric ones (and destabilizing them), in the AA and RR systems, respectively. In either case, bistability is predicted, with a nonbifurcating stable branch, either antisymmetric or symmetric, coexisting with asymmetric ones. Solitons destabilized by the bifurcation tend to rearrange themselves into their stable asymmetric counterparts. In addition to the fundamental solitons, branches of twisted (odd) solitons in the AA system, and twisted bound states of fundamental solitons in both AA and RR systems, are found too. The impact of a phase mismatch, $\ensuremath{\Delta}$, between the OLs in the two cores is also studied. It is concluded that $\ensuremath{\Delta}=\ensuremath{\pi}∕2$ only mildly deforms the picture, while $\ensuremath{\Delta}=\ensuremath{\pi}$ changes it drastically, replacing the symmetry-breaking bifurcations by pseudobifurcations, with the branch of asymmetric solutions asymptotically approaching its symmetric or antisymmetric counterpart (in the AA and RR system, respectively), rather than splitting off from it. Also considered is a related model, for a binary BEC in a single-core trap with the OL, assuming that the two species (representing different spin states of the same atom) are coupled by linear interconversion. In that case, the symmetry-breaking bifurcations in the AA and RR models switch their character, if the interspecies nonlinear interaction becomes stronger than the intraspecies nonlinearity.
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.
We present a systematic study of the generation of the array of optical or matter-wave kinks (dark solitons) in the ground state (GS) of binary systems. We consider quasi-one-dimensional systems described by a pair of nonlinear Schr\"odinger (NLSE's) or Gross-Pitaevskii equations (GPE's), which are coupled by the linear mixing, with local strength $\ensuremath{\Omega}$, and by nonlinear interactions. We assume the self-repulsive nonlinearity in both components, and include the effects of a harmonic trapping potential, while the nonlinear interaction between the components may be both repulsive and attractive. The model may be realized in terms of periodically modulated slab waveguides in nonlinear optics and also in Bose-Einstein condensates. Depending on the sign and strengths of the linear and nonlinear couplings between the components, the ground states in such binary systems may be symmetric, antisymmetric, or asymmetric. In this work, we introduce a periodic spatial modulation of the linear coupling, making $\ensuremath{\Omega}$ an odd or even function of the coordinate (x). The sign flips of $\ensuremath{\Omega}(x)$ strongly modify the structure of the GS in the binary system, as the relative sign of its components tends to lock to the local sign of $\ensuremath{\Omega}$. Using a systematic numerical analysis and an analytic approximation, we demonstrate that the GS of the trapped system contains one or several kinks (dark solitons) in one component, while the other component does not change its sign. The final results are presented in the form of maps showing the number of kinks in the GS as a function of the system's parameters, with the odd (even) modulation function giving rise to the odd (even) number of the kinks. The modulation of $\ensuremath{\Omega}(x)$ also produces a strong effect on the transition between states with nearly equal and strongly unequal amplitudes of the two components.