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New Journal of Physics

Oscillatory soliton pairs in spin-orbit coupled spin-1/2 Bose–Einstein condensates

Feiyan Liu, Qin Zhou

Published Jan 29, 2025• 13 citations

Abstract

Abstract We investigate oscillatory soliton pairs including the bright–bright solitons, dark–dark solitons, bright–dark solitons and beating solitons (beating soliton refers to a type of soliton characterized by components that are each composed of both bright and dark solitons, and the overall density is equivalent to that of a single soliton) in the spin-orbit coupled spin-1/2 Bose–Einstein condensates. Two important effects that are the spin orbit coupling (SOC) and Raman detuning resonance on these solitons are studied. The results show that the intensity of SOC affects the spatial oscillation of solitons, while Raman detuning resonance affects the temporal oscillation of solitons. Moreover, the oscillatory soliton pairs and their background are affected by gain/loss. Under different gain/loss modulator, the periodic, parabolic, dromion type and line solitons are constructed for bright–bright pairs, and the periodic and parabolic background planes are presented for dark–dark soliton pairs, as well as the bound state solitons on periodic and kink type background planes are obtained for bright–dark soliton pairs. For one-beating soliton pairs, two types of beating solitons including the petal and breather-like are discovered by adjusting the wave number. For two-beating soliton pairs, the interaction between different combinations of the two types of beating solitons mentioned above, especially the X- and Y-type beating solitons, are presented by controlling two wave numbers. Furthermore, we verify the stability of soliton pairs through numerical simulation and the physics-informed neural network methods.

DOI: 10.1088/1367-2630/adafd7

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We present families of beating dark-dark solitons in spin-orbit (SO) -coupled Bose-Einstein condensates. These families consist of solitons residing simultaneously in the two bands of the energy spectrum. The soliton components are characterized by two different spatial and temporal scales, which are identified by a multiscale expansion method. The solitons are ``beating'' ones, as they perform density oscillations. The characteristic frequency of the latter is relevant to Zitterbewegung (ZB) oscillations, which were recently observed in experiments with SO-coupled condensates [C. Qu et al., Phys. Rev. A 88, 021604(R) (2013); L. J. LeBlanc et al., New J. Phys. 15, 073011 (2013)]. We find that spin oscillations may occur, depending on the parity of each soliton branch, which consequently lead to ZB oscillations of the beating dark solitons. Analytical results are corroborated by numerical simulations, illustrating the robustness of the solitons.

We demonstrate that the model of a spatially nonuniform two-component Bose-Einstein condensates (BEC) featuring the helicoidal spin-orbit coupling (SOC), gives rise to dark-bright soliton complexes characterized by spatiotemporal periodic oscillations in each component. These solitons are formed by the superposition of dark and bright ones, and exhibit a beating state over time and a striped state across space, earning them the designation of beating stripe solitons. Our analysis demonstrates that helicoidal SOC significantly affects the formation and dynamical properties of these solitons, also serving as the primary driver for spin oscillations. Through the nonlinear superposition of the beating stripe solitons, a range of intricate scenarios of the interaction between multiple solitons emerges, including head-on collisions, bound states, and parallel states.

We theoretically investigate the dynamics of polar bright soliton under the influence of spin-orbit coupling (SOC) in spin-1 three component Bose-Einstein condensate (BEC). Periodically oscillating dark-bright-dark solitons are found to be induced by the SOC with their multifrequency oscillations jointly determined by the interparticle nonlinear interaction and the SOC strength. Explicit formulas are provided to identify these frequencies, which are in good agreement with the numerical simulation results. A linear Zeeman interaction is further introduced to break the time-reversal symmetry and change the polarity of the spinor BECs, which results in the periodic transition of the system from polar state to ferromagnetic state.

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.

The existence and dynamics of solitons in quasi-one-dimensional Bose-Einstein condensates with spin-orbit coupling (SOC) and attractive two-body interactions are described for two coupled atomic pseudospin components with slowly and rapidly varying time-dependent Raman frequency. By varying the Raman frequency linearly in time, it was shown that ordinary nonlinear Schr\"odinger-type bright solitons can be converted to striped bright solitons, and vice versa. The internal Josephson oscillations between atom number of the coupled soliton components and the corresponding center-of-mass motion are studied for different parameter configurations. In this case, a mechanism to control the soliton parameters is proposed by considering parametric resonances, which can emerge when using time-varying Raman frequencies. Full-numerical simulations confirm variational analysis predictions when applied to the region where regular solitons are expected. In the limit of high frequencies, the system is described by a time-averaged Gross-Pitaevskii formalism with renormalized nonlinear and SOC parameters and modified phase-dependent nonlinearities. By comparing full-numerical simulations with averaged results, we have also studied the lower limits for the frequency of Raman oscillations in order to obtain stable soliton solutions.

We investigate the matter-wave solitons in a spin–orbit-coupled spin-1 Bose–Einstein condensate using a multiscale perturbation method. Beginning with the one-dimensional spin–orbit-coupled threecomponent Gross–Pitaevskii equations, we derive a single nonlinear Schrödinger equation, which allows determination of the analytical soliton solutions of the system. Stationary and moving solitons in the system are derived. In particular, a parameter space for different existing soliton types is provided. It is shown that there exist only dark or bright solitons when the spin–orbit coupling is weak, with the solitons depending on the atomic interactions. However, when the spin–orbit coupling is strong, both dark and bright solitons exist, being determined by the Raman coupling. Our analytical solutions are confirmed by direct numerical simulations.