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arXiv:2503.15068 · Physical review. E

Beating stripe solitons arising from helicoidal spin-orbit coupling in Bose-Einstein condensates

Cui-Cui Ding, Qin Zhou, Boris A. Malomed

Published Apr 4, 2025• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.111.044203

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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.
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.
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.
The spinor Bose-Einstein condensate (BEC) provides an ideal platform for observing and manipulating topological structures, which arise from the spin degrees of freedom and the superfluid nature of the gas. Artificial helicoidal spin-orbit coupling (SOC) in the spinor BEC, owing to the spatially varying gauge potential and the more flexible adjustability, provides possibly an unprecedented opportunity to search for novel quantum states. The previous studies of the BEC with helicoidal SOC mainly focus on the two-component case. However, there are few reports on the studies of helicoidal SOC in three-component BEC. Especially considering one-dimensional three-component BEC, whether the helicoidal SOC can generate previously unknown types of topological excitations and phase diagrams is still an unsolved problem. In this work, by solving quasi one-dimensional Gross-Pitaevskii equations, we study the ground state structure of one-dimensional helicoidal spin-orbit coupled three-component BEC. The numerical results show that the helicoidal SOC can induce a phase separation among the components in ferromagnetic BEC. Through numerical calculations of the system, a phase diagram is obtained as a function of the helicoidal SOC strength and gauge potential, which shows the critical conditions for phase separation and phase miscibility in ferromagnetic BEC. Meanwhile, we also study the influences of the helicoidal SOC and the gauge potential on the antiferromagnetic BEC ground state. The numerical results show that the helicoidal SOC is beneficial for the miscibility in antiferromagnetic BEC. When the helicoidal SOC strength or gauge potential increases, the ground state of antiferromagnetic BEC exhibits a stripe soliton structure. Adjusting the strength of helicoidal SOC or gauge potential can control the transitions between a plane-wave soliton and a stripe soliton. In addition, we show the changes of the particle number density maximum and the number of peaks of stripe solitons for adjusting the helicoidal SOC strength or gauge potential. Our results show that helicoidal spin-orbit coupled BEC not only provides a controlled platform for investigating the exotic topological structures, but also is crucial for the transitions between different ground states. This work paves the way for exploring the topological defect and the corresponding dynamical stability in quantum systems subjected to the helicoidal SOC in future.
We investigate the existence and stability of higher-order bright solitons, stripe solitons, and bright-dark solitons in a Bose-Einstein condensate with helicoidal spin-orbit coupling under a Zeeman lattice using numerical methods. The higher-order bright solitons that exist in the first-finite energy gap are stable except near the edge. The stripe solitons with parity-time symmetry and pseudospin-parity symmetry have partially overlapping norm curves; they are stable in the lower edge of the first-finite energy gap. Additionally, the bright-dark solitons discovered in the system not only exist within energy gaps but also embed within energy bands as they have periodic backgrounds. These findings offer insights into the diversity and behavior of solitons within energy bands and contribute to a deeper understanding of their distribution and dynamics.
Abstract We show that attractive two-dimensional (2D) spinor Bose–Einstein condensates with helicoidal spatially periodic spin–orbit coupling (SOC) support a rich variety of stable fundamental solitons and bound soliton complexes. Such states exist with chemical potentials belonging to the semi-infinite gap in the band spectrum created by the periodically modulated SOC. All these states exist above a certain threshold value of the norm. The chemical potential of fundamental solitons attains the bottom of the lowest band, whose locus is a ring in the space of Bloch momenta, and the radius of the non-monotonous function of the SOC strength. The chemical potential of soliton complexes does not attain the band edge. The complexes are bound states of several out-of-phase fundamental solitons whose centers are placed at local maxima of the SOC-modulation phase. In this sense, the impact of the helicoidal SOC landscape on the solitons is similar to that of a periodic 2D potential. In particular, it can compensate repulsive forces between out-of-phase solitons, making their bound states stable. Extended stability domains are found for complexes built of two and four solitons (dipoles and quadrupoles, respectively). They are typically stable below a critical value of the chemical potential.
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.