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arXiv:2609.17092 · Fortschritte der Physik

Stable Three‐Dimensional Lattice Solitons in Spin‐Orbit‐Coupled Bose‐Einstein Condensates

Liangwei Zeng, Boris A. Malomed, Yaroslav V. Kartashov, Xing Zhu

Published Sep 1, 2026• 0 citations

Abstract

ABSTRACT We address three‐dimensional (3D) solitons maintained by spin‐orbit coupling (SOC) in the binary self‐interacting Bose‐Einstein condensate (BEC) held in the 3D optical lattice (OL). The analysis reveals that the SOC‐OL interplay results in the formation of stable full‐vortex (FV) solitons, built as sets of four density peaks residing in neighboring wells of the lattice potential, with the superimposed global vortical phase, and site‐centered semi‐vortices (SVs), in which the vorticity is present in only one component. The FV solitons, with their specific phase textures, do not exist in a uniform BEC with SOC. FV and SV states in the binary self‐attractive BEC are stable despite the possibility of the supercritical collapse in the 3D system. Such states also exist, as gap solitons, in the self‐repulsive 3D system. In terms of the chemical potential and number of particles, the stability regions of the 3D FV solitons and SVs expand with the increase of the SOC strength and OL depth. The results open the route to the creation of 3D complexes of vorticity‐carrying condensates that can be realized with existing experimental techniques.

DOI: 10.1002/prop.70153

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ABSTRACT We address three‐dimensional (3D) solitons maintained by spin‐orbit coupling (SOC) in the binary self‐interacting Bose‐Einstein condensate (BEC) held in the 3D optical lattice (OL). The analysis reveals that the SOC‐OL interplay results in the formation of stable full‐vortex (FV) solitons, built as sets of four density peaks residing in neighboring wells of the lattice potential, with the superimposed global vortical phase, and site‐centered semi‐vortices (SVs), in which the vorticity is present in only one component. The FV solitons, with their specific phase textures, do not exist in a uniform BEC with SOC. FV and SV states in the binary self‐attractive BEC are stable despite the possibility of the supercritical collapse in the 3D system. Such states also exist, as gap solitons, in the self‐repulsive 3D system. In terms of the chemical potential and number of particles, the stability regions of the 3D FV solitons and SVs expand with the increase of the SOC strength and OL depth. The results open the route to the creation of 3D complexes of vorticity‐carrying condensates that can be realized with existing experimental techniques.

We elaborate a mechanism for the creation of stable three-dimensional (3D) solitons in spin-orbit-coupled (SOC) atomic-molecular Bose-Einstein condensate, modeled by the mean-field equations with the quadratic three-wave interaction, characterized by mismatch

α

. The planar (effectively two-dimensional) SOC is applied to the soliton's atomic component, structuring it as a mixed mode (MM) or semi-vortex (SV). The molecular component of the SV soliton is shaped as a 3D vortex, while the molecular component in the MM soliton is an MM too. The solitons exist up to a critical value of

α

. The system demonstrates a relatively large norm share of the vortex components, exceeding

50\%

of the total norm, which is an essential feature of SOC-supported solitons. This is scheme for realizing stable vortex solitons in free space with the quadratic nonlinearity.

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.

Applying a variational approach and numerical analysis to the system of Gross-Pitaevskii equations, we find three-dimensional (3D) stable solitons in binary atomic Bose-Einstein condensates with spin-orbit coupling (SOC) and out-of-phase linear and nonlinear Bessel optical lattices. We discuss the stability of 3D solitons by utilizing their norm and energy. The introduction of Bessel potentials makes the evolution and collisions of solitons more stable and improves their resistance to collapse. Depending on the strength of the intra- and intercomponent spatial modulation of the nonlinearity and SOC, we find stable solitons of the semivortex and mixed-mode structures. Furthermore, we show that the solitons are stable against small perturbations in propagation and collisions.

By means of variational methods and systematic numerical analysis, we demonstrate the existence of metastable solitons in three dimensional (3D) free space, in the context of binary atomic condensates combining contact self-attraction and spin-orbit coupling, which can be engineered by available experimental techniques. Depending on the relative strength of the intra- and intercomponent attraction, the stable solitons feature a semivortex or mixed-mode structure. In spite of the fact that the local cubic self-attraction gives rise to the supercritical collapse in 3D, and hence the setting produces no true ground state, the solitons are stable against small perturbations, motion, and collisions.

Stable vortex solitons (VSs) are objects of great interest for fundamental studies and various applications, including particle trapping, microscopy, data encoding, and matter-wave gyroscopes. However, three-dimensional (3D) VSs with high topological charges, supported by self-attractive nonlinearities, are unstable against fragmentation, which eventually leads to internal blowup (supercritical collapse) of the fragments. Here, we propose a scheme for realizing stable 3D VSs with topological charges up to

5

and

6

in the two components of a binary, Rydberg-dressed Bose-Einstein condensate (BEC) with spin-orbit coupling (SOC). We show that, if the SOC strength exceeds a critical value, the rotational symmetry of the VSs in the transverse plane gets broken, resulting in separation of the two components. Nevertheless, the VSs with the broken symmetry remain stable. The VS stability domains are identified in the system's parameter space. Moreover, application of torque to the stable VSs sets them in the state of robust gyroscopic precession.

We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose-Einstein condensates of Rydberg atoms under the action of the spin-orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius.