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Stabilization of Multidimensional Solitons by Spin–Orbit Coupling (SOC)

Boris A. Malomed

Published Sep 8, 2022• 0 citations
Publisher

Abstract

The possibility of using spinorbit coupling (SOC) for the stabilization of 2D solitons in two-component BEC was first predicted by Sakaguchi et al. [Phys. Rev. E 89, 032920 (2014)]. Then, a possibility to create metastable 3D solitons in the same setting was reported by Zhang et al. [Phys. Rev. Lett. 115, 253902 (2015)]. Many interesting theoretical results were predicted in subsequent works, although they have not yet been realized in the experiment. In this chapter, the theoretical predictions are summarized, following, in particular, review articles by Sakaguchi and Malomed [Phys. Rev. A 97, 013607 (2018)] and by Malomed [Europhys. Lett. 122, 36001 (2018)]. Basic results included in the chapter represent absolutely stable 2D two-component solitons (ground states) of the semi-vortex (SV) and mixed-mode (MM) types, supported by the SOC in the Rashba and combined Rashba–Dresselhaus forms, as well as mobility of such solitons and collisions between them. The SVs are compound states with zero vorticity in one component and vorticity S = ±1 in the other, while MMs mix the zero-vorticity terms and ones with S = ±1 in both components. Also considered are effects of the Zeeman splitting (ZS) between the components—in particular, stable 2D gap solitons of the SV type, produced by the interplay of the ZS and SOC, while the kinetic-energy terms may be neglected. Finally, it is demonstrated that the SOC maintains three-dimensional metastable solitons, of both the SV and MM types, in two-component BEC with the attractive intrinsic nonlinearity.

DOI
10.1063/9780735425118_009

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The possibility of using spinorbit coupling (SOC) for the stabilization of 2D solitons in two-component BEC was first predicted by Sakaguchi et al. [Phys. Rev. E 89, 032920 (2014)]. Then, a possibility to create metastable 3D solitons in the same setting was reported by Zhang et al. [Phys. Rev. Lett. 115, 253902 (2015)]. Many interesting theoretical results were predicted in subsequent works, although they have not yet been realized in the experiment. In this chapter, the theoretical predictions are summarized, following, in particular, review articles by Sakaguchi and Malomed [Phys. Rev. A 97, 013607 (2018)] and by Malomed [Europhys. Lett. 122, 36001 (2018)]. Basic results included in the chapter represent absolutely stable 2D two-component solitons (ground states) of the semi-vortex (SV) and mixed-mode (MM) types, supported by the SOC in the Rashba and combined Rashba–Dresselhaus forms, as well as mobility of such solitons and collisions between them. The SVs are compound states with zero vorticity in one component and vorticity S = ±1 in the other, while MMs mix the zero-vorticity terms and ones with S = ±1 in both components. Also considered are effects of the Zeeman splitting (ZS) between the components—in particular, stable 2D gap solitons of the SV type, produced by the interplay of the ZS and SOC, while the kinetic-energy terms may be neglected. Finally, it is demonstrated that the SOC maintains three-dimensional metastable solitons, of both the SV and MM types, in two-component BEC with the attractive intrinsic nonlinearity.
We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.
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.
This mini-review collects results predicting the creation of matter-wave solitons by the spinor system of Gross-Pitaevskii equations (GPEs) with the self-attractive cubic nonlinearity and linear first-order-derivative terms accounting for the spin-orbit coupling (SOC). In 1D, the so-predicted bright solitons are similar to usual ones, supported by the GPE in the absence of SOC. Essentially new results were recently obtained for 2D and 3D systems: SOC suppresses the collapse instability in the multidimensional GPE, creating 2D ground-state solitons and metastable 3D ones of two types: semi-vortices (SVs), with vorticities m = 1 in one component and m = 0 in the other, and mixed modes (MMs), with m = 0 and present in both components. With the Galilean invariance broken by SOC, moving solitons exist up to a certain critical velocity. The latest result predicts stable 2D “quantum droplets” of the MM type in the presence of the Lee-Huang-Yang corrections to the GPE system, induced by quantum fluctuations, in the case when the inter-component attraction dominates over the self-repulsion in each component.
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 ααα. 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 ααα. The system demonstrates a relatively large norm share of the vortex components, exceeding 50%50\%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.
Excited states (ESs) of two- and three-dimensional (2D and 3D) solitons of the semivortex (SV) and mixed-mode (MM) types, supported by the interplay of the spin-orbit coupling (SOC) and local nonlinearity in binary Bose-Einstein condensates, are unstable, on the contrary to the stability of the SV and MM solitons in their fundamental states. We propose a stabilization strategy for these states in 3D, combining SOC and long-range Rydberg-Rydberg interactions (RRI), in the presence of a spatially-periodic potential, that may include a parity-time (PT)-symmetric component. ESs of the SV solitons, which carry integer vorticities S and S+1 in their two components, exhibit robustness up to S= 4. ESs of MM solitons feature an interwoven necklace-like structure, with the components carrying opposite fractional values of the orbital angular momentum. Regions of the effective stability of the 3D solitons of the SV and MM types (both fundamental ones and ESs), are identified as functions of the imaginary component of the PT-symmetric potential and strengths of the SOC and RRI terms.
This concise review aims to provide a summary of the most relevant recent experimental and theoretical results for solitons, i.e., self-trapped bound states of nonlinear waves, in two- and three-dimensional (2D and 3D) media. In comparison with commonly known one-dimensional solitons, which are, normally, stable modes, a challenging problem is the propensity of 2D and 3D solitons to instability, caused by the occurrence of the critical or supercritical wave collapse (catastrophic self-compression) in the same spatial dimension. A remarkable feature of multidimensional solitons is their ability to carry vorticity; however, 2D vortex rings and 3D vortex tori are subject to strong splitting instability. Therefore, it is natural to categorize the basic results according to physically relevant settings which make it possible to maintain stability of fundamental (non-topological) and vortex solitons against the collapse and splitting, respectively. The present review is focused on schemes that were recently elaborated in terms of Bose-Einstein condensates and similar photonic setups. These are two-component systems with spin-orbit coupling, and ones stabilized by the beyond-mean-field Lee-Huang-Yang effect. The latter setting has been implemented experimentally, giving rise to stable self-trapped quasi-2D and 3D "quantum droplets".