We report families of two-dimensional (2D) composite solitons in spinor dipolar Bose-Einstein condensates, with two localized components linearly mixed by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by the dipole-dipole interaction (DDI) between atomic magnetic moments polarized in plane by an external magnetic field. Recently, stable solitons were predicted in the form of semivortices (composites built of coupled fundamental and vortical components) in the 2D system combining the SOC and contact attractive interactions. Replacing the latter by the anisotropic long-range DDI, we demonstrate that, for a fixed norm of the soliton, the system supports a continuous family of stable spatially asymmetric vortex solitons (AVSs), parameterized by an offset of the pivot of the vortical component relative to its fundamental counterpart. The offset is limited by a certain maximum value, while the energy of the AVS practically does not depend on the offset. At small values of the norm, the vortex solitons are subject to a weak oscillatory instability. In the present system, with the Galilean invariance broken by the SOC, the composite solitons are set in motion by a kick the strength of which exceeds a certain depinning value. The kicked solitons feature a negative effective mass, drifting along a spiral trajectory opposite to the direction of the kick. A critical angular velocity, up to which the semivortices may follow rotation of the polarizing magnetic field, is found too.
We report families of two-dimensional (2D) composite solitons in spinor dipolar Bose-Einstein condensates, with two localized components linearly mixed by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by the dipole-dipole interaction (DDI) between atomic magnetic moments polarized in plane by an external magnetic field. Recently, stable solitons were predicted in the form of semivortices (composites built of coupled fundamental and vortical components) in the 2D system combining the SOC and contact attractive interactions. Replacing the latter by the anisotropic long-range DDI, we demonstrate that, for a fixed norm of the soliton, the system supports a continuous family of stable spatially asymmetric vortex solitons (AVSs), parameterized by an offset of the pivot of the vortical component relative to its fundamental counterpart. The offset is limited by a certain maximum value, while the energy of the AVS practically does not depend on the offset. At small values of the norm, the vortex solitons are subject to a weak oscillatory instability. In the present system, with the Galilean invariance broken by the SOC, the composite solitons are set in motion by a kick the strength of which exceeds a certain depinning value. The kicked solitons feature a negative effective mass, drifting along a spiral trajectory opposite to the direction of the kick. A critical angular velocity, up to which the semivortices may follow rotation of the polarizing magnetic field, is found too.
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.
Spatially anisotropic solitary vortices, i.e., bright anisotropic vortex solitons (AVSs), supported by anisotropic dipole-dipole interactions, were recently predicted in spin-orbit-coupled binary Bose--Einstein condensates (BECs), in the form of two-dimensional semivortices (complexes built of zero-vorticity and vortical components). We demonstrate that the shape of the AVSs---horizontal or vertical, with respect to the in-plane polarization of the atomic dipole moments in the underlying BEC---may be effectively controlled by the strength $\mathrm{\ensuremath{\Omega}}$ of the Zeeman splitting (ZS). A transition from the horizontal to vertical shape with the increase of $\mathrm{\ensuremath{\Omega}}$ is found numerically and explained analytically. At the transition point, the AVS assumes the shape of an elliptical ring. The mobility of horizontal AVSs is studied, too, with the conclusion that, with the increase of $\mathrm{\ensuremath{\Omega}}$, their negative effective mass changes the sign to positive via a point at which the effective mass diverges. Lastly, we report a new species of inverted AVSs, with the zero-vorticity and vortex component placed in lower- and higher-energy components, as defined by the ZS. They are excited states, with respect to the ground states provided by the usual AVSs. Quite surprisingly, inverted AVSs are stable in a large parameter region.
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.
It is commonly known that two-dimensional mean-field models of optical and matter waves with cubic self-attraction cannot produce stable solitons in free space because of the occurrence of collapse in the same setting. By means of numerical analysis and variational approximation, we demonstrate that the two-component model of the Bose-Einstein condensate with the spin-orbit Rashba coupling and cubic attractive interactions gives rise to solitary-vortex complexes of two types: semivortices (SVs, with a vortex in one component and a fundamental soliton in the other), and mixed modes (MMs, with topological charges 0 and ±1 mixed in both components). These two-dimensional composite modes can be created using the trapping harmonic-oscillator (HO) potential, but remain stable in free space, if the trap is gradually removed. The SVs and MMs realize the ground state of the system, provided that the self-attraction in the two components is, respectively, stronger or weaker than the cross attraction between them. The SVs and MMs which are not the ground states are subject to a drift instability. In free space (in the absence of the HO trap), modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower existence threshold for the stable modes. Moving free-space stable solitons are also found in the present non-Galilean-invariant system, up to a critical velocity. Collisions between two moving solitons lead to their merger into a single one.
We present gap solitons (GSs) that can be created in free nearly two-dimensional (2D) space in dipolar spinor Bose-Einstein condensates with the spin-orbit coupling (SOC), subject to tight confinement, with size ${a}_{\ensuremath{\perp}}$, in the third direction. For quasi-2D patterns, with lateral sizes $l\ensuremath{\gg}{a}_{\ensuremath{\perp}}$, the kinetic-energy terms in the respective spinor Gross-Pitaevskii equations may be neglected in comparison with SOC. This gives rise to a band gap in the system's spectrum, in the presence of the Zeeman splitting between the spinor components. While the present system with contact interactions does not produce 2D solitons, stable gap solitons (GSs), with vorticities 0 and 1 in the two components, are found, in quasianalytical and numerical forms, under the action of dipole-dipole interaction (DDI). Namely, isotropic and anisotropic 2D GSs are obtained when the dipoles are polarized, respectively, perpendicular or parallel to the 2D plane. The GS families extend, as embedded solitons (ESs), into spectral bands, a part of the ES branch being stable for isotropic solitons. The GSs remain stable if the competing contact interaction, with the sign opposite to that of the DDI, is included, while the addition of the contact term with the same sign destabilizes the GSs, at first replacing them by breathers, and eventually leading to destruction of the solitons. Mobility and collision of the GSs are studied too, revealing negative and positive effective masses of the isotropic and anisotropic solitons, respectively.
Abstract We study construction and dynamics of two-dimensional (2D) anisotropic vortex-bright (VB) soliton in spinor dipolar Bose-Einstein condensates confined in a 2D optical lattice (OL), with two localized components linearly mixed by the spin-orbit coupling and long-range dipole-dipole interaction (DDI). It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of norm N. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked VB solition is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance and OL, and they can realize the transition from the bright component to the vortex component. Our work may provides a convenient way to prepare and manipulate anisotropic VB soliton in high dimensional space.