Abstract It is known that the interplay of the spin–orbit-coupling (SOC) and mean-field self-attraction creates stable two-dimensional (2D) solitons (ground states) in spinor Bose–Einstein condensates. However, SOC destroys the system’s Galilean invariance, therefore moving solitons exist only in a narrow interval of velocities, outside of which the solitons suffer delocalization. We demonstrate that the application of a relatively weak moving optical lattice (OL), with the 2D or quasi-1D structure, makes it possible to greatly expand the velocity interval for stable motion of the solitons. The stability domain in the system’s parameter space is identified by means of numerical methods. In particular, the quasi-1D OL produces a stronger stabilizing effect than its full 2D counterpart. Some features of the domain are explained analytically.
Abstract It is known that the interplay of the spin–orbit-coupling (SOC) and mean-field self-attraction creates stable two-dimensional (2D) solitons (ground states) in spinor Bose–Einstein condensates. However, SOC destroys the system’s Galilean invariance, therefore moving solitons exist only in a narrow interval of velocities, outside of which the solitons suffer delocalization. We demonstrate that the application of a relatively weak moving optical lattice (OL), with the 2D or quasi-1D structure, makes it possible to greatly expand the velocity interval for stable motion of the solitons. The stability domain in the system’s parameter space is identified by means of numerical methods. In particular, the quasi-1D OL produces a stronger stabilizing effect than its full 2D counterpart. Some features of the domain are explained analytically.
In a basic physical model where two-dimensional (2D) matter-wave solitons may be stable, namely, the Gross-Pitaevskii equation with the self-attractive nonlinearity and quasi-one-dimensional (1D) optical-lattice (OL) potential, we test robustness of the solitons against periodic time modulation of the OL strength. Stability diagrams for the 2D solitons are presented in the plane of the modulation depth and frequency. Basic features of the diagrams are explained with the help of the variational approximation for the stationary counterpart of the model. In the Bose-Einstein condensate of 7Li atoms, the stable 2D solitons may contain the number of atoms in the range of 104−−105, relevant values of the OL strength and modulation frequency being, respectively, $\ensuremath{\lesssim}5$ recoil energies and $\ensuremath{\lesssim}10\text{ }\text{kHZ}$. Head-on collisions between stable 2D solitons moving in the unconfined direction are studied in detail too, for velocities up to $\ensuremath{\sim}5\text{ }\text{cm}∕\text{s}$. A border between quasi-elastic collisions and merger of the solitons into a single localized state is identified. In some cases, the soliton produced by the merger is stable against collapse, which was not observed before in the static OL potential either.
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.
Abstract We address effects of spin–orbit coupling (SOC), phenomenologically added to a two-component Bose–Einstein condensate composed of particles moving by Lévy flights, in one- and two-dimensional (1D) and (2D) settings. The corresponding system of coupled Gross–Pitaevskii equations includes fractional kinetic-energy operators, characterized by the Lévy index, α < 2 (the normal kinetic energy corresponds to α = 2). The SOC terms, with strength λ, produce strong effects in the 2D case: they create families of stable solitons of the semi-vortex and mixed-mode types in the interval of 1 < α < 2, where the supercritical collapse does not admit the existence of stable solitons in the absence of the SOC. At λ → 0, amplitudes of these solitons vanish ∼λ 1/(α−1).
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.
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.
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.