Using the parity and time reversal symmetries of a two-dimensional spin-orbit coupled Bose-Einstein condensate in a lattice created by the Zeeman field, we identify and find numerically various families of localized solutions, including multipole and half-vortex solitons. The obtained solutions may exist at any direction of the gauge field with respect to the lattice and can be found either in finite gaps (for repulsive interatomic interactions) or in a semi-infinite gap (for attractive interactions). The existence of half-vortices requires higher symmetry (the reflection with respect to the field direction). Stability of these modes makes them feasible for experimental observation.
Using the parity and time reversal symmetries of a two-dimensional spin-orbit coupled Bose-Einstein condensate in a lattice created by the Zeeman field, we identify and find numerically various families of localized solutions, including multipole and half-vortex solitons. The obtained solutions may exist at any direction of the gauge field with respect to the lattice and can be found either in finite gaps (for repulsive interatomic interactions) or in a semi-infinite gap (for attractive interactions). The existence of half-vortices requires higher symmetry (the reflection with respect to the field direction). Stability of these modes makes them feasible for experimental observation.
Periodic potentials with flat bands in their spectra support strongly localized nonlinear excitations. Although a perfectly flat band cannot exist in a continuous systems, a spin–orbit–coupled Bose–Einstein condensate loaded in a Zeeman lattice can realize the quasi-flat lowest band with an extremely narrow bandwidth. In such a quasi-flat band, half-vortex solitons become confined within a single lattice cell, enabling the formation of arrays of coupled half-vortex solitons arranged of various spatial geometries. In this work, we study the existence and stability of these lattices within the framework of the two-component Gross–Pitaevskii equation. We demonstrate that, near the quasi-flat band, half-vortex solitons and their arrays can be excited with a nearly negligible number of atoms and are constrained by their local symmetries, which are isomorphic to a dihedral group of order 8. This allows observation of the respective field patterns in the nearly linear regime where they exhibit enhanced stability. The constructed lattices may have diverse geometric profiles, and in particular create a composite super-half-vortex soliton with nonlinear symmetry breaking. • A two-dimensional Zeeman lattice with a quasi-flat is obtained. • Half-vortex solitons with a negligible threshold energy are found; they can be observed in a quasi-linear regime. • Stable half-vortex soliton lattices of different shapes, as well as super-half-vortex solitons are obtained.
Abstract In ultracold atomic gases, optical lattices and spin-orbit coupling are two important and useful linear means to study and control the nonlinear and quantum dynamics of matter waves, and both have been well realized in experiments. Here, nonlinear localization of spin-orbit-coupled Bose-Einstein condensates in moiré optical lattices is revealed theoretically, stressing the existence and stability of localized modes of diverse types including fundamental, half-vortex matter-wave gap solitons, and full-vortex ones (of both components). Notably, to our knowledge, a new type of vector vortex gap solitons, which is composed of a gap vortex soliton (singular mode) and a rhombic-array gap vortex (quadruple mode), is predicted in our two-component theoretical framework. It is also found that the model exhibits unique linear Bloch band structures including band splitting and extremely-flat bands, because of which all the predicted localized modes tend to be robustly stable, backed by our direct perturbed simulations. Our results demonstrate that the moiré pattern (and its joint cooperation with spin-orbit coupling) provides great possibilities to explore flat-band and moiré physics in ultracold atoms, and to further investigate the nonlinear dynamics.
While different ways to realize spin-orbit coupling in Bose-Einstein condensates exist, not all are currently experimentally implementable. Here we present a detailed study of gap solitons in a Bose-Einstein condensate with experimentally realizable spin-orbit coupling and discuss two cases relating to a spin-dependent parity symmetry. In the parity symmetric case, two families of fundamental gap solitons in the second linear energy gap are demonstrated with the opposite sign of the parity, with one family having single-humped densities and the other double-humped ones. In the case of broken parity symmetry, the fundamental modes manifest spin polarization. Both families possess an opposite sign of the spin polarization.
We elaborate a mechanism for the formation of stable solitons of the semivortex type (with vorticities 0 and 1 in their two components), populating a finite band gap in the spectrum of the spin-orbit-coupled binary Bose-Einstein condensate with the Zeeman splitting, in the two-dimensional (2D) free space, under conditions which make the kinetic-energy terms in the respective coupled Gross-Pitaevskii equations negligible. Unlike a recent work which used long-range dipole-dipole interactions to construct stable gap solitons in a similar setting, we here demonstrate that stable solitons are supported by generic local interactions of both attractive and repulsive signs, provided that the relative strength of the cross- and self-interactions in the two-component system does not exceed a critical value $\ensuremath{\approx}0.77$. A boundary between stable and unstable fundamental 2D gap solitons is precisely predicted by the Vakhitov-Kolokolov criterion, while all excited states of the 2D solitons, with vorticities (m,1+m) in the two components, m=1,2,..., are unstable. The analysis of the one-dimensional (1D) reduction of the system produces an exact analytical solution for the family of gap solitons which populate the entire band gap, the family being fully stable. Motion of the 1D solitons in the trapping potential is considered too, showing that their effective mass is positive or negative if the cubic nonlinearity is attractive or repulsive, respectively.
We report a diversity of stable gap solitons in a spin-orbit-coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons can be classified by the main physical symmetries they obey, i.e., symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e., spin (C), inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have antiferromagnetic structure and are characterized, respectively, by nonzero and zero total magnetizations.
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.