We study the spontaneous symmetry breaking of dipolar Bose–Einstein condensates trapped in stacks of two-well systems, which may be effectively built as one-dimensional trapping lattices sliced by a repelling laser sheet. If the potential wells are sufficiently deep, the system is modeled by coupled discrete Gross–Pitaevskii equations with nonlocal self- and cross-interaction terms representing dipole–dipole interactions. When the dipoles are not polarized perpendicular or parallel to the lattice, the crossinteraction is asymmetric, replacing the familiar symmetric two-component solitons with a new species of cross-symmetric or -asymmetric ones. The orientation of the dipole moments and the interwell hopping rate strongly affect the shapes of the discrete two-component solitons as well as the characteristics of the cross-symmetry breaking and the associated phase transition. The sub- and super-critical types of cross-symmetry breaking can be controlled by either the hopping rate between the components or the total norm of the solitons. The effect of the interplay between the contact nonlinearity and the dipole angle on the cross-symmetry breaking is also discussed.
We study the spontaneous symmetry breaking of dipolar Bose–Einstein condensates trapped in stacks of two-well systems, which may be effectively built as one-dimensional trapping lattices sliced by a repelling laser sheet. If the potential wells are sufficiently deep, the system is modeled by coupled discrete Gross–Pitaevskii equations with nonlocal self- and cross-interaction terms representing dipole–dipole interactions. When the dipoles are not polarized perpendicular or parallel to the lattice, the crossinteraction is asymmetric, replacing the familiar symmetric two-component solitons with a new species of cross-symmetric or -asymmetric ones. The orientation of the dipole moments and the interwell hopping rate strongly affect the shapes of the discrete two-component solitons as well as the characteristics of the cross-symmetry breaking and the associated phase transition. The sub- and super-critical types of cross-symmetry breaking can be controlled by either the hopping rate between the components or the total norm of the solitons. The effect of the interplay between the contact nonlinearity and the dipole angle on the cross-symmetry breaking is also discussed.
We consider a dipolar Bose-Einstein condensate trapped in an array of two-well systems with an arbitrary orientation of the dipoles relative to the system's axis. The system can be built as a chain of local traps sliced into two parallel lattices by a repelling laser sheet. It is modeled by a pair of coupled discrete Gross-Pitaevskii equations, with dipole-dipole self-interactions and cross interactions. When the dipoles are not polarized perpendicular or parallel to the lattice, the cross interaction is asymmetric, replacing the familiar symmetric two-component discrete solitons by two new species of cross-symmetric ones, viz., on-site- and off-site-centered solitons, which are strongly affected by the orientation of the dipoles and separation between the parallel lattices. A very narrow region of intermediate asymmetric discrete solitons is found at the boundary between the on- and off-site families. Two different types of solitons in the PT-symmetric version of the system are constructed too, and stability areas are identified for them.
We study the effects of spontaneous symmetry breaking (SSB) in solitons composed of a dipolar Bose-Einstein condensate trapped in a dual-core system with dipole-dipole interactions (DDIs) and hopping between the cores. Two realizations of such a matter-wave coupler are introduced: weakly and strongly coupled. The former is based on two parallel pipe-shaped traps, whereas the latter is represented by a single pipe sliced by an external field into parallel layers. The dipoles are oriented along the axes of the pipes. In these systems, the dual-core solitons feature SSB of the supercritical and subcritical types, respectively. Stability regions are identified for symmetric and asymmetric solitons and nonbifurcating antisymmetric solitons, as well as for symmetric flat states, which may also be stable in the strongly coupled system due to competition between the attractive and repulsive intracore and intercore DDIs. The effects of the contact interactions are considered too. Collisions between moving asymmetric solitons in the weakly symmetric system feature an elastic rebound, a merger into a single breather, and passage accompanied by excitation of intrinsic vibrations of the solitons for small, intermediate, and large collision velocities, respectively. A PT-symmetric version of the weakly coupled system is considered briefly, which may be relevant for matter-wave lasers. Stability boundaries for PT-symmetric and -antisymmetric solitons are identified.
We study spontaneous symmetry breaking in a system of two parallel quasi-one-dimensional traps (cores), equipped with optical lattices (OLs) and filled with a Bose-Einstein condensate (BEC). The cores are linearly coupled by tunneling (the model may also be interpreted in terms of spatial solitons in parallel planar optical waveguides with a periodic modulation of the refractive index). Analysis of the corresponding system of linearly coupled Gross-Pitaevskii equations (GPEs) reveals that spectral band gaps of the single GPE split into subgaps. Symmetry breaking in two-component BEC solitons is studied in cases of the attractive (AA) and repulsive (RR) nonlinearity in both traps; the mixed situation, with repulsion in one trap and attraction in the other (RA), is considered too. In all the cases, stable asymmetric solitons are found, bifurcating from symmetric or antisymmetric ones (and destabilizing them), in the AA and RR systems, respectively. In either case, bistability is predicted, with a nonbifurcating stable branch, either antisymmetric or symmetric, coexisting with asymmetric ones. Solitons destabilized by the bifurcation tend to rearrange themselves into their stable asymmetric counterparts. In addition to the fundamental solitons, branches of twisted (odd) solitons in the AA system, and twisted bound states of fundamental solitons in both AA and RR systems, are found too. The impact of a phase mismatch, $\ensuremath{\Delta}$, between the OLs in the two cores is also studied. It is concluded that $\ensuremath{\Delta}=\ensuremath{\pi}∕2$ only mildly deforms the picture, while $\ensuremath{\Delta}=\ensuremath{\pi}$ changes it drastically, replacing the symmetry-breaking bifurcations by pseudobifurcations, with the branch of asymmetric solutions asymptotically approaching its symmetric or antisymmetric counterpart (in the AA and RR system, respectively), rather than splitting off from it. Also considered is a related model, for a binary BEC in a single-core trap with the OL, assuming that the two species (representing different spin states of the same atom) are coupled by linear interconversion. In that case, the symmetry-breaking bifurcations in the AA and RR models switch their character, if the interspecies nonlinear interaction becomes stronger than the intraspecies nonlinearity.
We introduce a wave-propagation model for a pair of linearly coupled two-dimensional (2D) cores with intrinsic cubic nonlinearity—self-focusing (attractive) or self-defocusing (repulsive). Each core carries a periodic potential corresponding to a 1D lattice, the two lattices being mutually perpendicular. Localized states in the model are classified according to their skew-symmetry or skew-antisymmetry, that combine the interchange of the cores and the rotation of coordinates by 90°. Such a setting would be difficult to implement in ordinary optics, but it has a straightforward realization in terms of atom-wave optics, using a Bose–Einstein condensate confined to a pair of tunnel-coupled parallel 'pancake-shaped' traps, which are equipped with crossed 1D optical lattices (an example of a greater versatility of matter-wave optics, as compared to the optics of photons). For the attractive nonlinearity in both cores, symmetry-breaking bifurcations are found, with the help of the variational approximation and, chiefly, by means of numerical methods, in families of skew-symmetric fundamental solitons, dipoles, and vortices. A new species of topologically organized states, which is specific to the dual-core system with crossed 1D lattices, is reported in the form of crosses. They are built as crossed dipole pairs in the two cores, with a phase shift of π/2. Actually, the crosses represent another species of skew-symmetric vortices, different from the ordinary ones. All topologically structured modes in the system exist in pairs: straight and diagonal dipoles, rhombus- and square-shaped vortices (alias on- and off-site ones), and straight and oblique crosses. A noteworthy dynamical feature is the fact that the instability which breaks the skew-symmetry of square- and rhombus-shaped vortices does not lead to their transformation into stable asymmetric counterparts, even though such states are found as stationary solutions. Actually, unstable squares and rhombuses always transform themselves, respectively, into stable crosses of the oblique and straight types. Fundamental solitons and vortices, which do not feature the symmetry-breaking instability, are also found in the first bandgap of the repulsive system, and in the semi-infinite gap of the mixed attractive–repulsive model.
Models of two-dimensional (2D) traps, with double-well structure in the third direction, for Bose-Einstein condensates are introduced with attractive or repulsive interactions between atoms. The models are based on systems of linearly coupled 2D Gross-Pitaevskii equations, where the coupling accounts for tunneling between the wells. Each well carries an optical lattice (OL) (stable 2D solitons cannot exist without OLs). The linear coupling splits each finite band gap in the spectrum of the single-component model into two subgaps. The main subject of the work is spontaneous symmetry breaking (SSB) in two-component 2D solitons and localized vortices (SSB was not considered before in 2D settings). Using variational approximation (VA) and numerical methods, we demonstrate that, in a system with attraction or repulsion, SSB occurs in families of symmetric or antisymmetric solitons (or vortices), respectively. The corresponding bifurcation destabilizes the original solution branch and gives rise to a stable branch of asymmetric solitons or vortices. The VA provides for an accurate description of the emerging branch of asymmetric solitons. In the model with attraction, all stable branches eventually terminate due to the onset of collapse. Stable asymmetric solitons in higher finite band gaps and vortices with a multiple topological charge are found too. The models also give rise to first examples of embedded solitons and embedded vortices (the states located inside Bloch bands) in two dimensions. In the linearly coupled system with opposite signs of the nonlinearity in the two cores, two distinct types of stable solitons and vortices are found, dominated by either the self-attractive component or the self-repulsive one. In the system with a mismatch between the two OLs, a pseudobifurcation is found: when the mismatch attains its largest value $(\ensuremath{\pi})$, the bifurcation does not happen, as branches of different solutions asymptotically approach each other, but fail to merge.
We numerically and analytically investigate the formations and features of two-dimensional discrete Bose–Einstein condensate solitons, which are constructed by quadrupole–quadrupole interactional particles trapped in the tunable anisotropic discrete optical lattices. The square optical lattices in the model can be formed by two pairs of interfering plane waves with different intensities. Two hopping rates of the particles in the orthogonal directions are different, which gives rise to a linear anisotropic system. We find that if all of the pairs of dipole and anti-dipole are perpendicular to the lattice panel and the line connecting the dipole and anti-dipole which compose the quadrupole is parallel to horizontal direction, both the linear anisotropy and the nonlocal nonlinear one can strongly influence the formations of the solitons. There exist three patterns of stable solitons, namely horizontal elongation quasi-one-dimensional discrete solitons, disk-shape isotropic pattern solitons and vertical elongation quasi-continuous solitons. We systematically demonstrate the relationships of chemical potential, size and shape of the soliton with its total norm and vertical hopping rate and analytically reveal the linear dispersion relation for quasi-one-dimensional discrete solitons.