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.
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 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 investigate two-dimensional (2D) states in Bose-Einstein condensates with self-attraction or self-repulsion, trapped in an axially symmetric optical-lattice potential periodic along the radius. The states trapped both in the central potential well and in remote circular troughs are studied. In the repulsive mode, a new soliton species is found, in the form of radial gap solitons. The latter solitons are completely stable if they carry zero vorticity (l=0) , while with l not equal 0 they develop a weak azimuthal modulation, which makes them rotating patterns, that persist indefinitely long. In addition, annular gap solitons may support stable azimuthal dark-soliton pairs on their crests. In remote troughs of the attractive model, stable localized states may assume a ringlike shape with weak azimuthal modulation, or shrink into solitons strongly localized in the azimuthal direction, which is explained in the framework of an averaged 1D equation with the cyclic azimuthal coordinate. Numerical simulations of the attractive model also reveal stable necklacelike patterns, built of several strongly localized peaks. Dynamics of strongly localized solitons circulating in the troughs is studied too. While the solitons with sufficiently small velocities are completely stable, fast solitons gradually decay, due to the leakage of matter into the adjacent trough, under the action of the centrifugal force. Investigation of head-on collisions between strongly localized solitons traveling in circular troughs shows that collisions between in-phase solitons in a common trough lead to collapse, while pi-out-of-phase solitons bounce many times, but eventually merge into a single one, without collapsing. In-phase solitons colliding in adjacent circular troughs also tend to merge into a single soliton.
We report on results of a systematic analysis of two-dimensional solitons and localized vortices in models including a rotating periodic potential and the cubic nonlinearity, with the latter being both self-attractive and self-repulsive. The models apply to Bose-Einstein condensates stirred by rotating optical lattices and to twisted photonic-crystal fibers, or bundled arrays of waveguides, in nonlinear optics. In the case of the attractive nonlinearity, we construct compound states in the form of vortices, quadrupoles, and supervortices, all trapped in the slowly rotating lattice, and identify their stability limits (fundamental solitons in this setting were studied previously). In rapidly rotating potentials, vortices decouple from the lattice in the azimuthal direction and assume an annular shape. In the model with the repulsive nonlinearity, which was not previously explored in this setting, gap solitons and vortices are found in both cases of the slow and rapid rotations. It is again concluded that the increase in the rotation frequency leads to the transition from fully trapped corotating vortices to ring-shaped ones. We also study ``crater-shaped'' vortices in the attraction model, which, unlike their compound counterparts, are trapped, essentially, in one cell of the lattice. Previously, only unstable vortices of this type were reported. We demonstrate that they have a certain stability region. Solitons and vortices are found here in the numerical form, and, in parallel, by means of the variational approximation.
We study stationary states of a two-dimensional (2D) Bose-Einstein condensate with both attractive and repulsive nonlinearities in a combination of a double-square-well potential in one direction and a perpendicular optical lattice. We look for dual-core solitons in this configuration, focusing on their symmetry-breaking bifurcations. For attractive interactions, without the lattice, a similar analysis was performed [M. Matuszewski et al., Phys. Rev. A 75, 063621 (2007)], where subcritical bifurcation transforming antisymmetric gap solitons into asymmetric ones was found. Here we focus on the effect of an optical lattice and the so created gap solitons. We discover that a phase transition occurs when the lattice depth increases and the additional dimension becomes strongly suppressed. The bifurcation type changes from subcritical (typical for a 2D system, with hysteresis) to supercritical (typical for a one-dimensional system). An additional advantage of the lattice is that gap solitons exist even for repulsive interactions. In this case we also discover bifurcation of a supercritical type. The analysis is based on a variational approximation, which is surprisingly well verified by numerical results.
We introduce two- and one-dimensional (2D and 1D) systems of two linearly coupled Gross-Pitaevskii equations (GPEs) with the cubic self-attraction and harmonic-oscillator (HO) trapping potential in each GPE. The system models a Bose-Einstein condensate with a negative scattering length, loaded in a double-pancake trap, combined with the in-plane HO potential. In addition to that, the 1D version applies to the light transmission in a dual-core waveguide with the Kerr nonlinearity and in-core confinement represented by the HO potential. The subject of the analysis is spontaneous symmetry breaking in 2D and 1D ground-state (GS, alias fundamental) modes, as well as in 2D vortices and 1D dipole modes. (The latter ones do not exist without the HO potential.) By means of the variational approximation and numerical analysis, it is found that both the 2D and 1D systems give rise to a symmetry-breaking bifurcation (SBB) of the supercritical type. The stability of symmetric and asymmetric states, produced by the SBB, is analyzed through the computation of eigenvalues for perturbation modes and verified by direct simulations. The asymmetric GSs are always stable, while the stability region for vortices shrinks and eventually disappears with the increase of the linear-coupling constant, $\ensuremath{\kappa}$. The SBB in the 2D system does not occur if $\ensuremath{\kappa}$ is too large (at $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{max}$); in that case, the two-component system behaves, essentially, as its single-component counterpart. In the 1D system, both asymmetric and symmetric dipole modes feature an additional oscillatory instability, unrelated to the symmetry breaking. This instability occurs in several regions which expand with the increase of $\ensuremath{\kappa}$.