We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.
We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.
We consider a two-component linearly-coupled system with the intrinsic cubic nonlinearity and the harmonic-oscillator (HO) confining potential. The system models binary settings in BEC and optics. In the symmetric system, with the HO trap acting in both components, we consider Josephson oscillations (JO) initiated by an input in the form of the HO's ground state (GS) or dipole mode (DM), placed in one component. With the increase of the strength of the self-focusing nonlinearity, spontaneous symmetry breaking (SSB) between the components takes place in the dynamical JO state. Under still stronger nonlinearity, the regular JO initiated by the GS input carry over into a chaotic dynamical state. For the DM input, the chaotization happens at smaller powers than for the GS, which is followed by SSB at a slightly stronger nonlinearity. In the system with the defocusing nonlinearity, SSB does not take place, and dynamical chaos occurs in a small area of the parameter space. In the asymmetric half-trapped system, with the HO potential applied to a single component, we first focus on the spectrum of confined binary modes in the linearized system. The spectrum is found analytically in the limits of weak and strong inter-component coupling, and numerically in the general case. Under the action of the coupling, the existence region of the confined modes shrinks for GSs and expands for DMs. In the full nonlinear system, the existence region for confined modes is identified in the numerical form. They are constructed too by means of the Thomas-Fermi approximation, in the case of the defocusing nonlinearity. Lastly, particular (non-generic) exact analytical solutions for confined modes, including vortices, in one- and two-dimensional asymmetric linearized systems are found. They represent bound states in the continuum.
We consider a two-component linearly coupled system with the intrinsic cubic nonlinearity and the harmonic-oscillator (HO) confining potential. The system models binary settings in BEC and optics. In the symmetric system, with the HO trap acting in both components, we consider Josephson oscillations (JO) initiated by an input in the form of the HO’s ground state (GS) or dipole mode (DM), placed in one component. With the increase of the strength of the self-focusing nonlinearity, spontaneous symmetry breaking (SSB) between the components takes place in the dynamical JO state. Under still stronger nonlinearity, the regular JO initiated by the GS input carries over into a chaotic dynamical state. For the DM input, the chaotization happens at smaller powers than for the GS, which is followed by SSB at a slightly stronger nonlinearity. In the system with the defocusing nonlinearity, SSB does not take place, and dynamical chaos occurs in a small area of the parameter space. In the asymmetric half-trapped system, with the HO potential applied to a single component, we first focus on the spectrum of confined binary modes in the linearized system. The spectrum is found analytically in the limits of weak and strong inter-component coupling, and numerically in the general case. Under the action of the coupling, the existence region of the confined modes shrinks for GSs and expands for DMs. In the full nonlinear system, the existence region for confined modes is identified in the numerical form. They are constructed too by means of the Thomas–Fermi approximation, in the case of the defocusing nonlinearity. Lastly, particular (non-generic) exact analytical solutions for confined modes, including vortices, in one- and two-dimensional asymmetric linearized systems are found. They represent bound states in the continuum.
On the contrary to the common intuition, which suggests that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schroedinger equations, which include expulsive potentials that are steeper than the quadratic ones, give rise to normalizable eigenstates, which may be considered as a manifestation of effective self-trapping in the linear system. These states constitute full continuous spectra in both the 1D and 2D cases. In 1D, they are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic expressions for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with full numerical solutions. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are briefly considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability
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 $\ensuremath{\kappa}$.
The trapping potential is a necessary ingredient of any setup used in experiments with atomic gases in the BEC state (otherwise, the gas would escape). Similarly, the presence of an effective trap is necessary for building various optical and photonic cavities. Therefore, the analysis of models combining the intrinsic nonlinearity acting in multidimensional settings with the trapping potential is a natural step in the theoretical work. The present chapter aims to summarize basic theoretical results produced by the work in this direction. The results are presented for 2D fundamental (zero-voticity) and vortical states trapped in the HO potential and for similar 3D states trapped in the HO potential, which may be anisotropic. Also considered are two-component 2D states with hidden vorticity, i.e., with opposite vorticities if the two components have equal norms. Stability regions are identified for all these settings. The chapter additionally reports findings for two-component 2D fundamental and vortex states in a linearly coupled system, in which the potential acting on one component is trapping, while the other component is subjected to the action of an expulsive potential. As for experimental results demonstrating the creation of 2D or 3D solitons in trapping potentials, very few of them are available. A recent experimental finding that may be relevant in this respect is the creation of quasi-stable effectively two-dimensional Townes solitons in atomic BEC loaded in a trapping potential [Bakkali-Hassani et al., Phys. Rev. Lett. 127, 023603 (2021)].
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.