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arXiv:1008.2916 · Physical Review A

Generation of optical and matter-wave solitons in binary systems with a periodically modulated coupling

Armand Niederberger, Boris A. Malomed, Maciej Lewenstein

Published Oct 26, 2010• 12 citations
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Abstract

We present a systematic study of the generation of the array of optical or matter-wave kinks (dark solitons) in the ground state (GS) of binary systems. We consider quasi-one-dimensional systems described by a pair of nonlinear Schr\"odinger (NLSE's) or Gross-Pitaevskii equations (GPE's), which are coupled by the linear mixing, with local strength $\ensuremath{\Omega}$, and by nonlinear interactions. We assume the self-repulsive nonlinearity in both components, and include the effects of a harmonic trapping potential, while the nonlinear interaction between the components may be both repulsive and attractive. The model may be realized in terms of periodically modulated slab waveguides in nonlinear optics and also in Bose-Einstein condensates. Depending on the sign and strengths of the linear and nonlinear couplings between the components, the ground states in such binary systems may be symmetric, antisymmetric, or asymmetric. In this work, we introduce a periodic spatial modulation of the linear coupling, making $\ensuremath{\Omega}$ an odd or even function of the coordinate ($x$). The sign flips of $\ensuremath{\Omega}(x)$ strongly modify the structure of the GS in the binary system, as the relative sign of its components tends to lock to the local sign of $\ensuremath{\Omega}$. Using a systematic numerical analysis and an analytic approximation, we demonstrate that the GS of the trapped system contains one or several kinks (dark solitons) in one component, while the other component does not change its sign. The final results are presented in the form of maps showing the number of kinks in the GS as a function of the system's parameters, with the odd (even) modulation function giving rise to the odd (even) number of the kinks. The modulation of $\ensuremath{\Omega}(x)$ also produces a strong effect on the transition between states with nearly equal and strongly unequal amplitudes of the two components.

DOI
10.1103/physreva.82.043622

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We present a systematic study of the generation of the array of optical or matter-wave kinks (dark solitons) in the ground state (GS) of binary systems. We consider quasi-one-dimensional systems described by a pair of nonlinear Schr"odinger (NLSE's) or Gross-Pitaevskii equations (GPE's), which are coupled by the linear mixing, with local strength \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ, and by nonlinear interactions. We assume the self-repulsive nonlinearity in both components, and include the effects of a harmonic trapping potential, while the nonlinear interaction between the components may be both repulsive and attractive. The model may be realized in terms of periodically modulated slab waveguides in nonlinear optics and also in Bose-Einstein condensates. Depending on the sign and strengths of the linear and nonlinear couplings between the components, the ground states in such binary systems may be symmetric, antisymmetric, or asymmetric. In this work, we introduce a periodic spatial modulation of the linear coupling, making \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ an odd or even function of the coordinate (xxx). The sign flips of \ensuremathΩ(x)\ensuremath{\Omega}(x)\ensuremathΩ(x) strongly modify the structure of the GS in the binary system, as the relative sign of its components tends to lock to the local sign of \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ. Using a systematic numerical analysis and an analytic approximation, we demonstrate that the GS of the trapped system contains one or several kinks (dark solitons) in one component, while the other component does not change its sign. The final results are presented in the form of maps showing the number of kinks in the GS as a function of the system's parameters, with the odd (even) modulation function giving rise to the odd (even) number of the kinks. The modulation of \ensuremathΩ(x)\ensuremath{\Omega}(x)\ensuremathΩ(x) also produces a strong effect on the transition between states with nearly equal and strongly unequal amplitudes of the two components.
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.
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 consider an effectively one-dimensional binary Bose-Einstein condensate (BEC) with nonlinear repulsive interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the presence of the trapping harmonic-oscillator (HO) potential, we report the existence of even, odd, and asymmetric spatial modes. They feature alternating domains with opposite directions of the pseudospin, i.e., antiferromagnetic structures, which is explained by the interplay of the linear couplings, HO confinement, and repulsive self-interaction. The number of the domains is determined by the strength of the SO coupling. The modes are constructed analytically in the weakly nonlinear system. The dynamical stability of the modes is investigated by means of the Bogoliubov--de Gennes equations and direct simulations. A notable result is that the multi-domain-wall (DW) structures are stable, alternating between odd and even shapes, while the simplest single-DW structure is unstable. Thus, the system features a transition to the complex ground states under the action of the SO coupling. The addition of the Zeeman splitting transforms the odd modes into asymmetric ones via spontaneous symmetry breaking. The results suggest possibilities for switching the binary system between states with opposite (pseudo)magnetization by external fields, and realization of similar stable states and dynamical effects in solid-state and nonlinear-optical settings emulated by the SO-coupled BECs.
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.
In the framework of coupled 1D Gross-Pitaevskii equations, we explore the dynamics of a binary Bose-Einstein condensate where the intra-component interaction is repulsive, while the inter-component one is attractive. The existence regimes of stable self-trapped localized states in the form of symbiotic solitons have been analyzed. Imbalanced mixtures, where the number of atoms in one component exceeds the number of atoms in the other component, are considered in parabolic potential and box-like trap. When all the intra-species and inter-species interactions are repulsive, we numerically find a new type of symbiotic solitons resembling dark-bright solitons. A variational approach has been developed which allows us to find the stationary state of the system and frequency of small amplitude dynamics near the equilibrium. It is shown that the strength of inter-component coupling can be retrieved from the frequency of the localized state's vibrations.
We develop a systematic analytical approach to consider the dynamics of linear and nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. We show that, for a condensate strongly confined in two transverse directions, the ground state of the system involves the high-order eigenmodes of the transverse confining potential in the transverse directions and effective high-order Thomas-Fermi wave functions in the axial direction. The linear excitations of the system have a Bogoliubov-type spectrum with the excitation frequency varying slowly along the axial direction. We find that, in a weak nonlinear approximation, the amplitude of a nonlinear excitation is governed by a variable coefficient Korteweg--de Vries equation with additional terms contributed from the transverse structure and the inhomogeneity in the axial direction of the condensate, which results in varying amplitude, width, and velocity for dark solitons. Because of the inhomogeneity the dark solitons undergo deformation and emit radiations when traveling along the axial direction. We finally demonstrate that a dark soliton will disintegrate into several ones plus a residual wave train when passing over a steplike potential.