On-site Nonlinearity Research Articles

Nonlinear switch and spatial lattice solitons of photonic s-p orbitals.

We develop fs laser-fabricated asymmetric couplers and zig-zag arrays consisting of single- and two-mode waveguides with bipartite Kerr nonlinearity in borosilicate (BK7) glass substrates. The fundamental mode (s orbital) is near resonance with the neighboring higher-order p orbital, causing efficient light transfer at low power. Due to Kerr nonlinearity, the coupler works as an all-optical switch between s and p orbitals. Single- and double-peak spatial lattice solitons of s-p orbitals form due to the bipartite nature of the on-site nonlinearity. We probe highly localized bulk and edge solitons, peaked at the s and p orbitals, spectrally residing in the photonic bandgap. Our work will be important for exploring inter-orbital couplings and nonlinear interactions in intricate photonic devices.

Breathing discrete nonlinear Schrödinger vortices

Breathing discrete vortices are obtained as numerically exact and generally quasiperiodic, localized solutions to the discrete nonlinear Schrödinger equation with cubic (Kerr) on-site nonlinearity, on a two-dimensional square lattice with nearest-neighbor couplings. We identify and analyze three different types of solutions characterized by circulating currents and time-periodically oscillating intensity distributions, two of which have been discussed in earlier works while the third being, to our knowledge, presented here for the first time. (i) A vortex-breather, constructed from the anticontinuous limit as a superposition of a single-site breather and a discrete vortex surrounding it, where the breather and vortex are oscillating at different frequencies. (ii) A charge-flipping vortex, constructed from an anticontinuous solution with an even number of sites on a closed loop, with alternating sites oscillating at different frequencies. (iii) A breathing vortex, constructed by continuation of a non-resonating linear internal eigenmode of a stationary discrete vortex. We illustrate by examples, using numerical Floquet analysis for solutions obtained from a Newton scheme, that linearly stable solutions exist from all three categories, at sufficiently strong discreteness.

Solitary waves in electro-mechanical lattices

We introduce and study both analytically and numerically a class of microelectromechanical chains aiming to turn them into transmission lines of solitons. Mathematically, their analysis reduces to the study of a spatially one-dimensional nonlinear Klein–Gordon equation with a model dependent onsite nonlinearity induced by the electrical forces. Since the basic solitons appear to be unstable for most of the force regimes, we introduce a stabilizing algorithm and demonstrate that it enables a stable and persisting propagation of solitons. Among other fascinating nonlinear formations induced by the presented models, we mention the “meson”: a stable square shaped pulse with sharp fronts that expands with a sonic speed, and “flatons”: flat-top solitons of arbitrary width.

Wave-packet spreading in the disordered and nonlinear Su-Schrieffer-Heeger chain

We numerically investigate the characteristics of the long-time dynamics of a single-site wave-packet excitation in a disordered and nonlinear Su-Schrieffer-Heeger model. In the linear regime, as the parameters controlling the topology of the system are varied, we show that the transition between two different topological phases is preceded by an anomalous diffusion, in contrast to Anderson localization within these topological phases. In the presence of onsite nonlinearity this feature is lost due to mode-mode interactions. Direct numerical simulations reveal that the characteristics of the asymptotic nonlinear wave-packet spreading are the same across the whole studied parameter space. Our findings underline the importance of mode-mode interactions in nonlinear topological systems, which must be studied in order to define reliable nonlinear topological markers.

Nonlinear bandgap transmission with zero frequency in a cross-stitch lattice

We consider a model for a cross-stitch lattice with onsite nonlinearity. The linear analysis and the determination of the homoclinic threshold for this model are carried out theoretically. In the case of a self-focusing nonlinearity, we show that the traveling bandgap soliton is possible as a result from the periodic excitation of the edge of the chains. Contrary to the usual supratransmission phenomenon, the generation of traveling solitons is possible by driving the lattice with zero frequency and constant amplitude. In the case of defocusing nonlinearities, the heteroclinic orbit is obtained with the frequency within the phonon band. By exciting one component of complex waves, a traveling phonon kink is obtained. Meanwhile, in the case of the driving of two complex waves with zero phonon and nonzero phonon frequencies, respectively, the fly phonon breather is observed. The collision of the waves from the flat band and phonon band gives the traveling bright carry by the traveling kink. These results are obtained through computer simulations.

Topological edge breathers in a nonlinear Su-Schrieffer-Heeger lattice

We show the existence of breathing edge modes in the Su-Schrieffer-Heeger model with cubic (Kerr) on-site nonlinearity, bifurcating from stationary edge solitons with propagation constant inside the topological gap of the linear model. These edge breathers are exact solutions to the nonlinear equations of motion, with time-periodic intensity oscillations and tails exponentially decaying from the edge. They bifurcate from two localized internal eigenmodes of the stationary edge soliton, having eigenfrequencies inside the topological gap and all higher harmonics above the linear spectrum. Numerical Floquet analysis for solutions obtained from a Newton scheme shows that edge breathers may be linearly stable even in regimes of large-amplitude oscillations, mainly manifested as time-periodic power exchange between the edge site and its next-nearest neighbor.

Topological characteristics of gap closing points in nonlinear Weyl semimetals

In this work we explore the effects of nonlinearity on three-dimensional topological phases. Of particular interest are the so-called Weyl semimetals, known for their Weyl nodes, i.e., point-like topological charges which always exist in pairs and demonstrate remarkable robustness against general perturbations. It is found that the presence of onsite nonlinearity causes each of these Weyl nodes to break down into nodal lines and nodal surfaces at two different energies while preserving its topological charge. Depending on the system considered, additional nodal lines may further emerge at high nonlinearity strength. We propose two different ways to probe the observed nodal structures. First, the use of an adiabatic pumping process allows the detection of the nodal lines and surfaces arising from the original Weyl nodes. Second, an Aharonov-Bohm interference experiment is particularly fruitful to capture additional nodal lines that emerge at high nonlinearity.

Localization in a non-Hermitian flat band lattice with nonlinearity

In this article, we aim to report the competing role of non-Hermiticity and nonlinearity introduced in the flat band lattice. For this, we initially examine the dynamics of a non-Hermitian single chain diamond network and show a pure imaginary flat band between the complex dispersive bands. The presence of a pure imaginary flat band leads to light propagation either with amplification or attenuation in a confined manner. With this information, we shift to exploiting the non-Hermitian dual chain diamond lattice which shows PT-symmetric nature in a particular situation. In the linear case, the model has two flat bands in its spectrum, whereas its nature depends on the system parameters. This gives rise to the distinct nature of localized light propagation in the system with controllable intensity. Interestingly, in the case of a pure real flat band with broad-site excitation, large-site stable oscillations appear in the system and exist for a long propagation distance. Finally, we study the dual chain lattice with on-site cubic nonlinearity which shows transient localized propagation for initially localized excitation and stationary kovaton like localization for broad-site excitation. Our results reveal the possibility of controlling transport dynamics in non-Hermitian systems with nonlinearity. Our system also acts as a candidate for optical applications, as in the long-distance diffraction-less transmission of information.

Stabilization of Multimode Schrödinger Cat States Via Normal-Mode Dissipation Engineering

Non-Gaussian quantum states have been deterministically prepared and autonomously stabilized in single- and two-mode circuit quantum electrodynamics architectures via engineered dissipation. However, it is currently unknown how to scale up this technique to multi-mode non-Gaussian systems. Here, we upgrade dissipation engineering to collective (normal) modes of nonlinear resonator arrays and show how to stabilize multi-mode Schrodinger cat states. These states are multi-photon and multi-mode quantum superpositions of coherent states in a single normal mode delocalized over an arbitrary number of cavities. We consider tailored dissipative coupling between resonators that are parametrically driven and feature an on-site nonlinearity, which is either a Kerr-type nonlinearity or an engineered two-photon loss. For both types of nonlinearity, we find the same exact closed-form solutions for the two-dimensional steady-state manifold spanned by superpositions of multi-mode Schrodinger cat states. We further show that, in the Zeno limit of strong dissipative coupling, the even parity multi-mode cat state can be deterministically prepared from the vacuum. Remarkably, engineered two-photon loss gives rise to a fast relaxation towards the steady state, protecting the state preparation against decoherence due to intrinsic single-photon losses and imperfections in tailored dissipative coupling, which sets in at longer times. The relaxation time is independent of system size making the state preparation scalable. Multi-mode cat states are naturally endowed with a noise bias that increases exponentially with system size and can thus be exploited for enhanced robust encoding of quantum information.

Thermalization in the one-dimensional Salerno model lattice.

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

Nonlinearity induced topological physics in momentum space and real space

Nonlinearity induced topological properties in nonlinear lattice systems are studied in both momentum space and real space. Experimentally realizable through the Kerr effect on photonic waveguide systems, our working model depicts on-site nonlinearity added to the Su-Schrieffer-Heeger (SSH) model plus a chiral-symmetry breaking term. Under the periodic-boundary condition, two of the nonlinear energy bands approach the energy bands of the chiral-symmetric SSH model as nonlinearity strength increases. Further, we account for a correction to the Zak phase and obtain a general expression for nonlinear Zak phases. For sufficiently strong nonlinearity, the sum of all nonlinear Zak phases (not the sum of all conventional Zak phases) is found to be quantized. In real space, it is discovered that there is a strong interplay between nonlinear solitons and the topologically protected edge states of the associated chiral-symmetric linear system. Nonlinearity can recover the degeneracy between two edge soliton states, albeit a chiral-symmetry breaking term. We also reveal the topological origin of in-gap solitons even when the associated linear system is in the topological trivial regime. These momentum-space and real-space results have clearly demonstrated new topological features induced by nonlinearity, indicating that topological physics in nonlinear lattice systems is far richer than previously thought.

Multichannel asymmetric transmission through a dimer defect with saturable inter-site nonlinearity

We consider the asymmetric transmission properties of a discrete nonlinear Schrödinger type dimer with a saturable nonlinear intersite coupling between the dimer sites, in addition to a cubic onsite nonlinearity and asymmetric linear onsite potentials. In contrast to previously studied cases with pure onsite nonlinearities, the transmission coefficient for stationary transmission is shown to be a multivalued function of the transmitted intensity, in regimes of low saturability and small or moderate transmitted intensity. The corresponding backward transfer map is analyzed analytically and numerically, and shown to have either one or three distinct solutions for saturable coupling, and zero or two solutions for the purely cubic nonlinear coupling. As saturation strength is increased, the multi-solution regimes disappear through bifurcations and the single-valued regime of the onsite model is recovered. The existence of multiple solution branches yields novel mechanisms for asymmetric left/right stationary transmission: in addition to shifts of the positions of transmission peaks, peaks for transmission in one direction may correspond to nonexistence of stationary solutions propagating in the opposite direction, at the corresponding branch and transmitted intensity. Moreover, one of these transmission channels behaves as a nearly perfect mirror for incoming signals. The linear stability of the stationary solutions is analyzed, and instabilities are typically observed, and illustrated by direct numerical simulations, in regimes with large transmission coefficient. Intersite nonlinearities are found to prevent the formation of a localized dimer mode in the instability-induced dynamics. Finally, the partial reflection and transmission of a Gaussian excitation is studied, and the asymmetric transmission properties are compared to previously studied onsite models.

Hybrid matter-wave - microwave solitons on the lattice

We introduce a two-component system which models a pseudospinor Bose–Einstein condensate (BEC), with a microwave field coupling its two components. The feedback of BEC onto the field (the local-field effect) is taken into account by dint of the respective Poisson equation, which is solved using the Green’s function. This gives rise to an effective long-range self-trapping interaction, which may act alone, or be combined with the contact cubic nonlinearity. The system is made discrete by loading the BEC into a deep optical-lattice potential. Numerical solutions of the discrete system demonstrate that onsite-centered fundamental solitons are stable in the cases of attractive or zero contact interactions, while offsite-centered solitons are unstable. In the case of the repulsive onsite nonlinearity, offsite solitons are stable, while their onsite-centered counterparts are stable only at sufficiently small values of the norm, where bistability between the off- and onsite-centered mode takes place. The shape of the onsite-centered solitons is very accurately predicted by a variational approximation (which includes essential technical novelties), and their super-exponentially decaying tails are found by means of direct analytical consideration. Spatially-antisymmetric (“twisted”) solitons are stable at small values of the norm, being unstable at larger norms. In the strongly asymmetric version of the two-component system, which includes the Zeeman splitting, the system is reduced to a single discrete Gross–Pitaevskii equation, by eliminating the small higher-energy component.

Breathers and other time-periodic solutions in an array of cantilevers decorated with magnets

In this article, the existence, stability and bifurcation structure of time-periodic solutions (including ones that also have the property of spatial localization, i.e., breathers) are studied in an array of cantilevers that have magnetic tips. The repelling magnetic tips are responsible for the intersite nonlinearity of the system, whereas the cantilevers are responsible for the onsite (potentially nonlinear) force. The relevant model is of the mixed Fermi-Pasta-Ulam-Tsingou and Klein-Gordon type with both damping and driving. In the case of base excitation, we provide experimental results to validate the model. In particular, we identify regions of bistability in the model and in the experiment, which agree with minimal tuning of the system parameters. We carry out additional numerical explorations in order to contrast the base excitation problem with the boundary excitation problem and the problem with a single mass defect. We find that the base excitation problem is more stable than the boundary excitation problem and that breathers are possible in the defect system. The effect of an onsite nonlinearity is also considered, where it is shown that bistability is possible for both softening and hardening cubic nonlinearities.

Electrical Tuning of Nonlinearities in Exciton-Polariton Condensates.

A primary limitation of the intensively researched polaritonic systems compared to their atomic counterparts for the study of strongly correlated phenomena and many-body physics is their relatively weak two-particle interactions compared to disorder. Here, we show how new opportunities to enhance such on-site interactions and nonlinearities arise by tuning the exciton-polariton dipole moment in electrically biased semiconductor microcavities incorporating wide quantum wells. The applied field results in a twofold enhancement of exciton-exciton interactions as well as more efficiently driving relaxation towards low energy polariton states, thus, reducing condensation threshold.

Renormalized vibrations and normal energy transport in 1d FPU-like discrete nonlinear Schr\xf6dinger equations

For one-dimensional (1d) nonlinear atomic lattices, the models with on-site nonlinearities such as the Frenkel-Kontorova (FK) and ϕ4 lattices have normal energy transport while the models with inter-site nonlinearities such as the Fermi-Pasta-Ulam-β (FPU-β) lattice exhibit anomalous energy transport. The 1d Discrete Nonlinear Schrödinger (DNLS) equations with on-site nonlinearities has been previously studied and normal energy transport has also been found. Here, we investigate the energy transport of 1d FPU-like DNLS equations with inter-site nonlinearities. Extended from the FPU-β lattice, the renormalized vibration theory is developed for the FPU-like DNLS models and the predicted renormalized vibrations are verified by direct numerical simulations same as the FPU-β lattice. However, the energy diffusion processes are explored and normal energy transport is observed for the 1d FPU-like DNLS models, which is different from their atomic lattice counterpart of FPU-β lattice. The reason might be that, unlike nonlinear atomic lattices where models with on-site nonlinearities have one less conserved quantities than the models with inter-site nonlinearities, the DNLS models with on-site or inter-site nonlinearities have the same number of conserved quantities as the result of gauge transformation.

Localized gap modes in nonlinear dimerized Lieb lattices

Compact localized modes of ring type exist in many two-dimensional lattices with a flat linear band, such as the Lieb lattice. The uniform Lieb lattice is gapless, but gaps surrounding the flat band can be induced by various types of bond alternations (dimerizations) without destroying the compact linear eigenmodes. Here, we investigate the conditions under which such diffractionless modes can be formed and propagated also in the presence of a cubic on-site (Kerr) nonlinearity. For the simplest type of dimerization with a three-site unit cell, nonlinearity destroys the exact compactness, but strongly localized modes with frequencies inside the gap are still found to propagate stably for certain regimes of system parameters. By contrast, introducing a dimerization with a 12-site unit cell, compact (diffractionless) gap modes are found to exist as exact nonlinear solutions in continuation of flat band linear eigenmodes. These modes appear to be generally weakly unstable, but dynamical simulations show parameter regimes where localization would persist for propagation lengths much larger than the size of typical experimental waveguide array configurations. Our findings represent an attempt to realize conditions for full control of light propagation in photonic environments.

Single and double linear and nonlinear flatband chains: Spectra and modes.

We report results of systematic analysis of various modes in the flatband lattice, based on the diamond-chain model with the on-site cubic nonlinearity, and its double version with the linear on-site mixing between the two lattice fields. In the single-chain system, a full analysis is presented, first, for the single nonlinear cell, making it possible to find all stationary states, viz., antisymmetric, symmetric, and asymmetric ones, including an exactly investigated symmetry-breaking bifurcation of the subcritical type. In the nonlinear infinite single-component chain, compact localized states (CLSs) are found in an exact form too, as an extension of known compact eigenstates of the linear diamond chain. Their stability is studied by means of analytical and numerical methods, revealing a nontrivial stability boundary. In addition to the CLSs, various species of extended states and exponentially localized lattice solitons of symmetric and asymmetric types are also studied, by means of numerical calculations and variational approximation. As a result, existence and stability areas are identified for these modes. Finally, the linear version of the double diamond chain is solved in an exact form, producing two split flatbands in the system's spectrum.

Interaction-induced two-photon edge states in an extended Hubbard model realized in a cavity array

We study theoretically two-photon states in a periodic array of coupled cavities with both on-site and nonlocal Kerr-type nonlinearities. In the absence of nonlinearity the structure is topologically trivial and possesses no edge states. The interplay of two nonlinear interaction mechanisms described by the extended Hubbard model facilitates formation of edge states of bound photon pairs. Numerical and exact analytical results for the two-photon wave functions are presented. Our findings thus shed light onto the edge states of composite particles and their localization properties.

Nonlinear localized flat-band modes with spin-orbit coupling

We report the coexistence and properties of stable compact localized states (CLSs) and discrete solitons (DSs) for nonlinear spinor waves on a flatband network with spin-orbit coupling (SOC). The system can be implemented by means of a binary Bose-Einstein condensate loaded in the corresponding optical lattice. In the linear limit, the SOC opens a minigap between flat and dispersive bands in the system's bandgap structure, and preserves the existence of CLSs at the flatband frequency, simultaneously lowering their symmetry. Adding onsite cubic nonlinearity, the CLSs persist and remain available in an exact analytical form, with frequencies which are smoothly tuned into the minigap. Inside of the minigap, the CLS and DS families are stable in narrow areas adjacent to the FB. Deep inside the semi-infinite gap, both the CLSs and DSs are stable too.