Dissipative Lattice Research Articles

Dissipative structures in a parametrically driven dissipative lattice: Chimera, localized disorder, continuous-wave, and staggered states

Discrete dissipative coupled systems exhibit complex behavior such as chaos, spatiotemporal intermittence, chimera among others. We construct and investigate chimera states, in the form of confined stationary and dynamical states in a chain of parametrically driven sites with onsite damping and cubic nonlinearity. The system is modeled by the respective discrete parametrically driven damped nonlinear Schrodinger equation. Chimeras feature quasi-periodic or chaotic dynamic in the filled area, quantified by time dependence of the total norm (along with its power spectrum), and by the largest Lyapunov exponent. Systematic numerical simulations, in combination with some analytical results, reveal regions in the parameter space populated by stable localized states of different types. A phase transition from the stationary disorder states to spatially confined dynamical chaotic one is identified. Essential parameters of the system are the strength and detuning of the forcing, as well as the lattice's coupling constant.

Modulational instability and soliton propagation in an alternate right-handed and left-handed multi-coupled nonlinear dissipative transmission network

Abstract We investigate analytically the dynamics of modulated waves in an alternate right-handed and left-handed multi-coupled nonlinear dissipative discrete electrical lattice, made of several of the well-known modified Noguchi electrical transmission network that are transversely coupled to one another by a linear capacitor C 2 . The damped two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) governing slowly modulated wave propagation in this network system is derived. Considering modulated Stokes waves propagating in the network, we establish their criteria of instability, derive the analytical expression of the modulational instability (MI) growth rate, and show that the presence of the dispersive element C S softens the instability of the network system. Through the MI growth rate, we establish that the instability features of our network system show the famous donut Benjamin–Feir instability region. With the help of the linear dispersion law, we show that the behavior of our network model depends on that of the propagating frequency f p . It can adopt purely right-handed (corresponding to monotone increasing f p ), purely left-handed (associated to monotone decreasing f p ) or composite right-/left-handed (for composite monotone increasing and monotone decreasing f p ) behaviors without changing its structure. Also, the network system under consideration supports purely backward (when f p is positive), purely forward (when f p is negative) or composite backward/forward (when f p changes its sign) traveling waves. It appears that the network is right-handed (RH) and/or composite right-/left-handed (RH/LH) and supports backward traveling waves only for very low frequencies and becomes left-handed and supports only forward traveling waves for only high frequencies. Though the exact and approximative soliton-like solutions of the derived CGLE of the network, we investigate analytically RH, LH, as well as RH/LH behaviors of the system and show how to manipulate the transversal wave number to modify the behavior of the network, as well as the width and the motion of the bright solitary voltage signals in the network. We show that our network system supports RH and LH bright and kink soliton signals that simultaneously propagate at the same frequency; we also show that it supports backward and forward LH bright and kink soliton signals which simultaneously propagate at opposite frequency.

Wave redirection, localization, and non-reciprocity in a dissipative nonlinear lattice by macroscopic Landau–Zener tunneling: Theoretical results

We consider an asymmetric dissipative network of two semi-infinite nonlinear lattices with weak linear inter-lattice coupling and study its capacity for passive wave redirection and non-reciprocity. Each lattice is composed of linearly grounded oscillators with essentially nonlinear (i.e., non-linearizable) next-neighbor intra-lattice coupling, and it supports breather propagation. Irreversible breather redirection between lattices is governed by a macroscopic analog of the quantum Landau–Zener tunneling (LZT) effect, whereby impulsive energy initially induced to the “excited lattice” is passively and irreversibly redirected to the “absorbing lattice.” Moreover, this wave redirection is realized only in a specific range of impulse intensity (energy), otherwise motion localization occurs. In this work, we show that LZT breather redirection in the dissipative network occurs only when the normalized linear inter-coupling stiffness is larger than the viscous damping ratio of the individual lattice oscillators, with breather arrest and localization occurring otherwise. Then, through a reduced-order model, we provide guidance for selecting the system parameters of the lattice network for robust breather redirection despite the presence of dissipation. To this end, we study the acoustic non-reciprocity and formulate a quantitative measure for studying it based on measured time-series responses at the four free boundaries of the finite network. Then, we show the dependence of non-reciprocity in this system on the intensity (energy) of the applied impulse. These results pave the way for conceiving practical nonlinear lattice networks with inherent capacities for passive wave redirection and acoustic non-reciprocity that are tunable (self-adaptive) to the applied impulsive excitations.

Traveling fronts in dissipative granular chains and nonlinear lattices

We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

Phase-space methods for simulating the dissipative many-body dynamics of collective spin systems

We describe an efficient numerical method for simulating the dynamics and steady states of collective spin systems in the presence of dephasing and decay. The method is based on the Schwinger boson representation of spin operators and uses an extension of the truncated Wigner approximation to map the exact open system dynamics onto stochastic differential equations for the corresponding phase space distribution. This approach is most effective in the limit of very large spin quantum numbers, where exact numerical simulations and other approximation methods are no longer applicable. We benchmark this numerical technique for known superradiant decay and spin-squeezing processes and illustrate its application for the simulation of non-equilibrium phase transitions in dissipative spin lattice models.

Kolmogorov Entropy of Random Attractors for Stochastic Lattice Systems With White Noise

Based on the asymptotic behavior theory for solutions of dissipative stochastic lattice systems, the Kolmogorov entropy of random attractors in stochastic Klein-Gordon-Schrdinger lattice dynamic systems with white noise was studied with the element decomposition method and the topological properties of polyhedral sphere covering in the finite dimensional space, and an upper bound for the Kolmogorov entropy of random attractors was obtained.

Untying links through anti-parity-time-symmetric coupling

We reveal how the vector field links are untied under the influence of anti-parity-time-symmetric couplings in a dissipative sublattice-symmetric topological photonic crystal lattice. The topology of the quasi-one-dimensional two-band system is encoded in the geometric topology of the vector fields associated with the Bloch Hamiltonian. The linked vector fields reflect the topology of the nontrivial phase. The topological phase transition occurs concomitantly with the untying of the vector field link at the exceptional points. Counterintuitively, more dissipation constructively creates a nontrivial topology. The linking number predicts the number of topological photonic zero modes.

A bi-material concept for periodic dissipative lattices

Structural periodic lattices made of two or more dissimilar materials can be viewed conceptually in terms of a spectrum of structure types, bounded by stretch-dominated lattices of a single material at one end and tensegrities of tensile and compressive struts at the other. The present study probes the unexplored domain within this lattice-tensegrity spectrum with a focus on dissipative bi-material lattices. To begin, a primitive structural motif that exhibits the desired behavior is identified and its compressive response is analyzed. A 2D multi-cell lattice based on the primitive motif is designed and several material variants are fabricated and tested. Analysis of test results addresses effects of finite node dimensions, constraints on strut rotation at the nodes, free edges, and friction with the loading platens as well as limits dictated by rupture of tensile struts or buckling or yielding of compressive struts. The study culminates with guidelines on design of bi-material lattices with high strength and high straining capability.

Bandgap merging and widening of elastic metamaterial with heterogeneous resonator

Much effort has been devoted to exploring broad bandgap in low frequency with limited mass by various structural designs of elastic metamaterials (EMs). In this paper, a heterogeneous resonator configuration in 1-D dissipative lattice mass system without additional increase of mass and coupling complexity is presented to generate multiple bandgaps. Special attention is focused on the effect of parameters control to merge the multiple bandgaps, and the analytical expression of parameters setting is obtained for broad bandgap merging. Specifically, the merging of all bandgaps in EMs with two heterogeneous resonators achieves 30% lower and 135% wider bandgap than that of classical local resonance (LR) EMs with the same mass ratio and stiffness ratio, while the merging of three LR bandgaps in EMs with three heterogeneous resonators is 20% lower and 63% wider. The relative movements of masses in the unit cell at different frequencies are depicted to reveal the working mechanism. The effect of damping factor on broadening vibration attenuation regime is studied. Finally, the vibration transmission properties of finite system with heterogeneous resonators, mass graded resonators and mass detuning resonators are compared to ensure the broad vibration attenuation range. This work could be beneficial for the design of vibration attenuation application in industry.

A nonlinear dissipative elastic metamaterial for broadband wave mitigation

Nonlinearity and dissipation are two important aspects in elastic metamaterials that hold the potential to provide a novel method for the control of elastic waves. Here, we present a nonlinear dissipative elastic metamaterial in a triatomic mass-spring chain to explore the interplay between nonlinearity and dissipation for broadband wave attenuation. In the study, nonlinear stiffness is considered in the spring between the primary and secondary triatomic masses, and a damper is implemented in parallel to the tertiary spring. Various numerical tests on transient wave propagations are conducted, and show that the proposed design with properly selected nonlinear and damping parameters can generate broader wave attenuation regions compared with corresponding linear dissipative metamaterials and nonlinear non-dissipative metamaterials. We quantify the effects of nonlinearity and material damping in higher harmonic wave generations and wave energy absorption using narrow band incidences and demonstrate the application of the nonlinear dissipative triatomic lattice for blast wave mitigations. This work provides a novel approach to design materials capable of suppressing blast-induced shock waves or impact generated pulses that can cause severe local damage to nearby structures.

Disorder-aided pulse stabilization in dissipative synthetic photonic lattices

We consider a discrete time evolution of light in dissipative and disordered photonic lattice presenting a generalization of two popular non-Hermitian models in mathematical literature: Hatano-Nelson and random clock model and suggest a possible experimental implementation using coupled fiber loops. We show that if the model is treated as non-unitary Floquet operator rather than the effective Hamiltonian the combination of controlled photon loss and static phase disorder leads to pulse stabilization in the ring topology. We have also studied the topological invariant associated with the system and found additional evidence for the absence of Anderson transition.

Breather arrest, localization, and acoustic non-reciprocity in dissipative nonlinear lattices.

The effect of on-site damping on breather arrest, localization, and non-reciprocity in strongly nonlinear lattices is analytically and numerically studied. Breathers are localized oscillatory wavepackets formed by nonlinearity and dispersion. Breather arrest refers to breather disintegration over a finite "penetration depth" in a dissipative lattice. First, a simplified system of two nonlinearly coupled oscillators under impulsive excitation is considered. The exact relation between the number of beats (energy exchanges between oscillators), the excitation magnitude, and the on-site damping is derived. Then, these analytical results are correlated to those of the semi-infinite extension of the simplified system, where breather penetration depth is governed by a similar law to that of the finite beats in the simplified system. Finally, motivated by the experimental results of Bunyan, Moore, Mojahed, Fronk, Leamy, Tawfick, and Vakakis [Phys. Rev. E 97, 052211 (2018)], breather arrest, localization, and acoustic non-reciprocity in a non-symmetric, dissipative, strongly nonlinear lattice are studied. The lattice consists of repetitive cells of linearly grounded large-scale particles nonlinearly coupled to small-scale ones, and linear intra-cell coupling. Non-reciprocity in this lattice yields either energy localization or breather arrest depending on the position of excitation. The nonlinear acoustics governing non-reciprocity, and the surprising effects of existence of linear components in the coupling nonlinear stiffnesses, in the acoustics, are investigated.

Neural-Network Approach to Dissipative Quantum Many-Body Dynamics.

In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational MonteCarlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.

Experimental and numerical observation of dark and bright breathers in the band gap of a diatomic electrical lattice.

We observe dark and bright intrinsic localized modes (ILMs), also known as discrete breathers, experimentally and numerically in a diatomic-like electrical lattice. The experimental generation of dark ILMs by driving a dissipative lattice with spatially homogenous amplitude is, to our knowledge, unprecedented. In addition, the experimental manifestation of bright breathers within the band gap is also novel in this system. In experimental measurements the dark modes appear just below the bottom of the top branch in frequency. As the frequency is then lowered further into the band gap, the dark ILMs persist, until the nonlinear localization pattern reverses and bright ILMs appear on top of the finite background. Deep into the band gap, only a single bright structure survives in a lattice of 32 nodes. The vicinity of the bottom band also features bright and dark self-localized excitations. These results pave the way for a more systematic study of dark breathers and their bifurcations in diatomic-like chains.

Simultaneous Control of Multispecies Particle Transport and Segregation in Driven Lattices.

We provide a generic scheme to separate the particles of a mixture by their physical properties like mass, friction, or size. The scheme employs a periodically shaken two-dimensional dissipative lattice and hinges on a simultaneous transport of particles in species-specific directions. This selective transport is achieved by controlling the late-time nonlinear particle dynamics, via the attractors embedded in the phase space and their bifurcations. To illustrate the spectrum of possible applications of the scheme, we exemplarily demonstrate the separation of polydisperse colloids and mixtures of cold thermal alkali atoms in optical lattices.

Fully dissipative relativistic lattice Boltzmann method in two dimensions

In this paper, we develop and characterize the fully dissipative Lattice Boltzmann method for ultra-relativistic fluids in two dimensions using three equilibrium distribution functions: Maxwell–Jüttner, Fermi–Dirac and Bose–Einstein. Our results stem from the expansion of these distribution functions up to fifth order in relativistic polynomials. We also obtain new Gaussian quadratures for square lattices that preserve the spatial resolution. Our models are validated with the Riemann problem and the limitations of lower order expansions to calculate higher order moments are shown. The kinematic viscosity and the thermal conductivity are numerically obtained using the Taylor–Green vortex and the Fourier flow, respectively, and these transport coefficients are compared with the theoretical prediction from Grad’s theory. In order to compare different expansion orders, we analyze the temperature and heat flux fields on the time evolution of a hot spot.

Dynamical properties of dissipative XYZ Heisenberg lattices

We study dynamical properties of dissipative XYZ Heisenberg lattices where anisotropic spin-spin coupling competes with local incoherent spin flip processes. In particular, we explore a region of the parameter space where dissipative magnetic phase transitions for the steady state have been recently predicted by mean-field theories and exact numerical methods. We investigate the asymptotic decay rate towards the steady state both in 1D (up to the thermodynamical limit) and in finite-size 2D lattices, showing that critical dynamics does not occur in 1D, but it can emerge in 2D. We also analyze the behavior of individual homodyne quantum trajectories, which reveal the nature of the transition.

Exploring photonic topological insulator states in a circuit-QED lattice

We propose a simple protocol to explore the topological properties of photonic integer quantum Hall states in a one-dimensional circiut-QED lattice. By periodically modulating the on-site photonic energies in such a lattice, we demonstrate that this one-dimensional lattice model can be mapped into a two-dimensional integer quantum Hall insulator model. Based on the lattice-based cavity input–output theory, we show that both the photonic topological protected edge states and topological invariants can be clearly measured from the final steady state of the resonator lattice after taking into account cavity dissipation. Interestingly, we also find that the measurement signals associated with the above topological features are quite unambitious even in five coupled dissipative resonators. Our work opens up a new prospect of exploring topological states with a small-size dissipative quantum artificial lattice, which is quite attractive to the current quantum optics community.

Steady States of Infinite-Size Dissipative Quantum Chains via Imaginary Time Evolution.

Directly in the thermodynamic limit, we show how to combine local imaginary and real-time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESSs) of one-dimensional dissipative quantum lattices governed by a local Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the real-time evolution trajectory by directly converging to the weakly correlated subspace of the NESS, after which, real-time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the dissipative transverse field quantum Ising chain. We show that a crossover of an order parameter shown to be smooth in previous finite-size studies remains smooth in the thermodynamic limit.

Critical behavior of dissipative two-dimensional spin lattices

We explore critical properties of two-dimensional lattices of spins interacting via an anisotropic Heisenberg Hamiltonian and subject to incoherent spin flips. We determine the steady-state solution of the master equation for the density matrix via the corner-space renormalization method. We investigate the finite-size scaling and critical exponent of the magnetic linear susceptibility associated to a dissipative ferromagnetic transition. We show that the Von Neumann entropy increases across the critical point, revealing a strongly mixed character of the ferromagnetic phase. Entanglement is witnessed by the quantum Fisher information which exhibits a critical behavior at the transition point, showing that quantum correlations play a crucial role in the transition even though the system is in a mixed state.