Critical state generators from perturbed flatbands.

One-dimensional all-bands-flat lattices are networks with all bands being flat and highly degenerate. They can always be diagonalized by a finite sequence of local unitary transformations parameterized by a set of angles θi. In a previous work, we demonstrated that quasiperiodic perturbations of a specific one-dimensional all-bands-flat lattice give rise to a critical-to-insulator transition and fractality edges separating critical from localized states. In this study, we generalize these studies and results to the entire manifold of all-bands-flat models and study the effect of the quasiperiodic perturbation on the entire manifold. For weak perturbation, we derive an effective Hamiltonian and we identify the sets of manifold parameters for which the effective model maps to extended or off diagonal Harper models and hosts critical states. For all the other parameter values, the spectrum is localized. Upon increasing the perturbation strength, the extended Harper model evolves into a system with energy dependent critical-to-insulator transitions, which we dub fractality edges. Additionally, the fractality edges are perturbation-independent, i.e., remain constant as the perturbation strength varies. The case where the effective model maps onto the off diagonal Harper model features a tunable critical-to-insulator transition at a finite disorder strength.

Almost compact moving breathers with fine-tuned discrete time quantum walks.

Discrete time quantum walks are unitary maps defined on the Hilbert space of coupled two-level systems. We study the dynamics of excitations in a nonlinear discrete time quantum walk, whose fine-tuned linear counterpart has a flat band structure. The linear counterpart is, therefore, lacking transport, with exact solutions being compactly localized. A solitary entity of the nonlinear walk moving at velocity v would, therefore, not suffer from resonances with small amplitude plane waves with identical phase velocity, due to the absence of the latter. That solitary excitation would also have to be localized stronger than exponential, due to the absence of a linear dispersion. We report on the existence of a set of stationary and moving breathers with almost compact superexponential spatial tails. At the limit of the largest velocity v = 1 , the moving breather turns into a completely compact bullet.

Intermittent many-body dynamics at equilibrium.

The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. Long excursions arise from sticky dynamics close to q-breathers localized in normal mode space. Measuring the exponent allows one to predict the transition into nonergodic dynamics. We generalize our method to Klein-Gordon lattices where the sticky dynamics is due to discrete breathers localized in real space.

Quasiperiodic driving of Anderson localized waves in one dimension.

We consider a quantum particle in a one-dimensional disordered lattice with Anderson localization in the presence of multifrequency perturbations of the onsite energies. Using the Floquet representation, we transform the eigenvalue problem into a Wannier-Stark basis. Each frequency component contributes either to a single channel or a multichannel connectivity along the lattice, depending on the control parameters. The single-channel regime is essentially equivalent to the undriven case. The multichannel driving increases substantially the localization length for slow driving, showing two different scaling regimes of weak and strong driving, yet the localization length stays finite for a finite number of frequency components.

Frequency combs with weakly lasing exciton-polariton condensates.

We predict the spontaneous modulated emission from a pair of exciton-polariton condensates due to coherent (Josephson) and dissipative coupling. We show that strong polariton-polariton interaction generates complex dynamics in the weak-lasing domain way beyond Hopf bifurcations. As a result, the exciton-polariton condensates exhibit self-induced oscillations and emit an equidistant frequency comb light spectrum. A plethora of possible emission spectra with asymmetric peak distributions appears due to spontaneously broken time-reversal symmetry. The lasing dynamics is affected by the shot noise arising from the influx of polaritons. That results in a complex inhomogeneous line broadening.

Nonequilibrium chaos of disordered nonlinear waves.

Do nonlinear waves destroy Anderson localization? Computational and experimental studies yield subdiffusive nonequilibrium wave packet spreading. Chaotic dynamics and phase decoherence assumptions are used for explaining the data. We perform a quantitative analysis of the nonequilibrium chaos assumption and compute the time dependence of main chaos indicators--Lyapunov exponents and deviation vector distributions. We find a slowing down of chaotic dynamics, which does not cross over into regular dynamics up to the largest observed time scales, still being fast enough to allow for a thermalization of the spreading wave packet. Strongly localized chaotic spots meander through the system as time evolves. Our findings confirm for the first time that nonequilibrium chaos and phase decoherence persist, fueling the prediction of a complete delocalization.

Nonlinear waves in disordered chains: probing the limits of chaos and spreading.

We probe the limits of nonlinear wave spreading in disordered chains which are known to localize linear waves. We particularly extend recent studies on the regimes of strong and weak chaos during subdiffusive spreading of wave packets [Europhys. Lett. 91, 30001 (2010)] and consider strong disorder, which favors Anderson localization. We probe the limit of infinite disorder strength and study Fröhlich-Spencer-Wayne models. We find that the assumption of chaotic wave packet dynamics and its impact on spreading is in accord with all studied cases. Spreading appears to be asymptotic, without any observable slowing down. We also consider chains with spatially inhomogeneous nonlinearity, which give further support to our findings and conclusions.

The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior.

A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics, the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.

Anderson localization or nonlinear waves: a matter of probability.

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.

Spreading of wave packets in disordered systems with tunable nonlinearity.

We study the spreading of single-site excitations in one-dimensional disordered Klein-Gordon chains with tunable nonlinearity |u(l)|(σ)u(l) for different values of σ. We perform extensive numerical simulations where wave packets are evolved (a) without and (b) with dephasing in normal-mode space. Subdiffusive spreading is observed with the second moment of wave packets growing as t(α). The dependence of the numerically computed exponent α on σ is in very good agreement with our theoretical predictions both for the evolution of the wave packet with and without dephasing (for σ≥2 in the latter case). We discuss evidence of the existence of a regime of strong chaos and observe destruction of Anderson localization in the packet tails for small values of σ.

Statistics of wave interactions in nonlinear disordered systems.

We study the properties of mode-mode interactions for waves propagating in nonlinear disordered one-dimensional systems. We focus on (i) the localization volume of a mode which defines the number of interacting partner modes, (ii) the overlap integrals which determine the interaction strength, (iii) the average spacing between eigenvalues of interacting modes, which sets a scale for the nonlinearity strength, and (iv) resonance probabilities of interacting modes. Our results are discussed in the light of recent studies on spreading of wave packets in disordered nonlinear systems and are related to the quantum many-body problem in a random chain.

Delocalization of wave packets in disordered nonlinear chains.

We consider the spatiotemporal evolution of a wave packet in disordered nonlinear Schrödinger and anharmonic oscillator chains. In the absence of nonlinearity all eigenstates are spatially localized with an upper bound on the localization length (Anderson localization). Nonlinear terms in the equations of motion destroy the Anderson localization due to nonintegrability and deterministic chaos. At least a finite part of an initially localized wave packet will subdiffusively spread without limits. We analyze the details of this spreading process. We compare the evolution of single-site, single-mode, and general finite-size excitations and study the statistics of detrapping times. We investigate the properties of mode-mode resonances, which are responsible for the incoherent delocalization process.

Universal spreading of wave packets in disordered nonlinear systems.

In the absence of nonlinearity all eigenmodes of a chain with disorder are spatially localized (Anderson localization). The width of the eigenvalue spectrum and the average eigenvalue spacing inside the localization volume set two frequency scales. An initially localized wave packet spreads in the presence of nonlinearity. Nonlinearity introduces frequency shifts, which define three different evolution outcomes: (i) localization as a transient, with subsequent subdiffusion; (ii) the absence of the transient and immediate subdiffusion; (iii) self-trapping of a part of the packet and subdiffusion of the remainder. The subdiffusive spreading is due to a finite number of packet modes being resonant. This number does not change on average and depends only on the disorder strength. Spreading is due to corresponding weak chaos inside the packet, which slowly heats the cold exterior. The second moment of the packet grows as t;{alpha}. We find alpha=1/3.

Vortex and translational currents due to broken time-space symmetries.

We consider the classical dynamics of a particle in a (d=2,3)-dimensional space-periodic potential under the influence of time-periodic external fields with zero mean. We perform a general time-space symmetry analysis and identify conditions, when the particle will generate a nonzero averaged translational and vortex currents. We perform computational studies of the equations of motion and of corresponding Fokker-Planck equations, which confirm the symmetry predictions. We address the experimentally important issue of current control. Cold atoms in optical potentials and magnetic traps are among possible candidates to observe these findings experimentally.

Absence of wave packet diffusion in disordered nonlinear systems.

We study the spreading of an initially localized wave packet in two nonlinear chains (discrete nonlinear Schrödinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverges with time. We find that the participation number of a wave packet does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times we find a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wave packet, we rule out the possibility of slow energy diffusion. The dynamical state could approach a quasiperiodic solution (Kolmogorov-Arnold-Moser torus) in the long time limit.

q-Breathers in finite two- and three-dimensional nonlinear acoustic lattices.

In their celebrated experiment, Fermi, Pasta, and Ulam (FPU) [Los Alamos Report No. LA-1940, 1955] observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined q-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs are enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.

Energy flow of moving dissipative topological solitons.

We study the energy flow due to the motion of topological solitons in nonlinear extended systems in the presence of damping and driving. The total field momentum contribution to the energy flux, which reduces the soliton motion to that of a point particle, is insufficient. We identify an additional exchange energy flux channel mediated by the spatial and temporal inhomogeneity of the system state. In the well-known case of a dc external force the corresponding exchange current is shown to be small but nonzero. For the case of ac driving forces, which lead to a soliton ratchet, the exchange energy flux mediates the complete energy flow of the system. We also consider the case of combination of ac and dc external forces, as well as spatial discretization effects.

q-breathers in Fermi-Pasta-Ulam chains: existence, localization, and stability.

The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit, each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes, and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.

Compactlike discrete breathers in systems with nonlinear and nonlocal dispersive terms.

Discrete breathers with purely anharmonic short-range interaction potentials localize superexponentially becoming compactlike. We analyze their spatial localization properties and their dynamical stability. Several branches of solutions are identified. One of them connects to the well-known Page and Sievers-Takeno lattice modes, another one connects with the compacton solutions of Rosenau. The absence of linear dispersion allows for extremely long-lived time-quasiperiodic localized excitations. Adding long-range anharmonic interactions leads to an extreme case of competition between length scales defining the spatial breather localization. We show that short- and long-range interaction terms competition results in the appearance of several characteristic crossover lengths and essentially breaks the concept of compactness of the corresponding discrete breathers.