We reveal the generic characteristics of wave packet delocalization in two-dimensional nonlinear disordered lattices by performing extensive numerical simulations in two basic disordered models: the Klein-Gordon system and the discrete nonlinear Schrödinger equation. We find that in both models (a) the wave packet's second moment asymptotically evolves as $t^{a_m}$ with $a_m \approx 1/5$ ($1/3$) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S.~Flach, Chem.~Phys.~{\bf 375}, 548 (2010)], (b) chaos persists, but its strength decreases in time $t$ since the finite time maximum Lyapunov exponent $Λ$ decays as $Λ\propto t^{α_Λ}$, with $α_Λ \approx -0.37$ ($-0.46$) for the weak (strong) chaos case, and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattice's excited part, which induces the wave packet's thermalization. We also propose a dimension-independent scaling between the wave packet's spreading and chaoticity, which allows the prediction of the obtained $α_Λ$ values.
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 $σ\geq 2$…
We demonstrate for various systems that the variance of a wave packet $M(t)\propto t^ν$, can show a {\it superballistic} increase with $2<ν\le3$, for parametrically large time intervals. A model is constructed which explains this phenomenon and its predictions are verified numerically for various disordered and quasi-periodic systems.
We discuss the long time behaviour of a finite energy wave packet in nonlinear Hamiltonians on infinite lattices at arbitrary dimension, exhibiting linear Anderson localization. Strong arguments both mathematical and numerical, suggest for infinite models that small amplitude wave packets may generate stationary quasiperiodic solutions (KAM tori) almost undistinguishable from linear wave packets. The probability of this event is non vanishing at small enough amplitude and goes to unity at amplitude zero. Most other wave packets (non KAM tori) are chaotic. We discuss the Arnold diffusion conjec…
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 wavepacket 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) selftrapping of a part of the packet, and subdiffusion…
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 [EPL {\bf 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 observa…
We consider the spreading of the wave packet in the generalized Rosenzweig-Porter random matrix ensemble in the region of non-ergodic extended states $1<γ<2$. We show that despite non-trivial fractal dimensions $0 < D_{q}=2-γ<1$ characterize wave function statistics in this region, the wave packet spreading $\langle r^{2} \rangle \propto t^β$ is governed by the "diffusion" exponent $β=1$ outside the ballistic regime $t>τ\sim 1$ and $\langle r^{2}\rangle \propto t^{2}$ in the ballistic regime for $t<τ\sim 1$. This demonstrates that the multifractality exhibits itself only in {\it local} quantit…
Non-Hermiticity and dephasing, collaborating in an unusual wave packet dynamics, realizes unconventional entanglement evolution in a disordered, interacting and asymmetric (non-reciprocal) quantum medium. Taking the Hatano-Nelson model as a concrete example, we first consider how wave packet spreads in a non-Hermitian disordered system for demonstraing that it is very different from the Hermitian case. Interestingly, a cascade like wave packet spreading as in the Hermitian case is suppressed in the clean limit and at weak disorder, while it revives in the vicinity of the localization-delocaliz…
We analyze mechanisms and regimes of wave packet spreading in nonlinear disordered media. We predict that wave packets can spread in two regimes of strong and weak chaos. We discuss resonance probabilities, nonlinear diffusion equations, and predict a dynamical crossover from strong to weak chaos. The crossover is controlled by the ratio of nonlinear frequency shifts and the average eigenvalue spacing of eigenstates of the linear equations within one localization volume. We consider generalized models in higher lattice dimensions and obtain critical values for the nonlinearity power, the dimen…
We show, theoretically and experimentally, the counterintuitive result that an increase of disorder can result in an enhanced spreading of an initially localized excitation. Moreover, we find that adding a focusing nonlinearity facilitates the expansion of the wave-packet even further by increasing its effective size. We find a clear transition between between the regions of enhanced spreading (weak disorder) and localization (strong localization) described by a "diffusion peak".
We study the characteristics of chaos evolution of initially localized energy excitations in the one-dimensional nonlinear disordered Klein-Gordon lattice of anharmonic oscillators, by computing the time variation of the fundamental frequencies of the motion of each oscillator. We focus our attention on the dynamics of the so-called `weak' and `strong chaos' spreading regimes [2010, EPL 91 30001], for which Anderson localization is destroyed. Based on the fact that large variations of the fundamental frequencies denote strong chaotic behavior, we show that in both regimes chaos is more intense…
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 thermaliz…
To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to …
In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonli…
The nonlinear Schroedinger equation in the presence of disorder is considered. The dynamics of an initially localized wave packet is studied. A subdiffusive spreading of the wave packet is explained in the framework of a continuous time random walk. A probabilistic description of subdiffusion is suggested and a transport exponent of subdiffusion is obtained to be 2/5.
We study spreading wave packets in a disordered nonlinear ladder with broken time-reversal symmetry induced by synthetic gauge fields. The model describes the dynamics of interacting bosons in a disordered and driven optical ladder within a mean-field approximation. The second moment of the wave packet $m_{2} = g t^α$ grows subdiffusively with the universal exponent $α\simeq 1/3$ similar to the time-reversal case. However the prefactor $g$ is strongly modified by the field strength and shows a non-monotonic dependence. For a weak field, the prefactor increases since time-reversal enhanced back…
A periodically driven lattice with two commensurate spatial periodicities is found to exhibit super metallic states characterized by enhancements in wave packet spreading and entropy. These resonances occur at critical values of parameters where multi-band dispersion curves reduce to a universal function that is topologically a circle and the effective quantum dynamics describes free propagation. Sandwiching every resonant state are a pair of anti-resonant {\it trapped states} distinguished by dips in entropy where the transport, as seen in the spreading rate, is only somewhat inhibited. Exist…
We study a version of the mathematical Ruijsenaars-Schneider model, and reinterpret it physically in order to describe the spreading with time of quantum wave packets in a system where multifractality can be tuned by varying a parameter. We compare different methods to measure the multifractality of wave packets, and identify the best one. We find the multifractality to decrease with time until it reaches an asymptotic limit, different from the mulifractality of eigenvectors, but related to it, as is the rate of the decrease. Our results could guide the study of experimental situations where m…
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 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 …
In this paper the chaotic properties of the TCP congestion avoidance mechanism are investigated. The analysis focuses on the origin of the complex behavior appearing in deterministic TCP/IP networks. From the traffic modeling point of view the understanding of the mechanism generating chaos is essential, since present models are unable to cope with this phenomena. Using the basic tools of chaos theory in our study, the main characteristics of chaotic dynamics are revealed. The dynamics of packet loss events is studied by a simple symbolic description. The cellular structure of the phase space …
We study the spreading of initially localized states in a nonlinear disordered lattice described by the nonlinear Schrödinger equation with random on-site potentials - a nonlinear generalization of the Anderson model of localization. We use a nonlinear diffusion equation to describe the subdiffusive spreading. To confirm the self-similar nature of the evolution we characterize the peak structure of the spreading states with help of Rényi entropies and in particular with the structural entropy. The latter is shown to remain constant over a wide range of time. Furthermore, we report on the depen…
This thesis focuses on the mechanisms of energy transport in multidimensional heterogeneous lattice models, studying in particular the case of the Klein-Gordon model of coupled anharmonic oscillators in one and two spatial dimensions. We report the effects of the type of the impurity (heterogeneity) parameter on the systems' transport properties and classifies the transport mechanisms of the nonlinear versions of the models into various dynamical regimes. We also identify computationally efficient numerical integration techniques, including the so-called symplectic integrators, and implement t…
We investigate the damage spreading effect in the Fortuin-Kasteleyn random cluster model for 2- and 3-dimensional grids with periodic boundary. For 2D the damage function has a global maximum at $p=\sqrt{q}/(1+\sqrt{q})$ for all $q>0$ and also local maxima at $p=1/2$ and $p=q/(1+q)$ for $q\lesssim 0.75$. For 3D we observe a local maximum at $p=q/(1+q)$ for $q\lesssim 0.46$ and a global maximum at $p=1/2$ for $q\lesssim 4.5$. The chaotic phase of the model's $(p,q)$-parameter space is where the coupling time is of exponential order and we locate points on its boundary. For 3-dimensional grids t…
A quantum-mechanical analysis of hyper-fast (faster than ballistic) diffusion of a quantum wave packet in random optical lattices is presented. The main motivation of the presented analysis is experimental demonstrations of hyper-diffusive spreading of a wave packet in random photonic lattices [L. Levi \textit{et al.}, Nature Phys. \textbf{8}, 912 (2012)]. A rigorous quantum-mechanical calculation of the mean probability amplitude is suggested, and it is shown that the power law spreading of the mean squared displacement (MSD) is $< x^2(t)>\sim t^α$, where $2<α\leq 3$. The values of the transp…
We study numerically a spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time $ \propto t^α$, with the exponent $α$ being in the range $0.3 - 0.4$. For small nonlinearities the distribution remains localized in a way similar to the linear case.
We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as $H_{nm}\propto |n-m|^{-α}$. We consider the critical case ($α=1$). Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave-packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave-packet shows a diffusive-like spread developing a power-law tail and achieve…
In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems is under intense theoretical debate and experimental study. We resolve this dispute showing that 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 …
We use a novel unitary map toolbox -- discrete time quantum walks originally designed for quantum computing -- to implement ultrafast computer simulations of extremely slow dynamics in a nonlinear and disordered medium. Previous reports on wave packet spreading in Gross-Pitaevskii lattices observed subdiffusion with the second moment $m_2 \sim t^{1/3}$ (with time in units of a characteristic scale $t_0$) up to the largest computed times of the order of $10^8$. A fundamental question remained as to whether this process can continue ad infinitum, or has to slow down. Current experimental devices…
