Elsevier

Annals of Physics

Volume 326, Issue 7, July 2011, Pages 1577-1655
Annals of Physics

Weak chaos in the disordered nonlinear Schrödinger chain: Destruction of Anderson localization by Arnold diffusion

https://doi.org/10.1016/j.aop.2011.02.004Get rights and content

Abstract

The subject of this study is the long-time equilibration dynamics of a strongly disordered one-dimensional chain of coupled weakly anharmonic classical oscillators. It is shown that chaos in this system has a very particular spatial structure: it can be viewed as a dilute gas of chaotic spots. Each chaotic spot corresponds to a stochastic pump which drives the Arnold diffusion of the oscillators surrounding it, thus leading to their relaxation and thermalization. The most important mechanism of equilibration at long distances is provided by random migration of the chaotic spots along the chain, which bears analogy with variable-range hopping of electrons in strongly disordered solids. The corresponding macroscopic transport equations are obtained.

Research highlights

► In a one-dimensional disordered chain of oscillators all normal modes are localized. ► Nonlinearity leads to chaotic dynamics. ► Chaos is concentrated on rare chaotic spots. ► Chaotic spots drive energy exchange between oscillators. ► Macroscopic transport coefficients are obtained.

Introduction

Anderson localization [1] is a general phenomenon occurring in many linear wave-like systems subject to a disordered background (see Ref. [2] for a recent review). It is especially pronounced in one-dimensional systems, where even an arbitrarily weak disorder localizes all normal modes of the system [3], [4]. Anderson localization in linear systems (single-particle problems in quantum mechanics) has been thoroughly studied over the last 50 years; its physical picture is quite clear by now (although some open questions still remain), and even rigorous mathematical results have been established. The situation is much less clear in the presence of nonlinearities/interactions.

One of the simplest systems where the effect of a classical nonlinearity on the Anderson localization can be studied, is the disordered nonlinear Schrödinger equation (DNLSE) in one dimension, discrete or continuous. It has attracted a lot of interest in the last few years, including both stationary and non-stationary problems, as well as the problem of the dynamic stability of stationary solutions [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. This interest was especially motivated by the recent experimental observation of the Anderson localization of light in disordered photonic lattices [38], [39], [40], where DNLSE describes light propagation in the paraxial approximation, and of cold atoms in disordered optical lattices [41], [42], [43], for which DNLSE provides the mean-field description. Other systems have also been studied, such as a chain of anharmonic oscillators [5], [15], [44], [45], [20], [21], [32], [33], [35], [46], a chain of classical spins [5], [47], a nonlinear Stark ladder [48], [49].

In contrast to linear problems which can always be reduced to finding the normal modes and the corresponding eigenvalues, the problem of localization in a nonlinear system can be stated in different inequivalent ways. For example, one can study solutions of the stationary nonlinear Schrödinger equation with disorder [12], [14], [28]; however, in a nonlinear system they are not directly related to the dynamics. Another possible setting is the system subject to an external perturbation or probe; studies of the transmission of a finite-length sample where an external flow is imposed [6], [8], [9], [10], [11], [19], [25], [27], or dipolar oscillations in an external trap potential [18], fall into this category. Most attention has been paid to the problem of spreading of an initially localized wave packet of a finite norm [7], [8], [9], [13], [15], [16], [17], [50], [20], [21], [22], [23], [48], [49], [24], [29], [30], [31], [32], [33], [34] (in linear systems it remains exponentially suppressed at long distances for all times). This setting corresponds directly to experiments [38], [39], [40], [41], [42]. A closely related but not equivalent problem is that of thermalization of an initially out-of-equilibrium system (in a linear localized system thermalization does not occur). The difference between these two settings is that in the latter the total energy stored in the system is proportional to its (infinite) size, while in the former the infinite system initially receives a finite amount of energy. The problem of thermalization has also received attention [44], [45], [47], [26], and it is the main subject of the present work (although the problem of wave packet spreading will also be briefly discussed). Still, despite a large body of work, detailed understanding of the equilibration dynamics in one-dimensional disordered nonlinear systems is still lacking.

On the one hand, the direct numerical integration of the differential equation shows that an initially localized wave packet of a finite norm does spread indefinitely, its size growing with time as a sub-diffusive power law [7], [9], [15], [16], [20], [21], [32], [33], [35]. Existence of different regimes of spreading, depending on whether the system is in the regime of strong or weak chaos, has been discussed [33], [35]. Several authors suggested a macroscopic description of the long-time dynamics in terms of a nonlinear diffusion-type equation resulting in sub-diffusive spreading of the wave packet [33], [31], [34]; however, arguments used to justify this equations are based on an a priori assumption of spatially uniform chaos.

On the other hand, it was argued that among different initial conditions with a finite norm, at least some should exhibit regular quasi-periodic dynamics, and thus would not spread indefinitely [5], [13], [37]. For finite-size systems, scaling of the probability for the system to be in the chaotic or regular regime has been studied numerically [36]. When scaling arguments are applied to the results of the numerical integration of the equation of motion, they indicate slowing down of the power-law spreading [34]. Rigorous mathematical arguments support the conclusion that at long times the wave packet should spread (if spread at all) slower than any power law [50].

The aim of the present work is the analysis of statistical properties of weak chaos in the discrete one-dimensional disordered nonlinear Schrödinger equation with the initial conditions corresponding to finite norm and energy densities. The main focus is on long-time dynamics and relaxation at large distances, when no stationary superfluid flow can exist in the system [11], [27] and the dynamics is chaotic. It is often assumed that upon thermalization chaos has no spatial structure, and all sites of the chain are more or less equally chaotic; here it is argued not to be the case.1 Namely, in the regime of strong disorder and weak nonlinearity chaos is concentrated on a small number of rare chaotic spots (essentially the same picture was also proposed in Ref. [47] for a disordered chain of coupled classical spins, but it was not put on a quantitative basis). A chaotic spot is a collection of resonantly coupled oscillators, in which one can separate a collective degree of freedom (namely, their relative phase whose dynamics is slow because of the resonance), which performs chaotic motion. Under the conditions of weak coupling between neighboring oscillators and weak nonlinearity (i.e., Anderson localization being the strongest effect), the distance between different chaotic spots is much greater than the typical size of the spot. The stochastic motion of the collective degree of freedom of each chaotic spot acts as a stochastic pump, i.e., drives the exchange of energy between other oscillators, non-resonantly coupled to the spot, which corresponds to the Arnold diffusion. This represents the main mechanism for relaxation and thermalization of the oscillators, as well as for the transport of conserved quantities (energy and norm). An important role is played by the fact that chaotic spots can migrate along the chain, as the collective degree of freedom may get in and out of the chaotic region of its phase space. A similar phenomenon has recently been proven to exist in a chain of weakly coupled pendula [51]. This migration also bears analogy with the variable-range hopping of electrons in strongly disordered solids [52], [53]. Still, there is an important difference that the chaotic spot does not carry any energy or norm; it only drives relaxation of oscillators surrounding it.

The main technical challenge in this work is the analysis of high orders of the perturbation theory, which is necessary both to separate the collective degree of freedom performing the chaotic motion, and to couple other oscillators to this degree of freedom. The perturbation theory is divergent, as chaos is a non-perturbative phenomenon. The divergence occurs because of overlapping resonances, according to Chirikov’s criterion of chaos. In this work, the probability for resonances to occur is estimated in each order of perturbation theory. The subsequent treatment of each resonance and description of the associated chaotic motion is based on the stochastic pump model of the Arnold diffusion, which was proposed for systems with few degrees of freedom [54], [55]. Thus, the present work is not more rigorous than the stochastic pump model; it does not add anything new to the current understanding of how chaos is generated in nonlinear systems; it rather deals with the statistics of chaos in a spatially extended system with an infinite number of degrees of freedom and local coupling between them.

The main quantitative result of the present work is the system of macroscopic equations which describe transport of the conserved quantities (energy and norm) along the chain. Such tansport determines equilibration of the chain at large distances, which occurs at long time scales. These equations are of nonlinear-diffusion type, and explicit expressions for the transport coefficients are derived. The macroscopic equations are valid at distances exceeding a certain length scale, which is also explicitly estimated.

The paper is organized as follows. In Section 2 the model is formulated and the main assumptions are discussed. Section 3 summarizes the main results. In Section 4 we qualitatively describe the physical picture and the main ingredients of the solution; various implications and some related issues are discussed. In Section 5 we analyze the chaotic phase space in few-oscillator configurations. In Section 6 perturbation theory is developed and statistics of different terms is analyzed. In Section 7 this perturbation theory is used to study the statistics of the chaotic phase volume for an arbitrary number of oscillators. Section 8 is dedicated to the analysis of the coupling between chaotic spots and other oscillators, and to the description of Arnold diffusion. In Section 9 the macroscopic transport coefficients are found.

Section snippets

Model and assumptions

The main subject of this study is the one-dimensional discrete nonlinear Schrödinger equation with diagonal disorder:idψndt=ωnψn-τΔψn-1+ψn+1+gψnψn2.Here the integer n runs from −∞ to +∞ and labels sites of a one-dimensional lattice. To each site a pair of complex conjugate variables ψn,ψn is associated.

Eq. (2.1) together with its complex conjugate are the Hamilton equations:dψndt=Hiψn,diψndt=-Hψn,corresponding to the classical Hamiltonian:Hiψn,{ψn}=nωnψnψn-τΔψnψn+1+ψn+1ψn+g2ψn2ψn2,

Main results

The first statement is that for general initial conditions, satisfying the assumptions of the previous section, the system locally thermalizes with a finite relaxation time. Local thermalization means that on a finite, sufficiently long segment of the chain [the corresponding length scale L is given explicitly below, Eq. (3.7)], the actions and the phases are distributed according to the grand canonical distribution, e-β(H-μItot). To respect the two conserved quantities, the total energy H and

Search for chaos

Appearance of chaos in systems with a few degrees of freedom has been thoroughly studied in the past [54], [55], [59]; the main steps will be reviewed in Section 5. The essential ingredients are the following:

  • 1.

    When an integer linear combination of several frequencies ω˜n(In) of the integrable system vanishes for some values of In, that is m1ω˜n1++mNω˜nN=0 (guiding resonance, in Chirikov’s terminology [54]), one can separate a slow degree of freedom ϕ˜, which is the corresponding linear

Resonant triples

This section represents a detailed analysis of chaos in a system of three oscillators whose frequencies happen to be close to each other. Although such resonant triples do not contribute to the main result for the conductivity, Eq. (3.6a), they have the thickest stochastic layers. Estimation of their chaotic phase volume is the main task of the present section.

First, we discuss a resonant pair of oscillators where chaos is generated by an external perturbation, which can be studied by the

General remarks

One could formulate the peturbation theory for Eq. (2.1) by taking independent oscillations, ψn(0)(t)=Ine-i(ωn+gIn)t+iϕn0, as the zero approximation and solving Eq. (2.1) for ψn(t) by iterations. Clearly, such perturbation theory is divergent. Indeed, if it were convergent, there would be no chaos. Formally, its divergent character can be seen already from the exact solution for two oscillators, given in Section 5.1. The pendulum frequency depends on τ as Ω|τ| (Eq. (5.8)), while the dependence

General remarks on the procedure

Each site n of the chain can be characterized by a variable wn, the chaotic fraction of the thermally-weighted phase volume, summed over all possible guiding resonances involving the site n and sites to the right of it [to be formally defined below, Eq. (7.8)]. For a resonant nearest-neighbor triple at τ  ρ this quantity was defined in Eq. (5.29) and estimated in Section 5.4, the result being [see Eqs. (5.42a), (5.42b)]:wnτρe-En/T,2gEn=3maxk=n,n+1,n+2(ωk-μ)2-k=nn+2(ωk-μ)2.

Note that the

Stochastic pump and Arnold diffusion

The standard description of Arnold diffusion in multidimensional dynamical systems is based on the stochastic pump model [54], [55]. Namely, motion of the oscillators participating in the guiding resonance (the chaotic spot) is assumed to have a stochastic component with a continuous frequency spectrum. The continuous spectrum arises because the motion corresponds to successive passages of the separtrix of the effective pendulum at random instants of time and is analyzed in detail in Section 8.1

General remarks on the procedure

Global transport properties of random one-dimensional systems tend to be determined by rare strong obstacles [53]. The reason for this is that in one dimension such obstacles cannot be bypassed, in contrast to higher dimensions. Such rare obstacles were called “blockades” [64], “weak links” [65], or “breaks” [61], [66], and here the latter term will be used. In the present problem, a break is a region of the chain where the chaotic fraction w assumes anomalously small values on many consecutive

Acknowledgements

The author is grateful to I.L. Aleiner, B L. Altshuler, S. Flach, and O.M. Yevtushenko for helpful discussions.

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