Nonlinear science emerged in its present form following a series of decisive analytic, numerical and experimental developments that took place in close interaction in the last three decades. Its aim is to provide the concepts and the techniques necessary for a unified description of the particular, yet quite large, class of phenomena whereby simple deterministic systems give rise to complex behavior associated with the appearance of unexpected spatial structures or evolutionary events. Such systems are encountered in a great number of disciplines notably in classical mechanics, statistical physics, fluid dynamics, chemistry, optics, atomic and molecular physics, environmental sciences, engineering sciences or biology, in the context of both fundamental and applied investigations.

While the concern for unification is central in every attempt of man to explain the natural world, the particular approach followed by nonlinear science in the pursuit of this goal is characterized by a great originality that differentiates it from other disciplines belonging to the traditional realm of physical sciences. Nonlinear science introduces a new way of thinking based on a subtle interplay between qualitative and quantitative techniques, between topological, geometric and metric considerations, between deterministic and statistical aspects. It uses an extremely large variety of methods from very diverse disciplines, but through the process of continual switching between different views of the same reality these methods are cross-fertilized and blended into a unique combination that gives them a marked added value.

The present collection of papers is devoted to a rather new and very controversial topic, the so-called “quantum chaos”. Some researchers see nothing essentially new at all in this phenomenon (apart from a number of various examples and models), and they have good reason to believe so. Indeed, the problems in this field all belong to the traditional, “old-fashion”, and rather “simple” quantum mechanics of finite-dimensional systems with a given interaction and no quantized fields.

Nevertheless, many, including ourselves, consider quantum chaos to be a new discovery, though in an old field, of a great importance for fundamental physics. To understand this, the phenomenon of quantum chaos should be put into its proper perspective in recent developments in physics. The central point of this perspective is the conception of dynamical chaos (also a rather new topic) in classical mechanics (for a good review see, e.g., Refs.[l-3]). Thus before discussing the current understanding of quantum chaos we need briefly to describe classical dynamical chaos

Introduction

Dynamical localization is a variant of Anderson localization appearing in quantum systems whose classical limit is chaotic. It is a novel quantum coherence effect—perhaps the most important new physical effect appearing in the field of quantum chaos. Like the familiar Anderson localization it is based on destructive interference of waves in random systems. What is new in dynamical localization is the fact that the randomness is not externally imposed, e.g., by a random medium, but is produced dynamically by a simple and completely deterministic system. Here the parallel to chaos (i.e. stochasticity) in deterministic classical dynamical systems with few degrees of freedom is apparent.

Dynamical localization has been reviewed in depth in the lectures of Fishman, Shepelyansky and Izrailev. Another useful review has been given in [1]. In order to set the stage it is, therefore, enough to recall a few basic facts and to mention some physical examples where this effect appears.

The discussion will be restricted to Hamiltonian systems which are either autonomous with two degrees of freedom or externally driven periodically in time with one degree of freedom. In fact, extending phase space the latter case can be viewed just as a special case of the former [2]. Classically, under conditions of chaos, the two action variables, describing the system together with the canonically conjugate angles, will undergo a diffusion process. We shall always assume that the chaotic part of phase space is sufficiently large to neglect boundary effects on the diffusion.

INTRODUCTION

In this paper we provide a detailed analysis of the classical structures that we are able to recognize in single eigenfunctions of a strongly chaotic system, the quantized version of the baker transformation [1], thus contributing to an answer to the question “what are single eigenfunctions of classically chaotic systems made of?” If we were to ask this question for an integrable system the answer would be immediate: the wave functions peak on the invariant tori quantized by discrete values of the actions. Thus each wave function shows the imprint of one torus which becomes more and more sharply defined as the classical limit is approached.

The onset of chaotic motion in externally driven classical systems is a key problem in nonlinear dynamics that is now well understood in its essential aspects. In many physically interesting cases, as the perturbation strength increases beyond some critical value, the system starts absorbing energy in a diffusive way. The question whether “diffusive” excitation processes can take place also in externally driven quantum systems is then a very interesting one for the physics of atoms and molecules in external electromagnetic fields. This is a deep question involving the nature and the validity of the quasiclassical approximation when the underlying classical dynamics is chaotic.

The question of “quantum diffusion” was first addressed in a simple model system, the kicked rotator. It was chosen because the features of classical chaos in it were relatively clean and well understood. Besides that, the numerical simulation of its quantum dynamics could be easily accomplished. The major indication of this model was that quantum mechanics suppresses the classical chaotic diffusion in action via a destructive interference effect similar to that responsible for the Anderson localization well known in condensed matter physics.

Abstract

We study the quantum mechanics of a two-dimensional version of the Fermi accelerator. The model consists of a free particle constrained to move inside a disk with radius varying periodically in time. A complete quantum mechanical solution of the problem is possible for a specific choice of the time-periodic oscillation radius. The quasi-energy spectral (QES) properties of the model are obtained from direct evaluation of finite-dimensional approximations to the time evolution operator. As the effective ħ is changed from large to small the statistics of the QES eigenvalues change from Poisson to circular orthogonal ensemble (COE). Different statistical tests are used to characterize this transition. The transition of the QES eigenfunctions is studied using the X2 test with v degrees of freedom. The Porter-Thomas distribution is shown to apply in the COE regime, while the Poisson regime does not fit the X2 with v = 0. We find that the Poisson regime is associated with exponentially localized QES eigenfunctions whereas they are extended in the COE regime.

Introduction

Although a complete understanding of the quantum manifestations of classical chaos (QMCC) is not yet in sight, significant progress has been made in obtaining partial answers to this question. This progress has primarily been achieved from studies of lower-dimensional models: two-dimensional for energy conserving models and one-dimensional when energy is not conserved. In particular, very few studies have been carried out in two-dimensional time-dependent problems, for even in the classical limit the theoretical analysis is nontrivial.

Abstract

We study the correlations in the quasienergy spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. Two complementary approaches are taken: We first study the “local spectral density”. For level separations below the mean distance, its two-point correlations are dominated by level attraction rather than level repulsion, a feature known from the electron spectra in disordered solids. We then turn to the unbiased spectra for the QKR on a finite-dimensional Hilbert space (“finite-sample approach”). They are characterized by a novel universal statistics which depends on the ratio γ of the localization length to the basis size. We derive semiclassical expressions for the two-point correlations which interpolate between COE behaviour for γ → ∞ and Poissonian (lack of correlations) for γ → 0. We show how the diffusive nature of the classical dynamics finds its expression in the quantal spectral correlations.

Introduction

One of the most impressive results in “quantum chaology” was that as soon as the corresponding classical dynamics becomes chaotic, the spectral fluctuations, in the limit ħ → 0, obey universal distribution laws, as predicted by random-matrix theory [1–3]. Recently, this quantum–classical correspondence has been extended to the domain of chaotic scattering, where the fluctuations in the S-matrix eigenphases were shown to follow the statistics of Dyson's circular orthogonal ensemble (COE) [4]. The periodic-orbit theory was the main tool for the semiclassical investigation of spectra and their relation to random-matrix theory.

Abstract

Classical and semiclassical periodic orbit expansions are applied to the dynamics of a point particle scattering elastically off several disks in a plane. Fredholm determinants, zeta functions, and convergence of their cycle expansions are tested and applied to evaluation of classical escape rates and quantum resonances. The results demonstrate the applicability of the Ruelle and Gutzwiller type periodic orbit expressions for chaotic systems.

Introduction

At the heart of semiclassical descriptions of chaotic systems is the Gutzwiller trace formula which relates the eigenvalue spectrum of the Schrödinger operator to the periodic orbits of the underlying classical system [1]. This relationship between the classical and the quantum properties can be viewed as a generalization of the Selberg trace formula which relates the spectrum of the Laplace-Beltrami operator to geodesic motion on surfaces of constant negative curvature [2]. Whereas the Selberg trace is exact, the Gutzwiller trace, derived within a stationary phase approximation, is only approximate, valid in a suitable semiclassical limit.

In one-dimensional systems the trace formula recovers the standard WKB quantization rules, which yield easy and sometimes quite accurate estimates for the quantum eigenvalues [3]. For systems with more than one degree of freedom a classical system can exhibit chaos. The simple WKB quantization fails and evaluation of the trace formulas can become rather difficult; in fact, it is often easier to do the full quantum calculation and to obtain the periods of classical periodic orbits from the quantum data by a Fourier transform [4].

Abstract

Trace formulas provide the only general relations known connecting quantum mechanics with classical mechanics in the case that the classical motion is chaotic. In particular, they connect quantal objects such as the density of states with classical periodic orbits. In this chapter, several trace formulas, including those of Gutzwiller and Balian and Bloch, Tabor, and Berry, are examined from a geometrical standpoint. New forms of the amplitude determinant in asymptotic theory are developed as tools for this examination. The meaning of caustics in these formulas is revealed in terms of intersections of Lagrangian manifolds in phase space. The periodic orbits themselves appear as caustics of an unstable kind, lying on the intersection of two Lagrangian manifolds in the appropriate phase space. New insight is obtained into the Weyl correspondence and the Wigner function, especially their caustic structures.

Introduction

This chapter concerns the trace formulas of Gutzwiller[1] and Balian and Bloch[2], which express the density of states of a bound quantal system as a sum over the periodic orbits of the corresponding classical system, and closely related trace formulas, such as that of Tabor[3] for the density of quasistates in a time-periodic system, and that of Berry[4] for the scars of Wigner functions in phase space. The purpose of this chapter is to explore the semiclassical structures of such formulas, i.e., the geometrical objects in the classical phase space associated with them and the interplay between these objects and the corresponding wave fields.

INTRODUCTION

The phase space of nonlinear nonintegrable classical conservative systems exhibits extremely complex structure consisting of regular Kolmogorov-Arnold-Moser (KAM) tori intermixed with chaos. In some regions of the phase space the structure is self-similar to all length scales and exhibits scaling behavior in space and time. KAM tori are the remnants of global conserved quantities. For many systems, when some parameter which characterizes the size of the nonlinearity is small, KAM tori dominate the phase space. However, as the nonlinearity parameter is increased in size, nonlinear resonances in the system grow and overlap and destroy KAM tori lying between them. The existence of KAM tori can have a profound effect on the dynamics of a conservative system with two degrees of freedom because some KAM tori can divide the phase space into disjoint parts. When such a KAM torus is destroyed by nonlinear resonance overlap, the dynamics of the classical system may change dramatically.

The mechanism by which KAM surfaces are destroyed by nonlinear resonances has been studied extensively by Greene, Shenker and Kadanoff, MacKay, and others. KAM tori have irrational winding numbers. Each irrational winding number can be represented uniquely by a continued fraction.