Abstract

The known manifestation of quantum chaos is the so-called Wigner-Dyson distribution P(s) for spacings between neighbouring levels in the spectrum. In the other limiting case of completely integrable systems, the distribution P(s) turns out to be very close to the Poissonian one. In the present chapter, the influence of quantum effects on level statistics is studied for the case in which the corresponding classical systems are fully chaotic. The numerical study of the kicked rotator model with a finite number of states allows us to establish the link between the degree of localization and level repulsion. As a good model for this phenomenon of localization, the band random matrices are suggested. It is shown that such matrices can be used to describe statistical properties of localized quantum chaos.

Introduction

One of the important problems in the theory of quantum chaos is the description of statistical properties of systems using classical and quantum (semiclassical) parameters. Numerous studies (see the review [1] and references therein) have shown that the general situation is very complicated and no universal properties can be predicted when in the classical limit the motion is not fully chaotic. On the other hand, it has been proved numerically that for many classical models (for example, billiards, see [2]) with strong chaos, statistical properties both of energy (quasienergy) spectra and eigenfunctions are well described by the Random Matrix Theory (RMT) [3].

This chapter deals with the semiclassical analysis of the individual eigenfunctions in a quantum system, especially when the classical dynamics is chaotic and the quantum bound states are considered. The situation is still barely understood, but analytic methods relevant to the problem have been steadily developing [1-4]. On the one hand, quantum maps have emerged as ideal dynamical models for basic studies, with their ability to exhibit classical chaos within a single degree of freedom [5]. On the other hand, phase space techniques have become recognized as extremely powerful for describing quantum states; however, because these techniques concentrate routinely on the density operators (namely, the eigenprojectors in Wigner or Husimi representations [6]), they are still currently a long way from grasping the semiclassical shapes of the wavefunctions as such.

We argue that well-adapted representations of eigenfunctions are essential for semiclassical analysis and that they should incorporate all previous observations. First, the dynamical problem should be considered in the reduced form of a quantum map; then, its eigenstates should be analyzed in phase space; there, however, they should not be displayed as density operators but directly parametrized as wavefunctions.

This chapter essentially reviews an explicit realization of that program in one degree of freedom, in which the crucial ingredient is a phase-space parametrization of 1-d wavefunctions [7]. Every 1-d wavefunction is first expressed in a holomorphic (Bargmann) representation, then factorized over the zeros of its Husimi function, to end up being represented by a pattern of essentially N ∼ h−1 of those zeros in a 1–1 correspondence; at that point the semiclassical regime appears as a thermodynamic (N → ∞) limit.

INTRODUCTION

In recent years the study of the effects of quantization on the properties of classically nonintegrable systems has attracted increasing attention. The understanding of the nature of quantum behavior of these systems is not only of fundamental importance but it is also a problem of experimental relevance in fields as diverse as photochemistry, electron dynamics in microstructures, and other contexts. Understanding the relation between quantum problems and their classical limit may also shed light on the zero wavelength approximation to other wave equations, for example the eikonal approximation to the magnetohydrodynamic equations that are of great interest in plasma physics. In the context of applications to photochemistry various model systems described by timeindependent Hamiltonians have been recently studied.

Abstract

Continuous and discrete spectrum states near the ionization limit are considered. The origin of the continuous spectrum narrow resonances is elucidated. We suggest the semiclassical equation for the states near the ionization limit. The way of analytical continuation of the equation for discrete spectrum states into the region of resonances in continuum is pointed out. It is shown that the wave function structure is intermediate between the regular structure and the chaotic one.

Introduction

In the works [1, 2, 3] on the photoexcitation of an Li atom in a magnetic field a number of continuous spectrum narrow resonances as well as discrete spectrum states were observed. The energies of all these states were comparable with the cyclotron frequency ω. The magnetic field was 6.1 T, and corresponding ω = 5.7 cm−1. These are very high Rydberg excitations and therefore the spectrum of an Li atom probably does not differ substantially from that of an H atom. In recent years a number of theoretical works have been devoted to the investigation of an H atom in a magnetic field (see e.g. a review paper [4]). Progress has been achieved in numerical integration of the Schrödinger equation for the continuous spectrum [5, 6, 7]. In Ref. [8] a detailed comparison of experimental data with the results of numerical solution is carried out. There is good agreement after some averaging over the energy.

Abstract

Modern semiconductor technology has enabled the fabrication of solid-state analogues of one-dimensional atoms. These are electrons confined in quantum wells by a graded band gap. Such structures typically have energy level spacings between 1 and 100 meV, and depths up to 300 meV. The development of a free-electron laser that is tuneable between 0.5 and 20 meV has now made possible the study of such solid-state atoms in oscillating electromagnetic fields with amplitudes sufficient to ionize them at frequencies much smaller than their binding energies. Thus experiments analogous to those carried out on atoms in strong electromagnetic fields can be performed (for example, ionization, harmonic generation). This chapter first introduces the physics of quantum wells, then discusses preliminary experimental results on ionization and harmonic generation from electrons in quantum wells, and finally describes the results of recent computer simulations. The chapter concludes by discussing a number of new issues in the interaction of light with matter which are raised in the study of solid-state atoms.

Introduction

Much of the theoretical work on quantum chaos in periodically-driven systems has been motivated by classic experiments on the microwave ionization in hydrogen. In these experiments, Rydberg hydrogen atoms are driven by microwaves with photon energy hv ≪ ionization energy E1 of the Rydberg atom, and with electric field energies comparable to E1. For hv smaller than the separation between Rydberg levels (scaled frequency < 1), simple classical models predict remarkably well observed ionization thresholds. Ionization is associated with the destruction of classical invariant tori and the onset of global chaotic diffusion.

Introduction

The quantum treatment of systems capable of chaotic motion in the classical limit is interesting for several reasons. First, there is the desire to see how the classical distinguishability between regular and chaotic motion gradually arises as the system is turned more and more classical by changing a suitable parameter. By a controlled increase of quantum mechanical time scales such as wave packet spreading times or inverse level spacings, for instance, one would like to study the growth of the life time of effectively chaotic evolution of suitable observables.

Perhaps an even greater incentive for quantum mechanical investigations lies in the question whether quantum chaos can be more than a mere transient mimicry of classical chaos. If so, we need intrinsically quantum mechanical criteria to distinguish regular an irregular behavior; the relation of such quantum criteria to the traditional ones for classical chaos would have to be clarified.

The statistical study of spectra of quantum systems is almost as old as quantum mechanics itself. One distinguishes two types of properties: global ones and local ones. An example of the former is provided by the density of levels as a function of excitation energy. In this Letter we shall discuss local properties, or more precisely, fluctuations (departures of the energy-level distribution from uniformity). We shall deal with time-independent systems and energies of stationary states.

There exists a well established theory to describe fluctuation properties of quantal spectra, namely the random matrix theory (RMT) initiated by Wigner, developed mainly by Dyson and Mehta, and later extended by several authors. Recently, the predictions of RMT [specifically, the predictions of the Gaussian orthogonal ensemble (GOE)] have been compared in great detail with the whole body of available nuclear data coming mainly from compound-nucleus resonances. No discrepancy between theory and experiment has been detected. In particular the data have been shown to exhibit two of the salient phenomena predicted by the theory—the level repulsion (tendency of the levels to avoid clustering) and especially the spectral rigidity (very small fluctuation around its average of the number of levels found in an interval of given length), which is a property due to correlations between level spacings.

Semiclassical approximations to the Schrödinger equation remain important in a large variety of contexts. They play the dual role of computational tools (when exact calculations are too difficult or unnecessary) and sources of insight and intuition, even if numerical solutions are available. However, classical chaos often spoils the utility of semiclassical methods. Gutzwiller [1] gave a formal connection between periodic orbits (embedded in chaos) and eigenvalues (the trace formula). Although the trace formula is not a practical tool and even divergent, it has been the guiding light in the search for more servicable approaches. A large effort to “quantize chaos,” over many years, has begun to come to fruition. Recent progress has been dramatic, in both the time domain [2-4] and the energy domain [5-8]. Historically, however, the great bulk of the effort in semiclassical methods has taken place in the energy representation.

In 1928, Van Vleck [9] gave the time-dependent coordinate space propagator which was later modified by Gutzwiller to extend beyond caustics [1].

This volume presents a collection of basic papers, some already published others specially written for this volume, devoted to the study of a new phenomenon, the so-called quantum chaos. This problem arose from the by now, well known, classical dynamical chaos. However, unlike the latter, the study of quantum chaos is still in its early stages, attracting the ever growing interest of many physicists (but, unfortunately, of many fewer, as yet, mathematicians).

The original intention, of physicists at least, was mainly to understand the very important generic phenomenon of classical chaos from the viewpoint of the more deep and general quantum mechanics. At first sight it might seem that quantum chaos is simply a particular case of the general phenomenon of dynamical chaos in the well developed ergodic theory of dynamical systems; or it might be a trivial implication of the correspondence principle. Yet, Nature has turned out to be much more tricky, and more interesting!

As the present collection of papers clearly shows, there is no classical-like chaos at all in quantum mechanics. On the other hand since Nature, as is commonly accepted, obeys quantum mechanics, what is then the physical meaning of dynamical chaos? As a result of this surprising obstacle, the general situation in the study of quantum chaos, in the present state of research, might be characterized as some confusion and disorganization which is of course a typical situation in the early stages of a new field of scientific research.

Abstract

Until a short time ago, the study of the quantum-mechanical behaviour of classically chaotic systems was an exclusive domain of the theoretical physicist. Model systems amongst others were irregularly shaped billiards, such as the Sinai and the stadium billiard. An alternative experimental approach to study these systems has recently been demonstrated using the fact that the time-independent Schrödinger and wave equations are mathematically equivalent (though, in general, the boundary conditions are different). This chapter concentrates on the presentation of microwave studies of billiards, but experiments with vibrating plates and water surfaces waves are also briefly discussed. Topics are the statistical properties of spectra, level dynamics with respect to geometrical changes of the billiards, and the close connection between quantum-mechanical spectra and classical trajectories.

Introduction

One day in February 1808 Ernst F. Chladni who was then on a long stay in Paris received an invitation to the Tuileries to give a demonstration of his famous experiments with vibrating plates in the presence of Napoleon and an illustrious audience. The Emperor was impressed by the performance and offered a prize of 3000 francs for the correct mathematical explanation of the sound figures which was still lacking at that time. The prize was paid in 1816 to the French mathematician Sophie Germain, in spite of the fact that her solution was still incomplete. The correct explanation for circular plates was not given until 1850 by Robert Kirchhoff (all the above details are taken from the book of F. Melde on the life of Chladni [1]).

A quantum analogue of the Baker's transformation is constructed using a specially developed quantization procedure. We obtain a unitary operator acting on an N-dimensional Hilbert space, with N finite (and even), that has similar properties to the classical baker's map, and reduces to it in the classical limit, which corresponds here to N → ∞. The operator can be described as a very simple, fully explicit N×N matrix. Generalized Baker's maps are also quantized and studied. Numerical investigations confirm that this model has nontrivial features which ought to represent quantal manifestations of classical chaoticity. The quasi-energy spectrum is given by irrational eigenangles, leading to no recurrences. Most eigenfunctions look irregular, but some exhibit puzzling regular features, such as peaks at coordinate values belonging to periodic orbits of the classical Baker's map. We compare the quantal and classical time-evolutions, as applied to initially coherent quasi-classical states: the evolving states stay in close agreement for short times but seem to lose all relationship to each other beyond a critical time of the order of log2N ∼ − logh.

INTRODUCTION

It is commonly believed that the essential features of chaotic behaviour in the classical Hamiltonian systems are basically understood [1].

The situation is rather different if one turns to the quantal transcription of classical dynamical systems. Classical chaos appears in bound systems, which implies that the quantal Hamiltonian operator has a discrete spectrum.