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.

The dynamics of a time-dependent quantum system can be qualitatively different from that of its classical counterpart when the latter is chaotic. It is shown that small noise can strongly alter this situation.

What is the nature of a quantum system whose classical counterpart exhibits chaotic dynamics? The subfield dealing with this question has been called quantum chaos. A striking result in quantum chaos has been obtained by Casati et al. These authors considered a particular Hamiltonian and a potential representing periodic impulses kicking the system. If the strength of the kicks is large enough, then, in the classical description the motion is chaotic, and the momentum variable, p, behaves diffusively. That is, the average value of p2 apparently increases linearly with time. Casati et al. considered numerically the quantum mechanical version of the same problem with ħ small. They found that for early times, the average value of p2 increased linearly with time at roughly the classical diffusive rate, but that for long time this linear increase slowed and eventually appeared to cease. Thus, there was no numerically discernible diffusion in the quantum case.

The observed saturation of the growth of 〈p2〉 is understandable if the Schrödinger operator for this problem has an essentially discrete quasienergy level spectrum. Recently, Fishman, Grempel, and Prange have presented strong arguments supporting the idea that the quasienergy spectra for systems of the type studied by Casati et al. are essentially discrete. These arguments are based on an analogy with Anderson localization of an electron in a solid with a random lattice.

Advances in the study of the chaotic behavior of strongly coupled and strongly perturbed nonlinear oscillators has lead many researchers to the question of how quantum mechanics modifies the chaotic classical dynamics in corresponding quantum systems like atoms or molecules in strong fields. Until Heller's detailed studies of the stadium billiard, highly excited states of chaotic systems were expected to resemble random functions in configuration space. However, Heller found that typical eigenfunctions were often strongly peaked along the paths of unstable classical periodic orbits (PO). In the past year these so-called “scars” of the PO have also been identified in several other systems that are classically chaotic, where the peaking of the quantum probability near PO was found to be even more pronounced in the phase-space (PS) representation of the wave functions. These quantum PS distributions also revealed that the stable and unstable manifolds associated with the PO also play an important role in determining the regular structure of the eigenfunctions.

Abstract

We consider the conductance g of disordered systems where the electronic quantum coherence extends over a large scale. In the first part, we show characteristic conductance fluctuations driven by a variation of the applied magnetic field B or of the Fermi energy EF, which have been observed at very low temperature in a mesoscopic wire where the carrier density is controlled by a gate. Following the gate voltage, the wire is a conductor (g ≫ 1) or an insulator (g ≪ 1). The fluctuations of g have a normal distribution with a universal variance for conductors and a very large log-normal distribution for insulators. In a macroscopic insulator, the magnetoconductance is mostly governed by the field dependence of the localization length §. In the second part, we review a random matrix theory adapted to the transfer matrix. This macroscopic approach to quantum transmission allows us to describe in a unified and simple way the conductance fluctuations observed in conductors and insulators, and to predict new universal symmetry breaking effects on the variance of g and on §. This approach, based on symmetry considerations and on a maximum entropy criterion, gives the eigenvalue distribution of t.t (t is the transmission matrix) in terms of a simple Coulomb gas analogy. In quasi-one dimension, the analogy is valid for conductors and insulators. Outside quasi-one dimension, we derive analytically the eigenvalue correlation functions of our maximum entropy model that we compare to their direct numerical evaluations from microscopic hamiltonians.

As described in the preceding Letter, hereafter referred to as DBG, the hydrogen atom in a magnetic field has attracted unusual interest because it is among the simplest nonseparable systems that are physically realizable, because it is one of the small number of systems whose classical motion displays chaotic behavior in regimes where accurate quantum-mechanical calculations are possible, and because it can be studied experimentally with high precision. The simplicity of this problem is deceiving, however, for carrying forward theory and experiment have both proven to be formidable undertakings. DBG describes a breakthrough in the problem of calculating the positive-energy spectrum at laboratorysized magnetic fields. We report here the results of a comparison of calculated spectra with spectra observed experimentally by the MIT group who are the co-authors of this joint paper.

The most successful previous study of this kind was a comparison of the observed and computed spectrum for deuterium by Holle et al. for energy in the range of −190 to −20 cm−1. However, the experimental resolution was too low to achieve fully resolved spectra at the highest energies, and the computational method was limited to the negative-energy region. The work described here overcomes these limitations.

In this chapter experiments on rubidium Rydberg atoms in external fields are reviewed [1 – 8]. In the first part results on the interaction of microwave radiation with rubidium Rydberg atoms are described. The Rydberg atoms interact with a microwave pulse of a well-defined duration. When applying coherent microwave radiation with a frequency exceeding a certain value, the experimentally observed microwave field strength at which 10% of the atoms are ionized has to be much larger than the value found by classical trajectory calculations. The atoms are stabilized by the dynamical localization. An extremely weak microwave broadband noise which is added to the coherent wave at least partially destroys the localization. This phenomenon depends on the bandwidth of the added noise. In further experiments the dependence of the 10% ionization microwave field strength on the duration τ of the microwave pulse has been investigated. For increasing noise level the 10% ionization field strength ∈0.1 develops from a behavior ∈0.1 ∼ τ−1/4 to a ∈0.1 ∼ τ−1/2 dependence, which is characteristic for a classical diffusion process.

The second part of the chapter deals with experiments in strong static crossed electric and magnetic fields. Below as well as above the ionization energy the observed quasi-Landau(QL)-resonances are explained using classical trajectories. The ionization behavior of the QL-resonances is clearly influenced by the spatial structure of the associated classical orbits. In further experiments the dependence of the observed ionization energy on the magnetic field was investigated in the presence of a fixed static electric field.

Abstract

We present and review several results on the semiclassical quantization of classically chaotic systems. Using Feynman path integrals and the stationary phase method, we develop a new semiclassical theory for quantum trace formulas in classically hyperbolic systems. In this way, we obtain corrections to the Gutzwiller-Selberg trace formulas like asymptotic series in powers of the Planck constant. The coefficients of these series are expressed in terms of Feynman diagrams. We illustrate our method with the calculation of resonances for the wave scattering on hard disks.

Introduction

In recent years, important advances have been made in our understanding of the dynamical behaviours which are intermediate between the classical and the quantal regimes [1-7]. The general context of these advances is the current research in mesoscopic physics and chemistry, which is bridging the gap between the microscopic and the macroscopic worlds. Systems which have been particularly studied in this respect include highly excited or reacting atoms and molecules as well as nanometric semiconductor devices, among many other systems [8-19].

In this chapter, we consider quantum mechanics in the semiclassical regime where the actions of the dynamical processes in question are larger than the Planck constant. This eikonal or short-wavelength assumption allows us to solve the Schrödinger equation using information on the classical orbits of a system and more especially on their periodic orbits. In this way, we can understand in detail how classical mechanics emerges from quantum mechanics.

Introduction

In recent years a number of experiments on the ionization of Rydberg (highly excited) atoms and dissociation of molecules by a strong monochromatic field have been carried out [1-5]. A characteristic peculiarity of such processes is the large number of absorbed photons Nφ 100 and the excitation of many unperturbed levels. Due to this the dynamics of excitation may be described in the first approximation by the classical equations of motion. Such an approach was used for molecules in ref. 6 and for Rydberg atoms in ref. 7. The process of excitation obeys the diffusion law. The appearance of diffusion in the absence of any random forces is connected with the chaotic dynamics of the corresponding classical system. The nature and the properties of such chaotic motion in classical mechanics is now well understood [8–10]. At the same time an investigation of simple models has shown that the dynamics of classically chaotic quantum systems has a number of peculiarities (see, e.g., refs. 9, 11 and 12).

Abstract

We define a classical scattering process as a canonical map between the phase spaces of incoming and outgoing channels. Time-reversal invariance is introduced and the iterated scattering map as defined by Jung appears naturally. Using the theory of unitary representations of classical canonical transformations the transition to the quantum problem is achieved in the framework of a semiclassical approximation. Families of canonical scattering maps are defined by canonical transformations on channel space; the invariance properties of these families translate naturally into invariant ensembles of S-matrices that are unique according to Dyson. The concept of structural invariance of an iterated scattering map or Poincaré scattering map is introduced to take into account the generic features of the scattering system on hand. We shall argue that, if this invariance is present in our map, its unitary representation is a characteristic member of the ensemble and thus by way of ergodicity the statistical properties of the corresponding S-matrix are those of the circular ensemble known as COE.

Introduction

The statistical behaviour of scattering processes has been of interest for many years in nuclear physics [1], but recently it has acquired more general importance in the context of quantum manifestations of chaos [2]. In this context one question asked, refers to the statistical distribution of the eigenphases of the S-matrix. Others are of more practical interest, but are also more complicated, such as average fluctuating cross sections and their variances and correlations.

Introduction

Classically chaotic quantum systems in general can exhibit intrinsinc chaotic behaviour on a quite restricted time scale [1], In particular localization phenomena provide direct evidence exhibiting the suppression of quantum ergodicity [2]. On the other hand, recent studies reveal that a many-dimensional system can mimic some aspects peculiar to chaotic dynamics quite well on an unexpectedly long time scale [3]. A remarkable feature that distinguishes chaotic motion from integrable motion is the sensitive dependence of dynamics on external perturbations. For classical dynamical systems such a sensitivity can be well defined by measuring how two nearby trajectories in the phase space separate in time. However, there has been proposed no systematic way to examine the sensitivity of quantum dynamics to external perturbation. In quantum dynamics, it is not possible to trace a well defined trajectory in phase space, because both quantum uncertainty and chaotic instability make a localized wavepacket spread suddenly over the phase space [4]. There is, however, a simple way in which we may test the quantum sensitivity quantitively. This is the time reversal experiment described below. In this chapter I describe several remarkable characteristics of quantal instability which have been clarified by the time reversal experiment.

Abstract

A few years ago we found the existence of an unbounded quantum mechanical diffusion process which strongly contrasted with the dynamical localization of the kicked rotator and other previously studied quantum systems. Here we review how this phenomenon is related to multifractal properties of the spectrum, its level statistics, and to the algebraic decay of correlations. We investigate the influence of classical chaos on uncountable fractal spectra and demonstrate that the concepts of level statistics can be related to multifractal concepts in these cases. First we point out a new class of level statistics where the level spacing distribution follows inverse power laws p(s) ∼ s−β with 1 < β < 2 and β = 1 + D0 where D0 is the fractal dimension of the spectrum. It is characteristic of hierarchical level clustering rather than level repulsion and appears to be universal for systems exhibiting unbounded quantum diffusion where the mean square displacement increases as t2δ with δ = β − 1. A realization of this class with β = 3/2 is a model of Bloch electrons in a magnetic field (Harper's equation), for which lateral surface superlattices on semiconductor heterojunctions presently serve as experimental realizations. While here diffusion is linear in time (δ = 1/2), in the Fibonacci chain model the spread of wave packets also shows anomalous diffusion (δ ≠ 1/2) with 0 < δ < 1.

We present a short introduction to various types of spin models commonly used in statistical mechanics. Different types of interactions are discussed, and we report on some of the known results on the possibility of phase transitions for spin systems with a certain interaction. The equivalence between symbolic stochastic processes and one-dimensional spin systems is elucidated. Finally we present a short introduction to the transfer matrix method.

Various types of spin models

A ferromagnet can be regarded as a system of a large number of elementary magnets that are placed at the sites of a crystal lattice. To model and understand the magnetic properties of solids, various types of lattice spin models have been introduced in statistical mechanics. Such a model of a spin system is defined by the lattice type, the possible values of the spins at each lattice site, and the interaction between the spins.

In this context the name ‘spin’ is used in quite a general sense: the spin is a random variable with certain possible values. These may be either discrete or continuous; moreover, in this generalized sense, the spin can be a scalar, vector, or tensor. Also the lattice, originally introduced to model the crystal structure of the solid, sometimes does not possess a physical realization in nature.

Quite common choices of lattice structures are cubic, triangular, or hexagonal lattices in dimensions d = 1, 2, 3, 4,… But one also studies so called hierarchical lattices, which have a self-similar structure.