Ebook: New Directions in Quantum Chaos

The first Enrico Fermi summer school on “Quantum Chaos” was conducted in 1991, when quantum chaos just emerged as a recognized branch of research. The rapid progress of the field and its many applications called for a second school, in which the students could have a chance to keep abreast of the new developments, being at the same time provided with a solid basis in subjects which form the canon of the field. Because of obvious limitations, the school could not give due exposure to all the topics which were actively pursued in the past decade. Rather, it was limited to a few broad subjects: i) Spectral statistics and their semiclassical interpretation in terms of the Gutzwiller trace formula. Since its very early days, the research in quantum chaos tried to unravel the dynamical origin of the universal spectral statistics which are observed in generic quantum systems whose classical counterparts are chaotic. In particular, these fluctuations are very well reproduced by Random Matrix Theory, and therefore, Random Matrix Theory is a recurrent subject in quantum chaos. F. Haake reviewed Random Matrix Theory in one of the basic courses in this school, and kept a welcome balance between the need to cover background material, and the exposition of some of the new results obtained by him and his group.

The main theoretical tool for establishing the connection between spectral analysis and the underlying classical dynamics is the semiclassical (Gutzwiller) trace formula. A very important tool in this study is the spectral ζ function, whose semiclassical approximation is derived from Gutzwiller's trace formula. The periodic orbits which form the classical data base for the semiclassical theory were discussed by R. Artuso. In this set of lectures, the elements of periodic orbit theory for hyperbolic dynamics were reviewed, and a uniform description of the classical and the quantum ζ function was presented.

The connection between spectral statistics and the properties of classical periodic orbits was discussed by E. Bogomolny. His lecture emphasizes the close relation to a theory which was developed for the study of the distribution of the zeros of the Riemann ζ function. This presentation can be considered as a natural continuation and extension of the work described in J. Keating's contribution to the Proceedings of the 1991 summer school (the Riemann zeta-function and quantum chaology).

In introducing the subject of universal spectral statistics, we were careful about qualifying the statement to “generic” systems. Indeed, some of the best studied classically hyperbolic maps, the linear automorphisms of the two-torus (known as “cat maps”) are non-generic in this sense. The lecture of J. Keating discussed the quantization of this class of maps, and explained the reasons of their non-generic quantum statistics.

The leading semiclassical theory, on which Gutzwiller's trace formula is based, breaks down in the vicinity of classical bifurcations. The effects of bifurcations on various spectral statistics were discussed in M. Berry's lectures. This problem is of special significance since bifurcations are abundant in mixed systems which are neither integrable nor hyperbolic. The semiclassical trace formula for mixed systems is not known, and the systematic research is in its infancy.

A deterministic classical system reveals its chaotic nature only after it evolved during a sufficiently long time. One can construct chaotic systems for which this time scale can be made arbitrarily long. T. Prosen studied the effects of this classical feature on the corresponding quantum spectra and their statistics. His observations give an example of an increasing number of works where deviations from the predictions of Random Matrix Theory are discussed and explained on classical grounds. ii) Quantum chaos and its applications in mesoscopic physics—spectral statistics and conductance fluctuations. One of the most important developments in quantum chaos in the last decade was the application of its ideas in the field of mesoscopic physics. Both fields address similar issues and concepts, which pertain to quantum systems with few effective degrees of freedom. However, there is an important difference between their approaches: in quantum chaos one analyses the statistics of a single systems, using energy averaging to define the statistical ensemble. In mesoscopics the common practice is to consider an ensemble of systems which differ in the distribution of impurities in a single energy domain in the vicinity of the Fermi energy. Typically, the Fermi wavelength is much smaller than the size of the mesoscopic devices, which justifies the use of the semiclassical approximation in this context. Disorder averaging is performed using the powerful field-theoretical methods which are discussed in A. Mirlin's lectures. Mirlin introduces the non-linear σ model, and also discusses the extent to which it can be applied for the analysis of a single system. One can get an impression of the fertile exchange between quantum chaos and mesoscopics by comparing Bogomolny's periodic orbit approach and Mirlin's field-theoretical discussion of the spectral form factor.

Conductance in mesoscopic systems is one of the most prominent observables which show the fingerprints of the underlying classical chaotic dynamics. R. Jalabert discusses this issue at length, and makes use of the semiclassical theory of chaotic scattering in this context.

The quantum dynamics of extended systems is another issue of distinct relevance to mesoscopic physics. The motion of a quantum particle in a periodic, quasi-periodic, or disordered potential is the central problem of Localization Theory, which was historically the first meeting point between Quantum Chaos and Solid State Physics. Here, the central issue is the spectral type of the quantum Hamiltonian, which can be pure-point or continuous, leading to qualitatively different dynamics; it can also exhibit a metal-insulator transition from one spectral type to the other. The case when the spectrum is singular continuous, that is, multifractal, is of special interest, leading to anomalous diffusion. In T. Geisel's lectures some general results about the connection between the multifractal spectral structure and anomalous diffusion were presented; moreover, the role of classical chaos in determining the quantum spectral type was elucidated. It was in fact shown that the avoided level crossings, which characterize the spectra of bounded, classically chaotic systems, in the case of extended systems are replaced by avoided band crossings, which can sometimes lead to metal-insulator transitions, and to the appearance of singular continuous spectra. iii) Quantum chaos in systems with many degrees of freedom. The research in quantum chaos in the past years concentrated mostly on systems with the minimum number of degrees of freedom which allow classically chaotic dynamics. Only recently, systems with a larger number of freedoms are considered, and this will probably be one of the most important directions in future research. In systems composed of many particles, the identity of the constituents is an important symmetry which affects the quantum treatment through Pauli's principle. Its implementation in the semiclassical trace formula was discussed in A. Weidenmüller's talk as one of the special features which should be considered when the techniques of quantum chaos are to be extended to many-body systems. In most many-body systems, the particles interact pairwise, so that the Hamiltonian matrix is sparse in the Fock space representation. What will the resulting spectral statistics be? Will it approach the predictions of Random Matrix Theory in detail? Are there deviations which originate from the classical dynamics? These are issues which are addressed especially in F. Izrailev's lecture, using concepts and methods which where developed for the study of atomic and nuclear spectra.

Another important issue in the description of complex many-body systems is the possibility of reducing the problem by treating a few collective degrees of freedom which are singled out by the dynamics. A. Weidenmüller gave an example of a system where such a reduction seems to be possible. However, the reduced system which consists of the special degrees of freedom remains in interaction with the other degrees of freedom, inducing dissipation in the reduced dynamics. The treatment of quantum systems in interaction with a “bath” is an important problem which was discussed by D. Cohen. This series of lectures can be considered as an elaboration of R. Graham's lectures on “Dynamical localization, dissipation and noise” which was given in the 1991 school.

M. Raizen's lectures on the manifestation of quantum chaos in experiments with cold atoms demonstrated beautifully the applications of quantum chaos in the design and analysis of actual experiments. The rather schematic “kicked rotor” system (see S. Fishman's review on quantum localization in the Proceedings of the 1991 school) is modeled by cold atoms in interaction with crossed laser beams. A tour de force of experimental ingenuity which illustrates quantum chaos in a profound and elegant way.

In this short introduction we tried to explain the line of thought which guided us in planning this school—a line which connects and continues past and present achievements and prepares the ground for a future full of intriguining and important developments. We hope that this volume of lecture notes will be of use and help for the many young scientists which join this field.

G. Casati, I. Guarneri and U. Smilansky