In the preceding chapters we have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems, and this in spite of the oversimplifying assumptions applied. On the one hand it is very satisfactory that one single theory can cope with such a variety of systems as nuclei, mesoscopic structures, or microwave billiards, on the other hand this is a bit disappointing. If there is no possibility of discriminating between the spectra of a nucleus and a quantum dot, then there is little hope of learning anything of relevance about it. Fortunately, random matrix theory is only one side of the coin. We have already come across some examples demonstrating its limits of validity. Remember the spectral level dynamics where bouncing balls disturbed the otherwise universal Gaussian velocity distribution (see Section 5.2.2). Another example is the scarring phenomenon observed in many wave functions. Here obviously closed classical orbits have left their fingerprints in the amplitude patterns. We cannot expect that the universal random matrix theory can correctly account for individual features such as periodic orbits. We now come to an alternative approach to analysing the spectra. As we know from the correspondence principle, in the semiclassical limit quantum mechanics eventually turns into classical mechanics. That is why classical dynamics must be hidden somewhere in the spectra, at least in the limit of high quantum numbers. In the introduction we have already discussed this connection for a particle in a one-dimensional box.
In the preceding chapters we have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems, and this in spite of the oversimplifying assumptions applied. On the one hand it is very satisfactory that one single theory can cope with such a variety of systems as nuclei, mesoscopic structures, or microwave billiards, on the other hand this is a bit disappointing. If there is no possibility of discriminating between the spectra of a nucleus and a quantum dot, then there is little hope of learning anything of relevance about it. Fortunately, random matrix theory is only one side of the coin. We have already come across some examples demonstrating its limits of validity. Remember the spectral level dynamics where bouncing balls disturbed the otherwise universal Gaussian velocity distribution (see Section 5.2.2). Another example is the scarring phenomenon observed in many wave functions. Here obviously closed classical orbits have left their fingerprints in the amplitude patterns. We cannot expect that the universal random matrix theory can correctly account for individual features such as periodic orbits. We now come to an alternative approach to analysing the spectra. As we know from the correspondence principle, in the semiclassical limit quantum mechanics eventually turns into classical mechanics. That is why classical dynamics must be hidden somewhere in the spectra, at least in the limit of high quantum numbers. In the introduction we have already discussed this connection for a particle in a one-dimensional box.
This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of N indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit ℏeff=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.
From the very beginning classical nonlinear dynamics has enjoyed much popularity even among the noneducated public as is documented by numerous articles in well-renowned magazines, including nonscientific papers. For its nonclassical counterpart, the quantum mechanics of chaotic systems, termed in short ‘quantum chaos’, the situation is completely different. It has always been considered as a more or less mysterious topic, reserved to a small exclusive circle of theoreticians. Whereas the applicability of classical nonlinear dynamics to daily life is comprehensible for a complete outsider, quantum chaos, on the other hand, seems to be of no practical relevance at all. Moreover, in classical nonlinear dynamics the theory is supported by numerous experiments, mainly in hydrodynamics and laser physics, whereas quantum chaos at first sight seems to be the exclusive domain of theoreticians. In the beginning the only experimental contributions came from nuclear physics [Por65]. This preponderance of theory seems to have suppressed any experimental effort for nearly two decades. The situation gradually changed in the middle of the eighties, since when numerous experiments have been performed. An introductory presentation also suited to the experimentalist with no or only little basic knowledge is still missing. It is the intention of this monograph to demonstrate that there is no reason to be afraid of quantum chaos. The underlying ideas are very simple. It is essentially the mathematical apparatus that makes things difficult and often tends to obscure the physical background.
The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform(ρL) of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.
The fundamental correspondence between quantum chaotic single-particle systems and random matrix theory is well-understood via periodic orbit theory. In contrast, we show that many-body systems with explicit subsystem structure possess characteristics different from the single-particle theory. We present a periodic orbit theory for many-body systems with well defined semiclassical limit. For this we identify periodic orbit families arising exclusively in the many-body setting and implement a central limit theorem characterizing their correlations. Based on this we demonstrate that spectral correlations in chaotic quantum circuits are characterized by the breaking of individual time translation invariance of periodic orbits in the subsystems into residual synchronous time translations only. This provides a systematic approach to confirming random matrix universality in deterministic many-body systems.
Abstract Semiclassical mechanics is a short-wavelength link between classical and quantum mechanics, similar to that between wave and geometrical optics. Its object is to maintain contact with the structure imposed by classical mechanics, without sacrificing quantum mechanical accuracy, or, in the vivid words of Berry and Mount (1972), to ‘sew quantum mechanical flesh onto classical bones’. The overriding principles are that the classical phase space is divided into ‘accessible’ and ‘inaccessible’ regions characteristic of the type of motion, and that the boundaries of these regions impose a classical structure, which is evidenced in the quantum mechanics by interference in the bright accessible regions and exponential decay into the shadow regions. The strength of the classical skeleton is that this structure is maintained whether the observations are made in coordinate space, quantum number space, or some mixed representation. In all cases the central quantity is the classical action, whose dimensions are those of Planck’s constant; on one hand it guides the classical motion, according to the minimum action principle, and on the other it adds a phase to each classical trajectory, which gives rise both to quantum mechanical interference and, by a mathematical extension, to quantum mechanical tunnelling.