This project is concerned with the statistical properties of mesoscopic systems. Both simple, few-degree-of-freedom systems and many-body quantum systems are studied. Fundamental issues related to the advancement of classical, semiclassical, and statistical methods are addressed, impacting a broad spectrum of physical systems and providing a new approach to their understanding. The author proposes to investigate: i) ground state fluctuations due to electron-electron interactions; ii) analytic approximations to the Kolmogorov-Sinai entropy in many-body interacting systems; iii) the possibility of using the Herman-Kluk propagator in understanding localizing features and transport in systems with mixed phase spaces; iv) the statistical properties of extreme values of eigenstates; and v) fluctuations in classical sum rules. These problems are approached through dynamical system theory, semiclassical, statistical, and mean-field techniques that combine synergistically. The PI has made a number of significant contributions to these fields.
Broader Impacts: There is both a strong connection to experiments and a strong emphasis on multi-disciplinary studies. Techniques developed in the context of so-called quantum chaos have been applied in acoustics, nanophysics, nuclear physics, optics, and the study of many-body systems. In turn, new ideas are flowing back into quantum chaos, acting as a focal point for cross-fertilization of disparate research fields. Students show a great deal of interest in chaos studies, and this helps attract/retain bright students. Their classroom experiences incorporate some of the latest advances. The research training of postgraduate associates and/or graduate students in dynamical systems theory, statistical methods, and many-body physics prepares them for a wide variety of future pursuits. A member of an underrepresented group in physics will be supported as a Ph. D. student by this grant. Support for this work leads to dissemination of ideas between many European, Asian, former Soviet, and South American researchers.
This project was focussed on research into the statistical properties of quantum systems that are chaotic, and may contain many-interating particles. It resulted in the completion of a broad range of research topics. They include: i) long range propagation of sound in deep ocean water - a new statistical method was developed for simulating experiments that enable one to extract information from the statistical properties found in data. The methods turn out to be quite analogous to those found in the study of metal-insulator transitions, which may lead to importing results from one domain into the other; ii) theory of scanning gate microscopy - since the 1980's there has been a tremedous advance in the development of many new microscopies. Often, it is difficult to know exactly how to interpret experimental results. We gave a foundational theory for scanning gate microscopy, which is, for example, important for nanoscale semiconductor systems; iii) Friedel-like oscillations in microwave cavity - we applied new theoretical results on the wave oscillations of particles new boundaries to data taken in microwave cavities. They worked perfectly; iv) entanglement of two subsystems - quantum mechanics allows for strange correlations known as entanglement. We used statistical methods adopted both from the study of chaotic and strongly interacting systems, and the study of extreme values to identify an important entanglement transition with particle number in the two subsystems; v) wave transport in correlated disorder - many wave mechanical systems (electrons, sound, light) have particles traveling through random media. When that randomness has correlations new behaviors can exist. We developed a remarkably simple model that could be analyzed exactly with novel transparency behaviors in certain regimes; and vi) fluctuations and entropies in chaotic systems - the analytic structure of fluctuations in sum rule quantities and entropies were found and analyzed in a number of cases. The results were also used to locate small regions of stable motion embedded in otherwise chaotic motion. The project led to the professional/research training of several undergraduate and graduate students, and postdocs. The research led to the publication of 11 scientific papers in journals, a book chapter, and a book review. There were 26 invited colloquia, conference talks, and research seminars on this work, and 1 course given at an international school of physics.
