Quantum localization in few and many-body chaotic systems

The core of our research project will be the study of quantum or dynamical localization in chaotic quantum systems, either one-particle systems like billiards, or many-body systems like the famous and very important Dicke model. The latter one comprises coupling between the N two-level atoms with the electromagnetic field, the bosonic part of the field sector. It exhibits important and fascinating phenomena like transition from the radiant phase to the superradiant phase (laser), and the quantum phase transition at zero temperature. It is an example of a system without a classical limit. Nevertheless, a corresponding classical nonlinear Hamiltonian with smooth potential can be constructed by means of coherent states, and it is indeed a chaotic system. We want to generalize this approach to other many-body systems without direct classical limit. The central questions addressed in all above mentioned systems with classically chaotic counterpart, to be studied theoretically using the semiclassical methods, are as follows. Show the existence of localization of chaotic Husimi functions, if the Heiseberg time is shorter than classical transport time. Show that the localization measure (entropy localization measure or equivalent ones) obeys the beta distribution, if there are no stickiness effects, otherwise it is nonuniversal, typically bimodal or even multimodal distribution. Show that the level spacing distribution of chaotic levels obeys the Brody distribution and in particular exhibits the fractional power-law level repulsion with exponent beta. Show that beta is a linear function of the mean localization measure. Investigate the role of the Heisenberg time in time independent and time dependent picture. Use Berry's random function model to derive the beta distribution. Show that the entropy localization measure, the correlation localization measure and the normalized inverse localization measures are equivalent. Demonstrate these phenomena numerically in the above mentioned model systems with higher accuracy and statistical significance. Study the effects of stickiness in classical and quantum chaotic systems. We have a highly sophisticated numerical software package for the calculation of classical and quantum billiards, developed by Batistic and Lozej, which enables us to calculate millions of highly accurate levels and associated eigenfunctions and Poincare -Husimi functions. We shall also study quantum chaos in the time-dependent domain, especially the out-of-time order correlator (OTOC) and the role of Ehrenfest time and Heisenberg time, which is recently of great interest also in the quantum field theories.