Shpyrko S. Quantum chaos in electron-phonon systems

The object of the present work are quantum objects with complicated interaction between electron and phonon subsystems in them (for example, systems of electrons and heavy nuclei in molecules), whose essential behaviour cannot be described within the adiabatic approximation by artificially separating these two subsystems. In particular, this is the case of molecules with degenerate electron levels, interacting with vibrational degrees of freedom from nuclei, which, in their turn, can split the electron levels (Jahn-Teller effect). We investigate the generalized systems, combining two traditionally considered Jahn-Teller and Dicke model classes and characterizing by the interaction between the degenerated electron subsystem and two oscillatory modes of the optical (intramolecular) phonons with generally different interaction constants. We study analytically and numerically the ground state of such systems and reveal the domains of different behaviour of the system in the space of the model parameters depending on the relation between the coupling constants -- the regime of localization of the ground state function on the lower electron level (after level splitting) or tunneling between levels. We investigate the spectrum of the excited phonon states of this system. Using two variants of the level statistics, namely, the nearest-neighbour and long-range correlations (in the energy space) we establish the criteria for development of the quantum chaos in such systems and universal properties of a limiting statistical distribution, observed in a wide range of the coupling constants values. Further the system is generalized to the chain of identical Jahn-Teller molecules or to a molecule with several degenerated electron levels and the quantum phase transition between localization and radiant phases is studied.