Quantum chaos and its kinetic stage of evolution

Abstract

Usually reason of irreversibility in open quantum-mechanical system is interaction with a thermal bath, consisting form infinite number of degrees of freedom. Irreversibility in the system appears due to the averaging over all possible realizations of the environment states. But, in case of open quantum-mechanical system with few degrees of freedom situation is much more complicated.Should one still expect irreversibility, if external perturbation is just an adiabatic force without any random features? Problem is not clear yet. This is main question we address in this review paper. We prove that key point in the formation of irreversibility in chaotic quantum-mechanical systems with few degrees of freedom, is the complicated structure of energy spectrum. We shall consider quantum mechanical-system with parametrically dependent energy spectrum. In particular, we study energy spectrum of the Mathieu-Schrodinger equation. Structure of the spectrum is quite non-trivial, consists from the domains of non-degenerated and degenerated stats, separated from each other by branch points. Due to the modulation of the parameter, system will perform transitions from one domain to other one. For determination of eigenstates for each domain and transition probabilities between them, we utilize methods of abstract algebra. We shall show that peculiarity of parametrical dependence of energy terms, leads to the formation of mixed state and to the irreversibility, even for small number of levels involved into the process. This last statement is important. Meaning is that, we are going to investigate quantum chaos in essentially quantum domain.In the second part of the paper, we will introduce concept of random quantum phase approximation. Then along with the methods of random matrix theory, we will use this assumption, for derivation of muster equation in the formal and mathematically strict way.

Dynamics of the Dicke model close to the classical limit

“…The perceived irreversibility thus is a consequence of averaging over diverging trajectories starting from nearby phase space points. For quantum chaotic systems with few degrees of freedom this kind of irreversibility is linked to the complex energy spectrum [43], as revealed in random matrix theory [4]. True irreversibility, involving the approach to a stationary equilibrium state, requires coupling to an infinite number of degrees of freedom provided, e.g., by a bath or the environment [44,45].…”

Random matrix theory of the isospectral twirling

“…Such a measure of chaos is impractical for a quantum many-body system, as it requires a perfect knowledge of the spectrum of the Hamiltonian, and is quite indirect. After all, we still would not know what quantum chaos means [21][22][23][24][25][26][27][28].…”

Generation of coherence in an exactly solvable nonlinear nanomechanical system

“…ω , ϕ → 2ϕ, respectively. A detailed analysis of the Mathieu-Schrödinger equation (19) was done in numerous works, for example the references [43][44][45][46] . The energy spectrum of the Mathieu-Schrödinger equation depends parametrically on the potential barrier E n l and contains two degenerate G − , G + and one non-degenerate domain G 0 (See Fig.…”