Multicomponent loschmidt echo and mixing in the quantum dynamics of systems with dense discrete spectra

Abstract

A dynamic problem for a system coupled to a reservoir possessing a dense discrete spectrum of states has been analytically solved under two simplifying assumptions proposed by Zwanzig [15], according to which the reservoir spectrum is equidistant and all the system-reservoir coupling matrix elements are identical (i.e., independent of reservoir states). It is demonstrated that a multicomponent Loschmidt echo arises in each recurrence cycles, the number of components being equal to the cycle number. At a certain critical cycle number, the components of neighboring cycles exhibit mixing. As a result, the dynamics of the system transforms from a regular to stochastic-like dynamics, in which an arbitrarily small coarse graining (inherent in any real system) of the results of measurements or uncertainty in the initial conditions leads to (i) the loss of one-to-one correspondence between the discrete spectrum of eigenvalues and the state of the system and (ii) the loss of invariance with respect to the time reversal. Interrelation between the mixing of cycles with the entanglement of trajectories, on the one hand, and the overlap of resonances in classical systems with mixing, on the other hand, is discussed. The properties of the proposed model are consistent with a variety of the kinetic regimes of vibrational relaxation (from exponential decay to irregular, weakly damping oscillations) observed in various objects. Common features are average distances between neighboring levels on the order of 1–10 cm−1 and recurrence cycles on a time scale of 10–13–10−11 s, which is studied using femtosecond spectroscopy techniques.