Loschmidt echo and stochastic-like quantum dynamics of nano-particles

We investigate time evolution of prepared vibrational state (system) coupled to a reservoir with dense spectrum of its vibrational states. We assume that the reservoir has an equidistant spectrum, and the system – reservoir coupling matrix elements are independent of the reservoir states. The analytical solution manifests three regimes of the evolution for the system: (I) weakly damped oscillations; (II) multicomponent Loschmidt echo in recurrence cycles; (III) overlapping recurrence cycles. We find the characteristic critical values of the system - reservoir coupling constant for the transitions between these regimes. Stochastic dynamics occurs in the regime (III) due to inevoidably in any real system coarse graining of time or energy measurements, or initial condition uncertainty. At any finite accuracy one can always find the cycle number $k_c$ when dynamics of the system for $k>k_c$ can not be determined uniquely from the spectrum, and in this sense long time system evolution becomes chaotic. Even though a specific toy model is investigated here, when properly interpreted it yields quite reasonable description for a variety of physically relevant phenomena, such as complex vibrational dynamics of nano-particles, with characteristic inter-level spacing of the order of 10 cm$^{-1}$, observed by sub-picosecond spectroscopy methods.