Loschmidt echo and the stochastic quantum dynamics of nanoparticles

Abstract

The time evolution of a prepared vibrational state (system) coupled to a reservoir with a dense spectrum of its vibrational states has been investigated under the assumption 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) a multicomponent Loschmidt echo in recurrence cycles; and (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 regime (iii) due to an unavoidably, in any real system, coarse graining of the time or energy measurements or the initial condition uncertainty. At any finite accuracy, one can always find the cycle number k c when the dynamics of the system for k>k c cannot be determined uniquely from the spectrum, and, in this sense, the long-time system evolution becomes chaotic. Even though a specific toy model is investigated here, when properly interpreted, it yields quite a reasonable description for a variety of physically relevant phenomena, such as the complex vibrational dynamics of nanoparticles with characteristic interlevel spacing on the order of 10 cm−1 observed using subpicosecond spectroscopy methods.