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

V. A. Benderskii, L. A. Falkovsky^{+}, E.I.Kats^{*+}

*Institute of Problems of Chemical Physics RAS, 142432 Chernogolovka, Moscow Region, Russia*

^{+}L.D. Landau Institute for Theoretical Physics RAS, 117334 Moscow, Russia and

Institute of the High Pressure Physics RAS, 142190 Troitsk, Moscow Region, Russia

^{*}Laue-Langevin Institute, F-38042 Grenoble, France

PACS: 05.45.-a, 72.10.-d

**Abstract**

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.