  Semiclassical results are usually expected to be valid in the semiclassical
regime. An interesting question is, in models in which appropriate effective
Planck constants can be introduced, to what extent will a semiclassical
prediction stay valid when the effective Planck constant is increased? In this
paper, we numerically study this problem, focusing on semiclassical predictions
for the decay of the quantum Loschmidt echo in deep quantum regions. Our
numerical simulations, carried out in the chaotic regime in the sawtooth model
and in the kicked rotator model and also in the critical region of a 1D Ising
chain in transverse field, show that the semiclassical predictions may work
even in deep quantum regions, in particularly, for perturbation strength in the
so-called Fermi-Golden-rule regime.
