  We introduce the quantitative measures characterizing the rates of
decoherence and thermalization of quantum systems. We study the time evolution
of these measures in the case of a quantum harmonic oscillator whose relaxation
is described in the framework of the standard master equation, for various
initial states (coherent, `cat', squeezed and number). We establish the
conditions under which the true decoherence measure can be approximated by the
linear entropy $1-{Tr}\hat\rho^2$. We show that at low temperatures and for
highly excited initial states the decoherence process consists of three
distinct stages with quite different time scales. In particular, the `cat'
states preserve 50% of the initial coherence for a long time interval which
increases logarithmically with increase of the initial energy.
