  The decoherence of a two-state system coupled with a sub-Ohmic bath is
investigated theoretically by means of the perturbation approach based on a
unitary transformation. It is shown that the decoherence depends strongly and
sensitively on the structure of environment. Nonadiabatic effect is treated
through the introduction of a function $\xi_k$ which depends on the boson
frequency and renormalized tunneling. The results are as follows:(1) the
non-equilibrium correlation function $P(t)$, the dynamical susceptibility
$\chi''(\omega)$ and the equilibrium correlation function $C(t)$ are
analytically obtained for $s\leq 1$; (2) the phase diagram of thermodynamic
transition shows the delocalized-localized transition point $\alpha_l$ which
agrees with exact results and numerical data from the Numerical Renormalization
Group; (3) the dynamical transition point $\alpha_c$ between coherent and
incoherent phase is explicitly given for the first time. A crossover from the
coherent oscillation to incoherent relaxation appears with increasing coupling
(for $\alpha > \alpha_c $, the coherent dynamics disappear); (4) the Shiba's
relation and sum rule are exactly satisfied when $\alpha \leq \alpha_c $; (5)
an underdamping-overdamping transition point $\alpha_c^{*}$ exists in the
function $S(\omega)$. Consequently, the dynamical phase diagrams in both ohmic
and sub-Ohmic case are mapped out. For $\Delta \ll \omega_c$, the critical
couplings ($\alpha_l, \alpha_c$ and $\alpha_c^{*}$) are proportional to
$\Delta^{1-s}$.
