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: (i) the nonequilibrium correlation function P(t), the dynamical susceptibility chi(')(omega), and the equilibrium correlation function C(t) are analytically obtained for s alpha(c), the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when alpha <=alpha(c); (v) 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 展开▼
