Quantum dynamics of the dissipative two-state system coupled with a sub-Ohmic bath

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 $\Delta^{1-s}$.