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: (i) the nonequilibrium correlation function P(t), the dynamical susceptibility chi(')(omega), and the equilibrium correlation function C(t) are analytically obtained for s <= 1; (ii) 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; (iii) the dynamical transition point alpha(c) between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for alpha >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 <omega(c), the critical couplings (alpha(l),alpha(c), and alpha(*)(c)) are proportional to Delta(1-s).