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 ξk which depends on the boson frequency and renormalized tunneling. The results are as follows: (1) the nonequilibrium correlation function P (t), the dynamical susceptibility χ(ω) and the equilibrium correlation function C(t) are analytically obtained for s ≤ 1; (2) the phase diagram of thermodynamic transition shows the delocalizedlocalized transition point αl which agrees with exact results and numerical data from the Numerical Renormalization Group; (3) the dynamical transition point αc 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 α > αc, the coherent dynamics disappear); (4) the Shiba’s relation and sum rule are exactly satisfied when α ≤ αc; (5) an underdamping-overdamping transition point α c exists in the function S(ω). Consequently, the dynamical phase diagrams in both ohmic and subOhmic case are mapped out. For ∆ ≪ ωc, the critical couplings (αl, αc and α c) are proportional to ∆ . PACS numbers: 03.65.Yz, 05.30.-d, 72.20.Dp, 75.30.Ds