Non-markovian dynamics of a biased qubit coupled to a structured bath.

A new analytical approach, beyond the rotating wave approximation, based on unitary transformations and the non-markovian master equation for the density operator, is applied to treat the biased spin-boson model with a Lorentzian structured bath for arbitrary detunings at zero temperature. Compared to zero bias, we find that the non-equilibrium dynamics demonstrates two more damping oscillation frequencies and one additional relaxation frequency for non-zero bias. Analytical expressions for the non-markovian dynamics and the corresponding spectrum, the localized-delocalized transition point, the coherent-incoherent transition point, the analytical ground energy, the renormalized tunneling factor and the susceptibility are determined. The sum rule and the Shiba relation are checked in the coherent regime.