Quantum dynamics of a qubit coupled with structured bath

The dynamics of an unbiased spin-boson model with Lorentzian spectral density is investigated theoretically in terms of the perturbation theory based on a unitary transformation. The non-equilibrium correlation function $P(t)$ and susceptibility ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(\omega )$ are calculated for both the off-resonance case $\mathrm{\Delta }\lesssim 0.5\mathrm{\Omega }$ and the on-resonance case $\mathrm{\Delta }\sim \mathrm{\Omega }$. The approach is checked by the Shiba’s relation and the sum rule. Besides, the coherent-incoherent transition point ${\alpha }_c$ can be determined, which has not been demonstrated for the structured bath by previous authors up to our knowledge.