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We propose an approximation scheme to describe the dynamics of the spin-bosonmodel when the spectral density of the environment shows a peak at acharacteristic frequency $\Omega$ which can be very close (or even equal) tothe spin Zeeman frequency $\Delta$. Mapping the problem onto a two-state system(TSS) coupled to a harmonic oscillator (HO) with frequency $\omega_0$ we showthat the representation of displaced HO states provides an appropriate basis totruncate the Hilbert space of the TSS-HO system and therefore a better pictureof the system dynamics. We derive an effective Hamiltonian for the TSS-HOsystem, and show it furnishes a very good approximation for the system dynamicseven when its two subsystems are moderately coupled. Finally, assuming theregime of weak HO-bath coupling and low temperatures, we are able toanalytically evaluate the dissipative TSS dynamics.

Dissipative dynamics of a two - level system resonantly coupled to a harmonic mode