We investigate the dynamics of the spin-boson model when the spectral density of the boson bath shows a resonance at a characteristic frequency but behaves Ohmically at small frequencies. The time evolution of an initial state
is determined by making use of the mapping onto a system composed of a quantum mechanical two-state system (TSS) which is coupled to a harmonic oscillator (HO) with frequency . The HO itself is coupled to an Ohmic environment.
The dynamics is calculated by employing the numerically exact quasiadiabatic path-integral propagator technique. We find significant new properties compared to the Ohmic spin-boson model. By reducing the TSS-HO system in the dressed states picture to a three-level system for the special case at resonance, we calculate the dephasing rates for the TSS analytically. Finally, we apply our model to experimentally realized superconducting flux qubits coupled to an underdamped dc-SQUID detector.