  We study the asymptotic dynamics of a driven spin-boson system where the
environment is formed by a broadened localized mode. Upon exploiting an exact
mapping, an equivalent formulation of the problem in terms of a quantum
two-state system (qubit) coupled to a harmonic oscillator which is itself
Ohmically damped, is found. We calculate the asymptotic population difference
of the two states in two complementary parameter regimes. For weak damping and
low temperature, a perturbative Floquet-Born-Markovian master equation for the
qubit-oscillator system can be solved. We find multi-photon resonances
corresponding to transitions in the coupled quantum system and calculate their
line-shape analytically. In the complementary parameter regime of strong
damping and/or high temperatures, non-perturbative real-time path integral
techniques yield analytic results for the resonance line shape. In both
regimes, we find very good agreement with exact results obtained from a
numerical real-time path-integral approach. Finally, we show for the case of
strong detuning between qubit and oscillator that the width of the $n$-photon
resonance scales with the $n$-th Bessel function of the driving strength in the
weak-damping regime.
