  When an open quantum system is driven by an external time-dependent force,
the coupling of the driving to the central system is usually included whereas
the impact of the driving field on the bath is neglected. We investigate the
effect of a quantum bath of linearly driven harmonic oscillators on the
relaxation dynamics of a quantum two-level system which itself is not directly
driven. In particular, we calculate the frequency-dependent response of the
system when the bath is subject to a Dirac and a Gaussian driving-pulse. We
show that a time-retarded effective force on the system is induced by the
driven bath which depends on the full history of the perturbation and the
spectral characteristics of the underlying bath. In particular, when a
structured Ohmic bath with a pronounced Lorentzian peak is considered, the
dynamical response of the system to a driven bath is qualitatively different as
compared to the undriven bath. Specifically, additional resonances appear which
can be directly associated to a Jaynes-Cummings-like effective energy spectrum.
