  The problem of a driven quantum system coupled to a bath and coherently
driven is usually treated using either of two approaches: Employing the common
secular approximation in the lab frame (as usually done in the context of
atomic physics) or in the rotating frame (prevailing in, e.g., the treatment of
solid-state qubits). These approaches are applicable in different parts of the
parameter space and yield different results. We show how to bridge between
these two approaches by working in the rotating frame without employing the
secular approximation with respect to the driving amplitude. This allows us to
uncover novel behaviors in regimes which were previously inaccessible or
inaccurately treated. New features such as the qualitative different evolution
of the coherence, population inversion at a lower driving amplitude, and novel
structure in the resonance fluorescence spectrum of the system are found. We
argue that this generalized approach is essential for analyzing hybrid systems,
with components that come from distinctly different regimes which can now be
treated simultaneously, giving specific examples from recent experiments on
quantum dots coupled to optical cavities, and single-spin electron paramagnetic
resonance.
