  We present a systematic method to implement a perturbative Hamiltonian
diagonalization based on the time-dependent Schrieffer-Wolff transformation.
Applying our method to strong parametric interactions we show how, even in the
dispersive regime, full Rabi model physics is essential to describe the dressed
spectrum. Our results unveil several qualitatively new results including
realization of large energy-level shifts, tunable in magnitude and sign with
the frequency and amplitude of the pump mediating the parametric interaction.
Crucially Bloch-Siegert shifts, typically thought to be important only in the
ultra-strong or deep-strong coupling regimes, can be rendered large even for
weak dispersive interactions to realize points of exact cancellation of dressed
shifts (`blind spots') at specific pump frequencies. The framework developed
here highlights the rich physics accessible with time-dependent interactions
and serves to significantly expand the functionalities for control and readout
of strongly-interacting quantum systems.
