Perturbative Diagonalization for Time-Dependent Strong Interactions

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