Floquet expansion by counting pump photons

Periodically-driven systems engender a rich competition between the time scales of the drives and those of the system, leading to a limited ability to describe the system in full. We present a framework for the description of interacting bosonic driven systems via a Floquet expansion on top of a quantization that "counts" the drive photons, and provide compelling arguments for the superior performance of our method relative to standard Floquet approaches. Crucially, our approach extends beyond the rotating wave approximation and addresses the long-standing issue of mismatch between the quantum Floquet formalism and its classical counterpart. We, furthermore, pinpoint key corrections to the positions of multiphoton resonances, which are commonly used in the calibration and operation of qubit architectures.