  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.
