Perturbative Input-Output Theory of Floquet Cavity Magnonics and Magnon Energy Shifts

We develop a perturbative input-output formalism to compute the reflectance and transmittance spectra of cavity magnonics systems subject to a Floquet modulation. The method exploits the strong hierarchy between the magnetic-dipole couplings transverse (drive field) and parallel (modulation field) to the static bias field, which naturally introduces the small parameter ε= (2Ns)^-1/2 associated with the total spin Ns of the ferromagnet. By organizing the cavity and magnon fields in a systematic expansion in ε, we obtain compact analytic expressions for the spectra up to second order. Using these results, we reproduce the characteristic sideband structure observed in recent Floquet cavity electromagnonics experiments. Furthermore, accounting for the Zeeman interaction between the modulation field and the fully polarized ground state - a contribution typically neglected in previous treatments - we predict an additional magnon detuning of approximately 0.8 GHz, independent of both modulation frequency and sample size and determined solely by the spatial volume occupied by the modulation field. This identifies a measurable and previously overlooked shift relevant for the interpretation and design of cavity magnonics experiments.