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Optical drives at terahertz and mid-infrared frequencies in quantum materialsare increasingly used to reveal the nonlinear dynamics of collective modes incorrelated many-body systems and their interplay with electromagnetic waves.Recent experiments demonstrated several surprising optical properties oftransient states induced by driving, including the appearance of photo-inducededges in the reflectivity in cuprate superconductors, observed both below andabove the equilibrium transition temperature. Furthermore, in other drivenmaterials, reflection coefficients larger than unity have been observed. Inthis paper we demonstrate that unusual optical properties of photoexcitedsystems can be understood from the perspective of a Floquet system; a systemwith periodically modulated system parameters originating from pump-inducedoscillations of a collective mode. We present a general phenomenological modelof reflectivity from Floquet materials, which takes into account parametricgeneration of excitation pairs. We find a universal phase diagram of driveinduced features in reflectivity which evidence a competition between drivingand dissipation. To illustrate our general analysis we apply our formalism totwo concrete examples motivated by recent experiments: a single plasmon band,which describes Josephson plasmons in layered superconductors, and aphonon-polariton system, which describes upper and lower polaritons inmaterials such as insulating SiC. Finally we demonstrate that our model can beused to provide an accurate fit to results of phonon-pump - terahertz-probeexperiments in the high temperature superconductor $\rm{YBa_2Cu_3O_{6.5}}$. Ourmodel explains the appearance of a pump-induced edge, which is higher in energythan the equilibrium Josephson plasmon edge, even if the interlayer Josephsoncoupling is suppressed by the pump pulse.

Generalized Fresnel-Floquet equations for driven quantum materials