  Light-driven matter can exhibit qualitatively distinct electronic and optical
properties from those observed at equilibrium. We introduce generalized sum
rules for the optical properties of driven systems by both quantum and
classical light. For classical light, it shows that the sum of all Fourier
components, indexed by n, of the time-dependent dipole matrix elements between
dressed states weighted by the corresponding quasienergy difference in the
first Floquet Brillouin zone plus n driving frequency is a constant, determined
by the number of electrons. An analogous sum rule for quantum light-dressing is
also derived. These developments provide guidance for the control of effective
optical properties of matter by light fields.
