  For many-electron systems, we consider a nonequilibrium state (NES) that is
driven by a pump field(s), which is either an optical field or a longitudinal
electric field. For the differential optical conductivity describing the
differential response of the NES to a probe optical field, we derive exact sum
rules and asymptotic behaviors, which open wide possibilities for experiments.
In deriving these results, we have also derived universal properties of general
differential response functions of time-dependent NESs of general systems.
