  The optical conductivity is the basic defining property of materials
characterizing the current response toward time-dependent electric fields. In
this work, following the approach of Kubo's response theory, we study the
general properties of the nonlinear optical conductivities of quantum many-body
systems both in equilibrium and non-equilibrium. We obtain an expression of the
second- and the third-order optical conductivity in terms of correlation
functions and present a perturbative proof of the generalized Kohn formula
proposed recently. We also discuss a generalization of the $f$-sum rule to a
non-equilibrium setting by focusing on the instantaneous response.
