Derivation of nonlinear optical conductivity by using a reduction identity and a state-dependent projection method

A new nonlinear optical conductivity formula for a system of electrons interacting with phonons was derived using a reduction identity and a state-dependent projection technique introduced by the authors. The results include a general formula for the nonlinear optical conductivity of the general rank and the linear, first-order nonlinear and second-order nonlinear conductivity are calculated in terms of the linewidth. The linewidth term includes the electron and phonon distribution functions properly. Therefore, it is possible to explain the phonon emission and absorption in all electron transition processes in an organized manner.