Derivation of second-order nonlinear optical conductivity by the projection-diagram method

A projection-diagram method is introduced for optical conductivity with lineshape functions, which takes into account the population criterion that the electron and phonon distribution functions are multiplicatively combined along with the energy conservation factors for proper interpretation of emission and absorption of phonons and photons in all the processes of electron transitions. It is further shown that the second order nonlinear optical conductivity of the system of electrons interacting with phonons, obtained using this method, is identical with that derived by using the state dependent projectors and the KC reduction identities [J. Phys. A: Math. Theor. 43, 165203 (2010)]. We expect that this method can reduce the amount of many-body calculation and can be of help in providing physical intuition into solid state quantum dynamics and representing perturbation expressions for such systems