Third order susceptibility: general formalism for photoinduced current   density in semiconductors
  This paper contains a detailed derivation of the photoinduced current density at third order in the coupling between a semiconductor and a multifrequency photon field, starting from its standard textbook expression as a third order time integral of a triple commutator. Due to a major intrinsic problem linked to this triple commutator, such a derivation has been made possible quite recently only, thanks to the tools developed in the composite-boson many-body theory we have just constructed. The photoinduced current density is shown to ultimately read in a compact form, in terms of the Pauli and Coulomb scatterings for exciton-exciton interactions introduced in this theory. Representation in Shiva diagrams is also given to better grasp the physics of the various contributions. 
M. Combescot and O. Betbeder-Matibet