A new nonlinear optical conductivity formula for a system of electrons interacting with phonons was derived using a reduction identity and a statedependent 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.

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We derive a new optical conductivity formula for cyclotron transitions in the quantum limit in a 3-D system of electrons interacting with phonons utilizing a projection technique. The electron and phonon distribution functions are included in the line·shape factor, which differs from those in the theories of Badjou and Argyres [Phys. Rev. B35 (1987), 5964] and Cho and Choi [Phys. Rev. B47 (1993), 9273]. It thus becomes possible to explain, in an organized way, the phonon emissions and absorptions in all electron transition processes.at

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