A Loop Diagram Approach to the Nonlinear Optical Conductivity for an Electron-Phonon System

Abstract

A loop diagram approach to the nonlinear optical conductivity of an electron-phonon system is introduced. This approach can be categorized as another Feynman-like scheme because all contributions to the self-energy terms can be grouped into topologically-distinct loop diagrams. The results for up to the first order nonlinear conductivity are identical to those derived using the KC reduction identity (KCRI) and the state- dependent projection operator (SDPO) introduced by the present authors. The result satisfies the “population criterion” in that the population of electrons and phonons appear independently or the Fermi distributions are multiplied by the Planck distributions in the formalism. Therefore it is possible, in an organized manner, to present the phonon emissions and absorptions as well as photon absorptions in all electron transition processes. In additions, the calculation needed to obtain the line shape function appearing in the energy denominator of the conductivity can be reduced using this diagram method. This method shall be called the “KC loop diagram method”, since it originates from proper application of KCRI’s and SDPO’s.