Green's function approach to the dynamics-controlled truncation formalism:Derivation of the chi((3)) equations of motion

The dynamics-controlled truncation (DCT) formalism is a successful microsco pic approach that describes coherent correlations in optically excited semi conductors. For practical reasons (including numerical evaluations), its ap plication is limited to lowest-order nonlinearities, such as the chi((3)) r egime. Therefore, it is not convenient to use this formalism to examine the role played by incoherent many-body effects, such as carrier-carrier scatt ering and screening. Traditionally, the most powerful approach to study inc oherent effects and correlations in highly excited semiconductors is that o f nonequilibrium Green's functions (NGF). A combination of the insights and technical advantages provided by the two (NGF and DCT) approaches will lea d to a comprehensive microscopic theory for nonlinear optical phenomena in semiconductors. In this paper, we take a first step in this direction by pr esenting detailed one-to-one relations between the two formalisms within th e chi((3)) approximation. Starting from the standard perturbation theory of nonequilibrium Green's functions, we derive the essential minimal order fa ctorization theorems, to arbitrary order, of DCT and the equations of motio ns for the interband polarization and the "biexcitonic" correlation functio n. This lays the foundation for future diagrammatic high-intensity generali zations of the DCT formalism.