Hierarchy of density matrices in coherent semiconductor optics

The hierarchy of many-body density matrices describing the nonlinear optical response of semiconductors is studied. The analysis is restricted to the coherent electronic dynamics generated by the relevant Hamiltonian. The strength of the optical excitation is used as a perturbation parameter, allowing a controlled truncation of the hierarchy and a drastic reduction of the number of independent multipoint functions describing the response to any given order. Thus the many-body effects contributing to the nth order susceptibility ${\mathrm{\chi }}^{\left(\mathit{n}\right)}$ are described by [n+1/2] independent multipoint functions. As a consequence of the assumed coherence it turns out that all densitylike variables can be expressed in terms of transitionlike quantities. The proposed systematic treatment of Coulomb correlations is compared with the conventional random-phase approximation factorization of many-body matrices.