Influence of a phonon bath on the hierarchy of electronic densities in an optically excited semiconductor

The influence of a phonon bath on the dynamics of the many-body density matrices relevant for the nonlinear optical response of semiconductors is studied. It is shown that, as in the phonon-free case, only a finite set of electronic density matrices is needed to describe the optical response in a ${\mathrm{\chi }}^{\left(\mathit{n}\right)}$ experiment. However, to obtain a closed set of equations of motion on the ${\mathrm{\chi }}^{\left(\mathit{n}\right)}$ level additional truncation concepts have to be introduced in order to decouple the infinite hierarchy of phonon-assisted density matrices. Relations characteristic of intrinsic systems, that have previously been derived to reduce the number of independent dynamical variables in the phonon-free case, are generalized. While in a coherent system the optical response is completely determined by transition-type variables, under the influence of a phonon bath occupation densities can also become independent dynamical quantities. Numerical results for a simple one-dimensional model are presented. Memory effects and dephasing induced by the electron-phonon coupling are discussed for all ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ relevant electronic density matrices. The pump-probe signal for ${\mathrm{\sigma }}^{+}$-${\mathrm{\sigma }}^{\mathrm{-}}$ excitation in the spectral region of the biexciton is analyzed. The respective contributions of two processes for the generation of biexcitons, via two- photon transitions (TPT’s) or from previously generated exciton densities, to the microscopic dynamics are identified. The second process yields a ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ signal only due to the coupling to the phonon bath and turns out to contribute considerably less than the TPT under all conditions studied in this paper. © 1996 The American Physical Society.