Four-wave-mixing theory beyond the semiconductor Bloch equations

Four-wave-mixing (FWM) experiments using a dynamical density matrix model of the semiconductor band edge are discussed. Higher-order correlation functions are retained which arc neglected in the commonly used RPA treatment leading to the semiconductor Bloch equations. In order to terminate the hierarchy of the equations of motion for the higher-order density matrices systematically a truncation scheme controlled by orders in the driving field is applied. For any prescribed order n in the exciting field a closed set of equations is obtained from which the dielectric response up to order n can be calculated exactly. In addition it turns out that in a coherently driven system part of the remaining density matrices become redundant and can be eliminated. Four-wave-mixing experiments are dominated by third-order contributions. Applying the above-described results one ends up with only two functions in this case. These are the excitonic and the biexcitonic transition densities. As an applica tion of our method the example of a GaAs single quantum well is studied. Two pulses with finite lengths are assumed such that both heavy-and light-hole excitons are excited. The influence of the biexciton contribution on the polarization properties of the FWM signal is analyzed and compared with experimental results.