Many-body effects in a χ⁽³⁾ analysis of optically excited semiconductors

Recently, the importance of the so-called excitation-induced dephasing (EID) effects for the understanding of four-wave-mixing signals has been discussed. In its simplest formulation this effect means that the effective dephasing time T₂ of the excitonic resonance, which is measured in a four-wave-mixing experiment, depends on the intensity of the excitation light pulses. This phenomenon causes, for example, a coupling of the two spin-degenerate heavy-hole excitons in a strained GaAs sample. Whereas the general consequences of EID can be described within a partly phenomenological approach, a fully systematic microscopic approach is highly desirable. In general, the microscopic description of many-body effects in a system with Coulomb interaction is complicated by the fact that the interaction causes an infinite hierarchy of coupled equations of motion for the various expectation values. However, Axt and Stahl have recently shown that under certain circumstances this hierarchy can be truncated. Within a χ⁽ⁿ⁾ expansion for the description the optical response to an external field one needs only to consider a finite number of expectation values of operators {N, M} = a₁⁺a₂⁺ ... a_N⁺a₁ ... a_M, where 1, 2, . . . denote band and momentum variables. As an example, we illustrate in Fig. 1 the relevant operators in a χ⁽³⁾ expansion, together with the corresponding couplings due to the external field and the Coulomb interaction. We apply this theory to a strained GaAs system and study four-wave-mixing signals induced by linearly polarized pulses. A complete numerical solution of the full set of equations, which might be called genralized semiconductor Bloch equations (see, for example, Refs. 3 and 4), is, however, still infeasible. We therefore neglect, in a first approach, the biexciton contribution and factorize contributions from the carrier-carrier scattering. In Fig. 2 we show the computed time-resolved four-wave-mixing signal induced by colinearly-polarized and cross-linearly-polarized pump and probe pulses. The ratio of the two signals is of the order 10², indicating a strong EID effect in the low-excitation limit.