Third-order exciton-correlation and nonlinear cavity-polariton effects in semiconductor microcavities

We present a microscopic theory of the coherent third-order $\left({\chi }^{\mathrm{}\left(3\right)\mathrm{}}\right)$ optical response of semiconductor quantum well microcavities, specialized to the four-wave-mixing configuration in the spectral vicinity of the lowest exciton frequency. The theory is that of a quantum-mechanical many-electron system dipole coupled to a classical radiation field. The many-electron dynamics is treated within the dynamics-controlled-truncation formalism restricted to the $1s$-exciton subspace. Within this limitation, all Coulomb correlation effects are included, resulting in an effective theory of (virtual) exciton-polariton scattering. Good quantitative agreement of the theory is obtained in comparison to the experiments reported by Gonokami et al., Phys. Rev. Lett. 79, 1341 (1997). This comparison reveals the signatures of both the bound biexciton and the exciton-exciton scattering (continuum) correlations. Furthermore, a proper calculation of the scattering correlations is shown to be important: each of two common approximations, the Markov and the second Born, results in clear discrepancies from the data.