Effects of trapped states on spectra of four-wave mixing and differential transmission

Excitons are sometimes trapped at impurities in crystals and at defects on interfaces in the confined systems, such as quantum wells and microcrystallites embedded in a host matrix. We discuss these trapping effects on the third-order optical polarization under nearly resonant pumping of the exciton. First we predict an enhancement of ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$(Ω;Ω,-Ω,Ω) by a factor ${\mathrm{\Gamma }}_{\mathit{n}\to \mathit{T}}$/${\mathrm{\Gamma }}_{\mathit{T}\to \mathit{g}}$ multiplicatively on that enhanced by the large excitonic dipole moments. Here ${\mathrm{\Gamma }}_{\mathit{n}\to \mathit{T}}$ and ${\mathrm{\Gamma }}_{\mathit{T}\to \mathit{g}}$ are, respectively, the rate at which the exciton is trapped and the decay rate of the trapped electron into the ground state. In many cases, ${\mathrm{\Gamma }}_{\mathit{n}\to \mathit{T}}$ is much larger than ${\mathrm{\Gamma }}_{\mathit{T}\to \mathit{g}}$ so that a large enhancement of ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ is expected. Second, the four-wave mixing spectrum ‖${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$(2${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$;${\mathrm{\Omega }}_1$,-${\mathrm{\Omega }}_2$,${\mathrm{\Omega }}_1$)‖ under two beams ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ is shown to consist of three hierarchical structures as a function of ${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$, i.e., the sharpest spike around ${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$=0 with the linewidth ‖${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$‖=${\mathrm{\Gamma }}_{\mathit{T}\to \mathit{g}}$, the steep shoulder with the center also at ${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$=0 and the width ${\mathrm{\Gamma }}_{\mathit{n}}$ (the exciton decay rate), and the wide structure with the width ${\mathrm{\Gamma }}_{\mathit{n}\mathit{g}}$ around ${\mathrm{\Omega }}_1$=${\mathrm{\omega }}_{1\mathit{s}}$ and ${\mathrm{\Omega }}_2$=${\mathrm{\omega }}_{1\mathit{s}}$. Here ${\mathrm{\Gamma }}_{\mathit{n}\mathit{g}}$ is the relaxation rate of the exciton polarization and ħ${\mathrm{\omega }}_{1\mathit{s}}$ the lowest exciton energy. Third, the differential transmission spectrum is found to show the strong absorption saturation peak or the sharp induced absorption dip as a function of pump-probe detuning ${\mathrm{\Omega }}_2$-${\mathrm{\Omega }}_1$, depending upon the pump frequency ${\mathrm{\Omega }}_1$ relative to the exciton peak ${\mathrm{\omega }}_{1\mathit{s}}$.