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

Abstract

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 ${\mathrm{\ensuremath{\chi}}}^{(3)}$(\ensuremath{\Omega};\ensuremath{\Omega},-\ensuremath{\Omega},\ensuremath{\Omega}) by a factor ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\rightarrow}\mathit{T}}$/${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$ multiplicatively on that enhanced by the large excitonic dipole moments. Here ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\rightarrow}\mathit{T}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$ 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, ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\rightarrow}\mathit{T}}$ is much larger than ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$ so that a large enhancement of ${\mathrm{\ensuremath{\chi}}}^{(3)}$ is expected. Second, the four-wave mixing spectrum \ensuremath{\Vert}${\mathrm{\ensuremath{\chi}}}^{(3)}$(2${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$;${\mathrm{\ensuremath{\Omega}}}_{1}$,-${\mathrm{\ensuremath{\Omega}}}_{2}$,${\mathrm{\ensuremath{\Omega}}}_{1}$)\ensuremath{\Vert} under two beams ${\mathrm{\ensuremath{\Omega}}}_{1}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$ is shown to consist of three hierarchical structures as a function of ${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$, i.e., the sharpest spike around ${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$=0 with the linewidth \ensuremath{\Vert}${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$\ensuremath{\Vert}=${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$, the steep shoulder with the center also at ${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$=0 and the width ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}}$ (the exciton decay rate), and the wide structure with the width ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\mathit{g}}$ around ${\mathrm{\ensuremath{\Omega}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$. Here ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\mathit{g}}$ is the relaxation rate of the exciton polarization and \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$ 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 ${\mathrm{\ensuremath{\Omega}}}_{2}$-${\mathrm{\ensuremath{\Omega}}}_{1}$, depending upon the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{1}$ relative to the exciton peak ${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$.