Strong and weak coupling limits in optics of quantum well excitons

Abstract

A transition between the strong (coherent) and weak (incoherent) coupling limits of resonant interaction between quantum well (QW) excitons and bulk photons is analyzed and quantified as a function of the incoherent damping rate ${\ensuremath{\gamma}}_{x}$ caused by exciton-phonon and exciton-exciton scatterings. For confined QW polaritons, a second, anomalous, ${\ensuremath{\gamma}}_{x}$-induced dispersion branch arises and develops with increasing ${\ensuremath{\gamma}}_{x}$. In this case, the strong-weak coupling transition is attributed to ${\ensuremath{\gamma}}_{x}={\ensuremath{\gamma}}_{x}^{\mathrm{tr}}$ or ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{x}^{\mathrm{tr}}$, when the intersection of the normal and damping-induced dispersion branches occurs either in ${{k}_{\ensuremath{\parallel}},\mathrm{Im}[\ensuremath{\omega}],\mathrm{Re}[\ensuremath{\omega}]}$ coordinate space (in-plane wave vector ${k}_{\ensuremath{\parallel}}$ is real) or in ${\ensuremath{\omega},\mathrm{Im}[{k}_{\ensuremath{\parallel}}],\mathrm{Re}[{k}_{\ensuremath{\parallel}}]}$ coordinate space (frequency $\ensuremath{\omega}$ is real), respectively. For the radiative states of QW excitons, i.e., for radiative QW polaritons, the transition is described as a qualitative change of the photoluminescence spectrum at grazing angles along the QW structure. We show that the radiative corrections to the QW exciton states with in-plane wave vector ${k}_{\ensuremath{\parallel}}$ approaching the photon cone, i.e., at ${k}_{\ensuremath{\parallel}}\ensuremath{\rightarrow}{k}_{0}=({\ensuremath{\omega}}_{0}\sqrt{{\ensuremath{\epsilon}}_{b}})∕(\ensuremath{\hbar}c)$ (${\ensuremath{\epsilon}}_{b}$ is the background dielectric constant), are universally scaled by the energy parameter ${({\ensuremath{\Gamma}}_{0}^{2}{\ensuremath{\omega}}_{0})}^{1∕3}$ with ${\ensuremath{\Gamma}}_{0}$ the intrinsic radiative width and ${\ensuremath{\omega}}_{0}$ the exciton energy at ${k}_{\ensuremath{\parallel}}=0$, rather than diverge. Similarly, the strong-weak coupling transition rates ${\ensuremath{\gamma}}_{x}^{\mathrm{tr}}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{x}^{\mathrm{tr}}$ are also proportional to ${({\ensuremath{\Gamma}}_{0}^{2}{\ensuremath{\omega}}_{0})}^{1∕3}$. The numerical evaluations are given for a GaAs single quantum well with realistic parameters: ${\ensuremath{\Gamma}}_{0}=45.5\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{eV}$ and ${({\ensuremath{\Gamma}}_{0}^{2}{\ensuremath{\omega}}_{0})}^{1∕3}\ensuremath{\approx}1.5\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$.