Diffusion of bottlenecked 29-cm−1phonons in optically excited ruby

Spatial and, additionally, spectral diffusion of 29-${\mathrm{cm}}^{-1\mathrm{}}$ phonons trapped in an excited cylindrical region with diameter ∼0.1 mm has been investigated under continuous optical pumping of ruby. For 130- and 700-ppm ruby, phonon decay strictly limited by spatial diffusion is reflected in a quadratic dependence of the bottlenecking factor $\sigma$ on the excited-state concentration $N^{*}$ up to $N^{*}\sim 3\times {10}^{17}/{\mathrm{cm}}^3$. At higher $N^{*}$, attainable in 700- and 2500-ppm ruby, the observed reduction of $\sigma$ relative to the second-power dependence, as well as the increased width of the dependence of $\sigma$ on magnetic field, reveal that spectral diffusion from inhomogeneous packet to packet is operative. In the spatially diffusive regime, evidence for Zeeman-induced spectral diffusion is found in a characteristic narrowing of the magnetic-field dependence of the $R_2$ fluorescence upon raising $\sigma$. At full separation of the Zeeman transitions $2\overline{A}\left({}_{}{}^2{}_{}{}^{}E\right)\leftrightarrow \overline{E}\left({}_{}{}^2{}_{}{}^{}E\right)$, $\sigma$ is reduced to 15-40% relative to zero field, depending on $N^{*}$ and ground-state concentration. Rate equations governing the ${\mathrm{Cr}}^{3+}$ populations in the two-level scheme ($2\overline{A}$,$\overline{E}$) with inclusion of spatial diffusion and anharmonic decay of the phonons account for the second-power dependence. Diffusive phonon decay turns out to dominate from $N^{*}R\sim 4\times {10}^{13}/{\mathrm{cm}}^2$ up to at least $N^{*}{R}^2\sim {10}^{14}/\mathrm{cm}$, with $R$ the characteristic dimension of the excited region. Spectral diffusion, treated as a perturbation, satisfactorily fits the data at higher $\sigma$, yielding a diffusive step of ∼2% of the linewidth. At low bottlenecking, the functional form of the magnetic field dependence of $\sigma$, including its width and level at full separation, is closely described by use of rate equations, taking the line shape to be Gaussian-type, and assuming the phonons associated with spin flip and those with non spin flip to belong to distinct classes.