# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 3 Methods and Results
### Laser Cooling on an Electric-Dipole-Allowed Transition in Ti:Al\({}_{2}\)O\({}_{3}\)
Figure 13: Normalized absorption (blue), \(\pi\)-polarized emission (orange), and \(\sigma\)-polarized emission (green) spectra of Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\) (GTAT sample). Dashed vertical lines indicate the calculated mean fluorescent wavelengths (MFW) for emission spectra. Figure 14: TLS transient signals for excitation wavelengths in the (**a**) absorptive (heating) and (**b**) emissive (cooling) spectral ranges in 0.02% Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\). Black circles are the measured TLS signal, and red curves are best fits that account for only thermal effects. Figure 15: TLS signal versus wavelength in Ti:Al\({}_{2}\)O\({}_{3}\) for (**a**) \(\pi\)-polarization and (**b**) \(\sigma\)-polarization. Positive signals correspond to heating; negative signals correspond to net cooling of the sample. Wavelength tuning was not continuous but proceeded in \(\sim\)0.3 nm steps. Although the observed resonances were grouped into sets of four perturbed site transitions for the purpose of making transition assignments, the tabulations were based purely on unperturbed host vibrational modes and electronic splitting of the ground state. Absorption transitions at wavelengths on the low energy side of the mean fluorescence wavelength 760 nm were calculated by subtracting the vibrational mode frequencies [21] or electronic splitting [22] (in cm\({}^{-1}\)) for the listed transitions from the mean fluorescence wavenumber 13,158 cm\({}^{-1}\) (\(\pi\)-polarization only). The mean fluorescence wavelength for \(\sigma\)-polarization was 763 nm. Despite the lack of information from the literature on site perturbations, agreement between the observed and the calculated resonant wavelengths in the tables was found to be remarkably good. The width of the resonances was close to the step-size tuning of the laser, consistent with the low damping parameters of the optical modes in sapphire [21]. The number of observed cooling resonances in the range of 750-790 nm also decreased when the optical polarization was rotated from \(\pi\) to \(\sigma\), consistent with higher losses from the impurity background absorption for \(\sigma\)-polarization [34]. This quenching of resonances in \(\sigma\)-polarization was therefore ascribed to increased background heating over much of the spectral range. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\lambda_{\text{obs}}\) (nm) & \(k_{\text{obs}}\) (cm\({}^{-1}\)) & Initial Level & Site & \(k_{calc}\) (cm\({}^{-1}\)) & \(k_{obs}-k_{calc}\) (cm\({}^{-1}\)) \\ \hline
815.50 & 12,257.91 & (\({}^{2}\)A(0), v\({}_{\text{d}}\)) & 4 & 12,199.56 & \(-\)58.35 \\
832.14 & 12,017.21 & (\({}^{2}\)A(1), v = 0) & 4 & 11,906.16 & \(-\)111.05 \\
899.11 & 11,122.11 & (\({}^{2}\)A(1), v\({}_{\text{d}}\)) & 1 & 10,999.56 & \(-\)122.55 \\
906.95 & 11,025.97 & (\({}^{2}\)A(1), v\({}_{\text{d}}\)) & 2 & 10,999.56 & \(-\)26.41 \\
928.00 & 10,775.86 & (\({}^{2}\)A(1), v\({}_{\text{d}}\)) & 3 & 10,999.56 & 223.70 \\
933.84 & 10,708.47 & (\({}^{2}\)A(1), v\({}_{\text{d}}\)) & 4 & 10,999.56 & 291.09 \\
944.03 & 10,592.88 & (\({}^{2}\)A(2), v = 0) & 1 & 10,496.16 & \(-\)96.72 \\
950.08 & 10,525.43 & (\({}^{2}\)A(2), v = 0) & 2 & 10,496.16 & \(-\)29.27 \\
954.42 & 10,477.57 & (\({}^{2}\)A(2), v = 0) & 3 & 10,496.16 & 18.59 \\
957.90 & 10,439.50 & (\({}^{2}\)A(2), v = 0) & 4 & 10,496.16 & 56.66 \\ \hline \end{tabular}
\end{table} 
Table 3: Assignments of cooling resonances in Ti:Al\({}_{2}\)O\({}_{3}\) for \(\sigma\)-polarization (mean fluorescence wavelength 763 nm). Wavenumbers for observed and calculated resonances are subtracted for comparison in the last column on the right. The average discrepancy is given in the bottom row. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\lambda_{\text{obs}}\) (nm) & \(k_{\text{obs}}\) (cm\({}^{-1}\)) & Initial Level & Site & \(k_{calc}\) (cm\({}^{-1}\)) & \(k_{obs}-k_