## 2 Samples and Experimental Setup
The as-grown samples were mounted in a continuous He-flow cryostat, and the temperature was controlled. A continuous-wave tunable laser (1470\(-\)1560 nm, spectral width of 400 kHz) with an erbium-doped fiber amplifier was used to excite the \({}^{4}I_{13/2}\) manifold. Since the bandwidth of the excitation laser was much narrower than the energy intervals in the \({}^{4}I_{13/2}\) manifold, the resonant excitation of one Stark level was possible. In the TI-PL measurements, the PL peaks in the 1.54-\(\upmu\)m (visible) wavelength regions were detected by InGaAs-photodiode array (Si-CCD) through the monochromator under the \({}^{4}I_{13/2}\) manifold excitation by varying the sample temperature 4-150 K. The excitation laser was focused on a sample surface with the diameter of \(\sim\)15 \(\upmu\)m by an objective lens, and the incident angle was set to be 45\({}^{\circ}\) from the sample growth axis. Avoiding a strong stray light, the PL signal was collected perpendicularly by an another objective lens with a numerical aperture of 0.4. The total spectral resolution was about 60 \(\upmu\)eV for the TI-PL measurements. For the TR-PL measurements, a pulsed excitation formed from the continuous-wave laser light by an acousto-optic modulator and a streak camera (model C11293S, Hamamatsu Photonics, Hamamatsu, Japan) were used to observe the temporal evolution of PL signals from \({}^{4}I_{13/2}\) manifold, and the sample temperature was changed around 4-60 K. The spectral and temporal resolutions of the TR-PL measurement system were 500 \(\upmu\)eV and 20 ps (for a 1 ns time range), respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{\(\mathbf{x}\)} & \multirow{2}{*}{\(\mathbf{\text{Er}^{3+}}\) Density (\(\mathbf{\text{cm}^{-3}}\))} & \multirow{2}{*}{\(\mathbf{\text{Er}^{3+}}\)} & \multirow{2}{*}{**Averaged Inter-Er\({}^{3+}\)**} & \multirow{2}{*}{**Averaged Intra(inter)-Site Distance**} \\  & & & & & & & \\  & & & & & & & \\ \hline A & 1.000 & 2.7 \(\times\) 10\({}^{22}\) & 32 & 3.3 & 1.95 & 5.27 & 2.66 \\ B & 0.054 & 1.4 \(\times\) 10\({}^{21}\) & 1.6 & 8.8 & 5.15 & 13.94 & 7.03 \\ C & 0.012 & 3.4 \(\times\) 10\({}^{20}\) & 0.4 & 14.5 & 8.50 & 23.01 & 11.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the studied (\(\text{Er}_{x}\text{Sc}_{1-x}\))\({}_{2}\text{O}_{3}\) samples. The averaged distances are evaluated from the lattice constant and Er concentration. Figure 3: **(a)** Schematic of the energy levels of \(\text{Er}^{3+}\) and the closeup views of \({}^{4}I_{13/2}\) and \({}^{4}I_{15/2}\) manifolds (after reference [21]). **(b)** The upconversion photoluminescence (UC-PLs) from various manifolds in Sample A with \(P_{\text{exc}}\) = 30 mW (lower panel, red) and \(P_{\text{exc}}\) = 1 mW (upper panel, black) under \(\text{Y}_{3}^{\prime}\) resonant excitation (\(\lambda_{\text{exc}}\) = 1530.5 nm) at 4 K. The PL intensities are scaled relatively between both \(P_{\text{exc}}\). **(c)** Excitation power dependences of the UC-PL intensities corresponding to the three manifolds, \({}^{4}F_{9/2}\) (solid circles), \({}^{4}S_{3/2}\) (open circles), and \({}^{4}I_{11/2}\) (triangles) in Sample A.