Spin-orbital entanglement in Cr$^{3+}$-doped glasses

A framework for reconstructing the one-electron spinors, $\Gamma_7$ and $\Gamma_8$, of \ch{Cr^3+} ions embedded in glasses from optical measurements has been developed. These spinors provide the basis for calculating the spin-orbital von Neumann entropy, offering a quantitative measure of quantum entanglement within the electronic state. To illustrate the applicability of this concept, an aluminum phosphate glass doped with 1 mol$\%$ chromium was prepared and characterized via optical absorption spectroscopy. By extracting the fundamental electronic parameters, including the spin-orbit coupling constant $\xi_{\rm 3d}$, the crystal field strength $Dq$, and the Racah parameters $B$ and $C$, we demonstrate how the spin-orbital entanglement entropy, $\Delta S_{\rm vN}^{\rm SO}$, can be mapped across different chemical environments. Our analysis reveals that while individual crystal field parameters do not dictate the degree of entanglement, the dimensionless ratio between the spin-orbit coupling and the crystal field strength ($\xi_{\rm 3d}/Dq$) exhibits a robust linear correlation with the entropy. This relationship serves as a clear illustration of how the competition between relativistic effects and local symmetry governs the information content of the 3d($O_h$) electronic manifold.