Crystal-Field Paschen-Back Effect on Ruby in Ultrahigh Magnetic Fields

Zeeman spectra of the R lines of ruby (Cr$^{3+}$: $\alpha$-Al$_{2}$O$_{3}$) were studied in ultrahigh magnetic fields up to 230 T by magneto-photoluminescence measurements. The observed Zeeman patterns exhibit nonlinear behaviors above 100 T, evidencing the breakdown of the previously reported Paschen-Back effect for ${\bf B} \perp c$ geometry. We adopted the crystal-field multiplet theory including the cubic crystal field (${\mathcal H}_{\rm cubic}$), the trigonal crystal field (${\mathcal H}_{\rm trig}$), the spin-orbit interaction (${\mathcal H}_{\rm SO}$), and the Zeeman interaction (${\mathcal H}_{\rm Z}$). It is found that the nonlinear splitting of the R lines is owing to the hybridization between the $^{2}E$ and $^{2}T_{1}$ states, which leads to the quantization of these Zeeman levels with the orbital angular momentum. Our results suggest that the exquisite energy balance among ${\mathcal H}_{\rm cubic}$, ${\mathcal H}_{\rm trig}$, ${\mathcal H}_{\rm SO}$, and ${\mathcal H}_{\rm Z}$ realized in ruby offers a unique opportunity to observe the onset of the $crystal-field$ Paschen-Back effect toward the high-field extreme.