Zeeman-Hyperfine Measurements of a Pseudo-Degenerate Quadruplet in CaF${}_2$:Ho${}^{3+}$

We report Zeeman infra-red spectroscopy of electronic-nuclear levels of ${}^5$I${}_8{\to }^5$I${}_7$ transitions of Ho${}^{3+}$ in the C${}_{4\mathrm{v}}$(F${}^{-}$) centre in CaF${}_2$ with the magnetic field along the $⟨111⟩$ direction of the crystal. Transitions to the lowest ${}^5$I${}_7$ state, an isolated electronic doublet, and the next group of states, a pseudo-quadruplet consisting of a doublet and two nearby singlets, exhibit strongly non-linear Zeeman splittings and intensity variations. Simulated spectra based upon a crystal-field analysis give an excellent approximation to the data, illustrating the strong predictive ability of the parametrised crystal-field approach. Anti-crossings in the hyperfine splittings, the basis of quantum information storage in rare-earth doped insulating dielectrics, are also predicted.