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

TL;DR

The study addresses the Zeeman-hyperfine structure of Ho$^{3+}$ in CaF$_2$ at a ${C}_{4v}(F^-)$ site under a magnetic field along $\langle111\rangle$, focusing on a ground-state doublet and a nearby pseudo-quartet with strongly nonlinear splittings. It combines high-resolution Zeeman-infrared spectroscopy with a crystal-field model, using the full Hamiltonian $H=H_{FI}+H_{CF}+H_{HF}+H_Z$ and established crystal-field parameters to compute energies and transition intensities. The main findings show excellent agreement between experiment and simulations across fields up to 0.6 T, including the prediction of hyperfine anticrossings (e.g., up to ~0.06 cm$^{-1}$ or 70–150 MHz in certain manifolds), highlighting the model's predictive power. This work supports the use of low-symmetry rare-earth-doped insulators for quantum information storage by enabling the designs of ZEFOZ-like transitions at avoided crossings.

Abstract

We report Zeeman infra-red spectroscopy of electronic-nuclear levels of $^5$I$_8 \rightarrow ^5$I$_7$ transitions of Ho$^{3+}$ in the C$_{\rm 4v}$(F$^-$) centre in CaF$_2$ with the magnetic field along the $\langle 111\rangle$ 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.

Figures (7)

• Figure 1: 4.2 K spectra at zero magnetic field for (a) transitions from Z$_1\gamma_1$, $Z_2\gamma_2$ of $^5$I$_{8}$ to Y$_1\gamma_5$ of $^5$I$_{7}$, (b) transitions from Z$_1\gamma_1$, $Z_2\gamma_2$ of $^5$I$_{8}$ to the $Y_2\gamma_3$, $Y_3\gamma_5$, $Y_4\gamma_2$ states of $^5$I$_{7}$. The feature at 5254.8 cm$^{-1}$, labelled by an asterisk, is an atmospheric H$_{2}$O absorption line.
• Figure 2: Calculated Zeeman-hyperfine energies for a magnetic field along the $\langle 111 \rangle$ direction for (a) Z$_1\gamma_1$, $Z_2\gamma_2$ of $^5$I$_{8}$, (b) Y$_1\gamma_5$ of $^5$I$_{7}$, (c) $Y_2\gamma_3$, $Y_3\gamma_5$, $Y_4\gamma_2$ of $^5$I$_{7}$.
• Figure 3: Experimental and calculated 4.2 K $\langle 111 \rangle$ Zeeman spectra for the Z$_{1,2}\gamma_{1,2}\longrightarrow$Y$_1\gamma_5$ transitions with (a) 0.0 T, (b) 0.2 T, (c) 0.4 T, (d) 0.6 T.
• Figure 4: Experimental (a) and calculated (b) 4.2 K $\langle 111 \rangle$ Zeeman spectra for the Z$_{1,2}\gamma_{1,2}\longrightarrow$Y$_1\gamma_5$ transitions for magnetic fields up to 0.6 T.
• Figure 5: Experimental and calculated 4.2 K $\langle 111 \rangle$ Zeeman spectra for the Z$_{1,2}\gamma_{1,2}\longrightarrow$Y$_{2,3,4}\gamma_{3,5,2}$ transitions with (a) 0.0 T, (b) 0.2 T, (c) 0.4 T.