Nuclear quadrupole interaction and zero first-order Zeeman transitions of Er 167 3 + in CaWO 4

We report microwave spectroscopy of $^{167}$Er$^{3+}$ doped in CaWO$_4$ which reveals the hyperfine splitting of the erbium electronic ground state ($Z_1$, $J_\mathrm{eff.}$=15/2) induced by the $I$=7/2 nuclear spin. From spectra measured below$\sim$50 mK in magnetic fields up to 200 mT, we extract spin Hamiltonian parameters including the electron $\textbf{g}$, hyperfine $\textbf{A}$, and nuclear electric quadrupolar $\textbf{Q}$ tensors. Crucially, our analysis demonstrate unambiguously, that the previously unobserved nuclear electric quadrupolar moment is essential to reproduce the experimental data. With these refined parameters, we identify zero first-order Zeeman (ZEFOZ) transitions at zero magnetic field. Extending the analysis to finite fields, we uncover that ZEFOZ points lie either along the $c$ axis or within the $a$-$b$ plane. These results establish CaWO$_4$ as a promising host for long lifetime quantum memories.