Nuclear quadrupole interaction and zero first-order Zeeman transitions of ${}^{167}$Er${}^{3+}$ 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 ($Z_1$, $J_{\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{.}}$=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 $\mathbf{\text{g}}$, hyperfine $\mathbf{\text{A}}$, and nuclear electric quadrupolar $\mathbf{\text{Q}}$ 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.