Nuclear quadrupole interaction and zero first-order Zeeman transitions of $^{167}$Er$^{3+}$ in CaWO$_4$

TL;DR

The paper addresses magnetic-noise-induced decoherence in Er3+-doped CaWO4 by performing millikelvin microwave spectroscopy to extract a complete spin Hamiltonian, including electron g-tensor, hyperfine A, and nuclear quadrupole Q terms. It demonstrates that the nuclear electric quadrupole interaction is essential to reproduce the observed hyperfine structure, especially zero-field splittings, and uses this model to locate ZEFOZ transitions at zero and finite magnetic fields. ZEFOZ points are found to cluster along the c-axis or in rings in the a-b plane, with the strongest in-plane transitions predicted to yield coherence times exceeding several seconds at practical fields (∼2 T). These results position CaWO4 as a promising host for Er-based quantum memories, combining telecom-band access with robust magnetic-noise protection.

Abstract

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.

Figures (5)

• Figure 1: The hyperfine interaction in $^{167}$Er$^{3+}$ between the nuclear magnetic dipole moment ($I$=7/2) and the $Z_1$ electronic ground state of the $J$=15/2 crystal electric field manifold (which can be described as an effective $S_\mathrm{eff.}$=1/2 state). The resultant 16 hyperfine energy levels can harbor clock transitions, which are also known as ZEFOZ transitions.
• Figure 2: (a) The crystal structure of CaWO$_4$ with Er$^{3+}$ dopant ion residing on the Ca$^{2+}$ site. (b) Magnetic field distribution of ZEFOZ points based on our three-dimensional B field search. The points lie either along the crystal $c$-axis or within the $a$-$b$ plane.
• Figure 3: (a) Normalized $S_{21}$ transmission as a function of field along the crystal $c$ axis, overlaid with the spectrum calculated from the refined spin Hamiltonian parameters. To highlight the zero-field splitting due to the nuclear electric quadrupolar moment of $^{167}$Er$^{3+}$, we plot in panels (b)--(d) the measured spectrum in a smaller frequency-field window. Panels (e)--(g) plots the calculated susceptibilities, which agree well with the measurements. We observed all visible transitions predicted by susceptibility calculations, apart from the line cutting across group C, which we are not able to explain.
• Figure 4: The distribution of the ZEFOZ points as function of $|\mathbf{S}_2|$ and magnetic field strength, parallel and perpendicular to the crystal $c$ axis.
• Figure 5: The stability of the $|14\rangle$$\rightarrow$$|15\rangle$ transition at $\textbf{B}$=(1.980,0,0), with respect to magnetic fluctuations in the (a) $\delta B_x$-$\delta B_z$ and (b) $\delta B_y$-$\delta B_z$ directions, respectively. (insert) the definition of magnetic field deviations $\delta B_x$, $\delta B_y$ and $\delta B_z$ with respect to the ZEFOZ point. (a) The heatmap indicates that the coherence time is more resilient against misalignment in the z direction. (b) The coherence time is also invariant with respect to the misalignment along the ring.