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

1. Problem Statement

Rare-earth-ion doped crystals are leading candidates for scalable quantum memories due to their long coherence times and narrow optical linewidths. Erbium (Er³⁺) is particularly attractive because its intra-4f transition lies within the telecommunications C-band, allowing direct interfacing with fiber-optic infrastructure. However, Er³⁺ ions possess a large electronic magnetic moment ($J_{eff}=15/2$), making them highly susceptible to decoherence from external magnetic field fluctuations and the nuclear spin bath of the host crystal.

While calcium tungstate (CaWO₄) has emerged as a promising host with a relatively "quiet" magnetic environment, systematic identification of Zero First-order Zeeman (ZEFOZ) transitions—operating points where transition frequencies are insensitive to linear magnetic field fluctuations—had not been achieved for ¹⁶⁷Er³⁺:CaWO₄. Previous studies relied on high-field electron spin resonance (ESR) data that neglected the nuclear electric quadrupole interaction (NQI). The authors hypothesized that the NQI, arising from charge compensation at the substitutional Ca²⁺ site, is substantial and essential for accurately modeling the spin Hamiltonian, particularly near zero magnetic field.

2. Methodology

• Sample Preparation: Single-crystal CaWO₄ doped with naturally occurring erbium at a concentration of 100 ppm was grown via the hybrid Czochralski method. Samples were cut and polished along the tetragonal $a$ and $c$ axes.
• Experimental Setup: Microwave spectroscopy was performed at millikelvin temperatures ($T \sim 50$ mK) using a dilution refrigerator. The samples were coupled to a coplanar waveguide (CPW) and subjected to magnetic fields up to 200 mT (and up to 1 T for orientation studies) using a vector magnet. Transmission ($S_{21}$) was measured using a vector network analyzer (VNA).
• Theoretical Modeling:
  • The system was modeled using a spin Hamiltonian for the effective spin-1/2 ground state ($Z_1$) coupled to the ¹⁶⁷Er nuclear spin ($I=7/2$).
  • The Hamiltonian included terms for electronic Zeeman, nuclear Zeeman, hyperfine coupling ($A$), and the nuclear electric quadrupole interaction ($Q$).
  • Parameters ($g$, $A$, and $Q$ tensors) were extracted via global nonlinear least-squares fitting of the experimental spectra across multiple field orientations.
• ZEFOZ Search:
  • The Hamiltonian was numerically diagonalized to calculate eigenenergies and transition frequencies.
  • A search was conducted for points where the magnetic field gradient of the transition frequency ($S_1 = \nabla_B f$) is zero (ZEFOZ condition).
  • Second-order derivatives ($S_2$, curvature) were calculated to estimate residual quadratic coupling to noise.
  • Monte Carlo simulations were used to estimate ambient magnetic noise ($\Delta B$) from the host lattice and dopant spins to predict coherence times ($T_2^{hyp}$).

3. Key Contributions

• Discovery of Essential NQI: The paper provides the first unambiguous experimental evidence that the nuclear electric quadrupole moment is non-zero and critical for ¹⁶⁷Er³⁺ in CaWO₄. Previous models ignoring $Q$ failed to reproduce zero-field splittings observed in the spectra.
• Refined Spin Hamiltonian: The authors extracted precise tensor parameters for $g$, $A$, and $Q$, correcting previous high-field-only estimates.
• Systematic ZEFOZ Mapping: The study mapped ZEFOZ transitions not only at zero field but also across a 3D magnetic field space (up to 20 T), identifying specific geometric loci (along the $c$-axis and within the $a$-$b$ plane) where these transitions occur.
• Prediction of Long Coherence: Theoretical modeling predicts coherence times exceeding 3 seconds for specific finite-field configurations, a significant improvement over zero-field candidates.

4. Key Results

• Spectroscopic Observations:
  • Measured spectra revealed hyperfine splitting of the electronic ground state.
  • Crucially, distinct zero-field splittings of $\sim30$ MHz (Group B) and $\sim50$ MHz (Group C) were observed. These could only be reproduced by including a non-zero quadrupole tensor $Q$.
  • Fitted parameters:
    • $g$-tensor: Diagonal with $g_{\perp} \approx 8.3$, $g_{\parallel} \approx 1.26$.
    • $A$-tensor: $A_{\perp} \approx -871$ MHz, $A_{\parallel} \approx -128$ MHz.
    • $Q$-tensor: $Q_{\perp} \approx 1.68$ MHz, $Q_{\parallel} \approx -3.36$ MHz.
• Zero-Field ZEFOZ:
  • Only one true ZEFOZ transition ($|0\rangle \to |15\rangle$) was found at $B=0$ with frequency 3.4844 GHz.
  • However, its large curvature ($|S_2| \approx 1.28 \times 10^5$ GHz T⁻²) limits the predicted coherence time to $\sim 12.7$ µs.
• Finite-Field ZEFOZ:
  • Geometric Distribution: ZEFOZ points cluster in two distinct regions:
    1. Along the crystal $c$-axis (below 7 T).
    2. Within the $a$-$b$ plane (forming continuous rings below 2 T), a result of the Hamiltonian's axial symmetry.
  • Performance: Transitions perpendicular to the $c$-axis (in the $a$-$b$ plane) exhibit significantly lower curvature ($|S_2|$) than those along the $c$-axis.
  • Optimal Candidate: The $|14\rangle \to |15\rangle$ transition at $B \approx 1.98$ T (along the $a$-axis) shows the smallest curvature ($|S_2| \approx 1.24$ GHz T⁻²).
  • Predicted Coherence: Monte Carlo simulations estimate a coherence time ($T_2^{hyp}$) exceeding 3 seconds for the $a$-axis configuration, which is three orders of magnitude longer than the $c$-axis configuration.
• Robustness: The coherence time is highly sensitive to field magnitude deviations along the $x$-axis but invariant to angular deviations along the ring (azimuthal symmetry), suggesting that precise field strength control is more critical than angular alignment.

5. Significance

This work establishes CaWO₄ doped with ¹⁶⁷Er³⁺ as a premier platform for long-lifetime quantum memories. By identifying the critical role of the nuclear quadrupole interaction, the authors resolved discrepancies in previous models and unlocked the ability to predict ZEFOZ transitions accurately. The discovery of ZEFOZ points in the $a$-$b$ plane with predicted coherence times in the seconds range (compatible with telecom-band optical access) addresses the primary bottleneck of magnetic decoherence in rare-earth systems. This paves the way for the development of scalable, high-fidelity quantum repeaters and networks.