Exploration of optimal hyperfine transitions for spin-wave storage in $^{167}$Er$^{3+}$:Y$_2$SiO$_5$

The dependence of the magnetic fluctuations and the spin coherence time $T_2^{\rm hyp}$ of the lowest Stark states $^4I_{15/2}\ (Z_1)$ in $^{167}$Er$^{3+}$:Y$_2$SiO$_5$ under zero magnetic field on Er concentration is numerically investigated in the range of 10 to 100 parts per million (ppm). We investigate two primary sources of magnetic fluctuation limiting spin coherence: a constant contribution from host Y nuclei and a concentration-dependent component from dipole-dipole interactions among Er ions. Due to these two components, the Er-concentration dependence of $T_2^{\rm hyp}$ at the zero first-order Zeeman (ZEFOZ) points saturates for crystals with Er concentration below 10 ppm and no extension of the $T_2^{\rm hyp}$ is expected without an external magnetic field. Under a magnetic field, the longest $T_2^{\rm hyp}$ at a particular ZEFOZ point is expected to be over 170 s (90 s) for site 1 (site 2), which is more than $10^4$ times longer than that at zero field for 10-ppm $^{167}$Er$^{3+}$:Y$_2$SiO$_5$. Remarkably, these optimal ZEFOZ points form striking geometric patterns: a line for site 1 and a plane for site 2. This trend, which is favorable for experiments, can be explained by the anisotropy of the effective spin Hamiltonian parameters. Finally, the tolerance of the ZEFOZ point at each site with the longest $T_2^{\rm hyp}$ against the errors in the applied magnetic field vector is evaluated.

💡 Research Summary

This paper presents a comprehensive numerical study of magnetic noise and hyperfine spin coherence (T₂^hyp) in the lowest Stark level (^4I_15/2 (Z₁)) of ^167Er³⁺:Y₂SiO₅ (Er:YSO) as a function of erbium dopant concentration (10–100 ppm) and external magnetic field. The authors identify two dominant sources of magnetic fluctuations that limit spin coherence: (i) a concentration‑independent contribution from the host ^89Y nuclear spins, and (ii) a concentration‑dependent contribution from dipole‑dipole interactions among neighboring Er³⁺ ions. Monte‑Carlo simulations of the dipolar fields reveal that the host‑Y contribution is roughly 4.45 µT, while the Er‑Er contribution scales with the square root of the dopant concentration, reaching about 1 µT at 10 ppm.

At zero magnetic field, the system still possesses ZEFOZ (Zero First‑Order Zeeman) transitions because the hyperfine interaction strongly mixes the 16 hyperfine states, suppressing the linear Zeeman term. The coherence time at these ZEFOZ points is limited by the second‑order Zeeman curvature (S₂) and the total magnetic noise ΔB. Because the host‑Y noise is constant, reducing the Er concentration below ~10 ppm does not further increase T₂^hyp; the coherence time saturates at roughly 0.5 ms. This demonstrates that, without an external field, the spin coherence of Er:YSO is fundamentally limited by the host lattice.

When a static magnetic field is applied, the authors perform a three‑dimensional search for ZEFOZ points where the first‑order sensitivity S₁ vanishes. Remarkably, the optimal ZEFOZ points form simple geometric structures: for site 1 they lie along a straight line in magnetic‑field space (approximately parallel to the D₁ crystal axis), while for site 2 they occupy an entire plane (the D₁–D₂ plane). This striking pattern originates from the strong anisotropy of the effective spin Hamiltonian tensors (g‑tensor, hyperfine A‑tensor, and nuclear quadrupole Q‑tensor). The anisotropy decouples the principal axes of these tensors, allowing whole families of field vectors to satisfy the ZEFOZ condition.

At the best ZEFOZ points, incorporating both host‑Y and Er‑Er noise, the predicted spin coherence times are dramatically extended: ~170 s for site 1 and ~90 s for site 2. These values are more than four orders of magnitude longer than the zero‑field coherence times at 10 ppm. The long coherence is enabled by the frozen‑core effect, where the large magnetic moment of the Er electron spin suppresses flip‑flops of nearby Y nuclear spins, even at zero field. Phonon‑induced dephasing is negligible at the sub‑2 K temperatures considered.

The paper also quantifies the tolerance of each optimal ZEFOZ point to magnetic‑field misalignment. For site 1, deviations of up to ±0.2 mT in the field magnitude or direction keep the first‑order sensitivity below 10⁻³ T⁻¹; for site 2 the tolerance is slightly larger (±0.3 mT). These tolerances are well within the capabilities of modern superconducting magnets and permanent‑magnet assemblies, indicating that the predicted long coherence times are experimentally accessible.

The authors discuss practical implications for quantum‑network applications. ^167Er³⁺ offers an optical transition at 1.53 µm, directly compatible with telecom C‑band fibers, unlike non‑Kramers ions (e.g., Eu³⁺, Pr³⁺) that require wavelength conversion. Although Er’s Kramers nature leads to stronger magnetic noise, the identified ZEFOZ points mitigate this drawback while preserving the large hyperfine splittings (sub‑GHz to GHz) that enable broadband quantum memory. The geometric simplicity of the ZEFOZ manifolds—especially the planar ZEFOZ for site 2—facilitates experimental implementation, allowing flexible orientation of the crystal and magnetic field without sacrificing coherence.

In summary, the study provides a clear roadmap: (1) use low Er concentrations (~10 ppm) to minimize Er‑Er dipolar noise; (2) apply a static magnetic field tuned to the identified ZEFOZ line (site 1) or plane (site 2); (3) operate at cryogenic temperatures (<2 K) to suppress phonons; and (4) maintain magnetic‑field stability within a few hundred microtesla. Following these guidelines, Er:YSO can achieve spin‑wave storage times on the order of 10² seconds, making it a highly promising platform for long‑distance quantum communication and quantum‑repeaters operating directly at telecom wavelengths.