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

TL;DR

This study numerically investigates the spin coherence times of $^{167}$Er$^{3+}$:Y$_2$SiO$_5$ at zero and applied magnetic fields, identifying optimal hyperfine transition points and their geometric patterns to enhance quantum memory performance.

Contribution

It reveals the magnetic fluctuation sources affecting spin coherence and identifies geometric patterns of ZEFOZ points, providing insights for optimizing quantum memory in Er-doped crystals.

Findings

Maximum $T_2^{ m hyp}$ exceeds 170 s under magnetic field.

ZEFOZ points form line and plane patterns for different sites.

Coherence time saturation occurs below 10 ppm Er concentration.

Abstract

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.