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.