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

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.

Figures (12)

• Figure 1: (a) Schematic of the energy level structure of $^{167}$Er$^{3+}$ doped in YSO. The superhyperfine interaction, depicted in the gray shaded area, is neglected in the effective spin Hamiltonian of Eq. \ref{['eqSpinH']}. Its effect is not considered in the search for ZEFOZ points, as its magnitude is negligible compared to the hyperfine interaction. (b) The energy level structure of the ground states $^4 I_{15/2}\ (Z_1)$ of $^{167}$Er$^{3+}$:YSO at site 1. The external magnetic field $\bm B$ is applied along the $D_1$ axis. Each state can be expressed by $\ket{S_z, I_z}$ where $S_z=+1/2 \ (-1/2)$ is represented by $\uparrow \ (\downarrow)$ and is denoted from the lowest to highest as $\ket{i}$ ($i=0, 1, 2,\cdots, 15$) for simplicity.
• Figure 2: Histogram of magnetic fluctuation by (a) Y ions $|\Delta \bm B_{\rm Y}|$ and (b) doped Er ions ($n_{\rm Er}= 10$ ppm) $|\Delta \bm B_{\rm Er}|$. The black dashed lines are fitting by probability density function. Insets show the images of magnetic fluctuations acting on the target Er$^{3+}$ center. (c) Dependence of $|\Delta \bm B_{\rm Er}|$ on $n_{\rm Er}$. The black dashed lines are the fitting curves. The distributions are normalized at their respective peak values.
• Figure 3: Dependence of $T_2^{\rm hyp}$ on $n_{\rm Er}$ at 0 T for (a) site 1 and (b) site 2. The red and blue lines indicate the dependence for the ZEFOZ transition $\ket{7} \rightleftarrows \ket{9} \ (\ket{\downarrow,+7/2} \rightleftarrows \ket{\uparrow,+5/2})$ for site 1 and site 2 affected by only $\Delta \bm B_{\rm Er}$ and by $\Delta \bm B_{\rm Er}$ and $\Delta \bm B_{\rm Y}$, respectively. The black solid lines represent the dependence of $T_2^{\rm hyp}$ on $n_{\rm Er}$ for other ZEFOZ transitions affected by $\Delta \bm B_{\rm Er}$ and $\Delta \bm B_{\rm Y}$. (c) $T_2^{\rm hyp}$ mapping for the ZEFOZ transitions between the $i$th and $j$th HF levels affected by $\bm B_{\rm Er}$ and $\Delta \bm B_{\rm Y}$ for (c) site 1 and (d) site 2 with the same color scale at $n_{\rm Er}$=10 ppm.
• Figure 4: $T_2^{\rm hyp}$ and $|\bm B|$ of the obtained ZEFOZ points for (a) site 1 and (b) site 2. The color tables are scaled by $|\bm S_2|$. The transition with the longest $T_2^{\rm hyp}$ has the minimal $\bm S_2$ actually according to Eq. \ref{['eq8']}. The ZEFOZ transition with the longest $T_2^{\rm hyp}$ under our condition $|\bm B|\le 3$ T corresponds to $\ket{10}-\ket{11}$$[\ket{\uparrow,+1/2} \rightleftarrows \ket{\uparrow,+3/2}]$ ($\ket{14}-\ket{15}$$[\ket{\uparrow,-7/2} \rightleftarrows \ket{\uparrow,-5/2}]$) for site 1 (site 2) indicated by the arrow. The dashed vertical line indicates $|\bm B|=3$ T.
• Figure 5: (a) The magnetic field response of the ZEFOZ point with the longest $T_2^{\rm hyp}$ at site 2 is indicated by the red line, and the response of other ZEFOZ points are shown by the black lines. The magnetic field direction for the ZEFOZ point with the longest $T_2^{\rm hyp}$ is $(\theta, \phi)=$( $\pm$37.5383$^{\circ}$, -10.9417$^{\circ}$). (b) Enlarged view of the dashed circle in (a) around +0.634 T. The FWHM of the peak is $\sim$1 mT. (c) Magnetic field magnitude $B$ versus transition frequency $\nu$ for the ZEFOZ point with the longest $T_2^{\rm hyp}$ (red line). (d) $T_2^{\rm hyp}$ mapping for the transition between the $i$th and $j$th HF levels at $B$=+633.52 mT.