Monte Carlo simulation of the nuclear spin decoherence process in ${\mathrm{Eu}}^{3+}$:${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystals

${\mathrm{Eu}}^{3+}$:${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystals are promising candidates for quantum memories because their nuclear spin coherence lifetime can be extended to 47 seconds at a particular magnetic field. Although an extended coherence lifetime of 6 hours can be achieved with a large number of dynamic decoupling sequences, these pulses will also introduce unwanted noise. To realize quantum storage with a lifetime of hours, it is necessary to quantitatively characterize the spin decoherence process of ${\mathrm{Eu}}^{3+}$ ions to guide the experimental efforts. In practice, a number of limiting factors could impose a limit on the achievable spin coherence lifetime, which includes the spin inhomogeneous broadening of the ${\mathrm{Eu}}^{3+}$ ensemble and the homogeneity and stability of the external magnetic field. Here, we provide a Monte Carlo simulation model of the nuclear spin decoherence process of ${\mathrm{Eu}}^{3+}$ ions in ${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystals, and these limiting factors are considered to provide an accurate prediction of the spin coherence lifetimes. This method is universal and can be applied to other rare-earth-ion-doped crystals.

Figure 1

The normalized spin-echo signal of ${\mathrm{Eu}}^{3+}$:${\mathrm{Y}}_2{\mathrm{SiO}}_5$ in logarithmic coordinates depends on the total evolution time $\tau$. Here the ZEFOZ field is ${[-0.786264\mathrm{T},-0.685481\mathrm{T},0.714586\mathrm{T}]}_{[D_1,D_2,b]}$ in the $D_1{D}_{2}b$ coordinate of ${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystals. The black squares (line) represent the Monte Carlo simulation results (fitting results) at a magnetic field of 0.05 T away from the ZEFOZ field. The red circles (line) represent the Monte Carlo simulation results (fitting results) at a magnetic field of 0.005 T away from the ZEFOZ field. The fitting results show that the phase memory times are 0.6 and 2.8 s, respectively.

Figure 2

The nuclear spin coherence lifetime is plotted against the inhomogeneity of the magnetic field for the ${}^{151}\mathrm{Eu}^{3+}$:${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystal with a spin inhomogeneous broadening of 10 kHz. The definition of inhomogeneity is the peak-to-peak value divided by the mean of the magnetic field.

Figure 3

The nuclear spin coherence lifetime is plotted against the rotation angle of the magnetic field relative to the ideal direction for the ${}^{151}\mathrm{Eu}^{3+}$:${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystal with a spin inhomogeneous broadening of 10 kHz. Angular represents the direction of the axis of rotation, and radial represents the angle of rotation. The axis of rotation corresponding to $0^{\circ }\left({90}^{\circ }\right)$ is ${[-0.657,0.754,0]}_{[D_1,D_2,b]}$ (${[-0.426,-0.371,-0.825]}_{[D_1,D_2,b]}$).

Figure 4

The field perturbation generated by ${}^{29}\mathrm{Si}$ and ${}^{17}\mathrm{O}$. The calculations include 2000 nearest Si and 10 000 nearest O. The isotope distribution was randomized 20 000 times to give the average results.

Figure 5

The nuclear spin coherence lifetime is plotted against the deviation value of the magnetic field in the $[x,y,z]=[D_1,D_2,b]$ direction relative to the ideal ZEFOZ field for ${\mathrm{Pr}}^{3+}$ ions. $dBx$, $dBy$, and $dBz$ are the deviation values of the magnetic field in the $x$, $y$, and $z$ directions. The top and middle figures are calculated at dBz = 0 and 0.5 G. The bottom figure is calculated at $dBy=0$ G.