A Raman-Heterodyne Study of the Hyperfine Interaction of the Optically-Excited State ⁵D₀ of Eu³⁺:Y₂SiO₅

The spin coherence time of $^{151}$Eu$^{3+}$ which substitutes the yttrium at site 1 in Y$_2$SiO$_5$ crystal has been extended to 6 hours in a recent work [\textit{Nature} \textbf{517}, 177 (2015)]. To make this long-lived spin coherence useful for optical quantum memory applications, we experimentally characterize the hyperfine interaction of the optically-excited state $^5$D$_0$ using Raman-heterodyne-detected nuclear magnetic resonance. The effective spin Hamiltonians for excited and ground state are fitted based on the experimental spectra obtained in 200 magnetic fields with various orientations. To show the correctness of the fitted parameters and potential application in quantum memory protocols, we also characterize the ground-state hyperfine interaction and predict the critical magnetic field which produces the 6-hour-long coherence time. The complete energy level structure for both the $^7$F$_0$ ground state and $^5$D$_0$ excited state at the critical magnetic field are obtained. These results enable the design of quantum memory protocols and the optimization of optical pumping strategy for realization of photonic quantum memory with hour-long lifetime.