Hyperfine interaction and coherence time of praseodymium ions at the site 2 in yttrium orthosilicate

Praseodymium (Pr$^{3+}$) ions doped in the site 1 of yttrium orthosilicate (Y$_2$SiO$_5$) has been widely employed as the photonic quantum memory due to their excellent optical coherence and spin coherence. While praseodymium ions occupying the site 2 in Y$_2$SiO$_5$ crystal have better optical coherence as compared with those at site 1, which may enable better performance in quantum memory. Here we experimentally characterize the hyperfine interactions of the ground state $^3$H$_4$ and excited state $^1$D$_2$ of Pr$^{3+}$ at site 2 in Y$_2$SiO$_5$ using Raman heterodyne detected nuclear magnetic resonance. The Hamiltonians for the hyperfine interaction are reconstructed for both ground state $^3$H$_4$ and excited state $^1$D$_2$ based on the Raman heterodyne spectra in 201 magnetic fields. The two-pulse spin-echo coherence lifetime for the ground state is measured to be 2.6$\pm$0.1 ms at site 2 with zero magnetic field, which is more than five times longer than that at site 1. The magnetic fields with zero first order Zeeman shift in the hyperfine transition for Pr$^{3+}$ at site 2 in Y$_2$SiO$_5$ are identified.