Optical investigation of ultra-slow spin relaxation in $^{171}$Yb$^{3+}$:Y$_2$SiO$_5$ single crystals

We present a comprehensive study of spin relaxation dynamics at cryogenic temperatures in a rare-earth-doped crystal used for quantum memory applications: $^{171}$Yb:Y$_2$SiO$_5$. Spin relaxation is indeed a major limiting factor for both the efficiency and storage time of quantum memory protocols based on atomic frequency combs in rare-earth materials. The relaxation dynamics among the four ground-state hyperfine levels were simultaneously investigated by optically perturbing the spin population distribution and monitoring its return to thermal equilibrium through optical absorption spectroscopy. By applying different types of perturbations, we were also able to distinguish between two types of relaxation processes, induced by spin-phonon and spin-spin interactions. Below 1 K, we observed that the re-thermalization of the Yb$^{3+}$ ion population takes several hours, driven solely by direct phonon absorption or emission. However, the effective lifetime of individual spin states is much shorter - on the order of several seconds in low-doped (2 ppm) samples and of milliseconds in 10 ppm samples - due to spin-spin interactions. These findings provide valuable guidelines for optimizing doping levels and operating temperatures in rare-earth-doped crystals for quantum applications. Notably, they suggest that atomic frequency combs with lifetimes of several hours could be realized using $^{171}$Yb:Y$_2$SiO$_5$ crystals with slightly less than 2 ppm doping and operating near 1 K.