Hyperfine spectroscopy and fast, all-optical arbitrary state initialization and readout of a single, ten-level ${}^{73}$Ge vacancy nuclear spin qudit in diamond

A high-spin nucleus coupled to a color center can act as a long-lived memory qudit in a spin-photon interface. The germanium vacancy (GeV) in diamond has attracted recent attention due to its excellent spectral properties and provides access to the ten-dimensional Hilbert space of the $I=9\mathrm{/}2$ ${}^{73}$Ge nucleus. Here, we observe the ${}^{73}$GeV hyperfine structure, perform nuclear spin readout, and optically initialize the ${}^{73}$Ge spin into any eigenstate on a $\mu$s timescale and with a fidelity of up to $\sim 84\mathrm{\%}$. Our results establish the ${}^{73}$GeV as an optically addressable high-spin quantum platform for a high-efficiency spin-photon interface as well as for foundational quantum physics and metrology.