  Solid-state electronic spins are extensively studied in quantum information
science, as their large magnetic moments offer fast operations for computing
and communication, and high sensitivity for sensing. However, electronic spins
are more sensitive to magnetic noise, but engineering of their spectroscopic
properties, e.g. using clock transitions and isotopic engineering, can yield
remarkable spin coherence times, as for electronic spins in GaAs, donors in
silicon and vacancy centres in diamond. Here we demonstrate simultaneously
induced clock transitions for both microwave and optical domains in an
isotopically purified $^{171}$Yb$^{3+}$:Y$_2$SiO$_5$ crystal, reaching
coherence times of above 100 $\mu$s and 1 ms in the optical and microwave
domain, respectively. This effect is due to the highly anisotropic hyperfine
interaction, which makes each electronic-nuclear state an entangled Bell state.
Our results underline the potential of $^{171}$Yb$^{3+}$:Y$_2$SiO$_5$ for
quantum processing applications relying on both optical and spin manipulation,
such as optical quantum memories, microwave-tooptical quantum transducers, and
single spin detection, while they should also be observable in a range of
different materials with anisotropic hyperfine interaction.
