  Rare-earth ion dopants in solid-state hosts are ideal candidates for quantum
communication technologies such as quantum memory, due to the intrinsic
spin-photon interface of the rare-earth ion combined with the integration
methods available in the solid-state. Erbium-doped cerium oxide (Er:CeO$_2$) is
a particularly promising platform for such a quantum memory, as it combines the
telecom-wavelength (~1.5 $\mu$m) 4f-4f transition of erbium, a predicted long
electron spin coherence time supported by CeO$_2$, and is also near
lattice-matched to silicon for heteroepitaxial growth. In this work, we report
on the epitaxial growth of Er:CeO$_2$ thin films on silicon using molecular
beam epitaxy (MBE), with controlled erbium concentration down to 2 parts per
million (ppm). We carry out a detailed microstructural study to verify the
CeO$_2$ host structure, and characterize the spin and optical properties of the
embedded Er$^{3+}$ ions. In the 2-3 ppm Er regime, we identify EPR linewidths
of 245(1) MHz, optical inhomogeneous linewidths of 9.5(2) GHz, optical excited
state lifetimes of 3.5(1) ms, and spectral diffusion-limited homogenoeus
linewidths as narrow as 4.8(3) MHz in the as-grown material. We test annealing
of the Er:CeO$_2$ films up to 900 deg C, which yields modest narrowing of the
inhomogeneous linewidth by 20% and extension of the excited state lifetime by
40%. We have also studied the variation of the optical properties as a function
of Er doping and find that the results are consistent with the trends expected
from inter-dopant charge interactions.
