Electron-spin spectral diffusion in an erbium doped crystal at millikelvin temperatures

Erbium doped crystals offer a versatile platform for hybrid quantum devices because they combine magnetically sensitive electron-spin transitions with telecom-wavelength optical transitions. At the high doping concentrations necessary for many quantum applications, however, strong magnetic interactions of the electron-spin bath lead to excess spectral diffusion and rapid decoherence. Here we lithographically fabricate a 4.4 GHz superconducting planar microresonator on a ${\text{CaWO}}_{4}$ crystal doped with Er ions at a concentration of 20 ppm relative to Ca. Using the microwave resonator, we characterize the spectral diffusion processes that limit the electron-spin coherence of Er ions at millikelvin temperatures by applying two- and three-pulse echo sequences. The coherence time shows a strong temperature dependence, reaching 1.3 ms at 23 mK for an electron-spin transition of $^{167}\mathrm{Er}$.