Temperature-dependent mechanical losses of Eu${}^{3+}$:Y${}_2$SiO${}_5$ for spectral hole burning laser stabilization

We investigate the mechanical loss characteristics of Eu${}^{3+}$:Y${}_2$SiO${}_5$$\text{unicode}x2013$a promising candidate for ultra-low-noise frequency stabilization through the spectral hole burning technique. Three different mechanical oscillators with varying surface-to-volume ratios and crystal orientations are evaluated. In this context, we perform mechanical ringdown and spectral measurements spanning temperatures from room temperature down to $15\mathrm{K}$. By doing so, we measure a maximum mechanical quality factor of $Q=3676$, corresponding to a loss angle of $\phi =2.72\times {10}^{-4}$. For a spectral hole burning laser stabilization experiment at $300\mathrm{m}\mathrm{K}$, we can estimate the Allan deviation of the fractional frequency instability due to Brownian thermal noise to be below ${\sigma }_{\delta \nu \mathrm{/}{\nu }_0}=2.5\times {10}^{-18}$, a value lower than the estimated thermal-noise limit of any current cavity-referenced ultra-stable laser experiment.