We study spin-lattice relaxation times of electron spins in ${\mathrm{Er}}^{3+}:{\mathrm{CaWO}}_4$ coupled to a low mode-volume superconducting resonator at millikelvin temperatures in a regime where resonant phonons are cooled to near their quantum ground state. At large magnetic field supporting strong phonon-emission rates, we observe a noticeable increase in relaxation times with increasing spin spectral density $c/{\mathrm{\Gamma }}_{\mathrm{inh}}$, which exhibits a $[\tanh (\hslash {\omega }_0/2k_{\mathrm{B}}T){]}^2$ contribution. These observations are typical of a phonon-bottlenecked spin relaxation, and have implications for quantum technologies that exploit rare-earth spin ensembles as coherent resources.

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• Spontaneous emission
• Vibrational states