High cooperativity coupling between a phosphorus donor spin ensemble and a superconducting microwave resonator

We investigate the coupling of an ensemble of phosphorus donors in an isotopically purified ${}^{28}$Si host lattice interacting with a superconducting coplanar waveguide resonator. The microwave transmission spectrum of the resonator shows a normal mode splitting characteristic for high cooperativity. The evaluated collective coupling strength $g_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is of the same order as the loss rate of the spin system $\gamma$, indicating the onset of strong coupling. We develop a statistical model to describe the influence of temperature on the coupling strength from $50\mathrm{m}\mathrm{K}$ to $3.5\mathrm{K}$ and find a scaling of the coupling strength with the square root of the number of thermally polarized spins.