Waveguide quantum electrodynamics at the onset of spin-spin correlations

We explore the competition between light-mediated and intrinsic matter-matter interactions in waveguide quantum electrodynamics. For this, we couple a superconducting transmission line to a model magnetic material, made of organic free radical molecules with a spin $S=1/2$ and a $g_{S}$ factor very close to that of a free electron. The microwave transmission has been measured in a wide range of temperatures ($0.013$ K $\leq T \leq 2$ K), magnetic fields ($0\leq B \leq 0.5$ T) and frequencies ($0 \leq \omega/2 \pi \leq 14$ GHz). We find that molecules belonging to one of the two crystal sublattices form one-dimensional spin chains. Temperature then controls the intrinsic correlations along these chains in a continuous and monotonic way. In the paramagnetic region ($T > 0.7$ K), the microwave transmission shows evidences for the collective coupling of quasi-identical spins to the propagating photons, with coupling strengths that reach values close to the dissipation rates. As $T$ decreases, the growth of spin correlations, combined with the anisotropy in the spin-spin exchange constants, tend to suppress the collective spin-photon coupling. In this regime, the spin visibility in transmission reflects also a gradual change in the nature of the dominant spin excitations, from single spin flips to bosonic magnons.