Photon condensation in magnetic cavity QED

We show that a system of magnetic molecules coupled to microwave cavities (or LC resonators) undergoes the equilibrium superradiant transition when their mutual interaction exceeds a critical value and that this transition is experimentally observable. In order to do that, we calculate the equilibrium thermodynamics of a macroscopic number of spins (interacting or not) coupled to the quantized magnetic field of a cavity (multimode or not). The effects of the coupling are first illustrated by the vacuum-induced ferrromagnetic ordering in a quantum Ising model.Then, we compute how the coupling modifies the magnetic phase diagram of ${\rm Fe_8}$ dipolar crystals, which exemplifies the co-operation between intrinsic and photon induced spin-spin interactions. To conclude, we demonstrate that a simple transmission experiment can resolve the superradiant transition, and thus measure the quantum electrodynamical control of magnetism.