  We introduce a model of quantum magnetism induced by the non-perturbative
exchange of microwave photons between distant superconducting qubits. By
interconnecting qubits and cavities, we obtain a spin-boson lattice model that
exhibits a quantum phase transition where both qubits and cavities
spontaneously polarise. We present a many-body ansatz that captures this
phenomenon all the way, from a the perturbative dispersive regime where photons
can be traced out, to the non-perturbative ultra-strong coupling regime where
photons must be treated on the same footing as qubits. Our ansatz also
reproduces the low-energy excitations, which are described by hybridised
spin-photon quasiparticles, and can be probed spectroscopically from
transmission experiments in circuit-QED, as shown by simulating a possible
experiment by Matrix-Product-State methods.
