Two-photon coupling via Josephson element I: Breaking the symmetry with magnetic fields

We consider a coupling element based on a symmetric superconducting quantum interference device (SQUID) and show that it mediates a two-photon interaction. This and other inductive interactions can be switched off in situ. We derive the system Hamiltonian for coupled resonator and rf SQUID. The rf SQUID dwells in the vicinity of its metastable well holding a number of energy states and acts as an artificial atom. We discuss how the Josephson symmetry breaks owing to magnetic fields in the superconducting loops. We assess that the two-photon coupling strength reaches 18 MHz which can exceed the single-photon capacitive interaction in the coupler.