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Inducing interactions between individual photons is essential forapplications in photonic quantum information processing and fundamentalresearch on many-body photon states. A field that is well suited to combinestrong interactions and low losses is microwave quantum optics withsuperconducting circuits. Photons are typically stored in an $LC$ circuit, andinteractions appear when the circuit is shunted by a Josephson tunnel junction.Importantly, the zero-point fluctuations of the superconducting phase acrossthe junction control the strength and order of the induced interactions.Superconducting circuits have almost exclusively operated in the regime wherephase fluctuations are smaller than unity, and two-photon interactions, knownas the Kerr effect, dominate. In this experiment, we shunt a high-impedance$LC$ oscillator by a dipole that only allows pairs of Cooper pairs to tunnel.Phase fluctuations, which are effectively doubled by this pairing, reach thevalue of 3.4. In this regime of extreme fluctuations, we observe transitionfrequencies that shift non-monotonically as we climb the anharmonic ladder.From this spectroscopic measurement, we extract two-, three- and four-photoninteraction energies of comparable amplitude, and all exceeding the photon lossrate. This work explores a new regime of high-order photon interactions inmicrowave quantum optics, with applications ranging from multi-photon quantumlogic to the study of highly correlated microwave radiation.

Spectral signature of high-order photon processes mediated by Cooper-pair pairing