  Inducing interactions between individual photons is essential for
applications in photonic quantum information processing and fundamental
research on many-body photon states. A field that is well suited to combine
strong interactions and low losses is microwave quantum optics with
superconducting circuits. Photons are typically stored in an $LC$ circuit, and
interactions appear when the circuit is shunted by a Josephson tunnel junction.
Importantly, the zero-point fluctuations of the superconducting phase across
the junction control the strength and order of the induced interactions.
Superconducting circuits have almost exclusively operated in the regime where
phase fluctuations are smaller than unity, and two-photon interactions, known
as 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 the
value of 3.4. In this regime of extreme fluctuations, we observe transition
frequencies that shift non-monotonically as we climb the anharmonic ladder.
From this spectroscopic measurement, we extract two-, three- and four-photon
interaction energies of comparable amplitude, and all exceeding the photon loss
rate. This work explores a new regime of high-order photon interactions in
microwave quantum optics, with applications ranging from multi-photon quantum
logic to the study of highly correlated microwave radiation.
