Occupation-selective photon-pair interactions in a flux-pumped SQUID resonator
Flux-pumped Josephson circuits provide a versatile platform for parametric photon generation and multimode bosonic control. We identify an occupation-conditioned pair interaction that arises when two harmonic normal modes participate in the same driven SQUID coordinate. We consider a degenerate pair process in which two photons are created or annihilated in a signal mode while the occupation of a second harmonic mode remains unchanged. Although it does not exchange excitations, this controller mode modifies both the resonance frequency and the strength of the pair transition. The two effects have different physical origins: diagonal Josephson nonlinearities shift the pair resonance, whereas the nonlinear Josephson matrix element makes the avoided-crossing gap itself depend on the controller occupation. Starting from the full Josephson cosine, we develop complementary time-dependent Schrieffer--Wolff and Magnus descriptions that expose the direct and virtual processes responsible for this interaction, and benchmark the resulting effective theory against a numerically converged Floquet treatment of the unexpanded two-mode circuit Hamiltonian. The occupation dependence produces a two-dimensional pair spectrum that enables selective transitions within the bosonic Fock lattice. When the controller is prepared in a coherent superposition, the same interaction converts this conditional dynamics into signal--controller entanglement and controller-dependent non-Gaussian state generation. Observable-level comparisons further show that the variation of the pair-transition strength cannot be reproduced by occupation-dependent resonance shifts alone. These results establish a boson-controlled parametric interaction in which the state of one harmonic mode controls both the spectral position and the coupling strength of a nonlinear process in another.