Two-photon coupling via Josephson element II: Interaction dressing, cross-Kerr coupling, and limits of low-energy bosonic model

We study the interactions mediated by a symmetric superconducting quantum interference device (SQUID), their renormalizations, and the applicability of the anharmonic oscillator (bosonic) model for a coupled phase qubit. The latter dwells in its metastable well holding a number of anharmonic energy states. The coupling SQUID can switch between the single- and two-photon interactions in situ. We find that, in the bosonic two-photon regime, the cross-Kerr coupling never vanishes as it dresses due to asymmetry in the qubit potential and nonlinearity of the coupler. Our quantitative results also depend on the bosonic approximation. We approach determining its limits by finding the minimum number of coherent energy states required for a dressing. For that, we lay out diagrams of the dressing virtual processes that climb the qubit ladder as high as possible. Near the two-photon resonance with a coupled resonator, we systematically calculate other relevant renormalizations due to nonresonant interactions. We provide verifiable predictions for the coupling rates. Modified systems can be applied for two-photon detection and for quantum-nondemolition readout of a qubit with an asymmetrical potential.