Bose–Hubbard models with photon pairing in circuit-QED

In this work, we study a family of bosonic lattice models that combine an on-site repulsion term with a nearest-neighbour pairing term, ∑iai†ai+1†+H.c.?> Like the original Bose–Hubbard model, the nearest-neighbour term is responsible for the mobility of bosons and it competes with the local interaction, inducing two-mode squeezing. However, unlike a trivial hopping, the counter-rotating terms form a pairing that cannot be studied with a simple mean-field theory and does not present a quantum phase transition in phase space. Instead, we show that there is a cross-over from a pure insulator to long-range correlations that start up as soon as the two-mode squeezing is switched on. We also show how this model can be naturally implemented using coupled microwave resonators and superconducting qubits.