Controlling correlations of a polaritonic Luttinger liquid by engineered cross-Kerr nonlinearity

We study correlation properties of polaritons at zero temperature in a multiconnected Jaynes--Cummings (MCJC) lattice on a superconducting circuit quantum electrodynamics platform with engineered cross-Kerr nonlinearity that mimics attractive nearest-neighbour interaction. A multi-connected Jaynes--Cummings lattice is a one-dimensional lattice constructed from alternating qubits and resonators with different left and right couplings. The nearest-neighbour interaction or cross-Kerr coupling is implemented dispersively through ladder-type qutrits between each nearest neighboring pair of resonator modes. Projecting onto the lower-polaritonic manifold, we derive an extended two-mode (bipartite) Bose--Hubbard-like model featuring on-site and attractive nearest-neighbor interactions. Employing a continuum bosonization approach, we express the Hamiltonian in terms of symmetric (+) and antisymmetric (-) collective modes. In the regime where the (-) sector acquires a finite gap, one can reduce the system to an effective single-component Luttinger liquid model for the + sector. The cross-Kerr term reduces the compressibility of the (+) mode, thereby enhancing the corresponding Luttinger parameter K₊, resulting in the slower algebraic decay of single-particle correlations, G(x)propto|x|^-1/\(4K₊\).