We study correlation properties of polaritons at zero temperature in a multiconnected Jaynes-Cummings lattice on a superconducting circuit quantum electrodynamics platform with engineered cross-Kerr nonlinearity that mimics attractive nearest-neighbor interaction. A multiconnected Jaynes-Cummings lattice is a one-dimensional lattice constructed from alternating qubits and resonators with different left and right couplings. The nearest-neighbor 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, $G\left(x\right)\propto {\left|x\right|}^{-1/\left(4K_{+}\right)}$.

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• Cavity quantum electrodynamics
• Circuit quantum electrodynamics
• Kerr effect
• Luttinger liquid model