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A generalisation of the Jaynes-Cummings-Hubbard model for coupled-cavityarrays is introduced, where the embedded two-level system in each cavity isreplaced by a $\Xi$-type three-level system. We demonstrate that the resultingeffective polariton-polariton interactions at each site are both two-body andthree-body. By tuning the ratio of the two transition dipole matrix elements,we show that the strength and sign of the two-body interaction can becontrolled whilst maintaining a three-body repulsion. We then proceed todemonstrate how different two-body and three-body interactions alter the meanfield superfluid-Mott insulator phase diagram, with the possible emergence of apair superfluid phase in the two-body attractive regime.

Effective Three-Body Interactions in Jaynes-Cummings-Hubbard Systems