Quantum Phase Transition of Polaritonic Excitations in a Multi-Excitation Coupled Array

We analyze the quantum phase transition-like behavior in the lowest energy state of a two-site coupled atom-cavity system, where each cavity contains one atom but the total excitation number is not limited to two. Under the large-detuning condition, we identify an interesting coexisting phase involving characteristics of both photonic superfluid and atomic insulator, which has not been previously revealed. For small hopping, we find that the signature of the photonic superfluid state becomes more pronounced with the increase in total excitation number, and that the boundaries of the various phases shift with respect to the case of two excitations. In the limit of small atom-field interaction, the polaritonic superfluid region becomes broader as the total excitation number increases. We use alternative order parameters to characterize the nonclassical property in the lowest-energy state, and find that the entanglement of photons in the photonic superfluid state has an approximately quadratic-like dependence on the total excitation number within the large-detuning limits. The second-order cross-correlation function is demonstrated to become inversely proportional to the total excitation number in the large detuning limits.