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We analyze the quantum phase transition-like behavior in the lowest energystate of a two-site coupled atom-cavity system, where each cavity contains oneatom but the total excitation number is not limited to two. Utilizing thevariance of the total excitation number to distinguish the insulator andsuperfluid states, and the variance of the atomic excitation number to identifythe polaritonic characteristics of these states, we find that the totalexcitation number plays a significant role in the lowest-energy-state phasetransitions. In both the small hopping regime and the small atom-fieldinteraction regime, we identify an interesting coexisting phase involvingcharacteristics of both photonic superfluid and atomic insulator. For smallhopping, we find that the signature of the photonic superfluid state becomesmore pronounced with the increase in total excitation number, and that theboundaries of the various phases shift with respect to the case of $N=2$. Inthe limit of small atom-field interaction, the polaritonic superfluid regionbecomes broader as the total excitation number increases. We demonstrate thatthe variance of the total excitation number in a single site has a lineardependence on the total excitation number in the large-detuning limit.

Quantum phase transition of polaritonic excitations in a multi-excitation coupled array