Understanding the quantum Rabi ring using analogies to quantum magnetism

We map a quantum Rabi ring, consisting of $N$ cavities arranged in a ring geometry, into an effective magnetic model containing the XY exchange and the Dzyaloshinskii Moriya (DM) interactions. The analogue of the latter is induced by an artificial magnetic field, which modulates photon hopping between nearest-neighbor cavities with a phase. The mean-field behavior of both systems is almost identical, facilitating the description of the different phases in the quantum optical model through simple arguments of competing magnetic interactions. For the square geometry ($N=4$) the rich phase diagram exhibits three superradiant phases denoted as ferro-superradiant, antiferro-superradiant and chiral superradiant. In particular, the DM interaction is responsible for the chiral phase in which the energetically degenerate configurations of the order parameters are similar to the in-plane magnetizations of skyrmions with different helicities. The antiferro-superradiant phase is suppressed in the triangle geometry ($N=3$) as geometric frustration contributes to stabilize the chiral phase even for small values of the DM interaction. The chiral phases for odd and even $N$ show a different scaling behavior close to the phase transition. The equivalent behavior on both systems opens the possibility of simulating chiral magnetism in a few-body quantum optical platform, as well as understanding one system using the insights gained from the other.