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Analog quantum simulation - the technique of using one experimentallywell-controlled physical system to mimic the behavior of another - has quicklyemerged as one of the most promising near term strategies for studying stronglycorrelated quantum many-body systems. In particular, systems of interactingphotons, realizable in solid-state cavity and circuit QED frameworks, forexample, hold tremendous promise for the study of nonequilibrium many-bodyphenomena due to the capability to locally create and destroy photons. Thesesystems are typically modeled using a Jaynes-Cummings-Hubbard (JCH)Hamiltonian, named due to similarities with the Bose-Hubbard (BH) model. Here,we present a non-perturbative procedure for transforming the JC Hamiltonianinto a dressed operator representation that, in its most general form, admitsan infinite sum of bosonic $k$-body terms where $k$ is bound only by the numberof excitations in the system. We closely examine this result in both thedispersive and resonant coupling regimes, finding rapid convergence in theformer and contributions from $k\gg1$ in the latter. Through extension to atwo-site JCH system, we demonstrate that this approach facilitates closeinspection of the analogy between the JCH and BH models and its breakdown forresonant light-matter coupling. Finally, we use this framework to survey themany-body character of a two-site JCH for general system parameters,identifying four unique quantum phases and the parameter regimes in which theyare realized, thus highlighting phenomena realizable with finite JCH-basedquantum simulators beyond the BH model. More broadly, this work is intended toserve as a clear mathematical exposition of bosonic many-body interactionsunderlying JC-type systems, often postulated through analogy to Kerr-likenonlinear susceptibilities or by matching coefficients to obtain theappropriate eigenvalue spectrum.

Exact $k$-body representation of the Jaynes-Cummings interaction in the dressed basis: Insight into many-body phenomena with light