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

Analog quantum simulation-the technique of using one experimentally well-controlled physical system to mimic the behavior of another-has quickly emerged as one of the most promising near-term strategies for studying strongly correlated quantum many-body systems. In particular, systems of interacting photons, realizable in solid-state cavity and circuit QED frameworks, for example, hold tremendous promise for the study of nonequilibrium many-body phenomena, in part due to the capability to locally create and destroy photons. These systems are typically modeled using a Jaynes-Cummings-Hubbard (JCH) Hamiltonian, named due to similarities with the Bose-Hubbard Hamiltonian. While comparisons between the two are often made in literature, the JCH Hamiltonian comprises both bosonic and psuedospin operators, leading to physical deviations from the Bose-Hubbard model for particular parameter regimes. Here, we present a nonperturbative procedure for transforming the Jaynes-Cummings Hamiltonian into a dressed operator representation that, in its most general form, admits an infinite sum of bosonic k-body terms where k is bound only by the number of excitations in the system. We closely examine this result in both the dispersive and resonant coupling regimes, finding rapid convergence of this sum in the former and contributions from k >> 1 in the latter. Through extension to the simple case of a two-site JCH system, we demonstrate that this approach facilitates close inspection of the analogy between the JCH and Bose-Hubbard models and its breakdown for resonant light-matter coupling. Finally, we use this framework to survey the many-body character of a two-site JCH for general system parameters, identifying four unique quantum phases and the parameter regimes in which they are realized, thus highlighting phenomena realizable with finite JCH-based quantum simulators beyond the Bose-Hubbard model. More broadly, this paper is intended to serve as a clear mathematical exposition of bosonic many-body interactions underlying Jaynes-Cummings-type systems, often postulated either through analogy to Kerr-like nonlinear susceptibilities or by matching coefficients to obtain the appropriate eigenvalue spectrum.