Exact <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> -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 another---has quickly emerged as most promising near-term strategies for studying strongly correlated many-body systems. In particular, systems interacting photons, realizable in solid-state cavity and circuit QED frameworks, example, hold tremendous promise study nonequilibrium phenomena, part due capability locally create destroy photons. These are typically modeled a Jaynes-Cummings-Hubbard (JCH) Hamiltonian, named similarities with Bose-Hubbard Hamiltonian. While comparisons between two often made literature, JCH Hamiltonian comprises both bosonic psuedospin operators, leading deviations from model particular parameter regimes. Here, we present nonperturbative procedure transforming Jaynes-Cummings into dressed operator representation that, its general form, admits an infinite sum $k$-body terms where $k$ is bound only by number excitations system. We closely examine this result dispersive resonant coupling regimes, finding rapid convergence former contributions $k\ensuremath{\gg}1$ latter. Through extension simple case two-site system, demonstrate that approach facilitates close inspection analogy models breakdown light-matter coupling. Finally, use framework survey character parameters, identifying four unique phases regimes which they realized, thus highlighting phenomena finite JCH-based simulators beyond model. More broadly, paper intended serve clear mathematical exposition interactions underlying Jaynes-Cummings-type systems, postulated either through Kerr-like nonlinear susceptibilities or matching coefficients obtain appropriate eigenvalue spectrum.