Revealing the Hidden Bosonic Many-Body Nature of the Jaynes-Cummings Interaction

The Jaynes-Cummings Hamiltonian (JCH) is one of the simplest and most versatile models in quantum optics. The similarity of the JCH with the nonlinear Kerr Hamiltonain having two-body bosonic interaction terms has been previously explored in the literature. However, the many-body character has either been postulated or deduced by matching coefficients to obtain the correct eigenvalue spectrum, especially in the limit of a few quanta. In the present work, we have theoretically examined, in a systematic fashion, the many-body nature of the JCH. A procedure for obtaining the coefficients of the normally ordered many-body terms, describing the interactions between the bosonic dressed states, has been devised. We show how such terms originate in the case of a linear cavity coupled to a two-level system, in the absence of Kerr-like non-linear optical susceptibilities. The coefficients are independent of the number of quanta stored in the system similar to the Bose-Hubbard models in atomic physics. We have verified that the many-body version of the JCH has identical eigenvalues, eigenvectors and steady-state transmission response as the usual representation. Our work should better guide experiments on quantum simulation of few-body effects.