Construction of the Jaynes–Cummings interaction over the finite two-dimensional oscillator

The interaction between a two-level atom and the finite two-dimensional oscillator in the Cartesian coordinate system is addressed. The construction of the coupling between the degenerate energy states of the finite oscillator and the two levels of the atom allows the identification of an excitation-conserved eigenbasis. A dynamical approach to the evolution of mean values of the fields mediated by the atom is shown in the interaction picture. Furthermore, a quantitative characterization of the atom-field entanglement induced by the finite dynamics is presented. As an example, the eigenenergies states and 𝔰𝔲(2) coherent states are studied. Perspectives and prospects are presented to chart a path in the study of atomic systems coupled to finite versions of the harmonic oscillator.