Solutions and quantum dynamics of the quantum Rabi model with A-square terms

It is well known that the A-square term must be considered in both cavity and circuit quantum electrodynamics systems, because it arises in the derivation from the minimal coupling Hamiltonian at any finite coupling strength. In this paper, we study the quantum Rabi model with A-square terms using the Bogoliubov operator approach. After a unitary transformation, the A-square terms can be eliminated, resulting in a modified quantum Rabi model with renormalized parameters. A transcendental function responsible for the exact solution is then derived. The presence of the A-square terms is found to significantly alter the energy spectrum. The dynamics are also studied using the obtained exact wave function, which is sensitive to the strength of the A-square terms at strong coupling. We believe that these results could be observed in future light-matter interaction systems in the ultra-strong and deep strong coupling regimes.