Full spectrum of the two-photon and the two-mode quantum Rabi models (Q2928098)

The paper is concerned with the analytic diagonalization of two important spin-boson models. Their first model just as the original Rabi model describes the interaction of a single two-level system with a single photonic mode, where however, the photons are always created or annihilated in pairs by the interaction. Their second model also supports this property, but the two photons now belong to different modes.NEWLINENEWLINEThe paper starts with an introduction to the multi-photon Rabi model, discuss recent solutions of the single-photon version and then lead to the considered two-photon and two-mode Rabi models. In their second section, the two-photon Rabi model is solved. The authors first identify the generators of the \(\mathfrak{su}(1,1)\) algebra for the bosons and express the Hamiltonian in terms of these operators. A direct route to a Fock-Bargmann representation would lead to a fourth-order ordinary differential equation (ODE). Therefore, the authors first apply a unitary operation on the two-level system, which after subsequent representation in Fock-Bargmann space yields only a third-order ODE. The solution of this ODE in terms of confluent hypergeometric functions is worked out in the appendix, and combining their results the authors can present both the spectrum and the eigenvectors of the two-photon Rabi model in closed form. Afterwards, they turn towards the two-mode problem, where they apply a very similar formalism. On top of the unitary transformation they also use a similarity transformation involving the bosonic operators, but again a representation in Fock-Bargmann space by a third-order ODE yields explicit solutions for the spectrum and eigenvectors in closed form.NEWLINENEWLINEThe spectra presented in this paper have partially been obtained before with different methods. However, the paper presents an independent, elegant and rigorous derivation of both spectrum and eigenvectors in analytically closed form and should thus be of great interest to the community. Beyond this, the authors expect that their method may also be applicable to the standard Rabi model. The paper uses a formal mathematical style and is well written. Since it provides many references, it is also suitable to a broad range of readers familiar with bosonic creation and annihilation operators and the solutions of ODEs.