Equivalence between non-commutative harmonic oscillators and two-photon quantum Rabi models

We prove that the non-commutative harmonic oscillator on $L^2(\mathbb{R})\otimes {\mathbb{C}}^2$ introduced by Parmeggiani and Wakayama is equivalent to the two-photon quantum Rabi model, and they are also equivalent to a holomorphic differential equation on the unit disk. The confluence process of this differential equation and the relation with the one-photon quantum Rabi model are also discussed.