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

TL;DR

This paper establishes a mathematical equivalence between non-commutative harmonic oscillators, two-photon quantum Rabi models, and certain holomorphic differential equations, enhancing understanding of their interrelations in quantum physics.

Contribution

It proves the equivalence between non-commutative harmonic oscillators and two-photon quantum Rabi models, and explores their connection to holomorphic differential equations and the one-photon model.

Findings

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

Connection to holomorphic differential equations on the unit disk

Discussion of the confluence process and relation to one-photon models

Abstract

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.