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

Ryosuke Nakahama

TL;DR

The paper shows that the eigenproblem of $H_{NCHO}^{(\alpha,\beta,\eta)}$ is equivalent to an eigenproblem for $\widetilde{H}_{2QRM}^{(g,\Delta,\varepsilon)}$ up to the unitary factor $e^{\frac{\pi i}{4}H}$ and a similarity transform, and that on the even/odd subspaces this reduces to a holomorphic-disk differential equation. It builds explicit parameter correspondences and uses a sequence of unitary maps to translate the problem into disk-model terms. A confluence limit $\nu \to \infty$ yields a degeneration to the one-photon QRM, clarifying covering relations between the two- and one-photon models. The work thus provides a rigorous link between a mathematical non-commutative oscillator and quantum Rabi models, with potential number-theoretic insights and a framework for further spectral analysis.

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.

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

TL;DR

The paper shows that the eigenproblem of is equivalent to an eigenproblem for up to the unitary factor and a similarity transform, and that on the even/odd subspaces this reduces to a holomorphic-disk differential equation. It builds explicit parameter correspondences and uses a sequence of unitary maps to translate the problem into disk-model terms. A confluence limit yields a degeneration to the one-photon QRM, clarifying covering relations between the two- and one-photon models. The work thus provides a rigorous link between a mathematical non-commutative oscillator and quantum Rabi models, with potential number-theoretic insights and a framework for further spectral analysis.

Abstract

We prove that the non-commutative harmonic oscillator on 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.
Paper Structure (4 sections, 1 theorem, 42 equations)

This paper contains 4 sections, 1 theorem, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence between NCHO and two-photon QRM
  4. Covering of one-photon QRM

Key Result

Theorem 1

Let $\alpha,\beta\in\mathbb{R}_{>0}$, $\eta\sqrt{\alpha\beta-1},\lambda\in\mathbb{R}$, and let

Theorems & Definitions (3)

  • Theorem
  • Remark
  • Remark