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Perturbation Theory and the Quantum Rabi-model

Marcello Malagutti, Alberto Parmeggiani

Abstract

In the first part of the paper we study a perturbative model of the Rabi system of Quantum Optics. We therefore are able to describe, through Rellich's theory, an analytic expansion of finite families, but of any fixed length, of eigenvalues. In particular, we prove that for finite families of eigenvalues the Braak conjecture holds. In the second part we study the asymptotics of the Weyl spectral counting function of a class of systems that generalize the Quantum Rabi Model to an $N$-level atom ($N\geq3$) with $N-1$ cavity modes of the electromagnetic field.

Perturbation Theory and the Quantum Rabi-model

Abstract

In the first part of the paper we study a perturbative model of the Rabi system of Quantum Optics. We therefore are able to describe, through Rellich's theory, an analytic expansion of finite families, but of any fixed length, of eigenvalues. In particular, we prove that for finite families of eigenvalues the Braak conjecture holds. In the second part we study the asymptotics of the Weyl spectral counting function of a class of systems that generalize the Quantum Rabi Model to an -level atom () with cavity modes of the electromagnetic field.
Paper Structure (23 sections, 9 theorems, 111 equations)

This paper contains 23 sections, 9 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Physics of the Quantum Rabi Model
  3. Hamiltonian and basic structure
  4. Symmetry and integrability
  5. Coupling regimes
  6. Weak coupling: Jaynes-Cummings regime
  7. Ultrastrong coupling
  8. Deep strong coupling
  9. Physical interpretation
  10. Experimental realizations
  11. The setup
  12. Relation between $Q_{\mathrm{Rabi}}$ and $Q_R$
  13. Perturbation Theory
  14. Study of $(T_+\varphi,T_-\varphi)_0$
  15. The Braak conjecture
  16. ...and 8 more sections

Key Result

Lemma 3.6

There exists a metaplectic operator $U\colon L^2(\mathbb{R};\mathbb{C}^2)\longrightarrow L^2(\mathbb{R};\mathbb{C}^2)$ (which is an automorphism of $\mathscr{S}(\mathbb{R};\mathbb{C}^2)$ and $\mathscr{S}'(\mathbb{R};\mathbb{C}^2)$) such that where

Theorems & Definitions (20)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • proof
  • Remark 3.7
  • Lemma 4.1
  • proof
  • ...and 10 more