The Quantization of the Rabi Hamiltonian

TL;DR

This paper presents a novel method for analyzing the Rabi Hamiltonian using the Bargmann-Fock representation and Birkhoff transformation, leading to recurrence relations that determine eigenvalues and eigenvectors, and connecting solutions to Juddian baselines.

Contribution

It introduces a new analytical approach involving Birkhoff transformation and recurrence relations to solve the Rabi Hamiltonian, highlighting its relation to Juddian solutions and integrability.

Findings

Eigenvalues and eigenvectors obtained via recurrence relations.

Generation of integer quantum numbers linked to Juddian baselines.

Discussion of the relationship with Braak's integrability claim.

Abstract

The Bargmann-Fock representation of the Rabi Hamiltonian is expressed by a system of two coupled first-order differential equations in the complex field, which may be rewritten in a canonical form under the Birkhoff transformation. The transformation gives rise to leapfrog recurrence relations, from which the eigenvalues and eigenvectors could be obtained. The interesting feature of this approach is that it generates integer quantum numbers, which relate the solutions to the Juddian baselines. The relationship with Braak's integrability claim [PRL 107, 100401 (2011)] is discussed