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arXiv:1201.3843

An Exact Solution of a Generalization of the Rabi Model

Fred Moolekamp

Published Jan 18, 2012• 2 citations
arXivHTMLPDF

Abstract

There has been renewed theoretical interest recently in the Rabi model due to Braak's analytical solution and introduction of a new criterion for integrability. We focus not on the integrability of the system but rather why it is solvable in the first place. We show that the Rabi model is the limiting case of a more general finite dimensional system by use of a contraction and suggest that this is the reason for it's solvability, which still applies in the case of non-integrable but solvable variations.

DOI
10.48550/arxiv.1201.3843

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There has been renewed theoretical interest recently in the Rabi model due to Braak's analytical solution and introduction of a new criterion for integrability. We focus not on the integrability of the system but rather why it is solvable in the first place. We show that the Rabi model is the limiting case of a more general finite dimensional system by use of a contraction and suggest that this is the reason for it's solvability, which still applies in the case of non-integrable but solvable variations.
The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
The Rabi model is a paradigm for interacting quantum systems.It couples a bosonic mode to the smallest possible quantum model, a two-level system.I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit.A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry.Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
Using Hill's determinant method we show that the set of Judd's solutions is only a subset of all the eigenvalues with the form En=nω−g2/ωE_n=nω-g^2/ωEn​=nω−g2/ω in the spectrum of the Rabi model. Therefore Braak's solution of the quantum Rabi model is not complete.
The simplified two-qubit Rabi model is proposed and its analytical solution is presented. There are no level crossings in the spectral graph of the model, which indicates that it is not integrable. The criterion of integrability for the Rabi model proposed by Braak (2011 Phys. Rev. Lett. 107 100401) is also used for the simplified two-qubit Rabi model and the same conclusion, consistent with what the spectral graph shows, can be drawn, which indicates that the criterion remains valid when applied to the two-qubit case. The simplified two-qubit Rabi model is another example of a non-integrable but exactly solvable system except for the generalized Rabi model.