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arXiv:1205.3139

Comment on "Integrability of the Rabi model"

Alexander Moroz

Published May 14, 2012• 2 citations
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Abstract

In his recent letter, Braak suggested that a regular spectrum of the Rabi model was given by the zeros of a transcendental function $G_\pm(x)$ (cf Eqs. (3)-(5) of Ref. [1]) and highlighted the role of the discrete $\mathbb{Z}_2$-symmetry, or parity, in determining $G_\pm(x)$. We show here to the contrary that one can define a transcendental function $F_0(x)$ and obtain the regular spectrum of the Rabi model as the zeros of $F_0(x)$ (see Fig. 1) without ever making use of the underlying $\mathbb{Z}_2$-symmetry of the model.

DOI
10.48550/arxiv.1205.3139

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In his recent letter, Braak suggested that a regular spectrum of the Rabi model was given by the zeros of a transcendental function G±(x)G_\pm(x)G±​(x) (cf Eqs. (3)-(5) of Ref. [1]) and highlighted the role of the discrete Z2\mathbb{Z}_2Z2​-symmetry, or parity, in determining G±(x)G_\pm(x)G±​(x). We show here to the contrary that one can define a transcendental function F0(x)F_0(x)F0​(x) and obtain the regular spectrum of the Rabi model as the zeros of F0(x)F_0(x)F0​(x) (see Fig. 1) without ever making use of the underlying Z2\mathbb{Z}_2Z2​-symmetry of the model.
An implicit formula for the spectrum of the two-photon Rabi model was presented by Trav\ifmmode \check{e}\else \v{e}\fi{}nec in Phys. Rev. A 85, 043805 (2012), in analogy to the method of Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The spectrum is given by common zeros of four functions Gc(z,E){G}_{c}(z,E)Gc​(z,E) taken with a fixed value of $z\ensuremath{\in}\mathbb{C}$. We point out that all the Gc(z,E){G}_{c}(z,E)Gc​(z,E) functions defined by Trav\ifmmode \check{e}\else \v{e}\fi{}nec are identically zero, thus the spectrum is not in fact given by their roots.
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 quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum.
The spectrum of the two-photon Rabi Hamiltonian is found, proceeding in full analogy with the solution of the standard (one-photon) Rabi Hamiltonian, published by D. Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The Hamiltonian is rewritten as a set of two differential equations. Symmetries that get hidden after further treatment are found. One can plainly see how the Hilbert space splits into four disjunct subspaces, categorized by four values of the symmetry parameter $c=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}i$. There were only two values, $\ifmmode\pm\else\textpm\fi{}1$, for the standard Rabi model. The values of ccc are in fact the eigenvalues of the respective parity operators. Four analytic functions are introduced by a recurrence scheme for the coefficients of their series expansion. All their roots yield the complete spectrum of the Hamiltonian. Eigenstates in Bargmann space can also be found.