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arXiv:1510.00768

Comment on "Integrability of the Rabi Model"

Wang Qing-wei, Yu-Liang Liu

Published Oct 3, 2015• 0 citations
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Abstract

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 $E_n=nω-g^2/ω$ in the spectrum of the Rabi model. Therefore Braak's solution of the quantum Rabi model is not complete.

DOI
10.48550/arxiv.1510.00768

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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.
We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak [Phys. Rev. Lett. \textbf{107}, 100401 (2011)] are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.
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.
We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak (2011 Phys. Rev. Lett. 107 100401) are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.
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.
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.