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arXiv:1503.05636 · Physical Review A

Comment on “Solvability of the two-photon Rabi Hamiltonian”

Andrzej J. Maciejewski, Maria Przybylska, Tomasz Stachowiak

Published Mar 10, 2015• 32 citations
arXivHTMLPDF

Abstract

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 ${G}_{c}(z,E)$ taken with a fixed value of $z\ensuremath{\in}\mathbb{C}$. We point out that all the ${G}_{c}(z,E)$ 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.

DOI
10.1103/physreva.91.037801

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Solvability of the two-photon Rabi Hamiltonian

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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 $c$ 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.

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Reply to “Comment on ‘Solvability of the two-photon Rabi Hamiltonian’ ”

Igor Travěnec

In the Comment, it is shown that functions used to find the spectrum of two-photon Rabi Hamiltonian vanish. This is true; nevertheless, our method evidently works. The trick is based on a substitution making the coefficients of following truncated Taylor series explicitly dependent on energies, and then the roots of the above-mentioned functions approach the eigenenergies. Thus it is the combination of entire exponential functions and truncated Taylor series that gives nontrivial results.

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Comment on "Integrability of the Rabi model"

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Applying Bogoliubov transformations and Bargmann-Hilbert spaces, we obtain analytic representations of solutions to the 2-photon and two-mode quantum Rabi models. In each case, a transcendental function is analytically derived whose zeros give the energy spectrum of the model. The zeros can be numerically found by standard root-search techniques. We also present analytic solution to the driven Rabi model with broken ${\cal Z}_2$ symmetry.

•Apr 30, 2013•4 citations
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∈Cz\ensuremath{\in}\mathbb{C}z\ensuremath∈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.
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.
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.
Abstract The quantum Rabi model (QRM) is identified as the simplest model to study the interaction between light and matter. In recent years, Braak proposed an alternative more efficient method to obtain the exact solvability and spectrum of the QRM in the Bargmann space, which is appropriate frame for extended squeezed coherent states. Based on this feasible method, some study for the two-photon QRM is concentrated. In this paper we diagonalized the Hamiltonian and obtain the Spectrum of the system of two-photon QRM in the special case that the level splitting of the atom was equal to 0, and compared with Travĕnec’s results without diagonalizing the Hamiltonian. Finally, we confirm the validity of diagonalization for the Hamiltonian in the two-photon QRM.
Applying Bogoliubov transformations and Bargmann-Hilbert spaces, we obtain analytic representations of solutions to the 2-photon and two-mode quantum Rabi models. In each case, a transcendental function is analytically derived whose zeros give the energy spectrum of the model. The zeros can be numerically found by standard root-search techniques. We also present analytic solution to the driven Rabi model with broken Z2{\cal Z}_2Z2​ symmetry.