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Journal of Mathematical Physics

Full spectrum of the two-photon and the two-mode quantum Rabi models

Anselme F. Dossa, Gabriel Y. H. Avossevou

Published Oct 1, 2014• 11 citations
Publisher

Abstract

This paper is concerned with the rigorous analytical determination of the spectrum of the two-photon and the two-mode quantum Rabi models. To reach this goal, we exploit the hidden symmetries in these models by means of the unitary and similarity transformations in addition to the Bargmann-Fock space description. In each case, the purely quantum mechanical problem of the Rabi model studied is reduced to solutions for differential equations. This eventually gives a third-order differential equation for each of these models, which is reduced to a second-order differential equation by additional transformations. The analytical expressions of the wave functions describing the energy levels are obtained in terms of the confluent hypergeometric functions.

DOI
10.1063/1.4898765

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This paper is concerned with the rigorous analytical determination of the spectrum of the two-photon and the two-mode quantum Rabi models. To reach this goal, we exploit the hidden symmetries in these models by means of the unitary and similarity transformations in addition to the Bargmann-Fock space description. In each case, the purely quantum mechanical problem of the Rabi model studied is reduced to solutions for differential equations. This eventually gives a third-order differential equation for each of these models, which is reduced to a second-order differential equation by additional transformations. The analytical expressions of the wave functions describing the energy levels are obtained in terms of the confluent hypergeometric functions.
By applying the Bogoliubov transformations and through the introduction of the Bargmann-Hilbert spaces, we obtain analytic representations of solutions to the driven Rabi model without Z2{\cal Z}_2Z2​ symmetry, and the 2-photon and two-mode quantum Rabi models. In each case, 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.
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.
Within the Segal-Bargmann representation, a generalized Rabi model is considered that includes both two-photon and asymmetric terms. It is shown that, through a suitable transformation, nearly exact solutions can be obtained using the Bethe ansatz approach. Applying this approach to the meromorphic structure of the resulting differential equation, solutions in exact analytical form of the fourth-order problem are presented for both an arbitrary state and for the restriction between the parameters.
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.
By mapping the Hamiltonians of the 2-mode and 2-photon Rabi models to differential operators in suitable Hilbert spaces of entire functions, we prove that the two models possess entire and normalizable wave functions in the Bargmann–Hilbert spaces only if the frequency [Formula: see text] and coupling strength [Formula: see text] satisfy certain constraints. This is in sharp contrast to the quantum Rabi model for which entire wave functions always exist. For model parameters fulfilling the aforesaid constraints, we determine transcendental equations whose roots give the regular energy eigenvalues of the models. Furthermore, we show that for [Formula: see text], the [Formula: see text]-photon Rabi model does not possess wave functions which are elements of the Bargmann–Hilbert space for all non-trivial model parameters. This implies that the [Formula: see text] case is not diagonalizable, unlike its RWA cousin, the [Formula: see text]-photon Jaynes–Cummings model which can be completely diagonalized for all [Formula: see text].
Abstract We explore the spectral problem of the two photon Rabi model from the point of view of complex differential equations in the Bargmann representation. The wave-functions are automatically entire but to ensure finite norm one has to effectively construct asymptotic expansions. This is achieved by means of the Mellin transformation and convergent factorial series, which allow direct computation of the spectral determinant. By further analysing the differential equation satisfied by the Mellin transform, we obtain a new form of the spectral conditions—in terms of holonomy matrices and contour integrals.