Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1911.01965 · Journal of Physics A Mathematical and Theoretical

A novel approach to the spectral problem in the two photon Rabi model

Andrzej J. Maciejewski, Tomasz Stachowiak

Published Apr 27, 2017• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aa6fb8

Related papers

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

Anselme F. Dossa, Gabriel Y. H. Avossevou

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.

Journal of Mathematical Physics•Oct 1, 2014•11 citations

On the 2-mode and k-photon quantum Rabi models

Yao-Zhong Zhang

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].

arXiv:1507.03863•Reviews in Mathematical Physics•Apr 13, 2017•9 citations

Analytical Solutions to Asymmetric Two-Photon Rabi Model

M. Baradaran, L. M. Nieto, S. Zarrinkamar

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.

quant-pharXiv:2601.05421•Jan 12, 2026•0 citations

Solvability of the two-photon Rabi Hamiltonian

Igor Travěnec

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.

arXiv:1201.3717•Physical Review A•Apr 4, 2012•106 citations

Level crossings and new exact solutions of the two-photon Rabi model

Andrzej J Maciejewski, Tomasz Stachowiak

Abstract An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.

arXiv:1910.11037•Journal of Physics A Mathematical and Theoretical•Oct 22, 2019•14 citations

Contour-integral based quantum eigenvalue transformation: analysis and applications

Shan Jiang, Dong An

Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the efficiency of quantum algorithms based on contour integral representation for eigenvalue transformations from both theoretical and practical aspects. Theoretically, we establish a complete complexity analysis of the contour integral approach proposed in [Takahira, Ohashi, Sogabe, and Usuda. Quant. Inf. Comput., 22, 11\&12, 965--979 (2021)]. Moreover, we combine the contour integral approach and the sampling-based linear combination of unitaries to propose a quantum algorithm for estimating observables of eigenvalue transformations using only $3$ additional qubits. Practically, we design contour integral based quantum algorithms for linear ordinary differential equations, $\varphi$-functions, Fréchet derivatives of matrix functions, and matrix functions. In particular, we show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving asymptotically stable differential equations.

quant-pharXiv:2601.11959•Jan 20, 2026•0 citations
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.
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 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].
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.
Abstract An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.
Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the efficiency of quantum algorithms based on contour integral representation for eigenvalue transformations from both theoretical and practical aspects. Theoretically, we establish a complete complexity analysis of the contour integral approach proposed in [Takahira, Ohashi, Sogabe, and Usuda. Quant. Inf. Comput., 22, 11\&12, 965--979 (2021)]. Moreover, we combine the contour integral approach and the sampling-based linear combination of unitaries to propose a quantum algorithm for estimating observables of eigenvalue transformations using only 333 additional qubits. Practically, we design contour integral based quantum algorithms for linear ordinary differential equations, φ\varphiφ-functions, Fréchet derivatives of matrix functions, and matrix functions. In particular, we show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving asymptotically stable differential equations.