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arXiv:1507.03863 · Reviews in Mathematical Physics

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

Yao-Zhong Zhang

Published Apr 13, 2017• 9 citations
arXivHTMLPDF

Abstract

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].

DOI
10.1142/s0129055x17500131

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非対称量子ラビ模型の固有値に関する研究

アルマンド シッド

The quantum Rabi model is a model in quantum optics used to describe the interaction between a two-level atom and a single-level photon field.This model, described in 1963 by Jaynes and Cummings, is the fully quantized version of the original semi-classical model proposed in 1936 by Rabi.In previous years, the study of the properties of Hamiltonian of the quantum Rabi model and its energy levels (eigenvalues of the Hamiltonian) has been relegated mainly for two reasons.The first one was the belief that the quantum Rabi model was not an exactly solvable model, that is, that its energy levels could not be described analytically.The second reason is that in the parameter regime achievable at the time in experiments and applications, the quantum Rabi model could be approximated successfully by the Jaynes-Cummings model, a simpler model which is known to be exactly solvable.Both situations changed in recent years.In 2011, Daniel Braak proved the exact-solvability of the quantum Rabi model by constructing analytical solutions and defining a transcendental function, the G-function, whose zeros determine all the spectrum (with the exception of a (possible finite) set of exceptional eigenvalues).This pioneering technique has since then been successfully applied to show the exact-solvability of several models in quantum optics.On the other hand, due to the advances in experimental physics, starting from the first decade of this century, experiments have been steadily reaching parameter regimes where the Jaynes-Cummings model becomes unsuitable for approximation and in 2014, the experiments by Maissen et al reached regimes where it is imperative to consider the full Hamiltonian of the quantum Rabi model.Adding to this, recently there has also been proposals for applications of the quantum Rabi model in quantum information theory and quantum computing.For these reasons, there has been a large amount of research done in theoretical and experimental physics, and, more recently, mathematics on the subject of the quantum Rabi model, its generalizations and applications.In this thesis, we study the asymmetric quantum Rabi model, one of the generalizations of the quantum Rabi model.In this generalization, an additional non-trivial interaction term is introduced in the Hamiltonian, breaking the Z 2Z-symmetry of the quantum Rabi Hamiltonian.The presence of this symmetry in the quantum Rabi model explains the presence of crossings in the energy levels (degeneracies in the spectrum), so a priori there was no reason to expect degeneracies in the spectrum of the asymmetric version.In this work, we show that when the coefficient of the symmetry-breaking term is halfinteger, degeneracies appear in the spectrum of the asymmetric quantum Rabi model.This is done by studying the properties of the constraint polynomials, polynomials appearing in certain conditions that the parameters of the model must satisfy in order to have certain eigenvalues, called Juddian eigenvalues.The spectrum of the asymmetric quantum Rabi model is divided into different sets, an infinite set of regular eigenvalues, the zeros of the G-function and the exceptional eigenvalues, not captured as zeros of the G-function.The exceptional spectrum is further classified into Juddian and non-Juddian, according to the type of solution in the Segal-Bargmann space realization.These solutions are also described in terms of the sl 2 picture of the Hamiltonian in terms of irreducible representations of sl 2 .The case for Juddian and regular eigenvalues was already known, and in this thesis we complete the picture for the non-Juddian eigenvalues.Another main result of this work is to characterize the degeneracies of the asymmetric quantum Rabi model in terms of the types of eigenvalues and the parameters.Furthermore, by carefully studying the poles of the G-function, we define a new G-function whose zeros give the full spectrum of the asymmetric quantum Rabi model.Finally, we present a study on continued fractions expansions of integer powers of the Napier constant e with a nice representation and good convergence properties.This study is a byproduct of the study of certain orthogonal polynomial families related to the constraint polynomials of the AQRM.

•Jun 28, 2018•0 citations
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].
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.
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 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.
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.
The full two-photon quantum Rabi model describes the nonlinear coupling between a superconducting qubit and a resonator. It possesses [Formula: see text] symmetry, in contrast to the conventional two-photon quantum Rabi model in cavity quantum electrodynamics systems, which exhibits [Formula: see text] symmetry. Using the su(1,1) Lie algebra and [Formula: see text] symmetry, we derive a transcendental function whose zeros yield the exact eigenvalues. The spectral collapse phenomenon at the critical coupling [Formula: see text] is derived analytically. At [Formula: see text], the model can be mapped to an inverse-square potential well, where an infinite number of bound states approach the threshold energy exponentially. Our results may provide new insights into spectral phenomena in practical experimental realizations of superconducting circuits with nonlinear coupling.
The quantum Rabi model is a model in quantum optics used to describe the interaction between a two-level atom and a single-level photon field.This model, described in 1963 by Jaynes and Cummings, is the fully quantized version of the original semi-classical model proposed in 1936 by Rabi.In previous years, the study of the properties of Hamiltonian of the quantum Rabi model and its energy levels (eigenvalues of the Hamiltonian) has been relegated mainly for two reasons.The first one was the belief that the quantum Rabi model was not an exactly solvable model, that is, that its energy levels could not be described analytically.The second reason is that in the parameter regime achievable at the time in experiments and applications, the quantum Rabi model could be approximated successfully by the Jaynes-Cummings model, a simpler model which is known to be exactly solvable.Both situations changed in recent years.In 2011, Daniel Braak proved the exact-solvability of the quantum Rabi model by constructing analytical solutions and defining a transcendental function, the G-function, whose zeros determine all the spectrum (with the exception of a (possible finite) set of exceptional eigenvalues).This pioneering technique has since then been successfully applied to show the exact-solvability of several models in quantum optics.On the other hand, due to the advances in experimental physics, starting from the first decade of this century, experiments have been steadily reaching parameter regimes where the Jaynes-Cummings model becomes unsuitable for approximation and in 2014, the experiments by Maissen et al reached regimes where it is imperative to consider the full Hamiltonian of the quantum Rabi model.Adding to this, recently there has also been proposals for applications of the quantum Rabi model in quantum information theory and quantum computing.For these reasons, there has been a large amount of research done in theoretical and experimental physics, and, more recently, mathematics on the subject of the quantum Rabi model, its generalizations and applications.In this thesis, we study the asymmetric quantum Rabi model, one of the generalizations of the quantum Rabi model.In this generalization, an additional non-trivial interaction term is introduced in the Hamiltonian, breaking the Z 2Z-symmetry of the quantum Rabi Hamiltonian.The presence of this symmetry in the quantum Rabi model explains the presence of crossings in the energy levels (degeneracies in the spectrum), so a priori there was no reason to expect degeneracies in the spectrum of the asymmetric version.In this work, we show that when the coefficient of the symmetry-breaking term is halfinteger, degeneracies appear in the spectrum of the asymmetric quantum Rabi model.This is done by studying the properties of the constraint polynomials, polynomials appearing in certain conditions that the parameters of the model must satisfy in order to have certain eigenvalues, called Juddian eigenvalues.The spectrum of the asymmetric quantum Rabi model is divided into different sets, an infinite set of regular eigenvalues, the zeros of the G-function and the exceptional eigenvalues, not captured as zeros of the G-function.The exceptional spectrum is further classified into Juddian and non-Juddian, according to the type of solution in the Segal-Bargmann space realization.These solutions are also described in terms of the sl 2 picture of the Hamiltonian in terms of irreducible representations of sl 2 .The case for Juddian and regular eigenvalues was already known, and in this thesis we complete the picture for the non-Juddian eigenvalues.Another main result of this work is to characterize the degeneracies of the asymmetric quantum Rabi model in terms of the types of eigenvalues and the parameters.Furthermore, by carefully studying the poles of the G-function, we define a new G-function whose zeros give the full spectrum of the asymmetric quantum Rabi model.Finally, we present a study on continued fractions expansions of integer powers of the Napier constant e with a nice representation and good convergence properties.This study is a byproduct of the study of certain orthogonal polynomial families related to the constraint polynomials of the AQRM.