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arXiv:2401.05615

Analytical approximations for generalized quantum Rabi models

Chon‐Fai Kam, Chen Ning Yang

Published Jan 11, 2024• 0 citations
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Abstract

The quantum Rabi model is essential for understanding interacting quantum systems. It serves as the simplest non-integrable yet solvable model describing the interaction between a two-level system and a single mode of a bosonic field. In this study, we delve into the exploration of the generalized quantum Rabi model, wherein the bosonic mode of the field undergoes squeezing. Utilizing the Segal-Bargmann representation of the infinite-dimensional Hilbert space, we demonstrate that the energy spectrum of the generalized quantum Rabi model, when both the Rabi coupling strength and the squeezing strength are not significantly large compared to the field mode frequency, can be analytically determined by a bi-confluent Fuchsian equation with two regular singularities at 0 and 1 and an irregular singularity of rank two at infinity.

DOI
10.48550/arxiv.2401.05615

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The quantum Rabi model is essential for understanding interacting quantum systems. It serves as the simplest non-integrable yet solvable model describing the interaction between a two-level system and a single mode of a bosonic field. In this study, we delve into the exploration of the generalized quantum Rabi model, wherein the bosonic mode of the field undergoes squeezing. Utilizing the Segal-Bargmann representation of the infinite-dimensional Hilbert space, we demonstrate that the energy spectrum of the generalized quantum Rabi model, when both the Rabi coupling strength and the squeezing strength are not significantly large compared to the field mode frequency, can be analytically determined by a bi-confluent Fuchsian equation with two regular singularities at 0 and 1 and an irregular singularity of rank two at infinity.
The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
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.
The Rabi model is a paradigm for interacting quantum systems.It couples a bosonic mode to the smallest possible quantum model, a two-level system.I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit.A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry.Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
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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.
Abstract We present an exact analytical solution of a generic two-mode quantum Rabi model with non-identical bosonic modes. By means of a Bogoliubov transformation, the Hamiltonian is reduced to invariant subspaces labeled by a conserved quantity, and a transcendental G-function is derived, whose zeros determine the regular energy spectrum. We show that the spectra in different subspaces are related by a simple energy shift, revealing the crucial role of mode asymmetry in shaping the spectral structure. In the symmetric limit, the model exhibits an enhanced degeneracy, including fourfold degenerate energy levels arising from the merging of invariant subspaces. The structure of exceptional solutions is analyzed in detail. We identify finite-dimensional Juddian solutions associated with level crossings, as well as special nondegenerate solutions that give rise to isolated states. The degeneracy of these solutions is shown to be controlled by the underlying symmetry of the model. Furthermore, we investigate spectral collapse in the generic two-mode quantum Rabi model. Unlike in the two-photon quantum Rabi model, the collapse energy depends explicitly on the subspace, leading to a richer collapse scenario. We demonstrate that spectral collapse originates from the vanishing spacing between adjacent poles of the G-function, and that the collapse point supports an infinite sequence of bound states. These states exhibit exponential accumulation toward the collapse energy, revealing a hierarchical structure of collapse-induced bound states.