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Journal of Hunan University of Arts and Science

Parity symmetry and parity breaking in the quantum Rabi model

Wang Qion

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

The well-known quantum Rabi model strictly describes the simplest interaction between matter and the quantum light.We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced.We consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction.The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom.

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We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].

•Oct 28, 2026•0 citations
The well-known quantum Rabi model strictly describes the simplest interaction between matter and the quantum light.We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced.We consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction.The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom.
We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced. In contrast to a mathematical treatment in a previous publication [J. Phys. A 46(2013) 265302], we consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction, and then the parity symmetry is broken as well as the scaling behavior of the ground state by introducing a bias. The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom. Experimental feasibility of realizing the models under discussion is investigated.
Abstract We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced. In contrast to a mathematical treatment in a previous publication [ J. Phys. A 46 (2013) 265302 ], we consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction, and then the parity symmetry is broken as well as the scaling behavior of the ground state by introducing a bias. The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom. Experimental feasibility of realizing the models under discussion is investigated.
The quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum.
The light-matter interaction described by the Rabi model and Jaynes-Cummings (JC) model under local bias fields is investigated by parity breaking as well as the scaling behavior of the ground state. We show that the parity breaking leads to different scaling behaviors in the two models, where the biased Rabi model demonstrates scaling invariance, but the biased JC model behaves in a cusp-like way. Our study helps further understand the rotating-wave approximation and could present more subtle physics than other characteristic parameters for the difference between the two models. More importantly, the scaling behavior is observable using currently available techniques in light-matter interaction.
We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].
We investigate a controlled beyond-electric-dipole extension of the quantum Rabi model in which the leading electric-dipole interaction is supplemented by an electric-quadrupole contribution. A central issue is the parity selection rule: for two exact atomic parity eigenstates of opposite parity, an electric-quadrupole transition matrix element vanishes. We therefore do not interpret the additional longitudinal coupling as an E2 transition between an opposite-parity pair. Instead, we consider an inversion-asymmetric (or parity-mixed) two-level system for which the two retained states may have different diagonal quadrupole moments. Projection of the multipolar interaction then produces a state-dependent term proportional to σz(a + a†), while the ordinary transition dipole produces the transverse term proportional to σx(a + a†). A state-independent quadrupolar contribution is removed by a bosonic displacement, leaving the effective Hamiltonian H =ωa†a+∆σz/2+gσx(a+a†)+gQσz(a+a†). This construction connects the beyond-dipole physical problem to the well-studied class of Rabi models with simultaneous transverse and longitudinal coupling, including asymmetric or biased Rabi models. We make this relation explicit by a constant spin rotation, which maps the model to a biased Rabi Hamiltonian with effective coupling κ = √g2 +g2Q and bias δ = ∆gQ/(2κ). We then derive the corresponding second-order Bargmann differential equation and solve its exceptional polynomial sector using a Zhang-type functional Bethe ansatz. The resulting Bethe equations, root-sum constraint, exceptional energy branches, and the first nontrivial parameter condition are obtained explicitly. In the limit gQ → 0, the construction continuously recovers the ordinary Rabi exceptional sector. From an experimental perspective, the connection with demonstrated quadrupolar light--matter coupling in engineered quantum systems suggests that the present framework may have relevance for spectroscopic studies of beyond-electric-dipole interactions, although a direct experimental realization of the specific effective model considered here remains to be established [20].