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arXiv:1404.6918 · Communications in Theoretical Physics

Parity Symmetry and Parity Breaking in the Quantum Rabi Model with Addition of Ising Interaction*

Qiong Wang, Zhi He, Chunmei Yao

Published Apr 1, 2015• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0253-6102/63/4/510

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•Oct 28, 2026•0 citations
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.
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 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.
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 the critical phenomena of the asymmetric quantum Rabi model (AQRM), where parity symmetry is broken by an external bias. Through both analytical and numerical calculations, we identify second-order and first-order phase transitions, with the latter absent in the standard quantum Rabi model. We derive an analytical two-variable scaling function that describes the finite-frequency scaling behavior of the AQRM, and numerical results confirm this framework. The introduction of bias leads to additional critical exponents, including a bias-related critical exponent ν_{h} and susceptibility exponent γ. Moreover, we demonstrate that critical scaling persists even below the conventional critical coupling, indicating the emergence of field-induced quantum criticality. These findings establish a robust theoretical framework for understanding universal quantum criticality in light-matter systems.
The introduction of an asymmetric term into the quantum Rabi model generally lifts energy-level degeneracies. However, when the asymmetry parameter takes specific multiples of the bosonic mode frequency, level degeneracies reappear−-−a phenomenon referred to as the hidden symmetry in the asymmetric quantum Rabi model. Identifying the origin of this hidden symmetry and its explicit operator form constitutes two central tasks in studying this system. Here, we investigate the origin of this hidden symmetry using the method of two successive diagonalizations, with a focus on physics in the regime where the ratio between the two-level splitting ΔΔΔ and the mode frequency ωωω satisfies Δ/ω≫1Δ/ω\gg 1Δ/ω≫1. We find that the hidden symmetry stems from energy-level matching within the asymmetric double-well potential, a picture strongly supported by the wavefunctions of both the ground and excited states. Moreover, the emergence of an excited-state quantum phase transition is identified and qualitatively discussed, which arises from the breaking and restoration of this hidden symmetry across different coupling regimes. Our results provide deeper insight into the physics of the asymmetric quantum Rabi model, particularly in the previously less-explored strong-coupling regime where Δ/ω≫1Δ/ω\gg 1Δ/ω≫1.
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].