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arXiv:2402.09749 · Advanced Quantum Technologies

PT$\mathcal {PT}$‐Symmetric Quantum Rabi Model: Solutions and Exceptional Points

Jiong Li, Yi‐Cheng Wang, Liwei Duan, Qing‐Hu Chen

Published Mar 17, 2025• 6 citations
arXivHTMLPDF

Abstract

Abstract The ‐symmetric non‐Hermitian quantum Rabi model (QRM) with imaginary coupling is solved using the Bogoliubov operators approach. A transcendental function responsible for the exact solutions is derived, with its zeros yielding the regular spectrum. Two types of intersections: the exceptional points (EPs), which are well‐studied in the non‐Hermitian system; and another arising from doubly degenerate states due to the conserved QRM parity, which is well‐known in the Hermitian QRM, are found. These intersections are identified through this transcendental function. EPs emerge between pairs of adjacent excited energy levels, shifting toward lower coupling strengths as energy levels increase, and can also be predicted by a generalized rotating‐wave approximation approach. The fidelity susceptibility diverges to negative infinity at the EPs, consistent with recent findings in non‐Hermitian systems, while it diverges to positive infinity at the doubly degenerate points. The EPs are further confirmed by the vanishing c‐product in the biorthogonal basis.

DOI
10.1002/qute.202400609

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PTPT\mathcal {PT}PT‐Symmetric Quantum Rabi Model: Solutions and Exceptional Points
Abstract The ‐symmetric non‐Hermitian quantum Rabi model (QRM) with imaginary coupling is solved using the Bogoliubov operators approach. A transcendental function responsible for the exact solutions is derived, with its zeros yielding the regular spectrum. Two types of intersections: the exceptional points (EPs), which are well‐studied in the non‐Hermitian system; and another arising from doubly degenerate states due to the conserved QRM parity, which is well‐known in the Hermitian QRM, are found. These intersections are identified through this transcendental function. EPs emerge between pairs of adjacent excited energy levels, shifting toward lower coupling strengths as energy levels increase, and can also be predicted by a generalized rotating‐wave approximation approach. The fidelity susceptibility diverges to negative infinity at the EPs, consistent with recent findings in non‐Hermitian systems, while it diverges to positive infinity at the doubly degenerate points. The EPs are further confirmed by the vanishing c‐product in the biorthogonal basis.
Abstract Two non‐Hermitian two‐photon quantum Rabi models (tpQRM) that exhibit parity‐time () symmetry are investigated: the biased tpQRM (btpQRM), where the qubit bias is purely imaginary, and the dissipative tpQRM (dtpQRM), where the two‐photon coupling is made imaginary to introduce dissipation. For both models, exact solutions are derived using Bogoliubov transformations. In the btpQRM, spectral collapse is identified at a critical coupling strength, with accompanying symmetry breaking that correlates with exceptional points (EPs) arising from coalescing eigenstates. A direct correspondence is established between ‐broken regions and the doubly degenerate points of the Hermitian tpQRM, and the effect of qubit bias is analyzed via an adiabatic approximation. In the dtpQRM, although no spectral collapse occurs, both EPs and Juddian‐type degeneracies are present, with distinct behaviors governed by parity conservation. Through biorthogonal fidelity susceptibility and c‐product, the nature of these two types of level crossings is successfully identified and characterized. Finally, the dynamical evolution of both models is compared, revealing fundamentally different pathways to steady states governed by their respective non‐Hermitian spectral structures. These results provide exact characterizations of ‐symmetric non‐Hermitian tpQRMs and may offer theoretical insights for future experimental realizations.
The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.
In this work, we explore the PT\mathcal{PT}PT-symmetric quantum Rabi model, which describes a PT\mathcal{PT}PT-symmetric qubit coupled to a quantized light field. By employing the adiabatic approximation (AA), we are able to solve this model analytically in the parameter regime of interest and analyze various physical aspects. We investigate the static and dynamic properties of the model, using both the AA and numerical diagonalization. Our analysis reveals a multitude of exceptional points (EPs) that are closely connected with the exactly solvable points in the Hermitian counterpart of the model. Intriguingly, these EPs vanish and revive depending on the light-matter coupling strength. Furthermore, we discuss the time evolution of physical observables under the non-Hermitian Hamiltonian. Rich and exotic behaviors are observed in both strong and ultrastrong coupling regimes. Our work extends the theory of PT\mathcal{PT}PT symmetry into the full quantum light-matter interaction regime and provides insights that can be readily enlarged to a broad class of quantum optical systems.
The analytically exact solutions to the mixed quantum Rabi model including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4×44 \times 44×4 determinants responsible for the exact solutions are derived. These so-called GGG-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of GGG-functions reproduce completely the regular spectra. The exceptional eigenvalues can be also obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point 1/21/21/2. Surprisingly, all energy levels only collapse to the lower one, which diverges negatively. All the level crossings in the unmixed models are relaxed to avoided crossings in the present mixed model due to absence of parity symmetry. Very interestingly, the effective one-photon coupling is enhanced in the presence of the two-photon coupling, which paves a highly efficient and economic way to access the deep-strong coupling regime even though the original one-photon coupling is weak.
In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called GGG-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.
In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counterrotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding GGG-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities, and they are doubly degenerate. We find analytically that the ground state and the first excited state can cross several times, indicating multiple first-order phase transitions as a function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model, which are caused by a hidden symmetry.