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arXiv:2507.20508 · Advanced Quantum Technologies

PT‐Symmetric Two‐Photon Quantum Rabi Models

Yi‐Cheng Wang, Jiong Li, Qing‐Hu Chen

Published Oct 28, 2025• 1 citation
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Abstract

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.

DOI
10.1002/qute.202500622

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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.
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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.