$\mathcal{PT}$-symmetric two-photon quantum Rabi models

TL;DR

This work addresses PT-symmetric non-Hermitian extensions of the two-photon quantum Rabi model by constructing exact solutions via Bogoliubov transformations and SU(1,1) algebra to yield G-functions whose zeros determine the spectra. It demonstrates that spectral collapse persists in the biased model at $g_c=\tfrac{1}{2}$ with nonzero $\epsilon$, while exceptional points align with Juddian degeneracies of the Hermitian case; in the dissipative model, spectral collapse is absent but EPs and Juddian points still occur, with fidelity susceptibility and the c-product providing clear diagnostics. Dynamical analysis shows qualitatively different relaxation pathways: stepwise, plateau-like evolution dominated by PT-broken states in the biased case, versus rapid, monotonic relaxation toward a steady state in the dissipative case. The exact framework and diagnostic tools offered here deliver theoretical guidance for future experiments in circuit QED, trapped ions, and cold-atom setups exploring non-Hermitian quantum criticality and PT-symmetry breaking.

Abstract

We investigate two non-Hermitian two-photon quantum Rabi models (tpQRM) that exhibit $\mathcal{PT}$ symmetry: the biased tpQRM (btpQRM), in which 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, we derive exact solutions by employing Bogoliubov transformations. In the btpQRM, we identify spectral collapse at a critical coupling strength, with accompanying $\mathcal{PT}$ symmetry breaking that correlates with exceptional points (EPs) arising from coalescing eigenstates. We establish a direct correspondence between $\mathcal{PT}$-broken regions and the doubly degenerate points of the Hermitian tpQRM, and analyze the effects of qubit bias via an adiabatic approximation. In the dtpQRM, although no spectral collapse occurs, both EPs and Juddian-type degeneracies are present, with well-separated behaviors distinguished by parity conservation. Through biorthogonal fidelity susceptibility and c-product, we successfully identify and classify the nature of these two types of level crossings. Finally, we compare the dynamical evolution of both models, revealing fundamentally different pathways to steady states governed by their respective non-Hermitian spectral structures. Our results provide exact characterizations of $\mathcal{PT}$-symmetric non-Hermitian tpQRMs and may offer theoretical insights for future experimental realizations.

Figures (10)

• Figure 1: (a) $G$-function curves of the btpQRM in the real energy regime for $q = 1/4$ (left) and $q = 3/4$ (right). Blue lines represent the $G$-function, while black dashed lines indicate the pole positions $E_{n,0}^{(q,\mathrm{pole})}$. (b) Distribution of $\ln |G|^2$ in the complex energy plane for $q = 1/4$ (left) and $q = 3/4$ (right). In both panels, $\Delta = 0.50$, $g = 0.20$, and the open circles mark the zeros of the $G$-function.
• Figure 2: (a) $G$-function curves of the dtpQRM in the real energy regime for $q = 1/4$ (left) and $q = 3/4$ (right). Blue (red) lines represent the $G_{+}$ ($G_{-}$) functions. Black dashed lines indicate the pole positions $E_{n}^{(q,\mathrm{pole})}$. (b) Distribution of $\ln |G_{+}|^2$ (left) and $\ln |G_{-}|^2$ (right) in the complex energy plane for $q = 1/4$. In both panels, $\Delta = 0.50$, $g = 0.25$, and open circles mark the zeros of the $G$-function.
• Figure 3: Real (left) and imaginary (right) parts of the lowest few eigenvalues of the btpQRM as a function of the coupling strength $g$. Dashed lines indicate the pole positions $E_{n,0}^{(q,\mathrm{pole})}$. Here, $\Delta = 0.50$, $\epsilon = 0.10$, and $q = 1/4$.
• Figure 4: Scaled spectra $E' = (E + 1/2)/\beta$ at $\Delta = 0.50$ and $q = 1/4$. (a) The comparison between the btpQRM (left, $\epsilon = 0.10$) and the Hermitian tpQRM (right, $\epsilon = 0.00$). (b) is scaled spectra for the btpQRM at $\epsilon=0.40$. Blue (red) lines represent the real (imaginary) part of $E'$, while blue (red) dashed lines show the corresponding results obtained from the adiabatic approximation. Black dotted lines indicate $E_{n,0}^{(q,\mathrm{pole})}$. The green lines mark the parameters used in Sec. \ref{['sec_dyna_btp']}
• Figure 5: The real part of the fidelity susceptibility (left) and the c-product $\langle L' \vert R' \rangle$ (right) for the btpQRM, plotted as functions of the coupling strength $g$ using the same parameters as in Fig. \ref{['sepctra_bias_delta0.50_fig']}. Blue, red, and green lines correspond to the $1^{\text{st}}$, $3^{\text{rd}}$, and $5^{\text{th}}$ excited states, respectively, with $q = 1/4$, $\Delta = 0.50$, and $\epsilon = 0.10$.