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

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.