$\mathcal{PT}$-symmetric quantum Rabi model: Solutions and exceptional points

The $\mathcal{PT}$-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. We find two types of intersections: One is the exceptional point (EP), which is widely studied in the non-Hermitian system; another one is due to doubly degenerate states caused by the conserved QRM parity, which is well-known in the Hermitian QRM. 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. 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. All eigenstates are characterized by conserved energy and QRM parity. We conclude that the non-Hermitian QRM is integrable, analogous to its Hermitian counterpart.