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

The $\mathrm{PT}$-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. 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.