  The non-Hermitian one-photon and two-photon quantum Rabi models (QRMs) with
imaginary couplings are respectively solved through the Bogoliubov operators
approach. Transcendental functions responsible for exact solutions are derived,
whose zeros produce the complete spectra. Exceptional points (EPs) can be
identified in terms of the transcendental function. The EP is formed in the two
nearest-neighboring excited energy levels, and shifts towards lower coupling
strength at higher energy levels. Interestingly, under the resonant condition
in the non-Hermitian two-photon QRM, the lowest two excited states within the
same parity in the even photonic number subspace coalesce at an EP, but take
always the purely real energy, in sharp contrast to the conventional EP in the
non-Hermitian systems. For both non-Hermitian QRMs, the fidelity susceptibility
goes to negative infinity at the EPs, consistent with the recent observations
in non-Hermitian systems. All eigenstates can be labeled by the conserved
energy and the QRM parity, we argue that the non-Hermitian QRMs are also
integrable, similar to their Hermitian counterparts.
