  The analytical exact solutions to the mixed quantum Rabi model (QRM)
including both one- and two-photon terms are found by using Bogoliubov
operators. Transcendental functions in terms of $4 \times 4$ determinants
responsible for the exact solutions are derived. These so-called $G$-functions
with pole structures can be reduced to the previous ones in the unmixed QRMs.
The zeros of $G$-functions reproduce completely the regular spectra. The
exceptional eigenvalues can also be obtained by another transcendental
function. From the pole structure, we can derive two energy limits when the
two-photon coupling strength tends to the collapse point. All energy levels
only collapse to the lower one, which diverges negatively. The level crossings
in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM
due to absence of parity symmetry. In the weak two-photon coupling regime, the
mixed QRM is equivalent to an one-photon QRM with an effective positive bias,
suppressed photon frequency and enhanced one-photon coupling, which may pave a
highly efficient and economic way to access the deep-strong one-photon coupling
regime.
