  The asymmetric quantum Rabi model (AQRM) is a fundamental model in quantum
optics describing the interaction of light and matter. Besides its immediate
physical interest, the AQRM possesses an intriguing mathematical structure
which is far from being completely understood. In this paper, we focus on the
distribution of the level spacing, the difference between consecutive
eigenvalues of the AQRM in the limit of high energies, i.e. large quantum
numbers. In the symmetric case, that is the quantum Rabi model (QRM), the
spacing distribution for each parity (given by the $\mathbb{Z}_2$-symmetry) is
fully clarified by an asymptotic expression derived by de Monvel and Zielinski,
though some questions remain for the full spectrum spacing. However, in the
general AQRM case, there is no parity decomposition for the eigenvalues. In
connection with numerically exact studies for the first 40,000 eigenstates we
describe the spacing distribution for the AQRM which is characterized by a new
type of periodicity and symmetric behavior of the distribution with respect to
the bias parameter. The results reflects the hidden symmetry of the AQRM known
to appear for half-integer bias. In addition, we observe in the AQRM the
excited state quantum phase transition for large values of the bias parameter,
analogous to the QRM with large qubit energy, and an internal symmetry of the
level spacing distribution for fixed bias. This novel symmetry is independent
from the symmetry for half-integer bias and not explained by current
theoretical knowledge.
