  The asymmetric quantum Rabi model (AQRM), which describes the interaction
between a quantum harmonic oscillator and a biased qubit, arises naturally in
circuit quantum electrodynamic circuits and devices. The existence of hidden
symmetry in the AQRM leads to a rich energy landscape of conical intersections
(CIs) and thus to interesting topological properties. However, current
approximations to the AQRM fail to reproduce these CIs correctly. To overcome
these limitations we propose a generalized adiabatic approximation (GAA) to
describe the energy spectrum of the AQRM. This is achieved by combining the
perturbative adiabatic approximation and the exact exceptional solutions to the
AQRM. The GAA provides substantial improvement to the existing approaches and
pushes the limit of the perturbative treatment into non-perturbative regimes.
As a preliminary example of the application of the GAA we calculate the
geometric phases around CIs associated with the AQRM.
