Hidden Lie algebraic structures and exact exceptional spectrum of the two-photon quantum Rabi model with asymmetric static bias

Abstract We systematically investigate the quadratic and linear Lie algebraic structures of the two-photon quantum Rabi model (TPQRM) with an asymmetric static bias in the coordinate representation. The exact existence conditions for the two distinct Lie algebras are analytically derived and confirmed numerically. These algebraic structures result in two classes of exact eigenenergies parameterized by integers and half-integers, respectively. Furthermore, we investigate the role of the static bias term in modifying the spectral properties of the TPQRM. The level crossings corresponding to exact integer energies remain robust against static bias perturbations, while those associated with half-integer energies undergo a characteristic transition from level crossings to anti-crossings. The derived exact wave functions provide explicit evidence for interaction-controlled phase segregation phenomena. Finally, we demonstrate the versatility of our analytical framework through successful applications to the TPQRM in two alternative Fock-Bargmann representations.