Algebraic structure of the two-qubit quantum Rabi model and its solvability using Bogoliubov operators

We have found the algebraic structure of the two-qubit quantum Rabi model behind the possibility of its novel quasi-exact solutions with finite photon numbers by analyzing the Hamiltonian in the photon number space. The quasi-exact eigenstates with at most $1$ photon exist in the whole qubit-photon coupling regime with constant eigenenergy equal to single photon energy \hbar\omega, which can be clear demonstrated from the Hamiltonian structure. With similar method, we find these special "dark states"-like eigenstates commonly exist for the two-qubit Jaynes-Cummings model, with $E=N\hbar\omega$ (N=-1,0,1,...), and one of them is also the eigenstate of the two-qubit quantum Rabi model, which may provide some interesting application in a simper way. Besides, using Bogoliubov operators, we analytically retrieve the solution of the general two-qubit quantum Rabi model. In this more concise and physical way, without using Bargmann space, we clearly see how the eigenvalues of the infinite-dimensional two-qubit quantum Rabi Hamiltonian are determined by convergent power series, so that the solution can reach arbitrary accuracy reasonably because of the convergence property.