N-photon solutions to the two-qubit quantum Rabi model

We studied the two-qubit quantum Rabi model and found its dark state solutions with at most N photons. One peculiar case presents when N=3, which has constant eigenenergy in the whole coupling regime and leads to level crossings within the same parity subspace. We also discovered asymptotic solutions with at most N=2i+3 (i=1,2,3,…) photons, and constant eigenenergy Nħω when coupling g becomes much larger than photon frequency ω. Although generally all photon number states are involved in the two-qubit quantum Rabi model, such N-photon solutions exist and may have applications in quantum information processing with ultrastrong couplings.