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 $N=2i+3$ $(i=1,2,3,\dots )$ photons, and constant eigenenergy $N\mathrm{\hslash }\omega$ when coupling $g$ becomes much larger than photon frequency $\omega$. 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.