The quantum Rabi model for two qubits

We study a system composed of two nonidentical qubits coupled to a single-mode quantum field. We calculate the spectra of the system in the deep-strong-coupling regime via perturbation theory up to second-order corrections and show that it converges to two forced oscillator chains for cases well into that regime. Our predictions are confirmed by the numerical calculation of the spectra using a parity decomposition of the corresponding Hilbert space. The numerical results point to two interesting types of behavior in the ultra-strong-coupling regime: the rotating wave approximation is valid for some particular cases and there exist crossings in the spectra within each parity subspace. We also present the normal modes of the system and give an example of the time evolution of the mean photon number, population inversion, von Neumann entropy and Wootters concurrence in the ultra-strong- and deep-strong-coupling regimes.