The two-mode multi-photon intensity-dependent Rabi model

We have investigated the energy eigen-spectrum of the two-mode k-photon intensity-dependent Rabi (IDR) model for k ≥ 2. Our analysis shows that the model does not have eigenstates in the Hilbert space spanned by the eigenstates of the two-mode k-photon intensity-dependent Jaynes-Cummings (IDJC) model, which is obtained by applying the rotating-wave approximation (RWA) to the two-mode k-photon IDR model. That is, the two-mode k-photon IDR model is ill-defined for k ≥ 2, and it is qualitatively different from the RWA counterpart which is valid for all values of k, implying that the counter-rotating term does drastically alter the nature of the RWA counterpart. Hence, the previous study of the effect of the counter-rotating term in the two-mode k-photon IDJC model via the time-dependent perturbation expansion is completely invalid.