Exact eigenstates of the intensity-dependent Jaynes–Cummings model with the counter-rotating term

We have investigated the eigenenergy spectrum of the intensity-dependent Jaynes–Cummings (JC) model with and without the rotating-wave approximation (RWA). Our analysis has indicated that the counter-rotating term dramatically changes the nature of the RWA energy spectrum and that the non-RWA spectrum can be approximated by the RWA spectrum only in the range of a sufficiently small coupling constant. Furthermore, the intensity-dependent JC model without the RWA is well defined only if the coupling parameter is below a certain critical value. As a result, the dynamics of the intensity-dependent JC model without the RWA is significantly different from its RWA counterpart. For instance, the counter-rotating term can dramatically enhance the field squeezing effect. Furthermore, we believe that our results are of immediate relevance to the system of a trapped and laser-irradiated ion, which has been shown to exhibit an intensity-dependent JC dynamics recently.