Emulation of n-photon Jaynes Cummings and Anti-Jaynes-Cummings models via parametric modulation of cyclic qutrit

We study a circuit QED setup involving a single cavity mode and a cyclic qutrit whose parameters are time modulated externally. It is shown that in the dispersive regime this system behaves as a versatile platform to implement effective n-photon Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) models by suitably setting the modulation frequency. The atomic levels and the cavity Fock states involved in the effective Hamiltonians can be controlled through adjustment of the system parameters, and different JC and AJC interactions can be implemented simultaneously using multitone modulations. Moreover, one can implement some models that go beyond simple JC and AJC-like interaction, such as multiphoton coupling between certain entangled states. We estimate analytically the associated transition rates and study numerically the dynamics in the presence of Markovian dissipation, demonstrating that lower-order effective Hamiltonians can be implemented with current technology.