Heisenberg-picture operators for two-photon Jaynes-Cummings model

The dynamics of the degenerate two-photon Jaynes-Cummings model in Heisenberg picture is investigated. By solving the Heisenberg equations, the exact solutions of the photon annihilation operator a(t), the atomic dipole moment operator σ-(t), and the population difference operator σz(t) are obtained. We demonstrate that the correlation functions for the photon annihilation and dipole moment operators can be directly evaluated from these exact solutions. The solutions of the time-evolution operators are shown to be quite useful in the calculation of two-time correlation functions. Furthermore, some results of the dynamical behavior of the average values for the photon number 〈n(t)〉, the in-phase field component 〈X1(t)〉, and the population difference 〈σz(t)〉 are demonstrated by computer simulations.