Qualitative aspects of entanglement in the Jaynes-Cummings model with an external quantum field

We present a mathematical procedure which leads us to obtain analytical solutions for the atomic inversion and Wigner function in the framework of the Jaynes-Cummings model with an external quantum field, for any kind of cavity and driving fields. Such solutions are expressed in the integral form, with their integrands having a common term that describes the product of the Glauber-Sudarshan quasiprobability distribution functions for each field, and a kernel responsible for the entanglement. Considering two specific initial states of the tripartite system, the formalism is then applied to calculate the atomic inversion and Wigner function where, in particular, we show how the detuning and amplitude of the driving field modify the entanglement. In addition, we also obtain the correct quantum-mechanical marginal distributions in phase space