  In this work we start from a phenomenological Hamiltonian built from two
known systems: the Hamiltonian of a pumped optomechanical system and the Jaynes
Cummings Hamiltonian. Using algebraic techniques we construct an approximate
time evolution operator $\hat U_{opt}$ for the forced optomechanical system (as
a product of exponentials) and take the JC Hamiltonian as an interaction. We
transform the later with $\hat U_{opt}$ to obtain a generalized interaction
picture Hamiltonian which can be linearized and whose time evolution operator
is written in a product form. The analytic results are compared with purely
numerical calculations using the full Hamiltonian and the agreement between
them is remarkable.
