  In quantum interaction problems with explicitly time-dependent interaction
Hamiltonians, the time ordering plays a crucial role for describing the quantum
evolution of the system under consideration. In such complex scenarios, exact
solutions of the dynamics are rarely available. Here we study the nonlinear
vibronic dynamics of a trapped ion, driven in the resolved sideband regime with
some small frequency mismatch. By describing the pump field in a quantized
manner, we are able to derive exact solutions for the dynamics of the system.
This eventually allows us to provide analytical solutions for various types of
time-dependent quantities. In particular, we study in some detail the
electronic and the motional quantum dynamics of the ion, as well as the time
evolution of the nonclassicality of the motional quantum state.
