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Recently, in [Phys. Rev. A 97, 043806 (2018)], the detuned and nonlinearJaynes-Cummings model describing the quantized motion of a trapped ion wasintroduced and its corresponding dynamics was solved via considering thedriving laser in a quantized manner. In this work we reconsider this model andshow that it can likewise be solved with a classical driving laser field. Usingthe exact solution we investigate the quantum time-ordering effects of thesystem with respect to nonclassicality of the motional states of the ion.Furthermore, we use the Magnus expansion to analyze the impact of certainorders of the time ordering and derive and exact radius of convergence beyondthe established and only sufficient criterion. Finally, the differences of thesolution derived here and the previously found one using a quantized pump, arediscussed.

Quantum effects of operator time ordering in the nonlinear Jaynes-Cummings model