Non-Markovianity: Necessary and sufficient criteria for composition in time and energy domains

Given an open quantum mechanical process, whether the decoherence followed by a physical system is non-Markovian (and to what extent) is of the utmost importance since it defines the predictions that can be made for the physical observables. Furthermore, non-local correlations in time are the quintessential effect of Non-Markovian dynamics. Since in quantum mechanics, time and energy are canonical conjugate variables, it is natural to expect that those correlations in time can also be observed in the energy domain. In this paper, we resort to the principle of composition or divisibility to observe memory effects in the Wigner propagation function in complex phase spaces. In this sense, we show that the sudden coupling of the system to the bath contributes to non-Markovian effects, and we show the constraints for experimental accessibility. Furthermore, we incorporate sufficient and necessary criteria in the energy domain to compose it in the time domain. Finally, we indicate that the results are applicable to radiative quenching, ionization of atoms, intramolecular relaxation, and photon bath coupling based on the Weiskopf model.