We provide a detailed discussion about the unitary and reduced evolution
    induced by family of Hamiltonian models describing a multilevel system, with a
    ground state and a possibly multilevel excited sector, coupled to a multimode
    boson field via a rotating-wave interaction. We prove explicitly that the
    system, in the limit in which the coupling is flat with respect to the boson
    frequencies, is Markovian under sharp measurements in arbitrary bases; we also
    find necessary and sufficient conditions under which the process is classical,
    i.e. its family of multitime joint probability distributions satisfies the
    Kolmogorov consistency condition, and may thus be equivalently obtained by a
    classical stochastic process.

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