Consider the following short time propagation, from t to t + dt, of the reduced density operator ρ (Eq. 8) of a K ≥ 2 level system evolving under the dynamics of the quantum subsystem Hamiltonian Ĥ = ĤQ + ĤQC (R) and a single non-zero Lindblad jump operator L = |0⟩⟨1|. For a small dt, the time evolution can be approximated as