such that the expectation value will be treated as an approximation of the exponential function in order to extract γ R outside of the logarithm. This approximation is present due to the choice of a Poisson rate process in determining the distribution of the random phase in Eq. A28 which results in the decoherence being a first order approximation of an exponential decay. Thus, the chosen distribution for the random phase Φ requires γ R dt ≪ 1 for accuracy. Treating the expectation value as the exponential decay in Eq. B7, the leftmost non-zero element of Eq. B6 can be evaluated as where Eq. A22b has be used to substitute Γ/2 in for γ Eh + γ R . Comparing the effective Liouvillian in Eq. B6 and the actual Lindblad Liouvillian in Eq. A5, the terms k Eh + k R = Γ match. However, the term iE -γ Eh + ln β dt in Eq. B6 fails to match the term iE -Γ 2 in Eq. A5, by the difference of the extra last term in Eq. B8. The presence of this extraneous term is due to the modification of the magnitudes of the coefficients in Eqs. A27a and A27b. The expression inside of the logarithm in Eq. B8 is the ratio of the modified coefficient magnitudes to the original coefficient magnitudes. These magnitude modifications change the coherence between the ground and excited states (as well as all other states in larger dimensional systems) thus their impact must be taken into consideration in the off-diagonal decay procedure in order to match Lindblad dynamics.