Due to the presence of a random phase, the expectation value of Eq. B1 is what the trajectory-averaged density matrix will converge towards and will be treated as the result of operating the effective Liouvillian L R onto the density matrix, which can be written as where we have explicitly used the expressions in Eq. A27a-A27b to write down these coefficients. The superoperator e LRdt can be determined from Eq. B2 by identifying the coefficients |c 0 (t)| 2 , c 0 (t)c * 1 (t), c 1 (t)c * 0 (t), and |c 1 (t)| 2 on the left-hand side as the density matrix elements ρ 00 , ρ 01 , ρ 10 , and ρ 11 , respectively, and considering the terms that these density matrix elements multiply with as the rates that come from e LRdt . This superoperator can thus be identified as (B5)