Note that the dynamics for the quantum subsystem are not unitary due to coupling to the bath, and ρQ does not correspond to a pure state, and upon the trajectory average, Eq.11 describes reduced density matrix for mixed states. The time-dependent reduced density matrix of the system is obtained by averaging ρ from all trajectories. For the system with an initial state of ρQ (0) = c n0 |n 0 ⟩⟨ñ 0 |c ñ0 , the elements of the reduced system density matrix can now be obtained as