In this simplified picture, considering state |1⟩ to be the excited state and state |0⟩ to be the ground state, the jump operator L = |0⟩⟨1| causes the excited state to decay to the ground state and both excited and ground states to decohere from each other. When ĤQ = ĤQC = 0 and no other jump operators have non-zero interaction strength, the time evolution of the reduced density matrix (in Eq. 25) can be analytically expressed as e L Lt ρ 00 ρ 01 ρ 10 ρ 11 = ρ 00 + 1 -e -Γt ρ 11 e -Γt/2 ρ 01 e -Γt/2 ρ 10 e -Γt ρ 11 .