We solve Dyson’s equation using H$_{dot-leads}$ , as a perturbation. In the presence of the oscillator, the retarded and advanced Green’s functions on the dot, with the phonon states in the representation of the unoccupied dot, may be written as where g r ( a ) m ( k ) is the retarded (advanced) Green’s function on the occupied dot coupled to the leads may be written as, with ε$_{m}$ = ϵ$_{0}$ + ( m + 1 $_{2}$) ℏ ω − Δ, ε$_{k}$ = ϵ$_{0}$ + ( k + 1 $_{2}$) ℏ ω − Δ, and Δ = λ $^{2}$/ 2 mω $^{2}$. The above Eqs. (12, 13, 14, 15) will be the starting point of our examination of the time- dependent response of the coupled system. These functions are the essential ingredients for theoretical considerations of such diverse problems as low and high voltage, coupling of electron and phonons, transient and steady state phenomena. The density matrix is related to the dot population through ρ ( t ) = ∑ n ρ$_{n,n}$ ( t, t ), where the density matrix ρ$_{n,n}$ ( t, t ) = − i G < n,n ′ ( t, t $^{′}$), for t = t ′ and n = n $^{′}$. G < n,n ′ ( t, t $^{′}$) is the lesser Green’s function 24,25,35 on the dot including all the contributions from the leads. The lesser Green’s function for the dot in the presence of the nanomechanical oscillator is given by whereas, for t and t ′ < 0, the G < n,n ′ ( t, t $^{′}$) is equal to zero, and G < n,n ′ ( t, t $^{′}$) includes all the infor- mation of the nanomechanical oscillator and electronic leads of the system, and n$_{0}$, n ′ $_{0}$, n, n ′ are the oscillator indices. The lesser self-energy, Σ < n 0 ,n ′ $_{$_{0}$,α}$( t$_{1}$, t$_{2}$ ), contains electronic and os- cillator contributions. The electronic contributions are non-zero only when t$_{1}$ and t$_{2}$ > 0. As the oscillator is initially in its ground state, only the n$_{0}$ = n ′ 0 = 0 term gives a non-zero contribution to the lesser self-energy. The lesser self–energy for the dot may be written as 7 Σ < 0 , 0 $_{,α}$( t$_{1}$, t$_{2}$ ) = V ∗ $_{α}$( t$_{1}$ ) g < $_{α,α}$( t$_{1}$, t$_{2}$ ) V$_{α}$ ( t$_{2}$ ) , G < n,n ′ ( t, t $^{′}$) = ∑ n 0 ,n ′ $_{0}$,α ∫ ∫ d t$_{1}$ d t$_{2}$ G r n,n $_{0}$( t, t$_{1}$ )Σ < n 0 ,n ′ $_{$_{0}$,α}$( t$_{1}$, t$_{2}$ ) G a n ′ $_{0}$,n ′ ( t$_{2}$$_{,}$t $^{′}$) , t and t ′ > 0 (16) III. TIME-DEPENDENT DOT POPULATION ρ ( t ) g a $_{k}$( t$_{2}$, t $^{′}$) = +i θ ( t$_{2}$ − t $^{′}$) × exp[ − i( ε$_{k}$ + iΓ)( t$_{2}$ − t $^{′}$)] , t$_{2}$ > 0 (15) g r $_{m}$( t, t$_{1}$ ) = − i θ ( t − t$_{1}$ ) × exp[ − i( ε$_{m}$ − iΓ)( t − t$_{1}$ )] , t$_{1}$ > 0 (14) G r n,n ′ ( t, t$_{1}$ ) = ∑ m Φ$_{n,m}$ g r $_{m}$( t, t$_{1}$ )Φ ∗ n $^{′}$,m , G a n,n ′ ( t$_{2}$, t $^{′}$) = ∑ k Φ$_{n,k}$ g a $_{k}$( t$_{2}$, t $^{′}$)Φ∗ n $^{′}$,k (13)