In order to calculate the analytical solutions and to discuss the numerical results of the transient and steady state dynamics of the nanomechanical systems, our focus in this section is to derive an analytical relation for the time dependent effective self-energy and the Green’s functions. In obtaining these results we use the wide–band approximation only for simplicity, although the method we are using does not rely on this approximation, where the retarded self–energy of the dot due to each lead is given by 25,35 where α = L, R represent the left and right leads and the Green’s function in the leads for the uncoupled system is with the fact that ∑ j ↦→ + ∞ ∫ −∞ Nn$_{α}$ d ε$_{α}$, where j stands for every channel in each lead and n$_{α}$ is the constant number density of the leads. Now using the uncoupled Green’s function into equation (9), the retarded self energy may be written as Now we use the fact that V$_{α}$ ( t$_{1}$ ) = | V | × θ ( t$_{1}$ ), V$_{α}$ ( t$_{2}$ ) = | V | × θ ( t$_{2}$ ). Then the above expression can be written as where Γ$_{α}$ = 4 π | V | 2 n$_{α}$ is the damping factor (Γ$_{L}$ = Γ$_{R}$ = Γ). Similarly Σ a $_{α}$( t$_{1}$, t$_{2}$ ) = [Σ r $_{α}$( t$_{1}$, t$_{2}$ )] ∗ = + 1 $_{2}$iΓ$_{α}$ θ ( t$_{2}$ ) δ ( t$_{1}$ − t$_{2}$ ) . Σ r $_{α}$( t$_{1}$, t$_{2}$ ) = − 1 $_{2}$iΓ$_{α}$ θ ( t$_{2}$ ) δ ( t$_{1}$ − t$_{2}$ ) (12) = − i n$_{α}$V ∗ $_{α}$( t$_{1}$ ) V$_{α}$ ( t$_{2}$ ) θ ( t$_{1}$ − t$_{2}$ )$^{+}$$^{∞}$ ∫ −∞ d ε$_{α}$ exp[ − i ε$_{α}$ ( t$_{1}$ − t$_{2}$ )] , = − i n$_{α}$V ∗ $_{α}$( t$_{1}$ ) V$_{α}$ ( t$_{2}$ ) θ ( t$_{1}$ − t$_{2}$ ) × 2 πδ ( t$_{1}$ − t$_{2}$ ) , (11) Σ r $_{α}$( t$_{1}$, t$_{2}$ ) = − i n$_{α}$θ ( t$_{1}$ − t$_{2}$ )$^{+}$$^{∞}$ ∫ −∞ d ε$_{α}$V ∗ $_{α}$( t$_{1}$ ) exp[ − i ε$_{α}$ ( t$_{1}$ − t$_{2}$ )] V$_{α}$ ( t$_{2}$ ) , (10) g r $_{α,α}$( t$_{1}$, t$_{2}$ ) = 1 N ∑ j g r $_{α,j}$( t$_{1}$, t$_{2}$ ) = − i n$_{α}$θ ( t$_{1}$ − t$_{2}$ )$^{+}$$^{∞}$ ∫ −∞ d ε$_{α}$ exp[ − i ε$_{α}$ ( t$_{1}$ − t$_{2}$ )] , Σ r $_{α}$( t$_{1}$, t$_{2}$ ) = V ∗ $_{α}$( t$_{1}$ ) g r $_{α,α}$( t$_{1}$, t$_{2}$ ) V$_{α}$ ( t$_{2}$ ) , (9) 6