where in calculating the left current we need I 1 L m and both the contributions I 2 L m and I 2 R m and for the right current I 1 L m is replaced by I 1 R m . As before, special care is required in evaluating the Ei( x ) to choose the correct Riemann sheets in order to make sure that these functions are consistent with the initial conditions I$_{α}$ ( t ) = 0 and are continuous functions of time and chemical potential. To calculate the energy transferred from the electrons to the nanomechanical oscillator, we return to the density matrix ρ$_{n,n}$ ( t, t ) given in Eq. (18). We may therefore use the lesser Green’s function or density matrix to calculate the energy transferred to the oscillator as Note that the normalisation in equation (21) is required as the bare density matrix contians both electronic and oscillator contributions. The trace eliminates the oscillator part, leving the electronic part. In order to further characterize the state of the nanomechanical oscillator we investigate the Fano factor for the change of average occupation number, < n > as a function of time. The corresponding relation for the Fano factor is given by 46 The dot population, net current through the system, total current into the system, average energy and Fano factor of a resonant tunnel junction coupled to a nanomechanical oscillator are shown graphically as a function of time for different values of coupling strength, tunneling rate, and voltage bias. The following parameters 1–10,13,15–19,32 were employed: the single energy level of the dot ϵ$_{0}$ = 0 . 5, and the characteristic frequency of the oscillator ℏ ω = 0 . 1. These parameters will remain fixed for all further discussions and have same dimension as of ℏ ω . We are interested in small and large values of tunneling from the leads, different values VI. DISCUSSION OF RESULTS where 〈 n 〉 = ∑ n nρ$_{n,n}$ ( t, t ) / ∑ n ρ$_{n,n}$ ( t, t ) and 〈 n $^{2}$〉 = ∑ n n $^{2}$ρ$_{n,n}$ ( t, t ) / ∑ n ρ$_{n,n}$ ( t, t ), with the average evaluated using the diagonal element of the density matrix on the quantum dot. 10 F = 〈 n $^{2}$〉 − 〈 n 〉 2 〈 n 〉 , (22) E$_{ph}$ = 〈 n ℏ ω 〉 = ∑ n n ℏ ωρ$_{n,n}$ ( t, t ) n ρ$_{n,n}$ ( t, t ) . (21) V. AVERAGE ENERGY AND FANO FACTOR