17 Σ F e $_{-vib}$( τ$_{1}$, τ$_{2}$ ) = i γ 2 0 D$_{0}$ ( τ$_{1}$, τ$_{2}$ ) G ( τ$_{1}$, τ$_{2}$ ) , (A3) Σ F,ζ$_{1}$ζ$_{2}$ e -vib ( t$_{1}$, t$_{2}$ ) = i γ 2 0 D ζ$_{1}$ζ$_{2}$ 0 ( t$_{1}$, t$_{2}$ ) G ζ$_{1}$$^{ζ$_{2}$}$( t$_{1}$, t$_{2}$ ) , (A4) X r = X ++ − X + − = X − + − X −− (A5) G ( τ, τ $^{′}$) = − i 〈 T$_{C}$ K d$_{H}$ ( τ ) d † $_{H}$( τ $^{′}$) 〉 . (A1) X r = X t − X < = X > − X ˜ t (A6) X a = X ++ − X − + = X + − − X −− (A7) Σ H,x e $_{-vib}$( ω ) = − i γ 2 0 D x $_{0}$( ω = 0) ∫ d ω ′ 2 π G $^{<}$( ω $^{′}$) , (A8) Σ F,y e $_{-vib}$( ω ) = iγ 2 0 ∫ dω ′ 2 π D y $_{0}$( ω $^{′}$) G $^{y}$( ω − ω $^{′}$) , (A9) Σ F,r e $_{-vib}$( ω ) = i γ 2 0 ∫ dω ′ 2 π D r $_{0}$( ω − ω $^{′}$) G $^{<}$( ω $^{′}$) + D < 0 ( ω − ω $^{′}$) G $^{r}$( ω $^{′}$) + D r $_{0}$( ω − ω $^{′}$) G $^{r}$( ω ) . (A10) Σ H e $_{-vib}$( τ$_{1}$, τ$_{2}$ ) = − i γ 2 0 D$_{0}$ ( τ$_{1}$, τ$_{2}$ ) G ( τ$_{1}$, τ + 1 ) , (A2) 1. Lowest order self-energies Appendix A: Self-energies for the Born approximation and the second-order diagrams diagrams, shown in this paper for the spectral functions only, are also important in the full non-equilibrium trans- port properties of the nanojunctions. The study of these effects on the non-linear conductance are currently un- dertaken and will be considered in a forthcoming paper. and where τ$_{i}$ are times on the time-loop contour. The projections onto the real (physical) times are given by, for example, for the Fock self-energy . In this appendix, we show how to derive the expression for the self-energies corresponding to one- and two-vibron process diagrams in terms of electron-vibron coupling. Starting from the SSSM Hamiltonian, the general def- inition of the one-particle GF of the central region is ob- tained from the time-loop contour-ordered product of the creation and annihilation operators in the central region The index ζ$_{i}$ = ± labels the branch of the time-loop contour corresponding to the forward ( ζ$_{i}$ = +)/ back- ward ( ζ$_{i}$ = − ) time evolution respectively. There are several useful relationships between the different projections (or Keldysh components) of the Green’s functions X ≡ G, D : $^{66–70}$. They are or equivalently in terms of time-ordered ( t = ++), anti time-ordered ( ˜ t = −− ), greater ( > = − +) and lesser ( < = + − ) Green’s functions: The time-loop contour C$_{K}$ contains two branches, the up- per (+) and the lower ( − ) branch. On the upper branch, time starts in the infinitely remote past and evolves for- wards, then at the turning point, which can be placed at any arbitrary time, one passes onto the lower branch where the system evolves backwards in time back to the initially non-interacting starting point at t = −∞ . and Then any expectation value of products of opera- tor reduces to 〈 φ$_{0}$ | T$_{C}$ K ( ˆ A ( τ ) ˆ B ( τ $^{′}$) . . . S$_{C}$ K ) | φ$_{0}$ 〉 where 〈 φ$_{0}$ | . . . | φ$_{0}$ 〉 is the average over the non-interacting ground state. The operators are then given in the interaction picture, and S$_{C}$ K is the generalization of the time evolution operator on the Keldysh contour S$_{C}$ K = T$_{C}$ K ( exp {− i ∫ C K d τ ˆ V ( τ ) } ) where T$_{C}$ K is the time-ordering operator on the contour C$_{K}$ and ∫ C K d t implies integration over C$_{K}$ . And ˆ V is the “perturba- tion” to the reference Hamiltonian, which in our case would be the interaction electron-vibron as well as the coupling of the central region to the leads). Expanding S$_{C}$ K as a series in terms of the electron- vibron coupling Hamiltonian H$_{e}$$_{-vib}$ , one can derive the electron-vibron self-energies to any order of the electron- vibron coupling by calculating any time ordered products in the series using the usual rules of many-body perturba- tion theory, such as Feynmann diagrammatic expansion or Wick’s theorem $^{85–87}$. Using these relationships and taking the steady state limit, i.e. X ( t, t $^{′}$) = X ( t − t $^{′}$), and after Fourier trans- formation into an energy representation X ( ω ), we ob- tain the usual expressions 20,24,26,27,30–34,38,44–46 for the Hartree Σ H,x e $_{-vib}$( ω ) and Fock Σ F,x e $_{-vib}$( ω ) electron-vibron self- energies ( x = r, a, <, > ). For example, the Hartree and Fock self-energies are Similar relationships exist for the self-energies Σ x ↔ X x ( x = r, a, <, >, t, ˜ t ). (or equivalently X a = X t − X > = X < − X ˜ $^{t}$), and hence X r − X a = X > − X $^{<}$. where y represents one of the tree Keldysh compoentns y = t, <, > . Using the relationship Σ r = Σ t − Σ $^{<}$, we find The one-vibron process self-energies, corresponding to the Hartree and Fock-like diagrams (see Figure 2) are given by