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with only the Fock self-energy $^{92}$. 

## B. Non-equilibrium spectral functions 

In this s ection, we present results for the spectral func- tions A ( ω ) for the off-resonant transport regimes, for dif- ferent app lied biases, both with and without the Hartree contribut ion. We first present calculations for an asym- metric po tential drop with η$_{V}$ = 1, i.e. µ$_{L}$ = V and µ$_{R}$ = 0 ( µ e q = 0). 

Figure 6 (a) shows the spectral function calculated with just the F ock component at different applied biases. We then incre ase the bias V by increasing the chemical potential of the left contact while keeping that of the right cont act at zero. For low values of V ( V ≤ ω$_{0}$ ) there is lit tle change in the spectral function. However, once the value of V exceeds that of ω$_{0}$ , the spectral function b ecomes increasingly modified, especially when there is sig nificant spectral weight inside the bias window µ$_{R}$ < ω < µ$_{L}$ . In particular, the lineshapes of both the main peak and the vibron side-band peaks become de- formed, w ith a noticeable asymmetry of the main peak for biases where the main peak, but not the right-hand vibron sid e-band peak, lies within the bias window. 

There is a saturation regime once the vibron side band peaks, and thus nearly all of the spectral weight, is within the bias w indow (curves for µ$_{R}$ ≳ 1). Here the main peak becomes p inned around ω = 0 . 5 (effectively midway be- tween µ$_{L}$ a nd µ$_{R}$ ), and a symmetric lineshape is restored. We postu late that this is owing to the bias being large enough to achieve simultaneous electron and hole trans- port. 

Figure 6(b) shows the spectral function with both the 

## 1. Non-equilibrium off-resonant transport 

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Hartree and Fock diagrams included. As we have alread y noted, at zero bias there is no change from the Fock-on ly spectral function. For small biases ( V ≲ ω$_{0}$ ) ther e is little difference, but for larger biases the effect of th e Fock diagram becomes increasingly evident. This is be - cause once the bias exceeds the vibron frequency ω$_{0}$ , the non-equilibrium electron density becomes strongly per - turbed. The main peak is more stable in position tha n for the Fock-only spectral function, although it shift s slightly towards lower energies with increasing bias be - fore stabilising. The righthand vibron side-band pea k becomes strongly deformed, and moves to lower, rathe r than higher, energies as the bias is increased. The left - hand vibron side-band peak appears at a lower bias tha n for the Fock-only spectral function, at a frequency jus t above zero, then tends towards zero as the bias increases . As for the Fock-only case however, once both vibron side - band peaks are within the bias window, saturation is reached and the peak positions stabilize. For both the Fock-only and Hartree-Fock spectral functions, the sepa - 

