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ration between both side-band peaks and the main peak is ∼ 0 . 34 , > ω$_{0}$ for the parameters we used. 

Figure 7 shows the Hartree potential (i.e. the real part of Σ H,r e $_{-vib}$) plotted against applied bias for different values of the electronic level ε$_{0}$ . The solid line corresponds to the value of ε$_{0}$ used to calculate the spectral functions of the lower panel in Figure 6. The Hartree potential is small and almost constant in the low bias region. Hence, in the quasi-equilibrium regime, such a potential could be neglected in the calculations, because it simply corre- sponds to an energy reference shift. However when the bias window [ µ$_{L}$, µ$_{R}$ ] starts to encompass features in the spectral functions (either the main peak or vibron side- band peaks), the corresponding non-equilibrium electron density starts to vary substantially with the applied bias. Thus the Hartree potential shows a strong dependence on the value of the bias (as also shown in Ref. [60]), until the saturation regime is reached and the Hartree poten- tial is once again constant for very large biases. The asymptotic saturation value of the Hartree potential is dependent on the value of the electronic level ε$_{0}$ as one would expect. The first drop in the value of the Hartree potential happens at ˜ ε$_{0}$ , with the subsequent steps, which become progressively smaller and broader, at ˜ ε$_{0}$ + ∼ 0 . 34 and then ˜ ε$_{0}$ + nω$_{0}$ . 

This behaviour indicates that one cannot in principle neglect the Hartree diagram contribution in the calcu- lations, unless one is interested in calculating only the properties of the system for a bias range for which the electron density is (almost) constant $^{93}$. 

It is worth mentioning that it is not straightforward to relate the modification of the peak positions in the two panels in figure 6 to the Hartree potential alone. In self-consistent calculations, a highly non-linear sys- tem needs to be solved, since the Hartree potential is ob- 

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tained from one element of the Green’s functions which are themselves dependent on the values of the Hartree potential. 

We now consider what happens when we apply a bias in the resonant transport regime. We apply a symmetric potential drop (i.e. the potential of the left electrode is raised by an amount ηeV while that of the right elec- trode drops by the same amount). This allows us to keep the electron-hole symmetry and see under which circum- stances the electron-hole symmetry is broken. 

The spectral functions for the Fock diagram are shown in figure 8(a). As we have no Hartree term, the spec- tral function is electron-hole symmetric at equilibrium. Moreover, as we have chosen a symmetric potential drop ( η = 0 . 5) the spectral functions stay symmetric for all 

## 2. Non-equilibrium resonant transport 

