7 various many-body states involved in transport. In a first series of calculations we selected the chemical potential µ$_{R}$ = 0 . 2 and used two values of the chemical potential of the left lead µ$_{L}$ = 0 . 4 and µ$_{L}$ = 0 . 6. For µ$_{R}$ = 0 . 2 and µ$_{L}$ = 0 . 4 the bias window contains only the 1-st and the 2-nd excited configurations with N = 1, Fig. 5(b). The ground states for N = 1 and N = 2 are instead located below and above the bias window, respectively. Conse- quently the steady state current is very small. When µ$_{L}$ increases to 0.6 the ground state configuration with N = 2 enters the bias window and the current increases, Fig. 6(a). To analyze the transient regime we split the current into contributions given by the ground state and excited states with 1 electron (see Fig. 6(b)). When µ$_{L}$ = 0 . 4 the 1-st and 2-nd excited state carry currents exceeding the current associated to the ground state, which survive all the way to the steady state. The current corresponding to the 2-nd excited state is smaller than the current of the 1-st excited state, but comparable to that of the ground state. This is explained by the strength of the coupling coefficients shown in Fig. 5(a), the 2-nd single-particle state being stronger coupled to the leads. The remaining higher excited states give oscillating and fast decaying transient currents. In Fig. 6(c) µ$_{L}$ = 0 . 6 and therefore higher excited states enter the bias window; their tran- sient currents are still decaying but at a smaller rate. Comparing with Fig. 6(a) it in clear that the transient regime is dominated by excited states. Next we discuss currents associated with states having 2 and 3 electrons. We keep now fixed µ$_{R}$ = 0 . 35 and again increase µ$_{L}$ starting with 0.6. Fig. 7(a) shows the total currents in the left lead for N = 2 and N = 3. As the bias increases the transient currents are enhanced, but they become comparable as the system approaches the steady state. In Fig. 7(b) the total current on three particle states shows a different behavior: the steady states value increases drastically when µ$_{L}$ moves up. To explain this one can look again at the diagram of the chemical potentials, Fig. 5(b). At µ$_{L}$ = 0 . 6 the 3-particle configurations are above the bias window and as such they contribute less to the current. In contrast, as µ$_{L}$ increases the ground state configuration with N = 3 en- ters the bias window, the window is closer to the excited states, and thus the total current increases. Actually, for µ$_{L}$ = 0 . 8 and 0 . 9 the current for N = 3 does not reach the steady state in the time interval considered. Now we look at the contribution of the excited states with N = 2 for two cases, µ$_{L}$ = 0 . 6 and µ$_{L}$ = 0 . 9. Again, the inspection of the diagram in Fig. 5(b) predicts the results of Fig. 8. When µ$_{L}$ = 0 . 6 there is just one excited configuration within the bias window, in addition to the ground state. In Fig. 8(a) we see that in the steady state these two configurations give significant contributions to the current, whereas the higher excited states play a role only in the transient regime. Fig. 8(b) shows that at µ$_{L}$ = 0 . 9 the currents of the excited states and of the ground state are decreasing, some of them reaching even FIG. 6: (a) The total currents in the left and right leads for µ$_{L}$ = 0 . 6 and µ$_{L}$ = 0 . 4, while keeping µ$_{R}$ = 0 . 2. (b) The partial currents in the left lead for single-particle states when µ$_{L}$ = 0 . 4 and µ$_{R}$ = 0 . 2. (c) The partial currents in the left lead for single-particle states when µ$_{L}$ = 0 . 6 and µ$_{R}$ = 0 . 2 negative values towards the steady state. This happens because the bias window includes now the ground state with N = 3 and the excited states with N = 2 deplete in the favor of the ground state. The sign of the current carried by states with N par- ticles depends on the placement of the corresponding ground state chemical potential relatively to the bias win- dow. For example if we fix µ$_{L}$ = 1 . 5 and µ$_{R}$ = 0 . 65 we