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H$_{T}$ contains two important elements: (1) The time de- pendent switching functions χ$_{l}$ ( t ) which open the contact between the leads and the sample; these functions mimic the presence of a time dependent potential barrier. (2) The coupling T l qn between a state with momentum q of the lead l and the state n of the isolated sample, with wave function φ$_{n}$ . The coupling coefficients T l qn depend on the energies of the coupled states and, maybe more important, on the amplitude of the wave functions in the contact region. As we have shown in our previous work 7,8 this construction allows us to capture geometrical effects in the electronic transfer. A precise definition of the cou- pling coefficients is however model specific, and will be mentioned in the next section. 

The evolution of our system is completely determined by the statistical operator W ( t ) associated to the total Hamiltonian H ( t ) defined in Eq.(1). W ( t ) is the solution of the quantum Liouville equation with a known initial value, prior to the coupling of the sample and leads: 

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The isolated leads are described by equilibrium distribu- tions, 

and the isolated sample by the density operator ρ$_{S}$ . Af- ter the coupling moment the dynamics of the sample is conveniently described by the RDO which is defined by averaging the total statistical operator over those degrees of freedom belonging to the leads: 

In the absence of the electron-electron interaction the MES eigenvectors of H$_{S}$ are bit-strings of the form | ν 〉 = | i ν $_{1}$, i ν $_{2}$, .., i ν $_{n}$... 〉 , where i ν n = 0 , 1 is the occupation number of the n -th SES. The set { ν } is a basis in the Fock space of the isolated sample and the RDO can be seen as a matrix in this basis. From Eqs. (6)-(8) we obtain in the lowest (2-nd) order in the coupling parameters T l qn the GME (see Ref. 7 for details): 

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where the coupling operator T$_{ql}$ has matrix elements 

The operators Ω$_{ql}$ and Π$_{ql}$ are defined as 

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and f$_{l}$ is the Fermi function of the lead l . 

In the presence of the electron-electron interaction in the sample the MES which are eigenstates of H$_{S}$ are lin- ear combinations of bit-strings: H$_{S}$ | α ) = E$_{α}$| α ), where | α ) = ∑ ν C$_{αν}$ | ν 〉 , C$_{αν}$ being the mixing coefficients which can be found together with the energies E$_{α}$ by diagonalizing H$_{S}$ . (To distinguish better between the noninteracting and the interacting MES we use the right angular bracket for the former and the regular curved bracket for the later.) Using now the set { α } as a basis, i. e. the interacting MES, the GME has the same form as Eq. (9), where the matrix elements of all operators are now defined in the interacting basis and the matrix elements of the coupling operators are 

Because the sample is open the number of electrons N contained in the sample is not fixed. The Hamiltonian H$_{S}$ given in Eq. (3) commutes with the total “number” operator ∑ n d † $_{n}$d$_{n}$ . Thus N is a “good quantum number” such that any state | α ) has a fixed number of electrons. So the MES can also be labeled as | α ) = | N, i ) with i = 0 , 1 , 2 , ... an index for the ground and excited states of the MES subset with N electrons. The many-body energies can also be written as E$_{α}$ = E ( i ) N . In the practical calculations N varies between 0 (the vacuum state) and N$_{max}$ which is the total number of SES considered in the numerical diagonalization of H$_{S}$ . The total number of MES is thus 2 $^{N$_{max}$}$. 

If the coupling between the leads and the sample is not too strong we expect that only a limited number of MES participate effectively to the electronic transport. These states are naturally selected by the bias window [ µ$_{R}$, µ$_{L}$ ]. In the following examples, by selecting suitable values of the chemical potentials in the leads, we will truncate the basis of interacting MES to a reasonably small subset such that we can solve numerically Eq. (9) with our available computing resources. To relate the bias window with the effective MES we need to consider the chemical potential of the isolated sample containing N electrons, 

which is the energy required to add the N -th electron on top of the ground state with N − 1 to obtain the i -th MES with N particles. 23 We expect the current associated to the MES | N, i ) to depend on the location of the chemical potential µ ( i ) N relatively to the bias window. In particular it is clear that if at the coupling moment t$_{0}$ the sample is empty all MES with µ ( i ) N ≫ µ$_{L}$ will remain empty both during the transient and the steady states, so they can be safely ignored when solving the GME. 

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