content_image/343102.jpg content_image/343092.jpg content_image/343094.jpg content_image/343093.jpg content_image/343095.jpg
<img src='content_image/343102.jpg'>

with ϵ$_{F}$$_{α}$ being the right and the left Fermi levels and Ei( x ) the exponential integral function. Special care is required in evaluating the Ei( x ) to choose the correct Riemann sheets in order to make sure that these functions are consistent with the initial conditions ρ (0) = 0 and are continuous functions of time and chemical potential. The same applies to complex logarithms in the first, apparently simpler, form for Y α $_{mk}$. 

Now using equation (18), the dot population may be written as 

<img src='content_image/343092.jpg'>

## IV. TIME-DEPENDENT CURRENT FROM LEAD α 

The particle current I$_{α}$ into the interacting region from the lead is related to the expec- tation value of the time derivative of the number operator N$_{α}$ = ∑ αj c † $_{αj}$c$_{αj}$ , as 25,35–37 

and the final result for the current through each of the leads is written as (See appendix A) 

where 

<img src='content_image/343094.jpg'>

<img src='content_image/343093.jpg'>

<img src='content_image/343095.jpg'>

