ρ ( t ) = ∑ n ρ$_{n,n}$ ( t, t ) = Γ 2 π ∑ n,m,k Φ$_{n,m}$Φ ∗ m, $_{0}$Φ$_{0}$$_{,k}$ Φ ∗ n,k ε$_{k}$ − ε$_{m}$ + 2 i Γ [ Y L mk + Y R mk + Z L mk + Z R $_{mk}$] . I$_{α}$ = − e 〈 d d t x 〉 = − i e ℏ 〈 [ H, x ] 〉 (19) I$_{α}$ ( t ) = e Γ 2 π ℏ ∑ m Φ$_{0}$$_{,m}$Φ ∗ 0 ,m { I 1 α m + I 2 L m + I 2 R m } , (20) I 1 α m = 2 ( tan − 1 [ ϵ$_{F}$$_{α}$ − ε$_{m}$ Γ ] + π 2 ) − i { Ei[+i( ϵ$_{F}$$_{α}$ − ε$_{m}$ + iΓ) t ] − Ei[ − i( ϵ$_{F}$$_{α}$ − ε$_{m}$ − iΓ) t ] } , I 2 α m = − (1 + exp[ − 2Γ t ]) ( tan − 1 [ ε$_{Fα}$ ε$_{m}$ Γ ] + π 2 ) − 1 $_{2}$i exp[ − 2Γ t ] { Ei[+i( ϵ$_{F}$$_{α}$ − ε$_{m}$ − iΓ) t − Ei[ − i( ϵ$_{F}$$_{α}$ − ε$_{m}$ + iΓ) t ] } + 1 $_{2}$i { Ei[+i( ϵ$_{F}$$_{α}$ − ε$_{m}$ + iΓ) t ] − Ei[ − i( ϵ$_{F}$$_{α}$ − ε$_{m}$ − iΓ) t ] } , 9 where and the final result for the current through each of the leads is written as (See appendix A) The particle current I$_{α}$ into the interacting region from the lead is related to the expec- tation value of the time derivative of the number operator N$_{α}$ = ∑ αj c † $_{αj}$c$_{αj}$ , as 25,35–37 Now using equation (18), the dot population may be written as with ϵ$_{F}$$_{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 $_{mk}$. Y α mk = { 1 + exp[i( ε$_{k}$ − ε$_{m}$ + 2iΓ) t ] } { ln( ϵ$_{F}$$_{α}$ − ε$_{k}$ − iΓ) − ln( ϵ$_{F}$$_{α}$ − ε$_{m}$ + iΓ) } = { 1 + exp[i( ε$_{k}$ − ε$_{m}$ + 2iΓ) t ] } × { 1 2 ln[( ϵ$_{F}$$_{α}$ ε$_{k}$ ) 2 + Γ $^{2}$] ln[( ϵ$_{F}$$_{α}$ − ε$_{m}$ ) 2 + Γ $^{2}$] + i [ tan − 1 ( ε$_{Fα}$ − ε$_{k}$ Γ ) + tan − 1 ( ε$_{Fα}$ − ε$_{m}$ Γ ) + π ]} , Z α mk = exp[i( ε$_{k}$ − ε$_{m}$ + 2iΓ) t ] {− Ei[i( ϵ$_{F}$$_{α}$ − ε$_{k}$ − iΓ) t ] + Ei[ − i( ϵ$_{F}$$_{α}$ − ε$_{m}$ + iΓ) t ] } + { Ei[i( ϵ$_{F}$$_{α}$ − ε$_{m}$ + iΓ) t ] − Ei[ − i( ϵ$_{F}$$_{α}$ − ε$_{k}$ − iΓ) t ] } , IV. TIME-DEPENDENT CURRENT FROM LEAD α