with where f $_{α}$( ε$_{α}$ ) is the Fermi distribution functions of the left and right leads, which have different chemical potentials under a voltage bias. For the present case of zero temperature the lesser self–energy may be recast in terms of the Heaviside step function θ ( x ) as where Σ r, ( a ) , ( < ) 0 , 0 ,α ( t$_{1}$, t$_{2}$ ) are all non-zero only when both times ( t$_{1}$, t$_{2}$ ) are positive t$_{1}$, t$_{2}$ > 0 and ϵ$_{F}$$_{α}$ is the Fermi energy on each of leads. The density matrix ρ$_{n,n}$ ( t, t ) can be calculated by using Eqs. (12, 13, 14, 15, 17) in Eq. (16) at t = t ′ and n = n ′ as Although g r, ( a $^{)}$( t$_{1}$, t$_{2}$ ) is non-zero for t < 0, it is never required due to the way it combines with Σ r, ( a ) , ( < ) 0 , 0 ,α ( t$_{1}$, t$_{2}$ ). By carrying out the time integrations, the resulting expression is written as The integral over the energy in the above equation is carried out $^{45}$. The final result for the density matrix is written as where we have added the contribution from the right and the left leads, which can be written in terms of α as ρ$_{n,n}$ ( t, t ) = Γ 2 π ∑ α,m,k ∫ ϵ F α −∞ d ε$_{α}$ Φ$_{n,m}$Φ ∗ 0 $_{,m}$Φ$_{0}$$_{,k}$ Φ ∗ n,k ( ε$_{α}$ − ε$_{k}$ − iΓ)( ε$_{α}$ − ε$_{m}$ + iΓ) × { 1 + exp[i( ε$_{k}$ − ε$_{m}$ + 2iΓ) t ] − exp[ − i( ε$_{α}$ − ε$_{k}$ − iΓ) t ] − exp[i( ε$_{α}$ − ε$_{m}$ + iΓ) t ] } ρ$_{n,n}$ ( t, t ) = Γ 2 π ∑ m,k Φ$_{n,m}$Φ ∗ m, $_{0}$Φ$_{0}$$_{,k}$ Φ ∗ n,k ε$_{k}$ − ε$_{m}$ + 2iΓ [ Y L mk + Y R mk + Z L mk + Z R $_{mk}$] , (18) 8 ρ$_{n,n}$ ( t, t ) = − i ∑ α,m,k ∫ t 0 ∫ t 0 d t$_{1}$ d t$_{2}$ Φ$_{n,m}$Φ ∗ 0 $_{,m}$exp[ − i( ε$_{m}$ − iΓ)( t − t$_{1}$ )] ×{ iΓ ∫ ϵ F α −∞ d ε$_{α}$ 2 π exp[ − i ε$_{α}$ ( t$_{1}$ − t$_{2}$ ) } Φ$_{0}$$_{,k}$Φ ∗ n,k exp[ − i( ε$_{k}$ + iΓ)( t$_{2}$ − t )] , Σ < 0 , 0 $_{,α}$( t$_{1}$, t$_{2}$ ) = iΓ$_{α}$$^{+}$$^{∞}$ ∫ −∞ d ε$_{α}$ 2 π θ ( ϵ$_{F}$$_{α}$ + 1 2 ℏ ω − ε$_{α}$ ) θ ( t$_{1}$ ) θ ( t$_{2}$ ) exp[ − i ε$_{α}$ ( t$_{1}$ − t$_{2}$ )] , (17) g < $_{α,α}$( t$_{1}$, t$_{2}$ ) = 1 N ∑ j g < $_{α,j}$( t$_{1}$, t$_{2}$ ) = + ∞ ∫ −∞ d ε$_{α}$ f $_{α}$( ε$_{α}$ )2i n$_{α}$ exp[ − i ε$_{α}$ ( t$_{1}$ − t$_{2}$ )] ,