content_image/339124.jpg content_image/339123.jpg content_image/339122.jpg content_image/339119.jpg content_image/339120.jpg
with 

<img src='content_image/339124.jpg'>

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 

<img src='content_image/339123.jpg'>

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 

<img src='content_image/339122.jpg'>

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 

<img src='content_image/339119.jpg'>

The integral over the energy in the above equation is carried out $^{45}$. The final result for the density matrix is written as 

<img src='content_image/339120.jpg'>

where we have added the contribution from the right and the left leads, which can be written in terms of α as 

