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Time-dependent NEGF

In the time-dependent NEGF equation, given a sigma_in(t,t’) due to the dot, I am getting an I-V equation that is making it difficult for me to group terms. For instance, looking at the analogue of the first term of I = 2q/hbar. int dE TrGin I get I1 = q/ihbar. int dt1 TrSigma_in.G^dagger-G.Sigma_in

Now, because of the different order of the time arguments in the G and G^dagger terms, I cannot pool these to write it as a spectral function A = i(G-G^dagger). Similarly, I can’t seem to take Sigma – Sigma^dagger to get Gamma. Is this correct?

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    Supriyo Datta

    I think I(t,t) should be real except for numerical errors ..

    We can write

    I_L(t,t) = (1/i/hbar) int dt1 TraceSigin_LG+-GSigin_L

    = (1/i/hbar) int dt1 sum(i,j) Sigin_L (i,j;t,t1) G+(j,i;t1,t) – G(i,j;t,t1) sigin_L+(j,i;t1,t) = (1/i/hbar) int dt1 sum(i,j) Sigin_L (i,j;t,t1) G(i,j;t,t1)* – G(i,j;t,t1) sigin_L(i,j;t,t1)* =(1/i) * (X – X*) = Real.

    In the first step I made use of sigin_L = sigin_L+ and in the second step I made use of A+(i,j;t,t1) = A(j,i;t1,t)*

    Makes sense?

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