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...
S^{>,1}(\omega)= \frac{4e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{\beta\gamma\alpha\delta} [G^r_{\beta \alpha}(\epsilon) \Gamma_{\alpha\delta} G^a_{\delta \gamma}(\epsilon) \Gamma_{\alpha\delta} [F_{eh}+F_{hh}]
\end{eqnarray}
with the total noise as $S=S^A+S^B+S^C$
No we compute the second contribution to the noise from
$N(t,t')=(e^2/h)\sum_{k\beta,q\gamma} $N^>(t,t')=(e^2/h)\sum_{k\beta,q\gamma} V_{\beta k} V^*_{\gamma q}G_{\beta
q}>(t,t') q}^>(t,t') G^{h,<}_{\gamma
k}(t',t)$. These two Green functions read k}(t',t)$ and $N^<(t,t')=(e^2/h)\sum_{k\beta,q\gamma} V_{\beta k} V^*_{\gamma q}G_{\beta q}^<(t,t') G^{h,>}_{\gamma k}(t',t)$, the total one is then $N=N^>+N^<$. We start with $N^>$ that reads
\begin{eqnarray}
G^>_{\beta q}(t,t') = \frac{1}{h} \sum_\gamma \int dt_1 [G_{\beta\gamma}^r(t,t_1) V_{\gamma q} g^{>}_{q}(t_1,t')+ G_{\beta\gamma}^>(t,t_1) V_{\gamma q} g^{a}_{q}(t_1,t')
\end{eqnarray}
\begin{eqnarray}
G^{<,h}_{\gamma k}(t,t') = \frac{-1}{h} \sum_\beta \int dt_1 [G_{\gamma\beta}^r(t',t_1) V_{\beta k} g^{<,h}_{k}(t_1,t)+ G_{\gamma\beta}^<(t',t_1) V_{\beta k} g^{a,h}_{k}(t_1,t)
\end{eqnarray}
In the frequency domain the product of these two functions
read: becomes:
\begin{eqnarray}
N(\omega)=(e^2/h)\sum_{k\beta,q\gamma} N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma} \int \frac{d\epsilon}{2\pi} V_{\beta k} V^*_{\gamma q}G_{\beta q}>(\omega+\epsilon) G^{h,<}_{\gamma k}(\epsilon)
\end{eqnarray}
where
\begin{eqnarray}
...
\end{eqnarray}
Then, we have
\begin{eqnarray}
&&N(\omega)=(e^2/h)\sum_{k\beta,q\gamma, &&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} V_{\beta k} V^*_{\gamma q}
[G_{\beta\nu}^r(\omega+\epsilon) V_{\nu q} g^{>}_{q}(\omega+\epsilon)+ G_{\beta\nu}^>(\omega+\epsilon) V_{\nu q} g^{a}_{q}(\omega+\epsilon)]
\\ \nonumber
&&[G_{\gamma\mu}^r(\epsilon) V^*_{\mu k} g^{<,h}_{k}(\epsilon)+ G_{\gamma\mu}^<(\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon)]
\end{eqnarray}
\begin{eqnarray}
N(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, N^>(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\omega+\epsilon) V_{\nu q} g^{>}_{q}(\omega+\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\epslion) V^*_{\mu k} g^{<,h}_{k}(\epsilon) V_{\beta k}]
\\ \nonumber
&&[G_{\beta\nu}^r(\omega+\epsilon) V_{\nu q} g^{>}_{q}(\omega+\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon) V_{\beta k}]
...
\end{eqnarray}
Inserting the expressions for the self-energies we get
\begin{eqnarray}
N(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, N^>(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\omega+\epsilon) \Sigma_{0,\nu\gamma}^>(\omega+\epsilon) G_{\gamma\mu}^r(\epsilon) \Sigma_{0,\mu\beta}^{<,h}(\epsilon)]
\\ \nonumber
&&[G_{\beta\nu}^r(\omega+\epsilon) \Sigma_{0,\nu\gamma}^>(\omega+\epsilon) G_{\gamma\mu}^<(\epsilon) \Sigma_{0,\mu\beta}^{a,h}(\epsilon)]
...
\end{eqnarray}
\begin{eqnarray}
&&N(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, &&N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
[G_{\beta\nu}^r(\omega+\epsilon) \Gamma_{\nu\gamma} G_{\gamma\mu}^r(\epsilon) \Gamma_{\mu\beta} (1-f_e(\omega+\epsilon) f_{h}(\epsilon)] \\ \nonumber
&&\sum_{\lambda\delta}[G_{\beta\nu}^r(\omega+\epsilon) \Gamma_{\nu\gamma} G_{\gamma\lambda}^a(\omega) \Gamma_{\lambda\delta}G^a_{\delta\mu}(\epsilon)[i\Gamma_{\mu\beta}](1-f_e(\omega+\epsilon)(f_{h}(\epsilon)+f_e(\epsilon))] \\ \nonumber
&&\sum_{\lambda\delta} G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\mu}^r(\omega)(1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))f_h(\epsilon)\\ \nonumber
&&\sum_{\lambda\delta\theta\tau}[G_{\beta\lambda}^r(\omega+\epsilon)\Gamma_{\lambda\delta} G_{\delta\nu}^a(\omega+\epsilon)[i\Gamma_{\nu\gamma}] G_{\gamma\theta}^r(\epsilon)\Gamma_{\theta\tau} G_{\tau\mu}^a(\epsilon)[i\Gamma_{\mu\beta}](1-f_{e}(\omega+\epsilon)+1-f_{h}(\omega+\epsilon))(f_{e}(\epsilon)+f_{h}(\epsilon))] \Biggr\}
\end{eqnarray}
Finally, to obtain $N<(t,t)$ we just change $(1-f)\rightarrow f$, and $f\rightarrow (1-f)$.
The next term is $M(t,t') = M^>(t,t´) + M^<(t,t´)$ with
\begin{eqnarray}
M^>(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k}^{*}V_{\gamma q} [G^{h,>}_{kq}(t,t') G{<}_{\gamma\beta}(t',t)
\end{eqnarray}
