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G^>_{\beta\gamma}(\epsilon) = \sum_{p\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{\delta p} + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon)
\end{eqnarray}
We need to compute the following product of Green functions: $P^>(t,t')=e^2/\hbar^2\sum_{k\beta,q\gamma} G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\begin{eqnarray}
&&\sum_{k\alpha\delta} \begin{align*}
&\sum_{k\alpha\delta} V_{\gamma q}^* V_{\beta k} G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} V_{\gamma q}^* V_{\beta k}\Biggr\{ G^r_{\beta \alpha}(\epsilon) [V^*_{k\alpha} g^>_{k}(\omega)V_{\delta k} + V_{\alpha k} g^{h,>}_{k}(\omega) V^*_{\delta k}]G^a_{\delta \gamma}(\epsilon) g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} \\ \nonumber
&+& &+ \sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [ V_{\gamma q}^* g_{q}^{h,r}(\omega+\epsilon) V_{\tau q} G^{r}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,<}(\omega) V_{\beta k} \nonumber
\\
&&\sum_{p &\sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [ V_{\gamma q}^* g_{q}^{h,r}(\omega+\epsilon) V_{\tau q} G^{<}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,a}(\omega) V_{\beta k}
\nonumber
\\
&&+ &+ \sum_{p \alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p} g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p} g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [ V_{\gamma q}^* g_{q}^{h,<}(\omega+\epsilon) V_{\tau q} G^{a}_{\tau\theta}(\omega+\epsilon)V_{\theta k}^* g_{k}^{h,a}(\omega) V_{\beta k} \Biggr\}
\,,
\end{eqnarray} \end{align*}
We now compute separately the different parts of the previous expression for the ac noise
\begin{eqnarray}
P^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi}\sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) (\Sigma_{0,\alpha\delta}^{>,e}+ \Sigma_{0,\alpha\delta}^{>,h}) G^a_{\delta \gamma}(\epsilon)\Sigma_{\gamma\beta}^{<,h}(\omega+\epsilon)\delta_{kq}
