deletions | additions
diff --git a/untitled.tex b/untitled.tex
index fccf526..5e19899 100644
--- a/untitled.tex
+++ b/untitled.tex
...
\begin{equation}
G_{k\beta}^{h,t}(t,t') = \frac{-1}{\hbar} \sum_\beta\int dt_1 g_{k}^{h,t}(t,t_1) V_{\gamma k} G^{t}_{\gamma\beta}(t,t')
\end{equation}
Then, inserting the previous expression into the equation for
$G_{kq}^{h,t}(t,t')$, $G_{qk}^{h,t}(t,t')$, we obtain
\begin{equation}
G_{kq}^{h,t}(t,t') G_{qk}^{h,t}(t,t') =
G_{kq}^{h,t}(t,t')\delta_{kq} G_{qk}^{h,t}(t,t')\delta_{kq} + \sum_{\beta\gamma}\int \frac{-dt_1}{\hbar}\frac{-dt_2}{\hbar}
g_{k}^{h,t}(t,t_1) g_{q}^{h,t}(t,t_1) V_{\gamma
k} q} G^{t}_{\gamma\beta}(t_1,t_2)V_{\beta
q}^* g_{q}^{h,t}(t_2,t') k}^* g_{k}^{h,t}(t_2,t')
\end{equation}
The rest of equations for the Green functions that appear in the noise expression are already in J. S note.
Now we employ the following definition for the Fourier transform
...
Now we enter the expression for $G_{kq}^{h,t}(t,t')$ in the frequency domain
\begin{equation}
G_{kq}^{h,t}(\omega) G_{qk}^{h,t}(\omega) = g_{q}^{h,t}(\omega)\delta_{kq} + \sum_{\beta\gamma}
g_{k}^{h,t}(\omega) g_{q}^{h,t}(\omega) V_{\gamma
k} q} G^{t}_{\gamma\beta}(\omega)V_{\beta
q}^* g_{q}^{h,t}(\omega)\,, k}^* g_{k}^{h,t}(\omega)\,,
\end{equation}
\begin{multline}
...
Let us treat first the following term: $\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*} [G^>_{\beta\gamma}(\epsilon) G^{h,<}_{qk}(\omega+\epsilon) - G^{>}_{\beta q}(\epsilon)G^{h,<}_{\gamma k}(\epsilon+\omega)]$
Then,
\begin{eqnarray}
&&G_{kq}^{h,<}(\omega+\epsilon) &&G_{qk}^{h,<}(\omega+\epsilon) = g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} + \sum_{\beta\gamma}
[g_{k}^{h,r}(\omega+\epsilon) [g_{q}^{h,r}(\omega+\epsilon) V_{\gamma
k} q} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta
q}^* g_{q}^{h,<}(\omega)\nonumber k}^* g_{k}^{h,<}(\omega)\nonumber
\\
&&+g_{k}^{h,r}(\omega+\epsilon) &&+g_{q}^{h,r}(\omega+\epsilon) V_{\gamma
k} q} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta
q}^* g_{q}^{h,a}(\omega+\epsilon)+g_{k}^{h,<}(\omega+\epsilon) k}^* g_{k}^{h,a}(\omega+\epsilon)+g_{q}^{h,<}(\omega+\epsilon) V_{\gamma
k} q} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta
q}^* g_{q}^{h,a}(\omega+\epsilon)\,, k}^* g_{k}^{h,a}(\omega+\epsilon)\,,
\end{eqnarray}
On the other hand we have for $G^>_{\beta\gamma}(\epsilon)$ (accordingly with J.S note)
\begin{eqnarray}
...
We need to compute the following product of Green functions: $G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\begin{eqnarray}
&&G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} 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\beta\gamma\alpha\gamma} 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)
[g_{k}^{h,r}(\omega+\epsilon) [g_{q}^{h,r}(\omega+\epsilon) V_{\gamma
k} q} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta
q}^* g_{q}^{h,<}(\omega) k}^* g_{k}^{h,<}(\omega) \nonumber
\\
&&+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)
[g_{k}^{h,r}(\omega+\epsilon) [g_{q}^{h,r}(\omega+\epsilon) V_{\gamma
k} q} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta
q}^* g_{q}^{h,a}(\omega+\epsilon)]\\ k}^* g_{k}^{h,a}(\omega+\epsilon)]\\ \nonumber
&&+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)[g_{k}^{h,<}(\omega+\epsilon) \gamma}(\epsilon)[g_{q}^{h,<}(\omega+\epsilon) V_{\gamma
k} q} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta
q}^* g_{q}^{h,a}(\omega+\epsilon)]\,, k}^* g_{k}^{h,a}(\omega+\epsilon)]\,,
\end{eqnarray}
We now compute separately the different parts of the previous expression for the ac noise
\begin{eqnarray}
S^{>,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) [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_{\beta k} g_{q}^{h,<}(\omega+\epsilon)V^*_{\gamma q}\delta_{kq}
\end{eqnarray}
\begin{eqnarray}
S^{>,2}(\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) [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_{\beta k} g_{k}^{h,r}(\omega+\epsilon) [V^*_{\gamma q} g_{q}^{h,r}(\omega+\epsilon) V_{\gamma
q} G^{<}_{\gamma\beta}(\omega+\epsilon) V_{\beta k}
G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)V^*_{\gamma q}] g_{k}^{h,a}(\omega+\epsilon)V^_{\beta k}]
\end{eqnarray}
\begin{eqnarray}
S^{>,3}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \sum_{k,q,p\beta\gamma\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_{\beta k} g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta [V_{\gamma q}^*
g_{k}^{h,r}(\omega+\epsilon) V^*_{\gamma q} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\gamma q} g_{q}^{h,a}(\omega+\epsilon) V^*_{\gamma q}]
\end{eqnarray}
\begin{eqnarray}
S^{>,4}(\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)[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_{\beta k} g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)V^*_{\gamma q}]
