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AC Noise for the Majorana RC circuit
The current operator the the RC Majorana nanowire problem is defined as
\begin{equation}
I(t)=\frac{ie}{\hbar}\sum_{k\beta} V_{\beta k}\eta_\beta c_k - V_{\beta k}^{*}c_{k}^\dagger \eta_\beta\,,
\end{equation}
The charge noise can be expressed as
\begin{equation}
S(t,t') =
S(t,t')^>+S(t,t')^< S^>(t,t')+S^<(t,t')
\end{equation}
where $S(t,t')=\langle I(t),I(t')\rangle$. Let us consider now the time-ordered $S^t(t,t')^$, then
\begin{eqnarray}
&&S^t(t,t´)^t= &&S^t(t,t´)= \nonumber
\\
&&\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*}
\langle T \eta_\beta (t)c_k(t)c_{q}^\dagger(t') \eta_\gamma(t') \rangle
...
\end{eqnarray}
We apply Wick theorem to $S^t(t,t')$, the
\begin{eqnarray}
&&S^t(t,t´)^t=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} &&S^t(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*}
\\
\nonumber
&&\Biggr\{\langle T \eta_\beta (t)\eta_\gamma(t')\rangle \langle T c^\dagger_q(t')c_{k}(t)\rangle
...
\end{eqnarray}
Using the definition for the Green functions we can write down the expression for $S^t(t,t')$
\begin{eqnarray}
&&S^t(t,t´)^t=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} &&S^t(t,t´)=\frac{e^2}{\hbar^2}\sum_{k\beta,q\gamma} V_{\beta k} V_{\gamma q}^{*}
[G^t_{\beta\gamma}(t,t') G^{h,t}_{qk}(t',t) - G^{t}_{\beta q}(t,t')G^{h,t}_{\gamma k}(t',t)]\\ \nonumber
&& + V_{\beta k}^{*}V_{\gamma q} [G^{h,t}_{kq}(t,t') G{t}_{\gamma\beta}(t',t) - G^{h,t}_{k\gamma}(t,t')G^{t}_{q \beta}(t',t)]
\end{eqnarray}
...
\end{equation}
In a similar way we can obtain the mixed (hole) lead-Majorana Green function $G_{k\beta}^{h,t}{t,t'}$
\begin{equation}
G_{k\beta}^{h,t}{t,t'} 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 $G_{kq}^{h,t}(t,t')$, we obtain
\begin{equation}
