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\textit{Oh, an empty article!} AC Noise for the Majorana RC circuit
You 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
get started by \textbf{double clicking} this text block and begin editing. You be expressed as
\begin{equation}
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= \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
+ V_{\beta k}^{*}V_{\gamma q}
\langle T c_{k}^\dagger(t) \eta_\beta(t)\eta_\gamma(t')c_q(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} 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
-\langle T \eta_\beta (t)c^\dagger_q(t')\rangle \langle T \eta_\gamma(t') c_{k}(t)\rangle\Biggr\}
\\
\nonumber
&& + V_{\beta k}^{*}V_{\gamma q}\Biggr\{\langle T c_{k}^\dagger(t) c_q(t')\rangle \langle \eta_\gamma(t')\eta_\beta(t)\rangle -\langle T c_{k}^\dagger(t)\eta_\gamma(t') \rangle\langle c_q(t')\eta_\beta(t)\rangle\Biggr\}\,,
\end{eqnarray}
Using the definition for the Green functions we can
also click 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} 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}
Finally, we employ the
\textbf{Text} button below following relation to
add new block elements. Or you obtain $S^{<(>)(t,t')}$:
\begin{equation}
S(t,t') = A(t,t') B(t',t) \rightarrow S^{>(<)}(t,t') = A^{>(<)}(t,t') B^{<(>)}(t',t)
\end{equation}
Now we compute the lead-lead Green function $G^{h,t}_{kq}(t,t')=\langle T c_k^\dagger(t) c_q(t')\rangle$ that appears in the previous expression. We compute its equation-of-motion
\begin{equation}
i\hbar \partial_{t'} G_{kq}^{h,t}(t,t') = \epsilon_q G_{kq}^{h,t}(t,t') + \sum_\beta V_{\beta q}^* G^{h,t}_{k\beta}(t,t')
\end{equation}
Then we get
\begin{equation}
G_{kq}^{h,t}(t,t') = G_{kq}^{h,t}(t,t')\delta_{kq} + \frac{-1}{\hbar} \sum_\beta\int dt_1 G^{h,t}_{k\beta}(t,t')V_{\beta q}^* g_{q}^{h,t}(t_1,t')
\end{equation}
In a similar way we can
\textbf{drag and drop an image} right onto this text. Happy writing! obtain the mixed (hole) lead-Majorana Green function $G_{k\beta}^{h,t}{t,t'}$
\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 $G_{kq}^{h,t}(t,t')$, we obtain
\begin{equation}
G_{kq}^{h,t}(t,t') = G_{kq}^{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) V_{\gamma k} G^{t}_{\gamma\beta}(t_1,t_2)V_{\beta q}^* g_{q}^{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.