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\title{Notes on ewjn mag noise}

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The reproduction of the magnetic noise in \cite{QubitRelax} is below, in \cref{fig:lukemag}.
The important point for reproducing the figure in Luke's paper is that I did not use the expression for $r_s$ there.
Instead, I used eqs 2.20 and 2.26 in Ford-Weber\cite{Ford1984}:

\begin{align}
\zeta_s(u) &= 2 i \int_0^\infty \dd{y} \frac{1}{\epsilon_t(\sqrt{u^2 + y^2}) - u^2 - y^2} \\
r_s(u) &= \frac{\zeta_s(u) - \frac{\pi}{i u}}{\zeta_s(u) + \frac{\pi}{i u}},
\end{align}
up to a constant out front which I pulled to outer code.

Additionally, the other important thing is that every parameter in the caption of Luke's figure 3 are are actually in radians per second, except for $\nu$ (I think).
The collision frequency $\nu$ is the odd one, because in the expression for $\epsilon$, the local limit reduces to the Drude case precisely with $\tau = \frac{1}{\nu}$.
I was always under the impression that the Drude $\tau$ just carried units of seconds, rather than seconds per radian, but now I'm not sure (there are after all $\tau \omega$ terms which don't look ``unitless'').
That all implies that $\tau = \left(6 \pi \right)^{-1} \times \SI{e-12}{\s}$.


\begin{figure}[htp]
	\centering
	\includegraphics[width=\linewidth]{lukemagnoise}
	\caption{$T_1(z)$, dotted is the nonlocal case and solid is the local case.} \label{fig:lukemag}
\end{figure}

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