56 lines
1.9 KiB
TeX
56 lines
1.9 KiB
TeX
\documentclass{article}
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%other packages
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{physics}
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\usepackage[
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style=phys, articletitle=false, biblabel=brackets, chaptertitle=false, pageranges=false, url=true
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]{biblatex}
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\usepackage{graphicx}
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\usepackage{todonotes}
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\usepackage{siunitx}
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\usepackage{cleveref}
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\title{Notes on ewjn mag noise}
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\addbibresource{./bibliography.bib}
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\graphicspath{{./figures/}}
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\newcommand{\vf}{v_{\mathrm{F}}}
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\newcommand{\qf}{q_{\mathrm{F}}}
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\begin{document}
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\maketitle
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The reproduction of the magnetic noise in \cite{QubitRelax} is below, in \cref{fig:lukemag}.
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The important point for reproducing the figure in Luke's paper is that I did not use the expression for $r_s$ there.
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Instead, I used eqs 2.20 and 2.26 in Ford-Weber\cite{Ford1984}:
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\begin{align}
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\zeta_s(u) &= 2 i \int_0^\infty \dd{y} \frac{1}{\epsilon_t(\sqrt{u^2 + y^2}) - u^2 - y^2} \\
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r_s(u) &= \frac{\zeta_s(u) - \frac{\pi}{i u}}{\zeta_s(u) + \frac{\pi}{i u}},
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\end{align}
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up to a constant out front which I pulled to outer code.
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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).
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The collision frequency $\nu$ is the odd one, because in the expression for $\epsilon$, the local limit reduces to the Drude case precisely with $\tau = \frac{1}{\nu}$.
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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'').
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That all implies that $\tau = \left(6 \pi \right)^{-1} \times \SI{e-12}{\s}$.
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\begin{figure}[htp]
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\centering
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\includegraphics[width=\linewidth]{lukemagnoise}
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\caption{$T_1(z)$, dotted is the nonlocal case and solid is the local case.} \label{fig:lukemag}
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\end{figure}
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\printbibliography
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\end{document}
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