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small fixes in the io writeup

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Valentin Boettcher 2023-06-15 15:53:50 -04:00
parent d806dcb996
commit fdd1ab515e
1 changed files with 9 additions and 10 deletions
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roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex
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@ -17,8 +17,8 @@ headinclude=true,footinclude=false,BCOR=0mm]{scrartcl}
\newcommand{\inputf}[0]{\ensuremath{\mathrm{in}}} \newcommand{\inputf}[0]{\ensuremath{\mathrm{in}}}
\newcommand{\outputf}[0]{\ensuremath{\mathrm{out}}} \newcommand{\outputf}[0]{\ensuremath{\mathrm{out}}}
\usetikzlibrary{math} \usetikzlibrary{math}
\usetikzlibrary{external} % \usetikzlibrary{external}
\tikzexternalize[prefix=tikz/] % \tikzexternalize[prefix=tikz/]
\usepackage{pgfplots} \usepackage{pgfplots}
\begin{document} \begin{document}
@ -97,7 +97,7 @@ Transforming the \(h_{m}\) according to
where where
\begin{equation} \begin{equation}
\label{eq:35} \label{eq:35}
O_{nγ}(t)\equiv O_{nγ}\eu^{\iu ε_{n}t} O^\ast_{nγ}(t)\equiv O^\ast_{nγ}\eu^{-\iu ε_{n}t}
\end{equation} \end{equation}
leaves us with a very simple Hamiltonian. leaves us with a very simple Hamiltonian.
@ -465,9 +465,9 @@ first diagonalize \(V^{0}_{mn} + δ_{mn}\pqty{ε_{m}-i η_{m}}\)
to obtain \(O_{mγ}(t)\) and find to obtain \(O_{mγ}(t)\) and find
\begin{equation} \begin{equation}
\label{eq:32} \label{eq:32}
\dot{d}_{γ} = ∑_{m}O^{\ast}_{mγ}\dot{\tilde{c}}_{m} = \dot{d}_{γ} = ∑_{m}\pqty{O^{-1}(t)}_{γm}\dot{\tilde{c}}_{m} =
-\iu\bqty{\pqty{ω_{γ} - \iu \tilde{η}_{γ}}d_{γ} + -\iu\bqty{\pqty{ω_{γ} - \iu \tilde{η}_{γ}}d_{γ} +
∑_{σ=\pm}∑_{m}O^{\ast}_{mγ}(t)\frac{g_{m,σ}^\ast }{\sqrt{ω_{m}^{0}}} \eu^{\iu ω_{m}^{0}t} ∑_{σ=\pm}∑_{m}\pqty{O^{-1}(t)}_{γm}\frac{g_{m,σ}^\ast }{\sqrt{ω_{m}^{0}}} \eu^{\iu ω_{m}^{0}t}
b_{\inputf,σ}^{m}(t)}. b_{\inputf,σ}^{m}(t)}.
\end{equation} \end{equation}
@ -487,10 +487,10 @@ constant. With these considerations in mind we can simplify
and and
\begin{gather} \begin{gather}
\label{eq:34} \label{eq:34}
\dot{d}_{γ} = ∑_{m}O^{\ast}_{mγ}\dot{\tilde{c}}_{m} = \dot{d}_{γ} =
-\iu\bqty{\pqty{ω_{γ}-\iu \tilde{η}_{γ}}d_{γ} + \sqrt{κ} ∑_{σ=\pm} -\iu\bqty{\pqty{ω_{γ}-\iu \tilde{η}_{γ}}d_{γ} + \sqrt{κ} ∑_{σ=\pm}
U^{\pm}_{γ}(t) \frac{b_{\inputf}(t)}{\sqrt{ω_{0}}}}\\ U^{\pm}_{γ}(t) \frac{b_{\inputf}(t)}{\sqrt{ω_{0}}}}\\
U^{σ}_{γ}(t) = ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}O^\ast_{mγ}(t) \eu^{\iu ω_{m}^{0}t}= ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}O^\ast_{mγ} \eu^{\iu (ω_{m}^{0}-ε_{m})t}. U^{σ}_{γ}(t) = ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}(t)}_{γm} \eu^{\iu ω_{m}^{0}t}= ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}}_{γm}\eu^{\iu (ω_{m}^{0}-ε_{m})t}.
\end{gather} \end{gather}
These simplifications still capture the essence of the physics, as These simplifications still capture the essence of the physics, as
@ -729,7 +729,7 @@ To maximize the residual rotating terms, the minimum of the
Δ_{\max}\equiv \max_{δ}Δ_{\min}(δ) = \max_{δ}\min\Bqty{2δ, \abs{Ω_{B}-δ}, \abs{Ω_{B}-3δ}, Δ_{\max}\equiv \max_{δ}Δ_{\min}(δ) = \max_{δ}\min\Bqty{2δ, \abs{Ω_{B}-δ}, \abs{Ω_{B}-3δ},
\abs{2Ω_{B}-3δ}, \abs{Ω_{B}-2δ}}. \abs{2Ω_{B}-3δ}, \abs{Ω_{B}-2δ}}.
\end{equation} \end{equation}
We find that \(Δ_{\max}=Ω_{B}/4\) for We find that \(Δ_{\max}=2Ω_{B}/5\) for
\begin{equation} \begin{equation}
\label{eq:63} \label{eq:63}
δ_{\mathrm{opt}}=Ω_{B}/5, δ_{\mathrm{opt}}=Ω_{B}/5,
@ -748,7 +748,6 @@ as can be ascertained from \cref{fig:delta_choice}.
xtick = {0}, ytick = \empty, xtick = {0}, ytick = \empty,
clip = false, clip = false,
xtick={},ytick={}, xtick={},ytick={},
tick num = 10,
minor tick num=5, minor tick num=5,
grid=both, grid=both,
grid style={line width=.1pt, draw=gray!10}, grid style={line width=.1pt, draw=gray!10},
@ -760,7 +759,7 @@ as can be ascertained from \cref{fig:delta_choice}.
y label style={at={(axis description cs:-0.06,.5)},rotate=90,anchor=south}, y label style={at={(axis description cs:-0.06,.5)},rotate=90,anchor=south},
] ]
\addplot[domain = 0:1, restrict y to domain = 0:1, samples = \addplot[domain = 0:1, restrict y to domain = 0:1, samples =
1000, color = cerulean]{min(2*x, 1-x, abs(1-3*x), abs(2-3*x), 1000]{min(2*x, 1-x, abs(1-3*x), abs(2-3*x),
abs(1-2*x))}; abs(1-2*x))};
\addplot[color = black, mark = *, only marks, mark size = 3pt] \addplot[color = black, mark = *, only marks, mark size = 3pt]
coordinates {(.2, .4)}; coordinates {(.2, .4)};