mirror of
https://github.com/vale981/notes_io_loop
synced 2025-03-05 09:31:41 -05:00
add io-theory writeup
This commit is contained in:
commit
e38d137d52
13 changed files with 7161 additions and 0 deletions
1
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/.envrc
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1
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/.envrc
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@ -0,0 +1 @@
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||||||
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use flake
|
||||||
3
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/.gitignore
vendored
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3
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/.gitignore
vendored
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@ -0,0 +1,3 @@
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output
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||||||
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auto
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||||||
|
.direnv
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||||||
60
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/flake.lock
generated
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60
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/flake.lock
generated
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@ -0,0 +1,60 @@
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{
|
||||||
|
"nodes": {
|
||||||
|
"flake-utils": {
|
||||||
|
"inputs": {
|
||||||
|
"systems": "systems"
|
||||||
|
},
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1685518550,
|
||||||
|
"narHash": "sha256-o2d0KcvaXzTrPRIo0kOLV0/QXHhDQ5DTi+OxcjO8xqY=",
|
||||||
|
"owner": "numtide",
|
||||||
|
"repo": "flake-utils",
|
||||||
|
"rev": "a1720a10a6cfe8234c0e93907ffe81be440f4cef",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "numtide",
|
||||||
|
"repo": "flake-utils",
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||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nixpkgs": {
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1686135559,
|
||||||
|
"narHash": "sha256-pY8waAV8K/sbHBdLn5diPFnQKpNg0YS9w03MrD2lUGE=",
|
||||||
|
"owner": "NixOS",
|
||||||
|
"repo": "nixpkgs",
|
||||||
|
"rev": "381e92a35e2d196fdd6077680dca0cd0197e75cb",
|
||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"id": "nixpkgs",
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||||||
|
"ref": "nixos-unstable",
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||||||
|
"type": "indirect"
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||||||
|
}
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||||||
|
},
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||||||
|
"root": {
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||||||
|
"inputs": {
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||||||
|
"flake-utils": "flake-utils",
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||||||
|
"nixpkgs": "nixpkgs"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"systems": {
|
||||||
|
"locked": {
|
||||||
|
"lastModified": 1681028828,
|
||||||
|
"narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=",
|
||||||
|
"owner": "nix-systems",
|
||||||
|
"repo": "default",
|
||||||
|
"rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e",
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||||||
|
"type": "github"
|
||||||
|
},
|
||||||
|
"original": {
|
||||||
|
"owner": "nix-systems",
|
||||||
|
"repo": "default",
|
||||||
|
"type": "github"
|
||||||
|
}
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"root": "root",
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||||||
|
"version": 7
|
||||||
|
}
|
||||||
57
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/flake.nix
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roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/flake.nix
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{
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||||||
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description = "Typesetting for Report on the Reservoir Engineering";
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inputs = {
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||||||
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nixpkgs.url = "nixpkgs/nixos-unstable";
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||||||
|
flake-utils.url = "github:numtide/flake-utils";
|
||||||
|
};
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outputs = { self, nixpkgs, flake-utils }:
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with flake-utils.lib; eachSystem allSystems (system:
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let
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pkgs = nixpkgs.legacyPackages.${system};
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tex = pkgs.texlive.combine {
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inherit (pkgs.texlive) scheme-medium latexmk koma-script babel-english
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||||||
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physics mathtools amsmath fontspec booktabs siunitx caption biblatex float
|
||||||
|
pgfplots microtype fancyvrb csquotes setspace newunicodechar hyperref
|
||||||
|
cleveref multirow bbold unicode-math biblatex-phys xpatch beamerposter
|
||||||
|
type1cm changepage lualatex-math footmisc wrapfig2 curve2e pict2e wrapfig
|
||||||
|
appendixnumberbeamer sidecap appendix orcidlink ncctools bigfoot crop xcolor
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||||||
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acro translations tikzscale xstring pagesel biblatex-ieee;
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||||||
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};
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||||||
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in
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rec {
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packages = {
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document = pkgs.stdenvNoCC.mkDerivation rec {
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name = "reservoir_eng";
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src = self;
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||||||
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buildInputs = [ pkgs.coreutils tex pkgs.biber ];
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||||||
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phases = [ "unpackPhase" "buildPhase" "installPhase" ];
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||||||
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buildPhase = ''
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export PATH="${pkgs.lib.makeBinPath buildInputs}";
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export SOURCE_DATE_EPOCH="${toString self.sourceInfo.lastModified}"
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cd papers/Report/
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mkdir -p .cache/texmf-var
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mkdir -p output/src
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env TEXMFHOME=.cache TEXMFVAR=.cache/texmf-var \
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OSFONTDIR=${pkgs.tex-gyre-math.pagella}/share/fonts/opentype:${pkgs.gyre-fonts}/share/fonts:${pkgs.liberation_ttf}/share/fonts:${pkgs.lato}/share/fonts/lato:${pkgs.raleway}/share/fonts:${pkgs.lmodern}/share/fonts \
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latexmk index.tex
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'';
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installPhase = ''
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mkdir -p $out
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cp output/index.pdf $out/
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'';
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||||||
|
};
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||||||
|
};
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||||||
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defaultPackage = packages.document;
|
||||||
|
devShell = pkgs.mkShellNoCC {
|
||||||
|
buildInputs = packages.document.buildInputs;
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||||||
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shellHook = ''
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export SOURCE_DATE_EPOCH="${toString packages.document.src.lastModified}"
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export OSFONTDIR=${pkgs.tex-gyre-math.pagella}/share/fonts/opentype:${pkgs.gyre-fonts}/share/fonts:${pkgs.liberation_ttf}/share/fonts:${pkgs.lato}/share/fonts/lato:${pkgs.raleway}/share/fonts:${pkgs.lmodern}/share/fonts
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||||||
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'';
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||||||
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};
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||||||
|
});
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||||||
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}
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%% Creator: Inkscape 1.2.2 (b0a8486541, 2022-12-01), www.inkscape.org
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||||||
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%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
|
||||||
|
%% Accompanies image file 'rwa_illustr.pdf' (pdf, eps, ps)
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||||||
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%%
|
||||||
|
%% To include the image in your LaTeX document, write
|
||||||
|
%% \input{<filename>.pdf_tex}
|
||||||
|
%% instead of
|
||||||
|
%% \includegraphics{<filename>.pdf}
|
||||||
|
%% To scale the image, write
|
||||||
|
%% \def\svgwidth{<desired width>}
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||||||
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%% \input{<filename>.pdf_tex}
|
||||||
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%% instead of
|
||||||
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%% \includegraphics[width=<desired width>]{<filename>.pdf}
|
||||||
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%%
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||||||
|
%% Images with a different path to the parent latex file can
|
||||||
|
%% be accessed with the `import' package (which may need to be
|
||||||
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%% installed) using
|
||||||
|
%% \usepackage{import}
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||||||
|
%% in the preamble, and then including the image with
|
||||||
|
%% \import{<path to file>}{<filename>.pdf_tex}
|
||||||
|
%% Alternatively, one can specify
|
||||||
|
%% \graphicspath{{<path to file>/}}
|
||||||
|
%%
|
||||||
|
%% For more information, please see info/svg-inkscape on CTAN:
|
||||||
|
%% http://tug.ctan.org/tex-archive/info/svg-inkscape
|
||||||
|
%%
|
||||||
|
\begingroup%
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||||||
|
\makeatletter%
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||||||
|
\providecommand\color[2][]{%
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||||||
|
\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
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||||||
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\renewcommand\color[2][]{}%
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||||||
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}%
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||||||
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\providecommand\transparent[1]{%
|
||||||
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\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
|
||||||
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\renewcommand\transparent[1]{}%
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}%
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||||||
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\providecommand\rotatebox[2]{#2}%
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||||||
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\newcommand*\fsize{\dimexpr\f@size pt\relax}%
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||||||
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\newcommand*\lineheight[1]{\fontsize{\fsize}{#1\fsize}\selectfont}%
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||||||
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\ifx\svgwidth\undefined%
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||||||
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\setlength{\unitlength}{407.26289993bp}%
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||||||
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\ifx\svgscale\undefined%
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||||||
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\relax%
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||||||
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\else%
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||||||
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\setlength{\unitlength}{\unitlength * \real{\svgscale}}%
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||||||
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\fi%
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||||||
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\else%
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||||||
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\setlength{\unitlength}{\svgwidth}%
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||||||
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\fi%
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||||||
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\global\let\svgwidth\undefined%
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||||||
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\global\let\svgscale\undefined%
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||||||
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\makeatother%
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||||||
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\begin{picture}(1,0.0809482)%
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||||||
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\lineheight{1}%
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||||||
|
\setlength\tabcolsep{0pt}%
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||||||
|
\put(0,0){\includegraphics[width=\unitlength,page=1]{rwa_illustr.pdf}}%
|
||||||
|
\put(0.24904088,0.07454558){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m-1}+\lambda_{m-1}$\end{tabular}}}}%
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||||||
|
\put(0.18442909,0.00097159){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m-1}$\end{tabular}}}}%
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||||||
|
\put(0.48263967,0.00088302){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m}$\end{tabular}}}}%
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||||||
|
\put(0.5211689,0.07032182){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}{\fontsize{4pt}{1em}$\tilde{G'}_m(\omega)$}\end{tabular}}}}%
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||||||
|
\put(0.04112692,0.07454558){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m-1}-\lambda_{m-1}$\end{tabular}}}}%
|
||||||
|
\put(0.89070949,0.07640942){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m+1}+\lambda_{m+1}$\end{tabular}}}}%
|
||||||
|
\put(0.82978077,0.00283543){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m+1}$\end{tabular}}}}%
|
||||||
|
\put(0.71226032,0.07640942){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m+1}-\lambda_{m+1}$\end{tabular}}}}%
|
||||||
|
\put(0.55351781,0.00097159){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m}+\lambda_{m}$\end{tabular}}}}%
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||||||
|
\put(0.36770248,0.00097159){\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\omega_{m}-\lambda_{m}$\end{tabular}}}}%
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||||||
|
\end{picture}%
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||||||
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\endgroup%
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After Width: | Height: | Size: 22 KiB |
184
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hiromacros.sty
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\ProvidesPackage{hiromacros}
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\RequirePackage{ifdraft}
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||||||
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||||||
|
% Macros
|
||||||
|
|
||||||
|
%% qqgg
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||||||
|
\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
|
||||||
|
|
||||||
|
%% ppgg
|
||||||
|
\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma}
|
||||||
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|
||||||
|
%% Momenta and Polarization Vectors convenience
|
||||||
|
\DeclareMathOperator{\ps}{\slashed{p}}
|
||||||
|
|
||||||
|
\DeclareMathOperator{\pe}{\varepsilon}
|
||||||
|
\DeclareMathOperator{\pes}{\slashed{\pe}}
|
||||||
|
|
||||||
|
\DeclareMathOperator{\pse}{\varepsilon^{*}}
|
||||||
|
\DeclareMathOperator{\pses}{\slashed{\pe}^{*}}
|
||||||
|
|
||||||
|
%% Spinor convenience
|
||||||
|
\DeclareMathOperator{\us}{u}
|
||||||
|
\DeclareMathOperator{\usb}{\bar{u}}
|
||||||
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|
||||||
|
\DeclareMathOperator{\vs}{v}
|
||||||
|
\DeclareMathOperator*{\vsb}{\overline{v}}
|
||||||
|
|
||||||
|
%% Center of Mass energy
|
||||||
|
\DeclareMathOperator{\ecm}{E_{\text{CM}}}
|
||||||
|
|
||||||
|
%% area hyperbolicus
|
||||||
|
\DeclareMathOperator{\artanh}{artanh}
|
||||||
|
\DeclareMathOperator{\arcosh}{arcosh}
|
||||||
|
|
||||||
|
%% rectangle
|
||||||
|
\DeclareMathOperator{\rect}{rect}
|
||||||
|
|
||||||
|
%% sinc
|
||||||
|
\DeclareMathOperator{\sinc}{sinc}
|
||||||
|
|
||||||
|
%% sign
|
||||||
|
\DeclareMathOperator{\sgn}{sgn}
|
||||||
|
|
||||||
|
|
||||||
|
%% Fast Slash
|
||||||
|
\let\sl\slashed
|
||||||
|
|
||||||
|
%% hermitian/complex conjugate
|
||||||
|
\newcommand{\cc}{\ensuremath{\mathrm{c.c.}}}
|
||||||
|
\newcommand{\hc}{\ensuremath{\mathrm{h.c.}}}
|
||||||
|
|
||||||
|
%% eulers number
|
||||||
|
\def\eu{\ensuremath{\operatorname{e}}}
|
||||||
|
|
||||||
|
%% Notes on Equations
|
||||||
|
\newcommand{\shorteqnote}[1]{ & & \text{\small\llap{#1}}}
|
||||||
|
|
||||||
|
%% Typewriter Macros
|
||||||
|
\newcommand{\sherpa}{\texttt{Sherpa}}
|
||||||
|
\newcommand{\rivet}{\texttt{Rivet}}
|
||||||
|
\newcommand{\vegas}{\texttt{VEGAS}}
|
||||||
|
\newcommand{\lhapdf}{\texttt{LHAPDF6}}
|
||||||
|
\newcommand{\scipy}{\texttt{scipy}}
|
||||||
|
|
||||||
|
%% Sherpa Versions
|
||||||
|
\newcommand{\oldsherpa}{\texttt{2.2.10}}
|
||||||
|
\newcommand{\newsherpa}{\texttt{3.0.0} (unreleased)}
|
||||||
|
|
||||||
|
%% Special Names
|
||||||
|
\newcommand{\lhc}{\emph{LHC}}
|
||||||
|
|
||||||
|
%% Expected Value and Variance
|
||||||
|
\newcommand{\EX}[1]{\operatorname{E}\qty[#1]}
|
||||||
|
\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]}
|
||||||
|
|
||||||
|
%% Uppercase Rho
|
||||||
|
\newcommand{\Rho}{P}
|
||||||
|
|
||||||
|
%% Transverse Momentum
|
||||||
|
\newcommand{\pt}[0]{p_\mathrm{T}}
|
||||||
|
|
||||||
|
%% Sign Function
|
||||||
|
\DeclareMathOperator{\sign}{sgn}
|
||||||
|
|
||||||
|
%% Stages
|
||||||
|
\newcommand{\stone}{\texttt{LO}}
|
||||||
|
\newcommand{\sttwo}{\texttt{LO+PS}}
|
||||||
|
\newcommand{\stthree}{\texttt{LO+PS+pT}}
|
||||||
|
\newcommand{\stfour}{\texttt{LO+PS+pT+Hadr.}}
|
||||||
|
\newcommand{\stfive}{\texttt{LO+PS+pT+Hadr.+MI}}
|
||||||
|
|
||||||
|
%% GeV
|
||||||
|
\newcommand{\gev}[1]{\SI{#1}{\giga\electronvolt}}
|
||||||
|
|
||||||
|
\def\iu{\ensuremath{𝑖}}
|
||||||
|
\def\i{\iu}
|
||||||
|
\def\id{\ensuremath{\mathbb{1}}}
|
||||||
|
\def\NN{\ensuremath{\mathbb{N}}}
|
||||||
|
\def\RR{\ensuremath{\mathbb{R}}}
|
||||||
|
\def\CC{\ensuremath{\mathbb{C}}}
|
||||||
|
\def\ZZ{\ensuremath{\mathbb{Z}}}
|
||||||
|
\def\dim{\ensuremath{\operatorname{dim}}}
|
||||||
|
\def\hilb{\ensuremath{\mathcal{H}}}
|
||||||
|
|
||||||
|
% fixme
|
||||||
|
\newcommand{\fixme}[1]{\marginpar{\tiny\textcolor{red}{#1}}}
|
||||||
|
|
||||||
|
% HOPS/NMQSD
|
||||||
|
\def\sys{\ensuremath{\mathrm{S}}}
|
||||||
|
\def\bath{\ensuremath{\mathrm{B}}}
|
||||||
|
\def\inter{\ensuremath{\mathrm{I}}}
|
||||||
|
\def\nth{\ensuremath{^{(n)}}}
|
||||||
|
\def\target{\ensuremath{\mathrm{target}}}
|
||||||
|
\def\eff{\ensuremath{\mathrm{eff}}}
|
||||||
|
|
||||||
|
\newcommand{\mat}[1]{\ensuremath{{\underline{\vb{#1}}}}}
|
||||||
|
\def\kmat{{\mat{k}}}
|
||||||
|
|
||||||
|
|
||||||
|
% Thermo
|
||||||
|
\newcommand{\ergo}[1]{\ensuremath{\mathcal{W}\qty[#1]}}
|
||||||
|
\newcommand{\qrelent}[2]{\ensuremath{S\qty(#1\,\middle|\middle|\,#2)}}
|
||||||
|
\newcommand{\cyc}{\ensuremath{\mathrm{cyc}}}
|
||||||
|
|
||||||
|
% time evolution operator
|
||||||
|
\newcommand{\tevop}[2][t]{\ensuremath{\mathcal{U}_{#1}\bqty{#2}}}
|
||||||
|
|
||||||
|
% effective hamiltonian
|
||||||
|
\newcommand{\heff}[2][t]{\ensuremath{H_{\eff}\bqty{#2}(#1)}}
|
||||||
|
\makeatletter
|
||||||
|
\newsavebox\myboxA
|
||||||
|
\newsavebox\myboxB
|
||||||
|
\newlength\mylenA
|
||||||
|
|
||||||
|
\newcommand*\xoverline[2][0.75]{%
|
||||||
|
\sbox{\myboxA}{$\m@th#2$}%
|
||||||
|
\setbox\myboxB\null% Phantom box
|
||||||
|
\ht\myboxB=\ht\myboxA%
|
||||||
|
\dp\myboxB=\dp\myboxA%
|
||||||
|
\wd\myboxB=#1\wd\myboxA% Scale phantom
|
||||||
|
\sbox\myboxB{$\m@th\overline{\copy\myboxB}$}% Overlined phantom
|
||||||
|
\setlength\mylenA{\the\wd\myboxA}% calc width diff
|
||||||
|
\addtolength\mylenA{-\the\wd\myboxB}%
|
||||||
|
\ifdim\wd\myboxB<\wd\myboxA%
|
||||||
|
\rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA}%
|
||||||
|
\else
|
||||||
|
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB}%
|
||||||
|
\fi}
|
||||||
|
\makeatother
|
||||||
|
|
||||||
|
\DeclareMathOperator{\bosedist}{\xoverline{n}}
|
||||||
|
\DeclareDocumentCommand\bose{}{\opbraces{\bosedist}}
|
||||||
|
|
||||||
|
%% Including plots
|
||||||
|
% \newcommand{\plot}[1]{%
|
||||||
|
% \ifdraft{\includegraphics[draft=false]{./figs/#1.pdf}}{\input{./figs/#1.pgf}}}
|
||||||
|
\newcommand{\plot}[1]{%
|
||||||
|
\includegraphics[draft=false]{./figs/#1.pdf}}
|
||||||
|
\newcommand{\tval}[1]{{\input{./values/#1.tex}}}
|
||||||
|
|
||||||
|
%% citing "in ref"
|
||||||
|
\NewBibliographyString{refname}
|
||||||
|
\NewBibliographyString{refsname}
|
||||||
|
\DefineBibliographyStrings{english}{%
|
||||||
|
refname = {Ref\adddot},
|
||||||
|
refsname = {Refs\adddot}
|
||||||
|
}
|
||||||
|
|
||||||
|
\DeclareCiteCommand{\refcite}
|
||||||
|
{%
|
||||||
|
\ifnum\thecitetotal=1
|
||||||
|
\bibstring{refname}%
|
||||||
|
\else%
|
||||||
|
\bibstring{refsname}%
|
||||||
|
\fi%
|
||||||
|
\addspace\bibopenbracket%
|
||||||
|
\usebibmacro{cite:init}%
|
||||||
|
\usebibmacro{prenote}}
|
||||||
|
{\usebibmacro{citeindex}%
|
||||||
|
\usebibmacro{cite:comp}}
|
||||||
|
{}
|
||||||
|
{\usebibmacro{cite:dump}%
|
||||||
|
\usebibmacro{postnote}%
|
||||||
|
\bibclosebracket}
|
||||||
101
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hirostyle.sty
Normal file
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|
|
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|
||||||
|
\ProvidesPackage{hirostyle}
|
||||||
|
\usepackage[T1]{fontenc}
|
||||||
|
% load early
|
||||||
|
\usepackage{wrapfig2}
|
||||||
|
\usepackage[english]{babel}
|
||||||
|
\usepackage{physics}
|
||||||
|
\usepackage{graphicx, booktabs, float}
|
||||||
|
\usepackage[tbtags]{mathtools}
|
||||||
|
\mathtoolsset{mathic=true}
|
||||||
|
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage[backend=biber, language=english, style=ieee, sorting=none, backref=true]{biblatex}
|
||||||
|
\usepackage{siunitx}
|
||||||
|
\usepackage{caption}
|
||||||
|
\usepackage{sidecap}
|
||||||
|
\usepackage[list=true, font=footnotesize, labelformat=brace]{subcaption}
|
||||||
|
\usepackage[protrusion=true,expansion=true,tracking=true]{microtype}
|
||||||
|
\usepackage{fancyvrb}
|
||||||
|
\usepackage{longtable}
|
||||||
|
\usepackage{booktabs}
|
||||||
|
\usepackage[autostyle=true]{csquotes}
|
||||||
|
\usepackage{setspace}
|
||||||
|
\usepackage{newunicodechar}
|
||||||
|
\usepackage[pdfencoding=auto,hidelinks,colorlinks=true,
|
||||||
|
linkcolor=blue,
|
||||||
|
filecolor=blue,
|
||||||
|
citecolor = black,
|
||||||
|
urlcolor=cyan,]{hyperref} % load late
|
||||||
|
\usepackage[capitalize]{cleveref}
|
||||||
|
\usepackage{multirow,tabularx}
|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
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|
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|
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|
||||||
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|
||||||
|
]
|
||||||
|
\setmathfont{texgyrepagella-math.otf}
|
||||||
|
\KOMAoptions{DIV=last}
|
||||||
|
\usepackage[autooneside]{scrlayer-scrpage}
|
||||||
|
\usepackage[titletoc]{appendix}
|
||||||
|
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|
||||||
|
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|
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|
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|
||||||
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|
||||||
|
|
||||||
|
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|
||||||
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|
||||||
|
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|
||||||
|
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|
||||||
|
\usepackage[bottom]{footmisc}
|
||||||
|
\raggedbottom
|
||||||
|
|
||||||
|
\DeclareNewFootnote{default} % standard footnotes
|
||||||
|
\DeclareNewFootnote{URL}[roman] % href footnotes
|
||||||
|
\let\oldhref\href
|
||||||
|
\renewcommand{\href}[2]{\oldhref{#1}{#2}\footnoteURL{\url{#1}}}
|
||||||
|
|
||||||
|
%% Including Results
|
||||||
|
\newcommand{\result}[1]{\input{./results/#1}\!}
|
||||||
|
|
||||||
|
%% SI units
|
||||||
|
\sisetup{separate-uncertainty = true}
|
||||||
|
|
||||||
|
%% Captions
|
||||||
|
\captionsetup{font=small,format=plain}
|
||||||
|
\captionsetup[sub]{font=small,format=plain}
|
||||||
|
|
||||||
|
%% Labels
|
||||||
|
% \labelformat{chapter}{chapter~#1}
|
||||||
|
% \labelformat{section}{section~#1}
|
||||||
|
% \labelformat{figure}{figure~#1}
|
||||||
|
% \labelformat{table}{table~#1}
|
||||||
|
|
||||||
|
%% Cleverref
|
||||||
|
% \crefname{equation}{}{}
|
||||||
|
% \creflabelformat{equation}{(#2#1#3)}
|
||||||
|
|
||||||
|
%% Font for headings
|
||||||
|
\addtokomafont{disposition}{\rmfamily}
|
||||||
|
|
||||||
|
%% Minus Sign for Matplotlib
|
||||||
|
\newunicodechar{−}{-}
|
||||||
|
|
||||||
|
% Allow math page breaks
|
||||||
|
\allowdisplaybreaks
|
||||||
|
|
||||||
|
% cursive bold in maths
|
||||||
|
\unimathsetup{math-style=TeX,bold-style=ISO}
|
||||||
|
|
||||||
|
\recalctypearea
|
||||||
|
|
||||||
|
% aligned oversets, must be loaded last
|
||||||
|
\usepackage{aligned-overset}
|
||||||
|
|
||||||
|
% large integrals
|
||||||
|
\everydisplay{\Umathoperatorsize\displaystyle=5ex}
|
||||||
538
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex
Normal file
538
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex
Normal file
|
|
@ -0,0 +1,538 @@
|
||||||
|
\documentclass[fontsize=11pt,paper=a4,open=any,
|
||||||
|
twoside=no,toc=listof,toc=bibliography,headings=optiontohead,
|
||||||
|
captions=nooneline,captions=tableabove,english,DIV=15,numbers=noenddot,final,parskip=half-,
|
||||||
|
headinclude=true,footinclude=false,BCOR=0mm]{scrartcl}
|
||||||
|
\pdfvariable suppressoptionalinfo 512\relax
|
||||||
|
\synctex=1
|
||||||
|
|
||||||
|
\author{Valentin Boettcher}
|
||||||
|
\usepackage{hirostyle}
|
||||||
|
\usepackage{hiromacros}
|
||||||
|
\addbibresource{references.bib}
|
||||||
|
|
||||||
|
\title{Input-Output Theory for Modulated Fibre-Loops}
|
||||||
|
\date{2023}
|
||||||
|
\graphicspath{{graphics}}
|
||||||
|
|
||||||
|
\newcommand{\inputf}[0]{\ensuremath{\mathrm{in}}}
|
||||||
|
\newcommand{\outputf}[0]{\ensuremath{\mathrm{out}}}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
\maketitle
|
||||||
|
\tableofcontents
|
||||||
|
|
||||||
|
\section{Microscopic Derivation}
|
||||||
|
\label{sec:micr-deriv}
|
||||||
|
The setup we are describing consists of a general driven photonic
|
||||||
|
system \(A\) and a transmission line \(B\). The \(A\) system is
|
||||||
|
considered to have the Hamiltonian
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:1}
|
||||||
|
H_{A}=H_{0}+V(t) = ∑_{m} ω^{0}_{m} c_{m}^†c_{m} + V(t),
|
||||||
|
\end{equation}
|
||||||
|
where we are working in the basis that diagonalizes \(H_{0}\) and the
|
||||||
|
\(c_{m}\) are linear combinations of the bare modes in the photonic
|
||||||
|
system. We designate the bare modes of the EM field that are actually
|
||||||
|
in contact with the transmission line as \(a_{β}\) with
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:4}
|
||||||
|
E_{A}(x,t)= \iu \sqrt{\frac{\hbar}{2ε_{0}n_{A}^{2} L_{A,\perp}^{2} L_{A}}} ∑_{β}
|
||||||
|
\sqrt{ω_{k_β}} \pqty{a_{β}(t)
|
||||||
|
\eu^{\iu k_{β} x } - a_{β}^†(t) \eu^{-\iu k_{β} x}},
|
||||||
|
\end{equation}
|
||||||
|
where \(L_{A,\perp}\) is a length scale that can be interpreted as the
|
||||||
|
diameter of the transmission line~\cite{Jacobs} and \(L_{A}\) is the
|
||||||
|
length of the cavity/resonator that hosts the electric field. The
|
||||||
|
modes have wave numbers \(k_{β} = 2πβ/L_{A}\) for \(β \in \ZZ\) and
|
||||||
|
frequencies \(ω_{k_β} = c \abs{k_{β}}/n_{A}\), where \(n_{A}\) is the
|
||||||
|
refractive index inside the cavity. For simplicity we set \(\hbar
|
||||||
|
= 1\) such that time is measured in units of inverse energy.
|
||||||
|
|
||||||
|
The bare \(a_{β}\) modes are linear combinations of the \(c_{m}\) and
|
||||||
|
can be related through
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:5}
|
||||||
|
a_{β} = ∑_{m} U_{βm} c_{m},
|
||||||
|
\end{equation}
|
||||||
|
where \(U\) is a not necessarily square matrix that obeys the
|
||||||
|
unitarity relation \(U U^† = \id\).
|
||||||
|
Transitioning into a rotating frame with respect to \(H_{0}\) and
|
||||||
|
employing the rotating wave approximation removes the time dependence
|
||||||
|
from the interaction
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:12}
|
||||||
|
\tilde{c}(t)_{m} = c_{m}(t)\eu^{\iu ω^{0}_{m}t} \implies H_{A} \to \tilde{H}_{A}=
|
||||||
|
∑_{mn}V_{mn}(t) \eu^{-\iu t (ω^{0}_{n}-ω^{0}_{m})}
|
||||||
|
\tilde{c}_{m}^†\tilde{c}_{n} \approx ∑_{mn}V^{0}_{mn} \tilde{c}_{m}^†\tilde{c}_{n}.
|
||||||
|
\end{equation}
|
||||||
|
We can subsequently find a unitary transformation that diagonalizes
|
||||||
|
the RWA interaction
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:30}
|
||||||
|
∑_{mn}\pqty{O^{†}}_{im}V_{mn}^{0}O_{nj} = ω_{j} δ_{ij},
|
||||||
|
\end{equation}
|
||||||
|
where the columns of \(O\) are the normalized eigenvectors of
|
||||||
|
\(V_{mn}^{0}=\mel{m}{V^{0}}{n}\). So if \(\ket{ψ_{j}}\) is an
|
||||||
|
eigenvector of \(V\), then \(\braket{i}{ψ_{j}} = O_{ij}\)
|
||||||
|
\footnote{This is just a reminder for Valentin who can't seem to keep
|
||||||
|
this in his head.}.
|
||||||
|
|
||||||
|
Transforming the \(\tilde{c}_{m}\) according to
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:13}
|
||||||
|
d_{γ} = ∑_{n}O^{\ast}_{nγ} \tilde{c}_{n} \implies \tilde{H}_{A} = ∑_{γ}ω_{γ} d_{γ}^†d_{γ}
|
||||||
|
\end{equation}
|
||||||
|
leaves us with a very simple Hamiltonian.
|
||||||
|
Let us list the relation between the \(a\), \(c\) and \(d\) operators
|
||||||
|
for later reference
|
||||||
|
\begin{align}
|
||||||
|
\label{eq:15}
|
||||||
|
c_{n} &= \eu^{-\iu
|
||||||
|
ω^{0}_{n} t}\tilde{{c}}_{n} = \eu^{-\iu
|
||||||
|
ω^{0}_{n} t} ∑_{γ} O_{nγ}d_{γ} \\
|
||||||
|
a_{β} &= ∑_{m} U_{β,m} c_{m} = ∑_{m} U_{β,m} \tilde{c}_{m}\eu^{-\iu
|
||||||
|
ω^{0}_{m} t} = ∑_{mγ} U_{β,m} \eu^{-\iu
|
||||||
|
ω^{0}_{m} t} O_{mγ}d_{γ} \equiv ∑_{γ}\mathcal{U}(t)_{β,γ} d_{γ}.
|
||||||
|
\end{align}
|
||||||
|
|
||||||
|
|
||||||
|
The transmission line is considered to only
|
||||||
|
have one polarization direction and one dimension of
|
||||||
|
propagation, so that the vector potential is effectively scalar and we
|
||||||
|
have
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:2}
|
||||||
|
E_{B}(x, t) = \iu\sqrt{\frac{\hbar}{2ε_{0}n_{B}^{2}
|
||||||
|
(2π)^{3}L_{B,\perp}^{2}}} ∫{\sqrt{ω^{B}_{k}}} \pqty{b_{k}(t)
|
||||||
|
\eu^{\iu k x } - b_{k}^†(t) \eu^{-\iu k x}}\dd{k},
|
||||||
|
\end{equation}
|
||||||
|
with \(\comm{b_{k}}{b_{q}^†}=δ(k-q)\), \(ω^{B}_{k} = c \abs{k}/n_{B}\)
|
||||||
|
with \(n_{B}\) being the refractive index of the fibre and
|
||||||
|
\(L_{B,\perp}\) being the perpendicular length scale as discussed
|
||||||
|
above. Note that the \(b_{k}\) here have dimensions of \(\sqrt{[L]}\)
|
||||||
|
as opposed to \(\sqrt{[t]}\), as is the usual convention in
|
||||||
|
input-output theory. If a stochastic theory is desired, the latter
|
||||||
|
convention is preferrable and can be obtained through substituting
|
||||||
|
\(k\to \pm ω/c n_{B}\) and rescaling
|
||||||
|
\(b_{k}\to b_{k}/ \sqrt{c n_{B}^{-1}}\).
|
||||||
|
|
||||||
|
A phenomenological interaction between the transmission line and the
|
||||||
|
system \(A\) roughly inspired by coupled mode theory is
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:3}
|
||||||
|
H_{I} = g_{0} ∫ E_{A,+}(x,t)E_{B,-}(x,t) f(x) \dd{x} + \hc,
|
||||||
|
\end{equation}
|
||||||
|
where the subscripts \(\pm\) denote positive or negative frequency
|
||||||
|
portions of the fields and \(f(x)\) is a dimensionless weighting
|
||||||
|
function with compact support \([-Δx/2, Δx/2]\) whose maximum is
|
||||||
|
unity. Coupling only the positive/negative parts simplifies the
|
||||||
|
calculations and is consistent with the later application of the
|
||||||
|
rotating wave approximation. A possible phase shift between the fields
|
||||||
|
has been absorbed into the definition of the creation and annihilation
|
||||||
|
operators.
|
||||||
|
|
||||||
|
Expanding the fields in \cref{eq:3} we obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:6}
|
||||||
|
H_{I} = {g_{0}} \frac{\hbar Δx}{2 ε_{0}n_{A}n_{B} (2π)^{3}
|
||||||
|
L_{A,\perp}L_{B,\perp}\sqrt{L_{A}}} ∑_{β}∫
|
||||||
|
\sqrt{ω^{B}_{k}ω_{k_{β}}}\,\tilde{f}(k-k_{β})\, b^†_{k}
|
||||||
|
a_{β} \dd{k} + \hc
|
||||||
|
\end{equation}
|
||||||
|
The Fourier transform of the weighting function
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:7}
|
||||||
|
\tilde{f}(k) = \frac{1}{Δx} ∫f(x)\eu^{-\iu k x} \dd{x}
|
||||||
|
\end{equation}
|
||||||
|
controls how ``far'' the interaction reaches in \(k\)-space. In the
|
||||||
|
extreme case \(Δx\to 0\) every \(b_{k}\) couples to every \(a_{β}\),
|
||||||
|
whereas for \(Δx\to ∞\) only modes with matching wave-numbers
|
||||||
|
couple. As the \(b_{k}\) will contain both the coherent drive with a
|
||||||
|
laser and the output field amplitudes it is desirable to have this
|
||||||
|
coupling to be as local in \(k\)-space as possible for targeted
|
||||||
|
control and precise readout. In the limit of weak coupling between
|
||||||
|
transmission line and system, which we will assume in a short while,
|
||||||
|
the rotating wave approximation will ensure that our result won't
|
||||||
|
depend significantly on the choice of \(f\).
|
||||||
|
|
||||||
|
The coupling constant \(g_{0}\) in \cref{eq:6} has the dimensions of
|
||||||
|
\([L]^{2}\times [ε_{0}]\). We define a new coupling constant that has
|
||||||
|
units of energy as
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:8}
|
||||||
|
g_{0} = g\frac{n_{A}n_{B}ε_{0} L_{A,\perp}L_{B,\perp} 2(2π)^{3}}{\hbar ω_{0}},
|
||||||
|
\end{equation}
|
||||||
|
where \(ω_{0}\) is a typical frequency\footnote{For example, the
|
||||||
|
frequency of the drive laser.}.
|
||||||
|
Using this, \cref{eq:6} becomes
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:9}
|
||||||
|
\begin{aligned}
|
||||||
|
H_{I} &= \frac{gΔx}{
|
||||||
|
\sqrt{L_{A}}} ∑_{β}∫
|
||||||
|
G_{β}(k) b^†_{k}
|
||||||
|
a_{β} \dd{k} + \hc
|
||||||
|
&G_{β}(k) &= \frac{\sqrt{ω^{B}_{k}ω_{k_β}}}{ω_{0}} \tilde{f}(k-k_{β}).
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
We note that for
|
||||||
|
a \(ω_{k_{β}}= ω_{0} + δω\) with \(δω \ll ω_{0}\) the coupling factor
|
||||||
|
\(G_{β}(k)\) only depends on the difference \(k-k_{β}\). By defining
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:11}
|
||||||
|
\mathcal{O(k)} = \frac{Δx}{\sqrt{L_{A}}} ∑_{β} G_β(k)a_{β} =
|
||||||
|
\frac{Δx}{\sqrt{L_{A}}} ∑_{β,m} G_β(k)U_{β,m}c_{m}
|
||||||
|
\end{equation}
|
||||||
|
the interaction takes on the more familiar form
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:14}
|
||||||
|
H_{I} = {g} ∫
|
||||||
|
b^†_{k} \mathcal{O}(k)
|
||||||
|
\dd{k} + \hc
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Changing variables from \(k\) to\footnote{This is a bit
|
||||||
|
unconventional.} \(ω^{B}_{k}=k c / n_{B}\) in
|
||||||
|
\cref{eq:9} we obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:17}
|
||||||
|
H_{I} = \frac{gΔx}{\sqrt{L_{A}}} ∑_{β}∫_{-∞}^{∞}
|
||||||
|
G'_{β}(ω)f^†_{ω}
|
||||||
|
{a_{β}} \dd{ω} + \hc,
|
||||||
|
\end{equation}
|
||||||
|
where \(f_{ω}=\sqrt{\frac{n_{B}}{c}}b_{\frac{ω n_{B}}{c}}\) with
|
||||||
|
\(\comm{f_{ω}}{f_{ω'}^†}=δ(ω-ω')\) and
|
||||||
|
\(G'_{β}(ω)=G_{β}\pqty{\frac{ω n_{B}}{c}}\).
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Rotating Wave and First Markov Approximation}
|
||||||
|
\label{sec:rotating-wave-first}
|
||||||
|
Following the route taken in \cite{Jacobs}, the next step would be to
|
||||||
|
transition into a rotating frame so that
|
||||||
|
\(\tilde{H}_{A}=\tilde{H}_{B}=0\) and apply the rotating wave
|
||||||
|
approximation. Here, the rotating terms that would occur have the
|
||||||
|
frequencies of the form \(ω^{0}_m + ω_{γ}\) which are not guaranteed
|
||||||
|
to be spaced sufficiently far apart for the RWA to
|
||||||
|
apply\footnote{Consider, for example the SSH model where the
|
||||||
|
\(k\)-space density can be arbitrarily high depending on the length
|
||||||
|
of the chain.}. We therefore work in the frame of the
|
||||||
|
\(\tilde{c}_{m}\) and \(\tilde{f}_{ω} = f_{ω}\eu^{\iu \abs{ω}t}\) to
|
||||||
|
obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:10}
|
||||||
|
\tilde{H}_{I}= \frac{gΔx}{
|
||||||
|
\sqrt{L_{A}}} ∑_{β,m}∫
|
||||||
|
G'_{β}(ω) \eu^{-\iu
|
||||||
|
(ω^{0}_{m}-\abs{ω}) t}
|
||||||
|
U_{β,m} \tilde{f}_k^†\tilde{c}_{m} \dd{ω} + \hc
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\begin{figure}[H]
|
||||||
|
\centering
|
||||||
|
{\fontsize{8pt}{1em}
|
||||||
|
\input{graphics/rwa_illustr.pdf_tex}}
|
||||||
|
\caption{\label{fig:rwa_illustr} In the rotating wave approximation
|
||||||
|
The bare frequencies of the resonator only couple to the
|
||||||
|
transmission line in frequency sub-intervals
|
||||||
|
\([ω_{m}-λ_{m}, ω_{m}+λ_{m}]\). A second effect that comes into
|
||||||
|
play is the geometrically induced coupling amplitude \(\tilde{G'}_{m}(ω)\),
|
||||||
|
which is visualized around \(ω_{m}\) under the assumption \(ω_{β}
|
||||||
|
\approx ω_{0}^{m}\) for some small range of \(m\).}
|
||||||
|
\end{figure}
|
||||||
|
For \(g \ll ω_{m}^{0}\) each \(\tilde{c}_{m}\) in \cref{eq:10} only
|
||||||
|
interacts with non-overlapping sub-intervals
|
||||||
|
\([ω^{0}_{m}-λ_{m}, ω^{0}_{m}+λ_{m}]\) of the transmission frequency axis
|
||||||
|
(rotating wave approximation) with \(g\ll λ_{m} \ll ω_{m}^{0}\). This
|
||||||
|
situation is illustrated in \cref{fig:rwa_illustr}. Also, the coupling
|
||||||
|
amplitude \(G_{β}(ω)\) is local in frequency space and can assist the
|
||||||
|
RWA depending on the choice of parameters and how close the
|
||||||
|
\(ω^{0}_{m}\) are to the \(ω_{k_{β}}\). We obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:16}
|
||||||
|
\tilde{H}_{I}\approx \frac{gΔx}{
|
||||||
|
\sqrt{L_{A}}} ∑_{β,m}∫_{ω^{0}_{m}-λ_{m}}^{ω^{0}_{m}+λ_{m}}
|
||||||
|
\eu^{-\iu
|
||||||
|
(ω^{0}_{m}-\abs{ω}) t}
|
||||||
|
U_{β,m} \pqty{G'_{β}(ω) \tilde{f}_{ω}^† + G'_{β}(-ω)
|
||||||
|
\tilde{f}_{-ω}^†}\tilde{c}_{m} \dd{ω} + \hc
|
||||||
|
\end{equation}
|
||||||
|
For any finite \(Δx\) and
|
||||||
|
\(ω_{0}^{m},ω_{k_{β}}\gg \frac{2πc}{Δx n_{A}}\) we can assume
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:44}
|
||||||
|
G'_{β}\pqty{-\sgn(β) ω}\approx 0
|
||||||
|
\end{equation}
|
||||||
|
in \cref{eq:16}.
|
||||||
|
|
||||||
|
As each \(\tilde{c}_{m}\) is now interacting with non-overlapping
|
||||||
|
transmission-line field modes, we can introduce a separate field for
|
||||||
|
each \(\tilde{c}_{m}\) that commutes with all other fields and extend
|
||||||
|
the integration bounds to infinity again\footnote{This is called the
|
||||||
|
``First Markov Approximation'' in \refcite{Gardiner1985}.}.
|
||||||
|
Care has to be taken to maintain consistency with \cref{eq:44},
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:16}
|
||||||
|
\tilde{H}_{I}= \frac{gΔx}{
|
||||||
|
\sqrt{L_{A}}} ∑_{β,m}∫_{0}^{∞}
|
||||||
|
\eu^{-\iu
|
||||||
|
(ω^{0}_{m}-\abs{ω}) t}
|
||||||
|
U_{β,m} G'_{β}(\sgn({β})ω) \tilde{f}^{m,†}_{\sgn({β})ω}{c}_{m} \dd{ω} + \hc
|
||||||
|
\end{equation}
|
||||||
|
which becomes\footnote{A lot of discussion for a simple result :).}
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:18}
|
||||||
|
H_{I}= ∑_{m}∫_{-∞}^{∞}
|
||||||
|
\tilde{G}_{m}(k) {b}^{m,†}_{k}{c}_{m} \dd{k}
|
||||||
|
\end{equation}
|
||||||
|
upon transitioning out of the rotating frame with \(\tilde{G}_{m}(k) =
|
||||||
|
\frac{gΔx}{
|
||||||
|
\sqrt{L_{A}}} ∑_{β\gtrless 0}U_{β,m} G_{β}(k)δ_{\sgn(β),\sgn(k)}\). The equation of motion
|
||||||
|
for the transmission line modes become
|
||||||
|
\begin{gather}
|
||||||
|
\iu\dot{b}^{m}_{k} = ω^{B}_{k} b_{k}^{m,†} +
|
||||||
|
\tilde{G}_{m}(k) c_{m}\\
|
||||||
|
\label{eq:19}
|
||||||
|
\implies b^{m}_{k}(t) = b^{m}_{k}(0) \eu^{-\iu ω_{k}^{B}t} -\iu
|
||||||
|
\tilde{G}_{m}(k) ∫_{0}^{t}\eu^{-\iu
|
||||||
|
ω_{k}^{B}(t-s)} c_{m}(s)\dd{s}.
|
||||||
|
\end{gather}
|
||||||
|
The equation of motion for \(\tilde{c}_{m}\) is
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:21}
|
||||||
|
\iu\dot{\tilde{c}}_{m} = ∑_{n}V^{0}_{mn} \tilde{c}_n +
|
||||||
|
\underbrace{\eu^{\iu ω_{m}^{0}t}∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||||||
|
b_{k}^{m}(t)\dd{k}}_{\equiv I}.
|
||||||
|
\end{equation}
|
||||||
|
Further inspection of the rightmost term in \cref{eq:21} yields
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:22}
|
||||||
|
\begin{aligned}
|
||||||
|
I &= \eu^{\iu ω_{m}^{0}t} ∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||||||
|
b_{k}^{m}(t)\dd{k} \\
|
||||||
|
&= ∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||||||
|
b_{k}^{m}(0)\eu^{-\iu (ω^{B}_{k} - ω^{0}_{m})t}\dd{k} -\iu ∫_{0}^{t}∫_{-∞}^{∞}\abs{\tilde{G}_{m}(k)}^{2}
|
||||||
|
\tilde{c}_{m}(s)\eu^{-\iu ω^{B}_{k}(t-s)} \eu^{\iu
|
||||||
|
ω^{0}_{m}(t-s)}\dd{k}\dd{s}\\
|
||||||
|
&=II + III.
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
Inspired by the RWA, we now assume
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:23}
|
||||||
|
\begin{aligned}
|
||||||
|
\tilde{G}_{m}(k) &\approx
|
||||||
|
δ_{m}\tilde{G}_{m}\pqty{\sgn(k) ω_{m}^{0}\frac{n_{B}}{c n_{A}}} =
|
||||||
|
δ_{m}\frac{gΔx}{\sqrt{L_{A}}}∑_{β}U_{βm}G_{β}\pqty{\sgn(k) ω_{m}^{0}\frac{n_{B}}{c
|
||||||
|
n_{A}}} δ_{\sgn(β),\sgn(k)} \\
|
||||||
|
&\equiv ∑_{β}g^{0}_{β} U_{βm}δ_{\sgn(β),\sgn(k)} \equiv g_{m, \sgn(k)}
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
in the interval \([ω^{0}_{m}-λ_{m}, ω^{0}_{m}+λ_{m}]\) (see
|
||||||
|
\cref{eq:16}) where \(δ_{m}\) is a possible scaling factor to better approximate
|
||||||
|
\(\tilde{G}_{m}(k)\) as a constant in \cref{eq:16}.
|
||||||
|
|
||||||
|
Additionally we resurrect\footnote{Within
|
||||||
|
the RWA this is all equivalent, but I prefer having the input field
|
||||||
|
proportional to the electric field!} the \(ω_{k}^{B}\) dependence of
|
||||||
|
\(G_{m}(k)\) in \(I\) to obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:24}
|
||||||
|
\begin{aligned}
|
||||||
|
II &= \frac{\eu^{\iu ω_{m}^{0}t}}{\sqrt{ω_{m}^{0}}} \bqty{g_{m,+}^\ast ∫_{0}^{∞}\sqrt{ω^{B}_{k}}b^{m}_{k}(0)\eu^{-\iu
|
||||||
|
ω^{B}_{k}t}\dd{k} + g_{m,-}^\ast∫_{-∞}^{0}\sqrt{ω^{B}_{k}}b^{m}_{k}(0)\eu^{-\iu
|
||||||
|
ω^{B}_{k}t}\dd{k}}\\
|
||||||
|
&\equiv \frac{\eu^{\iu ω_{m}^{0}t}}{\sqrt{ω_{m}^{0}}}\pqty{
|
||||||
|
g_{m,+}^\ast b_{\inputf,+}^{m}(t) + g_{m,-}^\ast b_{\inputf,-}^{m}(t)},
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
where \(b_{\inputf,+(-)}^{m}(t)\) is identified as the
|
||||||
|
right(left)-moving input field and is proportional to the annihilation
|
||||||
|
part of the electric field. The second part of \cref{eq:22} becomes
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:25}
|
||||||
|
III= -\iu ∫_{0}^{t}\eu^{\iu ω^{0}_{m}(t-s)}\tilde{c}_{m}(s)
|
||||||
|
\bqty{ \abs{g_{m,+}}^{2} ∫_{0}^{∞}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k} + \abs{g_{m,-}}^{2} ∫_{-∞}^{0}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k}}\dd{s}.
|
||||||
|
\end{equation}
|
||||||
|
Now we use the identity
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:26}
|
||||||
|
∫_{0}^{∞}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k} = \frac{n_{B}}{c}
|
||||||
|
\bqty{\mathcal{P}\frac{-i}{t-s} + π δ(t-s)},
|
||||||
|
\end{equation}
|
||||||
|
but neglect the principal value, as it leads only to rapidly
|
||||||
|
oscillating terms that are inconsistent with the RWA, to obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:27}
|
||||||
|
III= -2\iu η_{m}∫_{0}^{t}\eu^{\iu ω^{0}_m(t-s)}\tilde{c}_{m}(s)
|
||||||
|
δ(t-s)\dd{s} = -\iu η_{m} \tilde{c}_{m}(t),
|
||||||
|
\end{equation}
|
||||||
|
where the factor \(1/2\) in the last equality stems from the fact that
|
||||||
|
we only use half of the delta function and
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:45}
|
||||||
|
η_{m}\equiv π\frac{n_{B}}{c}\bqty{\abs{g_{m,-}}^{2}+\abs{g_{m,+}}^{2}}.
|
||||||
|
\end{equation}
|
||||||
|
Note that \cref{eq:45} is an incoherent sum of the couplings to the
|
||||||
|
right moving and left moving fields in the transmission line.
|
||||||
|
Altogether we arrive at
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:28}
|
||||||
|
\dot{\tilde{c}}_{m} = -\iu\bqty{∑_{n}V^{0}_{mn} \tilde{c}_n +
|
||||||
|
\frac{\eu^{\iu ω_{m}^{0}t}}{\sqrt{ω_{m}^{0}}}
|
||||||
|
\pqty{
|
||||||
|
g_{m,+}^\ast b_{\inputf,+}^{m}(t) + g_{m,-}^\ast b_{\inputf,-}^{m}(t)}} - η_{m}\tilde{c}_{m}(t).
|
||||||
|
\end{equation}
|
||||||
|
The usual situation is that \(b^{m}_{\inputf, -} = 0\) and we can
|
||||||
|
restrict ourselves to the coupling to the right-moving input field.
|
||||||
|
|
||||||
|
\subsection{Input-Output Relation and further Simplifications}
|
||||||
|
\label{sec:input-outp-relat}
|
||||||
|
Integrating \cref{eq:19} over all \(k\) yields
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:29}
|
||||||
|
\begin{aligned}
|
||||||
|
\frac{b_{\outputf}^{m}(x,t)}{\sqrt{ω^{0}_{m}}} &\equiv
|
||||||
|
\frac{1}{\sqrt{ω_{m}^{0}}}∫ \sqrt{ω^{B}_{k}} b_{k}^{m}(t) \eu^{\iu k
|
||||||
|
t}\dd{k}\\
|
||||||
|
&=
|
||||||
|
\frac{1}{\sqrt{ω_{m}^{0}}} b_{\inputf}^{m}(x, t) -\iu
|
||||||
|
g_{m,\sgn(x)}\frac{π n_{B}}{c}
|
||||||
|
\tilde{c}_{m}(τ(x,t))\eu^{-i ω^{0}_{m}τ(x,t)}Θ(τ(x,t)),
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
which is the input-output relation with the retarded time
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:20}
|
||||||
|
τ(x,t)=t - \frac{\abs{x}n_{B}}{c}.
|
||||||
|
\end{equation}
|
||||||
|
The coupling constant accounts for the direction of propagation and
|
||||||
|
the time argument is properly retarded. We defined
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:48}
|
||||||
|
b_{\inputf}^{m}(x,t) = ∫\sqrt{ω^{B}_{k}} b_{k}^{m}(0)\eu^{\iu \pqty{kx -
|
||||||
|
ω_{k}^{B}t}}\dd{k}
|
||||||
|
\end{equation}
|
||||||
|
used that
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:42}
|
||||||
|
∫_{0}^{∞}\eu^{-\iu ω^{B}_{k}(t-s)}\eu^{\pm\iu k x} \dd{k} =
|
||||||
|
\frac{n_{B}}{c}
|
||||||
|
\bqty{\mathcal{P}\frac{-i}{t-s \pm \frac{x n_{B}}{c}} + π δ\pqty{t-s\mp
|
||||||
|
\frac{x n_{B}}{c}}}.
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
The case of \(x=0\) is recovered by defining
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:47}
|
||||||
|
\lim_{x\to0} g_{m,\sgn(x)=0} = \frac{1}{2} \pqty{g_{m,+} + g_{m,-}},
|
||||||
|
\end{equation}
|
||||||
|
which amounts to taking half of each delta function in
|
||||||
|
\cref{eq:42}. It shall be noted, that it is physical to assume
|
||||||
|
\(x>0\), as we necessarily measure outside the fibre-coupler between
|
||||||
|
transmission line and resonator. By neglecting the \(k\)-depnedence of
|
||||||
|
the coupling in \cref{eq:23} through invocation of the RWA we have
|
||||||
|
effectively ignored length \(Δx\), but to maintain consistency with
|
||||||
|
\cref{eq:44} we should assume it to be finite.
|
||||||
|
We can also neglect the retardation if \(x / v_{g}\) is
|
||||||
|
much smaller than a typical timescale we're interested in.
|
||||||
|
|
||||||
|
|
||||||
|
To integrate \cref{eq:28}, we
|
||||||
|
first diagonalize \(V^{0}_{mn}\)
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:32}
|
||||||
|
\dot{d}_{γ} = ∑_{m}O^{\ast}_{mγ}\dot{\tilde{c}}_{m} = -\iu\bqty{ω_{ω_{γ}}d_{γ} +
|
||||||
|
∑_{σ=\pm}∑_{m}O^{\ast}_{mγ}\frac{g_{m,σ}^\ast }{\sqrt{ω_{m}^{0}}} \eu^{\iu ω_{m}^{0}t}
|
||||||
|
b_{\inputf,σ}^{m}(t)} - π∑_{m}O^{\ast}_{mγ}η_{m}\tilde{c}_{m}(t).
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
We now introduce some additional simplifications. As the coupling to
|
||||||
|
the transmission line is likely not the only source of loss it is
|
||||||
|
justified to replace \(η_{m}\) with a constant \(η\) as the simplest
|
||||||
|
choice. Further, we equate all input fields \(b_{\inputf}^{m}\). This
|
||||||
|
is allowed, as we will transition to the classical picture later,
|
||||||
|
where the commutation relations do not matter. We also assume that
|
||||||
|
we're working in a region in \(m\) space, where the
|
||||||
|
\(g_{β}^{0}\approx \sqrt{κ}\) and
|
||||||
|
\(\sqrt{ω^{0}_{m}}\approx\sqrt{ω_{0}}\), where \(ω_{0}\) is a typical
|
||||||
|
frequency in the input field, can be assumed to be approximately
|
||||||
|
constant. With these considerations in mind we can simplify
|
||||||
|
\cref{eq:32} to
|
||||||
|
\begin{gather}
|
||||||
|
\label{eq:34}
|
||||||
|
\dot{d}_{γ} = ∑_{m}O^{\ast}_{mγ}\dot{\tilde{c}}_{m} =
|
||||||
|
-\iu\bqty{{ω_{γ}}d_{γ} + \sqrt{κ} ∑_{σ=\pm}
|
||||||
|
U^{\pm}_{γ}(t) \frac{b_{\inputf}(t)}{\sqrt{ω_{0}}}} - η d_{γ}\\
|
||||||
|
U^{σ}_{γ}(t) = ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}O^\ast_{mγ} \eu^{\iu ω_{m}^{0}t}
|
||||||
|
\end{gather}
|
||||||
|
These simplifications still capture the essence of the physics, as
|
||||||
|
demonstrated in the current long-range SSH experiment.
|
||||||
|
|
||||||
|
We can now proceed to integrate \cref{eq:34} to obtain
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:36}
|
||||||
|
d_{γ}(t)= d_{γ}(0) \eu^{-\pqty{\iu ω_{γ} + η}t} -
|
||||||
|
\frac{i}{\sqrt{κ}} Σ_{σ=\pm} ∫_{0}^{t}χ_{γ}(t-s) U^{σ}_{γ}(s)
|
||||||
|
\frac{b_{\inputf,σ}(t)}{\sqrt{ω_{0}}}\dd{s}
|
||||||
|
\end{equation}
|
||||||
|
with
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:37}
|
||||||
|
χ_{γ}(t) = κ \eu^{-\pqty{\iu ω_{γ} + η}t}.
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
When constructing the total output field, we have to remember how the
|
||||||
|
separate fields \(b_{\outputf,m}\) came about. We assumed that each
|
||||||
|
\(c_{m}\) only interacted with a finite range of modes (see
|
||||||
|
\cref{eq:16}) in the transmission line and then just extended the
|
||||||
|
resulting sub-fields back to full independent fields for
|
||||||
|
simplicity. Now, we have to perform the reverse process, which amounts
|
||||||
|
to summing together all system (resonator) contributions in
|
||||||
|
\cref{eq:34} as these only excite the sub-fields and we can safely
|
||||||
|
glue them back together. To be consistent, we have to sum together the
|
||||||
|
finite ranges of the input fields which amounts to having \emph{one}
|
||||||
|
whole copy of the input field.
|
||||||
|
This leads us to
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:38}
|
||||||
|
\frac{b_{\outputf}(x,t)}{\sqrt{ω_{0}}} \equiv
|
||||||
|
\frac{1}{\sqrt{ω_{0}}} b_{\inputf}(x, t) -i θ(τ(x,t)) \frac{\sqrt{κ}πn_{B}}{c}
|
||||||
|
∑_{γ}\bqty{U^{\sgn(x)}_{γ}\pqty{τ(x,t)}}^\ast d_{γ}(τ(x,t))
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Transitioning to expectation values and using \(\ev{d_{γ}(0)}=0\) we
|
||||||
|
find
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:39}
|
||||||
|
\ev{{b_{\outputf}(x,t)}} =
|
||||||
|
\ev{b_{\inputf}(x,t)} - ∑_{σ=\pm}∫_{0}^{τ(x,t)}χ_{σ,\sgn(x)}(τ(x,t),s) \ev{b_{\inputf,σ}(s)} \dd{s}
|
||||||
|
\end{equation}
|
||||||
|
with the time non-local susceptibility for the left and right moving
|
||||||
|
input fields
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:40}
|
||||||
|
χ_{σ,δ}(t,s) = \frac{π n_{B}}{c}Θ(t) ∑_{γ}\pqty{U^{δ}_{γ}(t)}^\astχ_{γ}(t-s)U^{σ}_{γ}(s).
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
For an input field with no left-moving components and a measurement
|
||||||
|
position \(x>0\) we have
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:31}
|
||||||
|
\ev{{b_{\outputf}(x>0,t)}} =
|
||||||
|
\ev{b_{\inputf}(x,t)} -∫_{0}^{τ(x,t)}χ_{++}(τ(x,t),s) \ev{b_{\inputf}(s)} \dd{s}.
|
||||||
|
\end{equation}
|
||||||
|
with \(b_{\inputf}(s) = b_{\inputf,+}(s) + b_{\inputf,-}(s) = b_{\inputf,+}(s)\).
|
||||||
|
|
||||||
|
|
||||||
|
\newpage
|
||||||
|
\printbibliography{}
|
||||||
|
\end{document}
|
||||||
|
|
||||||
|
|
||||||
|
%%% Local Variables:
|
||||||
|
%%% mode: latex
|
||||||
|
%%% TeX-master: t
|
||||||
|
%%% TeX-output-dir: "output"
|
||||||
|
%%% TeX-engine: luatex
|
||||||
|
%%% jinx-languages: "en_CA"
|
||||||
|
%%% End:
|
||||||
|
|
@ -0,0 +1,5 @@
|
||||||
|
$pdf_mode = 4;
|
||||||
|
@default_files = ('index.tex');
|
||||||
|
$out_dir = 'output';
|
||||||
|
$pdf_previewer = "zathura %O %S";
|
||||||
|
$lualatex = 'lualatex %O --shell-escape %S';
|
||||||
2891
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/references.bib
Normal file
2891
roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/references.bib
Normal file
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Add table
Reference in a new issue