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## Acknowledgements
# Many thanks to `https://gitignore.io/`, written and maintained by Joe Blau, which contributed much material to this gitignore file.
/.direnv/

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use flake

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output
auto
.direnv

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{
"nodes": {
"flake-utils": {
"locked": {
"lastModified": 1676283394,
"narHash": "sha256-XX2f9c3iySLCw54rJ/CZs+ZK6IQy7GXNY4nSOyu2QG4=",
"owner": "numtide",
"repo": "flake-utils",
"rev": "3db36a8b464d0c4532ba1c7dda728f4576d6d073",
"type": "github"
},
"original": {
"owner": "numtide",
"repo": "flake-utils",
"type": "github"
}
},
"nixpkgs": {
"locked": {
"lastModified": 1678380223,
"narHash": "sha256-HUxnK38iqrX84QdQxbFcosRKV3/koj1Zzp5b5aP4lIo=",
"owner": "NixOS",
"repo": "nixpkgs",
"rev": "1e2590679d0ed2cee2736e8b80373178d085d263",
"type": "github"
},
"original": {
"id": "nixpkgs",
"ref": "nixos-unstable",
"type": "indirect"
}
},
"root": {
"inputs": {
"flake-utils": "flake-utils",
"nixpkgs": "nixpkgs"
}
}
},
"root": "root",
"version": 7
}

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papers/Report/flake.nix Normal file
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{
description = "Typesetting for Report on the Reservoir Engineering";
inputs = {
nixpkgs.url = "nixpkgs/nixos-unstable";
flake-utils.url = "github:numtide/flake-utils";
};
outputs = { self, nixpkgs, flake-utils }:
with flake-utils.lib; eachSystem allSystems (system:
let
pkgs = nixpkgs.legacyPackages.${system};
tex = pkgs.texlive.combine {
inherit (pkgs.texlive) scheme-medium latexmk koma-script babel-english
physics mathtools amsmath fontspec booktabs siunitx caption biblatex float
pgfplots microtype fancyvrb csquotes setspace newunicodechar hyperref
cleveref multirow bbold unicode-math biblatex-phys xpatch beamerposter
type1cm changepage lualatex-math footmisc wrapfig2 curve2e pict2e wrapfig
appendixnumberbeamer sidecap appendix orcidlink ncctools bigfoot crop xcolor;
};
in
rec {
packages = {
document = pkgs.stdenvNoCC.mkDerivation rec {
name = "maxwell_time_scale_separation";
src = self;
buildInputs = [ pkgs.coreutils tex pkgs.biber ];
phases = [ "unpackPhase" "buildPhase" "installPhase" ];
buildPhase = ''
export PATH="${pkgs.lib.makeBinPath buildInputs}";
mkdir -p .cache/texmf-var
mkdir -p output/src
env TEXMFHOME=.cache TEXMFVAR=.cache/texmf-var \
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 \
latexmk ./index.tex
'';
installPhase = ''
mkdir -p $out
cp output/index.pdf $out/
'';
};
};
defaultPackage = packages.document;
devShell = pkgs.mkShellNoCC {
buildInputs = packages.document.buildInputs;
shellHook = ''
export OSFONTDIR=${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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\ProvidesPackage{hiromacros}
\RequirePackage{ifdraft}
% Macros
%% qqgg
\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
%% ppgg
\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma}
%% 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}}
\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}}}
\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}

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\ProvidesPackage{hirostyle}
\usepackage[T1]{fontenc}
% load early
\usepackage[english]{babel}
\usepackage{physics}
\usepackage{graphicx, booktabs, float}
\usepackage[tbtags]{mathtools}
\mathtoolsset{mathic=true}
\usepackage{amssymb}
\usepackage[backend=biber, language=english, style=phys, 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}
\usepackage{bbold}
\usepackage{scrhack}
\usepackage{fontspec}
\usepackage{unicode-math}
\setmainfont[Ligatures={Common,Rare,TeX,Required}]{texgyrepagella}[
Extension = .otf,
UprightFont = *-regular,
BoldFont = *-bold,
ItalicFont = *-italic,
BoldItalicFont = *-bolditalic,
]
\setmathfont{texgyrepagella-math.otf}
\KOMAoptions{DIV=last}
\usepackage[autooneside]{scrlayer-scrpage}
\usepackage[titletoc]{appendix}
\usepackage{bigfoot}
\usepackage{orcidlink}
%\usepackage[cmyk,hyperref]{xcolor}
\usepackage{wrapfig2}
\usepackage{xpatch}
\usepackage{lualatex-math}
\usepackage{manyfoot}
\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

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\documentclass[fontsize=10pt,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
\usepackage{hirostyle}
\usepackage{hiromacros}
\addbibresource{references.bib}
\title{Report on the Reservoir Engineering Efforts}
\date{2023}
\begin{document}
\maketitle
\tableofcontents
\section{Equations of Motion for a Modulated Fiber Loop}
\label{sec:equat-moti-modul}
\subsection{Introduction}
\label{sec:introduction}
To obtain an equation of motion for the electric field that can be
interpreted as a Hamiltonian, we have to reduce the wave equation to
first order in time. Here we work with the paraxial approximation,
ignoring any transverse fields. A more rigorous treatment, to be used
in numeric simulations, can be derived from wave guide theory as
in~\refcite{Yuan2018a,Haus1984}.
We ultimately want to treat a ring modulator with a space and time
dependent refractive index \(n = n_{0} + n_{1}(\vb{r}, t)\) with
\(n_{1}\ll n_{0}\). This situation is close to the case where
\(n_{1}=0\) where the wave equation can be solved by a plane-wave
ansatz.
To capture near-adiabatic deviations from these solutions, we split off
the fast time evolution of the electric field
\begin{equation}
\label{eq:1}
\vb{E}(\vb{r}, t) = \vb{E}_{0}(\vb{r}, t) \eu^{-\i ω t},
\end{equation}
where \(ω\) is as yet undetermined. Now, we \emph{assume} that
\(\dot{\vb{E}}_{0}\sim Ω \cdot\vb{E}_{0}\) with a characteristic
frequency \(Ω \ll ω\). This assumption will have to be verified in the
final result to guarantee consistency. We define the small parameter
\(δ = \frac{Ω}{ω} \ll 1\) for convenience.
For our purposes the magnetic permeability is constant \(μ=μ_{0}\),
whereas the permittivity \(ε(\vb{r}, t)\) is time dependent with
\(\dot{ε} \sim Ω ε\). As we are not taking spatial derivatives of
\(ε\), the spatial argument will be suppressed in the following.
The electric field is real valued, although we do not explicitly take
the real part for now.
\subsection{A Perturbative Maxwell Equation for a Slowy Changing
Envelope}
\label{sec:pert-maxw-equat}
Applying a second curl to the Maxwell equation\footnote{Which is the
canonical way to derive the wave equation.} \(\nabla \times\vb{E} =
- ∂_{t}\vb{{B}}\) leads to
\begin{equation}
\label{eq:2}
\begin{aligned}
\nabla \times (\nabla \times \vb{E}) &= -{\nabla}^{2} \vb{E} =
-∂_{t}\bqty{\mu ∂_{t}\pqty{ε\vb{E}}}=-\mu\pqty{\ddot{ε} \vb{E} + 2
\dot{ε}\dot{\vb{E}} + ε \ddot{\vb{E}}} \\
&= -μ ω^{2} \bqty{\underbrace{-ε\vb{E}_{0}}_{\sim δ^{0}} + \underbrace{2 \frac{\dot{ε}}{ω}
\vb{E}_{0} -
2\iu \frac{ε}{ω}\dot{\vb{E}}_{0}}_{\sim δ^{1}} + \underbrace{2 \frac{\dot{ε}}{ω^{2}}
\dot{\vb{E}}_{0} + \frac{ε}{ω^{2}}\ddot{\vb{E}}_{0}}_{\sim δ^{2}}}\eu^{-\iu ω t}
\end{aligned}
\end{equation}
Up to this point we have not made any approximation. We now proceed to
drop the terms of second order in \(δ\).
This leaves us with
\begin{equation}
\label{eq:3}
\nabla^{2}\vb{E}_{0} = μ ω^{2} \bqty{\pqty{\frac{2\dot{ε}}{ω} - ε}
\vb{E}_{0} - \frac{2 \iu ε}{ω} \dot{\vb{E}}_{0}} = \frac{n^{2} ω^{2}}{c^{2}} \bqty{\pqty{\frac{4\dot{n}}{nω} - 1}
\vb{E}_{0} - \frac{2 \iu }{ω} \dot{\vb{E}}_{0}},
\end{equation}
with \(n=\sqrt{ε μ} c\) which can be rearranged into a form that resembles the
Schr\"odinger equation
\begin{equation}
\label{eq:4}
\iu ∂_{t}\vb{E}_{0} = - \frac{c^{2}}{2 n^{2} ω} \nabla^{2}\vb{E}_{0} +
\frac{1}{2}\pqty{\frac{4\dot{n}}{n}-ω}
\vb{E}_{0}.
\end{equation}
Note however, that the ``mass'' \(ωn^{2}/c^{2}\) in the kinetic term is not
constant, giving rise to non hermitian dynamics. This is an artifact
of neglecting orders of \(δ\). We will find however, that in the
situation investigated in \cref{sec:modul-small-port} the violation of
Hermiticity is negligible.
In contrast to the result in \cite{Dutt2019}, \cref{eq:4} is still
second order in space and, provided \(ε\) is real, hermitian.
\subsection{Application to a Ring Resonator}
\label{sec:appl-ring-reson}
As we are describing a fiber ring of radius \(R\) it is convenient to
work in cylindrical coordinates \((ρ, ϕ, z)\). We are interested in
the electric field at the center of the fiber and assume that it does
not vary much in the transversal directions. Thus, we neglect the
dependence of the field on the \(z\) and \(ρ\) directions and make an
ansatz \(\vb{E}_{0} = E(ϕ, t, ρ=R)\hat{\vb{z}}\).
To satisfy the boundary condition \(E(t, 0) = E(t, 2π)\) \(\forall
t\), we expand the field into a Fourier series
\begin{equation}
\label{eq:6}
E(ϕ, t) = ∑_{m=-∞}^{+∞} C_{m} a_{m}(t) \eu^{\iu m ϕ},
\end{equation}
with \(C_{m}\) chosen appropriately later, so as to make the \(a_{m}\)
dimensionless. Note that to obtain the \(a_{m}\) of \cite{Dutt2019}
one has to make the substitution \(a_{m}(t) \to a_{-m}(t)\).
In the case of a constant refractive index \(n_{0}\), the modes solve
the ordinary wave equations so that
\(a_{m}\sim \eu^{\iu (mϕ - ω_{m}t)}\) with\footnote{This also deviates
from \cite{Dutt2019}. There \(m\geq 0\) and also the negative \(ω\)
solutions are missing. One has to include either one or the other to
capture all solutions. The reason for this lies their explicit
construction of a real solution. However, the fact that frequencies
with a different sign relative to the wave vector exist is not
captured by their first order differential equation.}
\(ω_{m} = \pm \frac{m c}{R n_{0}}\). Applying the procedure of
\cref{sec:pert-maxw-equat}, but to each of the mode amplitudes
\(a_{m}=b_{m}\eu^{-\iu ω_{m}t}\) (with \(ω_{m}\) as yet unspecified)
we find
\begin{equation}
\label{eq:7}
\begin{aligned}
\frac{1}{R^{2}} ∑_{m=-∞}^{+∞} C_{m} b_{m}(t)\eu^{-iω_{m}t} ∂_{ϕ}^{2}\eu^{\iu m ϕ}
&= \frac{-1}{R^{2}} ∑_{m=-∞}^{+∞} C_{m}m^{2} b_{m}(t) \eu^{\iu (m ϕ-ω_{m}t)}\\
&= ∑_{m=-∞}^{+∞}C_{m}\frac{n^{2} ω_{m}^{2}}{c^{2}} \bqty{\pqty{\frac{4\dot{n}}{nω_{m}} - 1}
b_{m}(t) - \frac{2 \iu }{ω_{m}} \dot{b}_{m}(t)}\eu^{\iu (m ϕ-ω_{m}t)}.
\end{aligned}
\end{equation}
In the limit \(\dot{n}\to 0 \implies n(ϕ,t) = n_{0}=\mathrm{const}\)
we should recover \(\dot{b}_{m}=0\), which implies
\begin{equation}
\label{eq:8}
ω_{m}^{2}=\pqty{\frac{mc}{Rn_{0}}}^{2} \implies ω_{m} = \pm
\abs{\frac{mc}{Rn_{0}}},
\end{equation}
which defines the \(ω_{m}\) in this case. However, to correctly
determine the \(ω_{m}\), a slightly more delicate argument has to be
made.
Evaluating the \(∂_{ϕ}\) derivative, rearranging, applying
\((2π)^{-1}\int_{0}^{2π}\dd{ϕ} \eu^{-\iu l ϕ}\) and defining \(n(ϕ,t)
= n_{0} + n_{1}(ϕ, t)\) yields
\begin{equation}
\label{eq:9}
\dot{b}_{l}=-\iu ∑_{m}\bqty{κ_{lm} + γ_{lm}}\eu^{-\iu (ω_{m}-ω_{l}) t}b_{m},
\end{equation}
with
\begin{align}
κ_{lm}&= \frac{C_{m}}{4π
ω_{l}C_{l}}∫_{0}^{2π}\pqty{\frac{m^{2}c^{2}}{n^{2}R^{2}} - ω_{m}^{2}}\eu^{\iu(m-l) ϕ}\dd{ϕ} \overset{\cref{eq:8}}{=}
A_{lm} ∫_{0}^{2π}\pqty{\frac{n_{0}^{2}}{n^{2}} - 1}\eu^{\iu(m-l)ϕ}\dd{ϕ} \label{eq:10}\\
\label{eq:11}
γ_{lm}&= A_{lm}∫_{0}^{2π}\pqty{\frac{4\dot{n}(ϕ, t)}{ω_{m}n(ϕ,
t)}}\eu^{\iu(m-l) ϕ}\dd{ϕ}\\
\label{eq:12}
A_{lm}&=\frac{C_{m}ω_{m}^{2}}{4π
ω_{l}C_{l}}=\frac{1}{4π}\frac{C_{m}}{C_{l}}\frac{m^{2}}{l}\frac{c}{Rn_{0}} =\frac{1}{4π}\frac{C_{m}}{C_{l}}\frac{m^{2}}{l}Ω_{R}
\end{align}
which is a Sch\"odinger equation with the Hamiltonian
\(H_{lm}=\bqty{κ_{lm} + γ_{lm}}\). The denominator of \cref{eq:12} may
be cause for concern in the case that \(l=0\). This would imply
\(ω_{l}=0\) which breaks our assumption \(δ\ll 1\). The sum over \(m\)
in \cref{eq:6} should therefore exclude small \(m\).
Note also that even in the \(\dot{n}=0\) case \cref{eq:10} does not
vanish. The coupling of the modes originates from the (spatially
modulated) deviation of \(n\) from \(n_{0}\). It is also clear, that
the choice of \(n_{1}(ϕ,t)\) has to be made so that
\(\int_{0}^{1}\abs{n_{0}^{2}/n_{1}^{2}-1}\dd{ϕ}\) is minimized to best
approximate the condition \(δ\ll 1\) by minimizing \(κ_{lm}\) and
justify the definition of the \(ω_{m}\). Remember that the \(ω_{m}\)
are still free parameters and have to be chosen so that
\({∂_{t}{b}_{m}}/{b_{m}}\sim δ \ll 1\). In particular
\(n_{1}(ϕ, t) = \mathrm{const}\) is not a valid choice. Preferably,
one should define the \(ω_{m}\) to minimize the \(κ_{lm}\) in
\cref{eq:10}. This also yields the exact solution in the
\(n(ϕ,t)=\mathrm{const}\) case.
For time independent \(n_{1}\) we can find suitable \(ω_{m}\) by
minimizing
\begin{equation}
\label{eq:15}
\bqty{∫_{0}^{2π}\pqty{\frac{m^{2}c^{2}}{n^{2}R^{2}} - ω_{m}^{2}}\dd{ϕ}}^{2}
\end{equation}
giving
\begin{equation}
\label{eq:14}
ω_{m}^{2}= \frac{1}{2π} \frac{m^{2}c^{2}}{R^{2}} ∫_{0}^{2π}
n(ϕ)^{-2}\dd{ϕ}=\pqty{\frac{m c}{R n_{\mathrm{eff}}}}^{2},
\end{equation}
where \(n_{eff}^{-2}=(2π)^{-1}∫_{0}^{2π}n^{-2}\dd{ϕ}\). If \(n_{1}\)
depends on time, one may use the long time average of \cref{eq:14}.
\subsection{Modulation of a small portion of the ring.}
\label{sec:modul-small-port}
We now turn to the case of the modulation of a small angular portion
\(ϕ_{W}\) of the ring. In such a case
\begin{equation}
\label{eq:5}
n=n_{0}+n_{1}(t)\rect\pqty{\frac{ϕ}{ϕ_{W}}},
\end{equation}
where \(\rect(x)=Θ(1/4-x^{2})\) and \(Θ\) is the Heaviside step
function. As the tangential components of the electric field are
continuous, we face no problems on this front.
If we further demand \(\abs{\max_{t} n_{1}(t)}\ll 1\) and \(\lim_{T\to
∞} T^{-1} ∫_{0}^{T}n_{1}(t)\dd{t} = 0\) we can choose
the \(ω_{n}\) as in \cref{eq:8}, as follows from \cref{eq:12}
\begin{equation}
\label{eq:16}
ω_{m}^{2}=\lim_{T\to
∞}\frac{1}{T}∫_{0}^{T}\frac{m^{2}c^{2}}{R^{2}}\bqty{\frac{1}{n_{0}^{2}}+ϕ_{W}\pqty{\frac{1}{n^{2}}-\frac{1}{n_{0}^{2}}}}\dd{t}
\approx \lim_{T\to
∞}\frac{1}{T}∫_{0}^{T}\frac{m^{2}c^{2}}{R^{2}}\bqty{\frac{1}{n_{0}^{2}}
- 2ϕ_{W}\pqty{\frac{n_{1}(t)}{n_{0}}}}\dd{t} =\pqty{\frac{mc}{Rn_{0}}}^{2}.
\end{equation}
To connect to the result in \cite{Dutt2019}, we can then further
evaluate \cref{eq:10}, using
\(C_{m}=\sqrt{\frac{\hbar \abs{ω_{m}}}{4π R ε_{0}n_{0}^{2}}}\) to find
\begin{equation}
\label{eq:17}
κ_{lm}=\frac{Ω_{R}ϕ_{W}\abs{l}^{-\frac{3}{2}}\abs{m}^{\frac{5}{2}}}{4π}
\pqty{\frac{n_{0}^{2}}{(n_{0}+n_{1}(t))^{2}}-1}\sinc\pqty{(m-l)\frac{ϕ_{W}}{2}}\sgn(l).
\end{equation}
Note that here, we introduced an additional sign compared to
\cite{Dutt2019}, and we already transformed to a rotating frame.
A slightly different choice of normalization
\begin{equation}
\label{eq:13}
C_{m}=\frac{1}{m^{2}}\sqrt{\frac{\hbar \abs{ω_{m}}}{4π R
ε_{0}n_{0}^{2}}}
\end{equation}
makes \(κ_{lm}\) hermitian and reproduces the result of
\cite{Dutt2019}
\begin{equation}
\label{eq:18}
\begin{aligned}
κ_{lm}&=\frac{Ω_{R}ϕ_{W}\sqrt{ml}}{4π}
\pqty{\frac{n_{0}^{2}}{(n_{0}+n_{1}(t))^{2}}-1}\sinc\pqty{(m-l)\frac{ϕ_{W}}{2}}\sgn(l)\\
&\approx
\frac{Ω_{R}ϕ_{W}\sqrt{ml}}{2π}
\frac{n_{1}(t)}{n_{0}}\sinc\pqty{(m-l)\frac{ϕ_{W}}{2}}\sgn(l).
\end{aligned}
\end{equation}
As we are interested in the case where \(m,l\gg 1\) and
\(m=M+δm\), \(l=M+δl\) with \(δm,δl\ll M\), the pre-factor of \(κ_{lm}\)
can be regarded as constant either way, guaranteeing hermiticity.
Note that compared to \cite{Dutt2019} we added an additional sign in
\cref{eq:9} and swapped the index \(m,l\) with \(-m,-l\). This leads to
\cref{eq:18} having the same sign as in this publication, rather than
the opposite as one would expect from \cref{eq:9} alone.
This constitutes a rigorous derivation of the result in that paper.
Evaluating \cref{eq:11} yields
\fixme{to be done}.
\subsection{Open Questions}
\label{sec:open-questions}
It would be interesting to compute an explicit expression for the
magnetic field as well. To do so would allow us to check the
continuity for \(μ\vb{H}=\vb{B}\) and to compute the Poynting vector
and give the modes a propagation direction. Of course this direction
can be inferred from the \(n=n_{0}=\mathrm{const}\) case.
It would also be of interest, to find out how good the paraxial
approximation captures the real field at the center of the fiber.
Lastly, the case of time independent \(n\) can likely be solved
exactly fairly easily. It would be interesting to see how this
solution relates to the equation found here.
\section{Engineering Hamiltonians}
\label{sec:engin-hamilt}
Having obtained the basic equations of motion in
\cref{sec:equat-moti-modul}, we now proceed to explore how to engineer
model Hamiltonians out of this equation.
\subsection{Notation and Preliminaries}
\label{sec:notat-prel}
Before we begin to detail procedures to obtain engineered
Hamiltonians, a few notions concerning notation and the general
assumptions will be introduced.
On the physical level, we work with Fourier components \(b_{m}(t)
\eu^{\iu (mϕ-ω_{m}t)}\) of
the electric Field in the ring resonators in an appropriate frame.
The amplitudes \(b_{m}\) can then be identified with a quantum state
by defining
\begin{equation}
\label{eq:19}
\ket{ψ} \equiv ∑_{m} b_{m}\ket{m}
\end{equation}
with \(\ket{m}\) being orthonormal unit vectors
(\(\braket{m}{n}=δ_{mn}\)) in a Hilbert space \(\hilb\). This defines
the fiducial basis in which the state can be measured by recording the
slowly changing envelopes of the modes.
In this language \cref{eq:9} (in the non-rotating frame) becomes
\begin{equation}
\label{eq:20}
\iu ∂_{t}\ket{ψ} = H(t) \ket{ψ},
\end{equation}
with \(H_{mn}(t)= κ_{mn}(t)\eu^{-\iu{ω_{n}-ω_{m}}t}\) where we've neglected the \(γ_{mn}\)
contribution from \cref{eq:11}. Let us also define
\begin{equation}
\label{eq:21}
\begin{aligned}
[D(ω)]_{mn} &\equiv ω_{m} δ_{mn} & κ_{mn}(t) & \equiv Δ_{mn}
\frac{n_{1}(t)}{n_{0}} \equiv Δ_{mn}\, f(t),
\end{aligned}
\end{equation}
where \(n(t) = n_{0} + n_{1}(t)\) with \(n_{1}\ll n_{0}\) as discussed
in \cref{sec:modul-small-port}.
The above may seem rather obvious, but a clarification of conventions
is crucial.
In that vein the Hamiltonian of \cref{eq:20} can be expressed as
\begin{equation}
\label{eq:22}
H = H_{0} + V(t)
\end{equation}
with \(H_{0}=D(ω)\) and \(V(t)=Δ\, f(t)\).
For practical reasons and in order for the assumptions of
\cref{sec:pert-maxw-equat} to hold, we will always work with finite
set of resonator modes, making our effective Hamiltonians finite
dimensional. Further, it is assumed that we stimulate and modulate the
system in such a way, that the boundary conditions in mode space are
not important, so that they can be chosen at our convenience. This
means that we're only concerned with a finite subspace
\(\hilb_{\mathrm{phys}} = \qty{\ket{m} \colon m\in
\bqty{{m_{0}-(N-1)/2},{m_{0} + (N-1)/2}}} \subset \hilb\).
The goal is now to choose the geometry and the modulation so that
the time evolution operator
\begin{equation}
\label{eq:23}
\mathcal{U}_{t}[H] = \mathcal{T}\,\exp(-\iu ∫_{0}^{t}H(t))
\end{equation}
for the Hamiltonian \(H\) of \cref{eq:20} matches the time evolution
operator for some reference Hamiltonian \(H_{\target}\)
\begin{equation}
\label{eq:24}
\norm{\mathcal{U}_{t}[U H U^\dag]-\mathcal{U}_{t}[H_{\target}]} < ε
\iff \norm{\bqty{U_{t}[U H U^†]-U_{t}[H_{\target}]}\ket{ψ}} \leq ε.
\end{equation}
for \(0\leq t\leq T\), where \(\norm{\cdot}\) on the left side is the
operator norm restricted \(\hilb_{\mathrm{phys}}\), \(U\) is some
unitary and \(ε>0\). The unitary \(U\) is allowing for a basis change
relative to the physical basis \cref{eq:19}. We require unitary
equivalence with the same unitary for all times. As
\(\mathcal{U}_{t}[H]\) is invertible, it would be trivial to achieve
perfect equivalence using a time dependent unitary
\begin{equation}
\label{eq:25}
U(t)=\mathcal{U}_{t}[H_{\target}] \mathcal{U}_{t}[H_{\target}]^†.
\end{equation}
As transformations into rotating frames are necessary we have to
loosen the requirement and allow those transformations as well. These
transformations then amount to considering oscillatory linear
combinations of the oscillator mode amplitudes.
This ``problem'' does not occur in \cite{Dutt2019}, as there the
response to a certain input is measured, yielding a band
structure. This requires the actual Floquet energies \emph{in the
rotating frame} to match the eigenenergies of the target
Hamiltonian. But if observables such as the mean displacement of a
walker such as in \refcite{Ricottone2020} are to be computed, the
measured quantity should be the amplitudes, or equivalently the state
\(\ket{ψ}\). These should be obtained directly using only simple
transformations such linear combinations (constant unitaries) and
translation of the signals in frequency space (rotating frame). For if
we knew the detailed dynamics induced \(H_{target}\) already, there
would be little point in trying to simulate them in the first place
and using an elaborate transformation such as \cref{eq:25} would
defeat the point.
It is useful to express the above in terms of an effective Hamiltonian
\begin{equation}
\label{eq:26}
H_{\eff}[H](t)\equiv \frac{1}{-\iu t} \log[U_{t}[H]].
\end{equation}
This Hamiltonian, similar to the Floquet Hamiltonian, has a spectrum
limited by the branch cut of the complex logarithm, which however has
no influence on the dynamics it generates. By continuity of the
operator exponential the closeness
\(\norm{H_{\eff}[H](t) -H_{\eff}[H_{\target}](t)} \leq ε/t\) of the
effective Hamiltonians implies the condition \cref{eq:24}. This
representation lends itself particularly well to visualizations and
numerical studies.
If one is only interested in specific initial states, then one could
specify the less strict requirement, that the evolution of these
specific states should not deviate from the target.
Another possibility to loosen restrictions would be to only demand the
coincidence of the time evolution operators or the effective
Hamiltonians for a specific time \(t\).
\section{A Single Fiber Loop}
\label{sec:single-fiber-loop}
This section mostly follows~\refcite{Dutt2019}, focusing on how to
engineer a one-dimensional tight-binding Hamiltonians with one orbital
per unit cell,
\begin{equation}
\label{eq:27}
H = ∑_{m,n=-M}^{M}t_{mn} \ketbra{m_{0}+m}{m_{0}+n}.
\end{equation}
There we employ periodic boundary conditions to simplify the
calculations, so that \(\ket{m+N} = \ket{m}\) and
\(t_{m,n}=\min_{l\in \ZZ}{\abs{m-n + l N}}\). Here, we choose
\(N=2M +1\) for some \(M\in\NN\) and \(m_{0}\) so, that the relevant
subspace \(\hilb_{phys}\) is contained in it so that states within
this subspace don't ``see'' the boundary at the relevant time scales.
\section{Measuring the State Amplitudes}
\label{sec:meas-state-ampl}
TBD
\section{Motivational Issues}
\label{sec:motiv-probl}
\emph{This section reflects the personal thoughts of Valentin Boettcher and
is correspondingly formulated in the first person.}
Along with the issue concerning arbitrary unitary transformations on
the amplitudes mentioned in \cref{sec:notat-prel}, I also wonder how
we could consider the fiber loop experiment a ``realization'' of a
certain model.
In a puristic sense, it is of course. We have a system that follows
dynamics induced by a linear differential equation resembling the
Schr\"odinger equation. More to the point\footnote{In my eyes.} would
be, to realize a quantum mechanical model in a \emph{physical quantum
mechanical system}. By this, I mean using a system whose dynamics
are given by a Hamiltonian we know and can engineer, so that the
measurement of its \emph{quantum state} matches our engineering
goals. This has been done in \cite{Roushan2014} for example. The
crucial difference to our fiber loop experiment is, that the actual
physics taking place are quantum mechanical in nature. Each
realization of a model in such a system would constitute a test of
quantum mechanics and thus be of interest from a physical
standpoint. It would also constitute a physical implementation of some
(hopefully) interesting behavior, for example Mojorana modes, or
super-conductivity, or some topological effects.
One can of course interject now that light, or any physical system, is
quantum mechanical in nature and by measuring the amplitude and phase
of the resonator modes, we also perform a quantum measurement albeit
neglecting the fluctuations. In that sense however, a numerical simulation
would then also constitute an equally valid ``realization''.
The one merit of the fiber loop setup (or any similar setup) that I
can think of is then to solve problems that are not tractable
analytically or numerically and that are nevertheless of practical
interest. The main goal of this project in the short term should then
be to identify such problems and asses them with respect to their
eligibility for treatment using the analog computer.
I must admit that this prospect seems daunting to me, as I'm lacking
the experience in the field. This of-course is no reason to throw the
towel. It just puts me off.
Working on something that has some chance of turning out to be of no
immediate practical use is also nothing that should frighten a
physicist. The development of quantum mechanics itself answered an
interesting physical questions and much later paid off by giving us
the technology that enables the information age.
Odd enough, I'm lacking the angle to find the question ``Can we coax
the fiber loop setup into simulating the Hamiltonian \(H\)?''
interesting. This is a subjective phenomenon, but how can't it be? Of
course, I can work on it and even achieve progress, but if work is all
I wanted, I could become a programmer and earn a much better living.
My mental machinery is easily blocked by doubt and anxiety. When I am
sitting down to work I am constantly questioning whether I'm doing the
right thing and that leads to situations in which I am too petrified
to think properly or I am taking every opportunity to procrastinate
(such as the Dresden Project and Course Work and Chatting with others
or acquiring stuff on Kiji or dawdling in the morning).
Also, I'm rather afraid to have to present my work in a motivated
manner, e.g. for the prelims and in applications for scholarships.
If this renders me unfit for the pursuit of a PhD in physics, then I
am perfectly happy to accept that. I am not doing it for the title,
but because I thought I would enjoy the experience.
All this complaining is not very constructive, but I hope it
illuminates how I feel and points a way towards the resolution of the
issue.
The next step after reaching the immediate goal of realizing the model
in \refcite{Ricottone2020}, as mentioned above, is to be constructive
and work on scoping out the future work to reduce the uncertainty. I
am currently preparing a list of things that interest me along with
reasons for that interest, but I'm not sure which of these things are
open to me and in which context. Do they have to be related to the
fiber loop setup? Do they have to be related to synthetic dimensions?
What role do open systems play? My hope is that I can rely, at least
in part, on the guidance of more qualified and experience persons to
choose a research direction. Otherwise, the responsibility may be too
much for me.
I may also just be overlooking something that alleviates the issue
discussed at the beginning, which would be the optimal situation.
\printbibliography{}
\end{document}
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