From f888305cb5a92376eb040c4a77d3563949061ef6 Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Wed, 14 Jun 2023 11:59:35 -0400 Subject: [PATCH] add some notes on non-markov. quantum walk --- .../hirostyle.sty | 17 +- .../index.tex | 241 +++++++++++++++++- 2 files changed, 250 insertions(+), 8 deletions(-) diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hirostyle.sty b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hirostyle.sty index 293576a..445da17 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hirostyle.sty +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hirostyle.sty @@ -33,12 +33,12 @@ linkcolor=blue, \usepackage{fontspec} \usepackage{unicode-math} \setmainfont[Ligatures={Common,Rare,TeX,Required}]{texgyrepagella}[ - Extension = .otf, - UprightFont = *-regular, - BoldFont = *-bold, - ItalicFont = *-italic, - BoldItalicFont = *-bolditalic, - ] +Extension = .otf, +UprightFont = *-regular, +BoldFont = *-bold, +ItalicFont = *-italic, +BoldItalicFont = *-bolditalic, +] \setmathfont{texgyrepagella-math.otf} \KOMAoptions{DIV=last} \usepackage[autooneside]{scrlayer-scrpage} @@ -99,3 +99,8 @@ linkcolor=blue, % large integrals \everydisplay{\Umathoperatorsize\displaystyle=5ex} + +% working setminus +\AtBeginDocument{% to do this after unicode-math has done its work + \renewcommand{\setminus}{\mathbin{\backslash}}% +} diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex index 488eac6..33fb794 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex @@ -1,6 +1,6 @@ \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-, +captions=nooneline,captions=tableabove,english,DIV=12,numbers=noenddot,final,parskip=half-, headinclude=true,footinclude=false,BCOR=0mm]{scrartcl} \pdfvariable suppressoptionalinfo 512\relax \synctex=1 @@ -10,12 +10,16 @@ headinclude=true,footinclude=false,BCOR=0mm]{scrartcl} \usepackage{hiromacros} \addbibresource{references.bib} -\title{Input-Output Theory for Modulated Fibre-Loops} +\title{Input-Output Theory for Modulated Optical Fibre Resonators} \date{2023} \graphicspath{{graphics}} \newcommand{\inputf}[0]{\ensuremath{\mathrm{in}}} \newcommand{\outputf}[0]{\ensuremath{\mathrm{out}}} +\usetikzlibrary{math} +\usetikzlibrary{external} +\tikzexternalize[prefix=tikz/] +\usepackage{pgfplots} \begin{document} \maketitle @@ -537,6 +541,239 @@ position \(x>0\) we have \end{equation} with \(b_{\inputf}(s) = b_{\inputf,+}(s) + b_{\inputf,-}(s) = b_{\inputf,+}(s)\). +\section{Application to the Non-Markovian Quantum Walk} +\label{sec:appl-non-mark} +The experimental setup for implementing the non-Markovian quantum walk +discussed in~\cite{Ricottone2020,Kitagawa2010} is illustrated in +\cref{fig:schematic}. The abstract system introduced in +\cref{sec:micr-deriv} is replaced by a small \(S\) and a large \(B\) +fibre loop with lengths \(L_{B}\gg L_{S}\). The resonant modes of the +loops do have the free spectral ranges +\begin{equation} + \label{eq:41} + \begin{aligned} + Ω_{B} &= \frac{2πc}{n_{B}} & Ω_{S} &= \frac{2πc}{L_{S}}, + \end{aligned} +\end{equation} +where \(n\) is the refractive index of the respective fibres. +\begin{figure}[htp] + \centering + \includegraphics{walker_setup} + \caption{\label{fig:schematic}Schematic of the experimental setup.} +\end{figure} +Attaching the transmission line to the smaller loop has the advantage +that we only excite and detect what we will later identify as the +\(A\) level of the non-Markovian quantum walk in momentum space. + +We assume that the loops share some common eigenfrequency +\begin{equation} + \label{eq:46} + ω_{s} = n_{0}^{S}Ω_{S} = n_{0}^{B}Ω_{B}\iff \frac{n_{0}^{B}}{L_{B}} + = \frac{n_{0}^{S}}{L_{S}} +\end{equation} +with \(n,m\gg 1\) and \(ω_{s}\) close to the frequency of the +laser. We choose to index the eigenfrequencies of the loops relative +to \(ω_{s}\) so that +\begin{equation} + \label{eq:49} + \begin{aligned} + ω_{n}^{X} &= ω_{s} + n Ω_{X} & k_{n}^{X} = \frac{2π}{L_{B}} + n_{0}^{X} + \frac{2π}{L_{B}} n = k_{0} + \frac{2π}{L_{B}} n, + \end{aligned} +\end{equation} +where \(X=S,B\). + +These loops are coupled to each other with amplitude \(δ\eu^{\iu ϕ}\) which can +possibly contain a phase due to the choice of the coordinate +origin. Let us now assume that +\begin{equation} + \label{eq:50} + \frac{Ω_{S}}{Ω_{B}} = 2N+1 +\end{equation} +with \(N\in \NN\). We denote by \(a\) the annihilation operator for the +mode with \(ω=ω_{s}\) in the small loop and suppress all other modes +in the small loop, as we will not populate them. The annihilation +operators \(f_{n}\) destroy modes with frequencies \(ω^{B}_{n}\) in +the big loop, where we limit \(n\) to the range \([-N, N]\) for the +same reasons as above. +This leads to the Hamiltonian +\begin{equation} + \label{eq:51} + H_{A}= ω_{s}a^†a + ∑_{n=-N}^{N} ω_{n}^{B}f_{n}^†f_{n} + δ \pqty{\eu^{\iu + ϕ}f_{0}^†a + \eu^{-\iu ϕ}a^†f_{0}} + ∑_{mn}J_{mn}(t) f_{m}^†f_{n}, +\end{equation} +where \(J_{mn}(t)\) is the coupling mediated by the EOM in the big +loop. The phase \(ϕ=k_{0}L_{S}/2\) of the coupling stems from the +location of the fibre coupler along the loop but can potentially also +contain other contributions. + +To be bring the Hamiltonian into the form \cref{eq:1}, we define +\begin{equation} + \label{eq:52} + \begin{aligned} + c_{\pm} &= \frac{1}{\sqrt{2}}\pqty{f_{0}\eu^{-\iu ϕ} \pm a} & + c_{n\neq + 0}&=f_{n}\\ + ω_{\pm}^{0}&=ω_{s}\pm δ & ω_{n\neq 0}&= ω_{s} + Ω_{B} n\\ + V_{\pm,\pm}&=\frac{J_{00}}{2} & V_{\pm, n\neq0} &= + \frac{J_{0n}}{\sqrt{2}}\eu^{\iu + ϕ}\\ + V_{n\neq \pm, m\neq \pm} &= J_{nm} + \end{aligned} +\end{equation} +and obtain +\begin{equation} + \label{eq:54} + H_{A} = ∑_{n} ω_{n}^{0}c_{n}^†c_{n} + ∑_{nm} V_{nm}(t) c_{n}^†c_{m}, +\end{equation} +where the index \(n\) can take on the values in +\(\pqty{[-N,N]\setminus \{0\}} \cap \{+, -\}\). The spectra of the +coupled and uncoupled systems are visualized in \cref{fig:spectra}. + +\tikzmath{ + \Nmodes = 5; + integer \NmodesBetween; + \NmodesBetween = \Nmodes - 1; + \delta = .2; +} +\begin{figure}[p] + \centering +\begin{tikzpicture}[y=-1.5cm] + \draw[->] (-1.5, -1) -- node[left] {\(ω\)} ++(0,2); + + \foreach \y in {-\Nmodes,...,\Nmodes} + \draw (0, \y) node[left] {\(ω^{B}_{\y}\)} -- ++(1, 0); + + \foreach \y/\name in {-\Nmodes/-1,0,\Nmodes/1} + \draw (1.1, \y) -- ++(1, 0) node[right] {\(ω^{S}_{\name}\)}; +\end{tikzpicture} +\begin{tikzpicture}[y=-1.5cm] + \foreach \y in {-\NmodesBetween,...,-1} + \draw (0, \y) node[left] {\(ω^{0}_{\y}\)} -- ++(1, 0); + + \foreach \y in {1,...,\NmodesBetween} + \draw (0, \y) node[left] {\(ω^{0}_{\y}\)} -- ++(.5, 0) node(mc\y){} + -- ++(.5, 0) node(mr\y){}; + + \foreach \y in {-\Nmodes,\Nmodes} { + \foreach \sub/\name in {-\delta/-, \delta/+} + \draw (0, \sub+\y) -- ++(.5, 0) -- ++(.5, 0); + } + + \foreach \y in {0} { + \foreach \sub/\name in {-\delta/-, \delta/+} + \draw (0, \sub+\y) node[left] {\(ω^{0}_{\name}\)} -- ++(.5, 0) node(pmc\name){} -- ++(.5, 0) node[right] (pmr\name){}; + } + + \foreach \y in {-\Nmodes,0,\Nmodes} + \draw[dashed,color=gray] (0, \y) -- ++(1, 0); + + \draw[<->] (pmc+) -- node[right] {\(2δ\)} (pmc-); + \draw[<->] (pmc+) -- node[right] {\(Ω_{B} - δ\)} (mc1); + \draw[<->] (mc1) -- node[right] {\(Ω_{B}\)} (mc2); +\end{tikzpicture} +\caption{\label{fig:spectra}The spectra of the uncoupled loops (left) \(B,S\) and the resulting + spectrum after coupling the loops (right) for \(N=2\) and \(δ=Ω_{B}/5\).} +\end{figure} + +\subsection{The Choice of the Hybridization Amplitude} +\label{sec:choice-hybridization} + +It remains to be discussed which choice of \(δ\) is suitable. By +modulating \(V(t)\) and applying the RWA we can couple certain levels +in the spectrum of the system. To be able to couple one level to many +and implement the non-Markovian quantum walk, we have to select out +unique level-spacings. At the same time, we want to maximize the +frequency of the residual rotating terms. + +A suitable mode for this one-to-many coupling is the \(c_{+}\) mode +with frequency \(ω_{+}^{0}=ω_{s}+δ\). We identify the \(A\) site of +the non-Markovian Quantum Walk in \(k\)-space with \(c_{+}\) and the +\(j\)th bath level with \(c_{j}\) (\(j\in [1, N]\)). + +Let us denote the frequency +difference of the \(m\)th and \(n\)th mode by +\(Δ_{nm}=ω^{0}_{n}-ω^{0}_{m}\). +For \(n>m\) we have +\begin{align} + \label{eq:56} + Δ_{+,-}&=2δ\\ + \label{eq:57}Δ_{n>0,-} &= n Ω_{B} + δ\\ + \label{eq:58}Δ_{n>0,+} &= n Ω_{B} - δ\\ + \label{eq:59}Δ_{n>0,m>0} &= (n-m) Ω_{B} - δ. +\end{align} +To couple exactly one other mode with \(n>0\) to the \(+\) mode, the +RWA requires that \(\abs{Δ_{n,+}}\neq \abs{Δ_{kl}}\) for \(k\neq n\), \(l\neq +\). +This requirement yields the following restrictions on the value of +\(δ\) +\begin{align} + \label{eq:61} + δ&\neq \frac{n}{2}Ω_{B} & \text{\cref{eq:58,eq:57}}\\ + δ&\neq \frac{n}{3}Ω_{B} & \text{\cref{eq:58,eq:56}}\\ + δ&\neq {n}Ω_{B} & \text{\cref{eq:58,eq:59}\footnote{Duh!}}. +\end{align} +To maximize the residual rotating terms, the minimum of the +\cref{eq:56,eq:57,eq:58,eq:59} has to be maximized for \(δ \in [0, Ω_{B}]\) +\begin{equation} + \label{eq:62} + 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}}. +\end{equation} +We find that \(Δ_{\max}=Ω_{B}/4\) for +\begin{equation} + \label{eq:63} + δ_{\mathrm{opt}}=Ω_{B}/5, +\end{equation} +as can be ascertained from \cref{fig:delta_choice}. +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + scale only axis=true, + width=.8\columnwidth, + height=.2\columnwidth, + xmin = 0, xmax = 1, + ymin = 0, ymax = .5, + axis lines* = left, + xtick = {0}, ytick = \empty, + clip = false, + xtick={},ytick={}, + tick num = 10, + minor tick num=5, + grid=both, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + ylabel = {\(Δ_{\min}/Ω_{B}\)}, + xlabel = {\(δ/Ω_{B}\)}, + x label style={at={(axis description cs:0.5,-0.2)},anchor=north}, + 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 = + 1000, color = cerulean]{min(2*x, 1-x, abs(1-3*x), abs(2-3*x), + abs(1-2*x))}; + \addplot[color = black, mark = *, only marks, mark size = 3pt] + coordinates {(.2, .4)}; + \addplot[color = black, dashed, thick] coordinates {(.2, 0) (.2, + .4) (0, .4)}; + + \addplot[color = gray, mark = *, only marks, mark size = 3pt] + coordinates {(.25, .25)}; + \addplot[color = gray, dashed, thick] coordinates {(.25, 0) (.25, + .25) (0, .25)}; + \end{axis} + \end{tikzpicture} + \caption{\label{fig:delta_choice} The minimal rotating frequencies + \cref{eq:62} for the range of possible \(δ\). The black marker + highlights \(δ=Ω_{B}/5\) and the grey marker marks \(δ=Ω_{B}/4\).} +\end{figure} + +\subsection{Steady-state Transmission} +\label{sec:steadyst-transm} + +TBD (already manually) + + \newpage \printbibliography{}