add some notes on non-markov. quantum walk

roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/hirostyle.sty

@@ -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}}%
+}

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{}