mirror of
https://github.com/vale981/master-thesis-tex
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898 lines
28 KiB
TeX
898 lines
28 KiB
TeX
\documentclass[10pt, aspectratio=169]{beamer}
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\usepackage{ifdraft}
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\usepackage{../hiromacros}
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\usepackage{unicode-math}
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\usepackage[backend=biber, language=english, style=authortitle-comp]{biblatex}
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\usepackage[list=true, font=small,
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labelformat=brace, position=top]{subcaption}
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\usepackage[tbtags]{mathtools}
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\usepackage{physics}
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\usepackage{cleveref}
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\usepackage{graphicx}
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\usepackage{appendixnumberbeamer}
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\graphicspath{ {./figs/} }
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\usepackage{tikz}
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\usepackage{pgfplots}
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\usetikzlibrary{calc,matrix,intersections,fillbetween}
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\usetheme{default}
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\usecolortheme{dolphin}
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\usefonttheme{professionalfonts}
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%\usepackage{newmathpx}
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\institute[TUD] % (optional)
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{
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TU Dresden
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}
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\setbeamertemplate{itemize items}[default]
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\setbeamertemplate{enumerate items}[default]
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\setbeamercolor{block body}{use=structure,bg=structure.fg!10}
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\AtBeginSection[]
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{
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\begin{frame}
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\tableofcontents[currentsection]
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\end{frame}
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}
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\AtBeginSubsection[]
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{
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\begin{frame}
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\tableofcontents[currentsubsection]
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\end{frame}
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}
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\setbeamertemplate{footline}[frame number]
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% \setbeamertemplate{bibliography item}{\insertbiblabel}
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\usepackage{pgfpages}
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% \setbeameroption{show notes on second screen}
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\makeatletter
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\def\beamer@framenotesbegin{% at beginning of slide
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\usebeamercolor[fg]{normal text}
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\gdef\beamer@noteitems{}%
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\gdef\beamer@notes{}%
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}
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\makeatother
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% Plots
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% \newcommand{\plot}[1]{%
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% \ifdraft{\includegraphics[draft=false]{#1.pdf}}{\input{./figs/#1.pgf}}}
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\addbibresource{references.bib}
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\synctex=1
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\title{Bath Observables with HOPS}
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\subtitle{Energy Flow in Strongly Coupled Open Quantum Systems}
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\author{\underline{Valentin Boettcher}, Richard Hartmann,
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Konstantin Beyer, Walter Strunz}
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\titlegraphic{
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\includegraphics[width=2cm]{figs/Logo_TU_Dresden.pdf}
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}
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\institute{Institute for Theoretical Physics, Dresden, Germany}
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\date{17.08.2022}
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\beamertemplatenavigationsymbolsempty
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\begin{document}
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\hypersetup{pageanchor=false}
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\begin{frame}[plain]
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\titlepage
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%\includegraphics[width=3cm]{figs/Logo_TU_Dresden.pdf}
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\end{frame}
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\hypersetup{pageanchor=true} \pagenumbering{arabic}
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\begin{frame}
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\tableofcontents
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\end{frame}
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\section{Introduction}
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\label{sec:intro}
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\subsection{Motivation}
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\begin{frame}
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\begin{block}{Situation}
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Consider an open quantum system
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\begin{equation}
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\label{eq:openthermo}
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H=\underbrace{H_\sys}_{\text{"small"}}\; +\;
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\underbrace{H_\inter}_{\mathrm{?}} \;+\;
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\underbrace{H_\bath}_{\text{"big", simple}}
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\end{equation}
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with \([H_S, H_B] = 0\).
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\end{block}
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\pause
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\begin{itemize}[<+->]
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\item weak coupling \(H_\inter\approx 0\)
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thermodynamics of open systems
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are somewhat understood \footcite{Rivas2019Oct,Talkner2020Oct}
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\item strong coupling: \(\ev{H_\inter} \sim \ev{H_\sys}\)
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\(\implies\) we can't neglect the interaction \(\implies\)
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thermodynamics?
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\note[item]{we do quantum mechanics \(\implies\) can't separate bath
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and system, especially not dynamics!}
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\item but what is clear: \emph{need to get access to exact dynamics
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of \(H_\inter,H_\bath\)}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\begin{center}
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\huge{Is that possible? \pause{}Yes.}\\\pause{}
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\huge{Using HOPS :)}
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\pause{}
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\end{center}
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\begin{block}{Sneak Peek}
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We will be able to calculate
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\(\dv{\ev{H_\bath}}{t}\) (and \(\ev{H_\inter}\)).
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\begin{itemize}
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\item more general: \(O_\sys\otimes (B^a)^\dag B^b\) with \(B=∑_{λ}g_{λ}a_{λ}\)
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\end{itemize}
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\end{block}
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\pause{}
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\begin{itemize}[<+->]
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\item won't call this \emph{heat-flow} because it isn't
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\emph{the} thermodynamic heat flow
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\item nevertheless: may be interesting \emph{qualitative} measure
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for energy flow
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\end{itemize}
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\end{frame}
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\subsection{Technical Basics}
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\begin{frame}{Standard Model of Open Systems}
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In the following we will work with models of the form\footnote{Sometimes this
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is called the ``Standard Model of Open Systems''.}
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\begin{equation}
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\label{eq:openmodel}
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H = H_\sys(t) + ∑_{n=1}^N \qty[H_\bath\nth + \qty(L_n^†(t)B_n + \hc)],
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\end{equation}
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where
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\note[item]{mention dimension system h}
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\note[item]{special because of separation of the bath couplings}
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\begin{itemize}
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\item \(H_\sys\) is the System Hamiltonian
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\item \(H_B\nth = ∑_λω_λ\nth a_λ^{(n),†}a_λ\nth\)
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\item \(B_n=∑_{λ} g_λ\nth a_λ\nth\).
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\end{itemize}
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\end{frame}
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\begin{frame}{What remains of the Bath?}
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\begin{block}{Bath Correlation Function}
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\[α(t-s) = \ev{B(t)B(s)} \qty(\overset{T=0}{=} ∑_λ
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\abs{g_λ}^2\,\eu^{-\iu ω_λ (t-s)})= \frac{1}{π} ∫J(ω) \eu^{-\iu ω
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t}\dd{ω}\]
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\end{block}
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\pause
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\begin{block}{Spectral Density}
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\[J(ω) = π ∑_{λ} \abs{g_{λ}}^{2}δ(ω-ω_{λ})\]
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\begin{itemize}
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\item in thermodynamic limit \(\to\) smooth function
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\item here usually: Ohmic SD \(J(ω)=η ω \eu^{-ω/ω_c}\) (think phonons)
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\end{itemize}
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\end{block}
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\end{frame}
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% \begin{frame}
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% \begin{tikzpicture}
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% \def\xmin{-.9}
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% \def\xmax{.9}
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% \def\spacing{1/4}
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% \def\maincolor{black}
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% \foreach \n in {0,...,4}{
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% \node (.2\n,.5\n) {test};
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% \draw[\maincolor,domain=\xmin:\xmax,samples=100,smooth,name
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% path=harmonic]
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% plot (\x,{2*(\x)^2});
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% \foreach \i in {0,...,5}{
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% \draw[name path=level\i,draw=none] (\xmin,{\spacing * (\i + 1/2)}) --
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% (\xmax,{\spacing * (\i + 1/2)});
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% \draw[name intersections={of=harmonic and level\i}];
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% \draw[\maincolor] (intersection-1) -- (intersection-2);
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% }
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% }
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% \end{tikzpicture}
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% \end{frame}
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\begin{frame}{NMQSD (Zero Temperature)}
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Open system dynamics formulated as a \emph{stochastic} differential equation:
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\begin{equation}
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\label{eq:multinmqsd}
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∂_tψ_t(\vb{η}^\ast_t) = -\iu H(t) ψ_t(\vb{η}^\ast_t) +
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\vb{L}\cdot\vb{η}^\ast_tψ_t(\vb{η}^\ast_t) - ∑_{n=1}^N L_n^†(t)∫_0^t\dd{s}α_n(t-s)\fdv{ψ_t(\vb{η}^\ast_t)}{η^\ast_n(s)},
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\end{equation}
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with
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\begin{equation}
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\label{eq:processescorr}
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\begin{aligned}
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\mathcal{M}(η_n(t)) &=0, & \mathcal{M}(η_n(t)η_m(s)) &= 0,
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& \mathcal{M}(η_n(t)η_m(s)^\ast) &= δ_{nm}α_n(t-s),
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\end{aligned}
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\end{equation}
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by projecting on coherent bath states.\footnote{For details see: \cite{Diosi1998Mar}}
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\end{frame}
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\begin{frame}{HOPS}
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Using \(α_n(τ)=∑_{\mu}^{M_n}G_μ\nth\eu^{-W_μ\nth τ}\) we define
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\begin{equation}
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\label{eq:dops}
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D_μ\nth(t) \equiv ∫_0^t\dd{s}G_μ\nth\eu^{-W_μ\nth (t-s)}\fdv{η^\ast_n(s)}
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\end{equation}
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and
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\(
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D^{\underline{\vb{k}}} \equiv
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∏_{n=1}^N∏_{μ=1}^{M_n}
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{\sqrt{\frac{\underline{\vb{k}}_{n,μ}!}{\qty(G\nth_μ)^{\underline{\vb{k}}_{n,μ}}}}
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\frac{1}{i^{\underline{\vb{k}}_{n,μ}}}}\qty(D_μ\nth)^{\underline{\vb{k}}_{n,μ}}\),
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\(
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ψ_{t}^{\kmat} \equiv D^\kmatψ_{t}\)
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we find
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\begin{multline}
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\label{eq:multihops}
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\dot{ψ}_{t}^\kmat = \qty[-\iu H_\sys(t) + \vb{L}(t)\cdot\vb{η}_{t}^\ast -
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∑_{n=1}^N∑_{μ=1}^{M_n}\kmat_{n,μ}W\nth_μ]ψ_{t}^\kmat \\+
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\iu ∑_{n=1}^N∑_{μ=1}^{M_n}\sqrt{G\nth_μ}\qty[\sqrt{\kmat_{n,μ}} L_n(t)ψ_{t}^{\kmat -
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\mat{e}_{n,μ}} + \sqrt{\qty(\kmat_{n,μ} + 1)} L^†_n(t)ψ_{t}^{\kmat +
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\mat{e}_{n,μ}} ].
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\end{multline}
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\end{frame}
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\section{Bath and Interaction Energy}
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\subsection{A Little (more) Theory}
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\begin{frame}{Zero Temperature, One Bath, Linear NMQSD}
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We want to calculate
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\begin{equation}
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\label{eq:heatflowdef}
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J = - \dv{\ev{H_\bath}}{t} = \ev{L^†∂_t B(t) + L∂_t B^†(t)}_\inter.
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\end{equation}
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\pause{} \ldots some manipulations \ldots{}\pause{}
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\begin{block}{Result (NMQSD)}
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\begin{equation}
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\label{eq:final_flow_nmqsd}
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J(t) = -\i \mathcal{M}_{η^\ast}\bra{\psi(η,
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t)}L^†\dot{D}_t\ket{\psi(η^\ast,t)} + \cc
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\end{equation}
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\end{block}
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with \(\dot{D}_t = ∫_0^t\dd{s} \dot{\alpha}(t-s)\fdv{η^\ast_s}\).\pause{}
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\begin{block}{Result (HOPS)}
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\begin{equation}
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\label{eq:hopsflowfock}
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J(t) = - ∑_\mu\sqrt{G_\mu}W_\mu
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\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,
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t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc
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\end{equation}
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\end{block}
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\end{frame}
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\begin{frame}{Generalizations}
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\begin{block}{Finite Temperature}
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\begin{equation}
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\label{eq:nonlinhopsflowfock}
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J(t) = J_0(t) + \qty[\ev{L^†∂_t ξ(t)} + \cc]
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\end{equation}
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with \(\mathcal{M}(ξ(t))=0=\mathcal{M}(ξ(t) ξ(s))\),
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\(
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\mathcal{M}\left(ξ(t) ξ^{*}(s)\right)=\frac{1}{\pi} ∫_{0}^{∞} \mathrm{d} ω \bar{n}(\beta ω) J(ω) e^{-\mathrm{i} ω(t-s)}
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\)
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and \(J(ω)=π\sum_λ\abs{g_λ}^2δ(w-ω_λ)\).\footnote{\(∂_t ξ(t)\) exists if correlation function is smooth}
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\end{block}
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\begin{itemize}
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\item nonlinear NMQSD/HOPS
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\item multiple baths straight forward\note[item]{due to simple
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structure of coupling}
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\item interaction energy: ``removing the dot''\ldots{}
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\item general ``collective'' bath observables \(O_\sys\otimes
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(B^a)^\dag B^b\) with \(B=∑_{λ}g_{λ}a_{λ}\)
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\end{itemize}
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\note[item]{can be calculated along with sample trajectory with little effort}
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\end{frame}
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\begin{frame}
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\begin{center}
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\huge{Is this any good?}
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\end{center}
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\end{frame}
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\subsection{Analytic Verification}
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\begin{frame}{Model}
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\begin{equation}
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\label{eq:qbmhamiltonian}
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H = \frac{Ω}{4}\qty(p^2+q^2) + \frac{1}{2} q
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\sum_λ\qty(g_λ^\ast b_λ + g_λ
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b^†_λ)+\sum_λ\omega_λ b^†_λ b_λ,
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\end{equation}
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\pause
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\ldots leading to \ldots
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\begin{align}
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\dot{q} &=Ω p \label{eq:qdot}\\
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\dot{p} &= -Ω q - \int_0^t \Im[α_0(t-s)] q(s)\dd{s} + W(t) \label{eq:pdot}
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\\
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\dot{b}_λ &= -\iu g_λ \frac{q}{2} - \iu\omega_λ b_λ
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\end{align}
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with the operator noise
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\(W(t)=-\sum_λ \qty(g_λ^\ast b_λ(0)
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\eu^{-\iu\omega_λ t } + g_λ b_λ^†(0)
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\eu^{\iu\omega_λ t })\),
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\(\ev{W(t)W(s)}=α(t-s)\) and \(α_0 \equiv \eval{α}_{T=0}\).
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\end{frame}
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\begin{frame}
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Solution through a matrix \(G(t)\) with \(G(0)=\id\) and
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\begin{equation}
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\label{eq:eqmotprop}
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\dot{G}(t) = A G(t) - \int_0^t K(t-s) G(s)\dd{s},\quad A=\mqty(0 &
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Ω \\ -Ω & 0), \quad K(t)=\mqty(0 & 0\\ \Im[α_0(t)] & 0).
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\end{equation}
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\pause
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Then
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\begin{equation}
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\label{eq:qpsol}
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\mqty(q(t)\\ p(t)) = G(t)\mqty(q(0)\\ p(0)) + \int_0^tG(t-s)
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\mqty(0\\ W(s))\dd{s}.
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\end{equation}
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\begin{itemize}
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\item ``exact'' solution via laplace transform and BCF expansion +
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residue theorem
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\end{itemize}
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\end{frame}
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\begin{frame}{Result}
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\begin{block}{Solution}
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\begin{equation}
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\label{eq:gfinal}
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G(t) = \sum_{l=1}^{N+1}\qty[R_l \mqty(\tilde{z}_l & Ω \\ \frac{\tilde{z}_l^2}{Ω} & \tilde{z}_l)\eu^{\tilde{z}_l \cdot
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t} + \cc]
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\end{equation}
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with \(R_l={f_0(\tilde{z}_l)}/{p'(\tilde{z}_l)}\), \(f_0,p\)
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polynomials, \(\tilde{z}_l\) roots of \(p\).
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\note[item]{Simple Structure, but lots of algebra -> computer}
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\end{block}
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\pause
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\begin{itemize}
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\item note: \(G\) doesn't depend on temperature
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\item solution very sensitive to precision of the fits and roots
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\end{itemize}
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\end{frame}
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\begin{frame}{Bath Energy Derivative}
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\begin{equation}
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\label{eq:bathderiv_1}
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\begin{aligned}
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\ev{\dot{H}_B} &= ∑_λ ω_λ \qty(\ev{b_λ^†\dot{b}_λ} + \cc) \\
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&=
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\begin{aligned}[t]
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-\frac{1}{2}\Im&\qty[∫_0^t\dd{s}\ev{q(t)q(s)}\dot{α}_0(t-s)] \\&+
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\frac12 G_{12}(t)[α(t)-α_0(t)]
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-\frac{Ω}{2}∫_0^t\dd{s} G_{11}(s)\qty[α(s)-α_0(s)]
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\end{aligned}
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\end{aligned}
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\end{equation}
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\begin{itemize}
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\item becomes huge sum of exponentials (thanks Mathematica)
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\note[item]{initially forgot last term, took some time to find}
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\end{itemize}
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\end{frame}
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\begin{frame}{One Bath, Finite Temperature}
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\begin{block}{Parameters}
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\tval{04/omega}, Ohmic BCF
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\(\frac{η}{π}\qty({ω_c}/({1+\iu ω_c τ}))^2\) with
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(\(α(0)=0.64,\, ω_c=2\)), \(N=10^{5}\) samples, \(15\) Hilbert
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space dimensions, \(\ket{ψ(0)}_{\sys} = \ket{1}_{\sys}\), \(T=1\)
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\end{block}
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\note[item]{left out zero temperature}
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\begin{figure}[t]
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\centering
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\includegraphics{figs/analytic_comp/flow_comp_nonzero.pdf}
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% \caption{\label{fig:comp_finite_t} The bath energy flow \(-J\) of
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% the quantum Brownian motion model for different parameters of the
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% ohmic bath correlation function \cref{eq:ohmic_bcf} in the finite
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% temperature \(T=1\) case. The presentation is equivalent to
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% \cref{fig:comp_zero_t}.}
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\end{figure}
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\end{frame}
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\begin{frame}{Two Baths, Finite Temperature (Gradient)}
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\begin{block}{Parameters}
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\(Ω=Λ=1\), symmetric Ohmic BCFs with
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(\(α(0)=0.25,\, ω_c=2\)), \(N=10^{4}\) samples, \(9\) Hilbert
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space dimensions, \(\ket{ψ(0)}_{\sys} = \ket{0,0}_{\sys}\),
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\(T=0.6\), \(γ=0.5\)
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||
\end{block}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\begin{subfigure}[t]{.49\linewidth}
|
||
\includegraphics{figs/analytic_comp/comparison_two_5bcf_5ho.pdf}
|
||
\end{subfigure}
|
||
\begin{subfigure}[t]{.49\linewidth}
|
||
\includegraphics{figs/analytic_comp/comparison_two_ho.pdf}
|
||
\end{subfigure}
|
||
\end{figure}
|
||
\end{frame}
|
||
|
||
% \subsection{First Experiments}
|
||
% \begin{frame}{One Qubit, One Bath, Zero Temperature}
|
||
% \begin{columns}
|
||
% \begin{column}{.39\textwidth}
|
||
% \begin{block}{Model}
|
||
% \begin{equation}
|
||
% \label{eq:01model}
|
||
% \begin{aligned}
|
||
% H_\sys &= \frac12 σ_z & L &= \frac12 σ_x & \ket{ψ(0)} &= \ket{\uparrow}
|
||
% \end{aligned}
|
||
% \end{equation}
|
||
% \begin{itemize}
|
||
% \item Ohmic BCF with cutoff frequency
|
||
% \tval{01/cutoff_freq} and
|
||
% scaling \tval{01/bcf_scale}.
|
||
% \item \tval{01/samples} samples
|
||
% \note[item]{the error bars are almost certainly
|
||
% underestimated}
|
||
% \note[item]{interaction energy of similar oom}
|
||
% \note[item]{\(ω_c=2\) is pretty non-Markovian, usual limit \(ω_c\rightarrow\infty\)}
|
||
% \end{itemize}
|
||
% \end{block}
|
||
% \end{column}
|
||
% \begin{column}{.6\textwidth}
|
||
% \only<1>{\plot{01/system_energy}}%
|
||
% \only<2>{\plot{01/flow}}%
|
||
% \only<3>{\plot{01/interaction}
|
||
% }%
|
||
% \end{column}
|
||
% \end{columns}
|
||
% \end{frame}
|
||
|
||
% \begin{frame}{One Qubit, One Bath, Finite Temperature}
|
||
% \begin{columns}
|
||
% \begin{column}{.39\textwidth}
|
||
% \begin{block}{Model}
|
||
% \begin{equation}
|
||
% \label{eq:01model}
|
||
% \begin{aligned}
|
||
% H_\sys &= \frac12 σ_z & L &= \frac12 σ_x & \ket{ψ(0)} &= \ket{\uparrow}
|
||
% \end{aligned}
|
||
% \end{equation}
|
||
% \begin{itemize}
|
||
% \item Ohmic BCF with cutoff frequency \tval{02/cutoff_freq} and
|
||
% scaling \tval{02/bcf_scale}
|
||
% \item Temperature \tval{02/temp}
|
||
% \item \tval{02/samples} samples
|
||
% \note[item]{didn't have many resources, just quick test}
|
||
% % \item<2-> initial gain, then steady loss
|
||
% % \item<3-> gain compensated by interaction
|
||
% \end{itemize}
|
||
% \end{block}
|
||
% \end{column}
|
||
% \begin{column}{.6\textwidth}
|
||
% \only<1>{\plot{02/system_energy}}%
|
||
% \only<2>{\plot{02/flow}}%
|
||
% \only<3>{\plot{02/interaction}}%
|
||
% \end{column}
|
||
% \end{columns}
|
||
% \end{frame}
|
||
|
||
% \begin{frame}
|
||
% \begin{center}
|
||
% Consistency check with interaction energy is difficult and not
|
||
% totally conclusive.
|
||
|
||
% \note[item]{convergence..., only indirect,
|
||
% numerical integration}
|
||
|
||
% \pause \(\implies\) Analytic Verification
|
||
% \end{center}
|
||
% \end{frame}
|
||
|
||
\section{Applications}
|
||
|
||
\subsection{One Bath}
|
||
\begin{frame}
|
||
\frametitle{One Bath, Zero Temperature}
|
||
\begin{block}{Model: Spin-Boson}
|
||
\begin{equation}
|
||
\label{eq:one_qubit_model}
|
||
H = \frac{1}{2} σ_z + \frac{1}{2} ∑_λ\qty(g_λ σ_x^† a_λ + g_λ^\ast
|
||
σ_x a_λ^†) + ∑_λ ω_λ a_λ^\dag a_λ,\;\ket{ψ_{0}}_{\sys} = \ket{\uparrow}
|
||
\end{equation}
|
||
\end{block}
|
||
\begin{itemize}
|
||
\item<+-> how do we check convergence:\pause
|
||
\begin{itemize}
|
||
\item old: difference of results to some ``good'' configuration\pause
|
||
\item new: consistency with energy conservation
|
||
\end{itemize}
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{Example: Dependence of the Interaction Energy on Stochastic Process
|
||
Sampling}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics{figs/one_bath_syst/stocproc_systematics_interaction}
|
||
\end{figure}
|
||
\begin{itemize}
|
||
\item \(α(0)=1.6\) and \(ω_{c}=4\) \(\implies\) stress HOPS through
|
||
fast decaying BCF
|
||
\item ``perfect'' results only with very high
|
||
accuracy\footnote{smaller \(\varsigma\) is better} \(\varsigma\)
|
||
\item good qualitative results for less extreme configurations
|
||
(common theme)
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{Various Cutoff Frequencies}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics[height=.9\textheight]{figs/one_bath_syst/omega_energy_overview}
|
||
\note[item]{marked differences between trajcetories -> non-markov,
|
||
strong coup}
|
||
\end{figure}
|
||
\end{frame}
|
||
|
||
\begin{frame}{Non-Markovian Dynamics}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\only<1>{\includegraphics{figs/one_bath_syst/markov_analysis}}
|
||
\only<2>{\includegraphics{figs/one_bath_syst/markov_analysis_longer}}
|
||
\end{figure}
|
||
\begin{itemize}
|
||
\item<+-> interaction strengths chosen for approx. same interaction
|
||
energy
|
||
\item<+-> timing important for energy transfer ``performance''
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\begin{center}
|
||
\begin{alertblock}{Beware :)}
|
||
The following is WIP and has not been written up properly yet.
|
||
\end{alertblock}
|
||
\end{center}
|
||
\end{frame}
|
||
|
||
|
||
\subsection{Energy Shovel}
|
||
\begin{frame}
|
||
\frametitle{Extracting Energy from One Bath}
|
||
\begin{itemize}
|
||
\item same model as above \cref{eq:one_qubit_model}, but with \(L(τ) =
|
||
\sin^2(\frac{Δ}{2} τ)σ_x\)
|
||
\item how much energy can be \emph{unitarily} extracted?
|
||
\(\implies\) \(ΔE_{\mathrm{max}}=\frac{1}{β}\qrelent{ρ_{\sys}}{ρ_{\sys}^{β}}\)
|
||
\end{itemize}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\only<1>{\includegraphics[height=.65\textheight]{figs/modcoup/omegas_total}}
|
||
\only<2>{\includegraphics[height=.65\textheight]{figs/modcoup/delta_dependence}}
|
||
\end{figure}
|
||
\note[item]{energy normalized to ergo}
|
||
\note[item]{vert lines bath memory time}
|
||
\end{frame}
|
||
|
||
|
||
\subsection{Otto Cycle}
|
||
\begin{frame}
|
||
\frametitle{Otto Cycle (proof of concept)}
|
||
\begin{block}{Model}
|
||
Spin-Boson model with compression of \(H_{\sys}\) and modulation
|
||
of \(L\).
|
||
\end{block}
|
||
\begin{itemize}
|
||
\item classical toy model of the quantum heat engine community\footcite{Geva1992Feb}
|
||
\end{itemize}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics{figs/otto/modulation}
|
||
\end{figure}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics{figs/otto/energy_cont}
|
||
\end{figure}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics{figs/otto/energy_strobe}
|
||
\end{figure}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics{figs/otto/power}
|
||
\end{figure}
|
||
\begin{itemize}
|
||
\item \(\bar{P} = 1.98 \cdot 10^{-3} \pm 2.3 \cdot 10^{-5}\),
|
||
\(η\approx 20\%\), \(T_{c}=1\), \(T_{h}=20\)
|
||
\item no tuning of parameters, except for resonant coupling
|
||
\item long bath memory \(ω_{c}=1\), but weak-ish coupling
|
||
\end{itemize}
|
||
\end{frame}
|
||
\begin{frame}{Questions (for the future)}
|
||
\begin{itemize}
|
||
\item better performance through ``overlapping'' phases?
|
||
\item strong coupling any good?
|
||
\item non-Markovianity + strong coupling any good?
|
||
\item what is the optimal efficiency and power? (probably no
|
||
advantage here)
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\subsection{Anti-Zeno Engine}
|
||
\begin{frame}
|
||
\frametitle{Anti-Zeno Engine}
|
||
\begin{block}{Question}
|
||
Is there a use for non-Markovianity in quantum heat engines?
|
||
\end{block}
|
||
\begin{itemize}
|
||
\item \cite{Mukherjee2020Jan} claims that one can exploit the
|
||
time-energy uncertainty for quantum advantage\footnote{I'd be
|
||
careful to call this quantum advantage.}
|
||
\end{itemize}
|
||
|
||
\begin{block}{Model}
|
||
Qubit coupled to two baths of different
|
||
temperatures (\(T_c, T_h\))
|
||
\begin{equation}
|
||
\label{eq:antizenomodel}
|
||
H_\sys= \frac12 \qty[ω_0 + γ Δ\sin(Δ t)]σ_z,\, L_{c,h}=\frac12 σ_x
|
||
\end{equation}
|
||
\note[item]{frictionless dynamics}
|
||
\end{block}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\begin{figure}[h]
|
||
\centering
|
||
\includegraphics{figs/anti_zeno/with_gap_coupling_diagram}
|
||
\end{figure}
|
||
\begin{itemize}
|
||
\item couple for \(n\) modulation periods slightly of resonance
|
||
\item for smaller \(n\) the \(\sin((ω-(ω_{0}\pm Δ))τ)/((ω-(ω_{0}\pm
|
||
Δ)) τ)\) has a greater overlap
|
||
\(\implies\) controls power output
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\begin{figure}
|
||
\centering
|
||
\begin{subfigure}[t]{.49\linewidth}
|
||
\centering
|
||
\includegraphics[width=\linewidth]{anti_zeno/anti_zeno_without_cool}
|
||
\caption{\tval{anti_zeno/power_without_cool}}
|
||
\end{subfigure}
|
||
\begin{subfigure}[t]{.49\linewidth}
|
||
\centering
|
||
\includegraphics[width=\linewidth]{anti_zeno/anti_zeno_with_cool}
|
||
\caption{
|
||
\tval{anti_zeno/power_with_cool}}
|
||
\end{subfigure}
|
||
\end{figure}
|
||
\begin{exampleblock}{Parameters}
|
||
{\tval{anti_zeno/delta}, \tval{anti_zeno/gamma}, \tval{anti_zeno/omega_alpha},
|
||
\tval{anti_zeno/omega_zero}, \tval{anti_zeno/tc}, \tval{anti_zeno/th}}
|
||
\end{exampleblock}
|
||
\begin{itemize}
|
||
\item this is not properly converged yet \(\rightarrow\) newer
|
||
results: no advantage at these temperatures / coupling strengths
|
||
\item new simulations with temperatures from paper
|
||
(\(β_{h(c)}=0.0005(0.005)\)) are promising
|
||
\begin{itemize}
|
||
\item interesting \(\rightarrow\) no good steady state power in this case
|
||
(insufficient samples?)
|
||
\end{itemize}
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
|
||
\section{Outlook}
|
||
\label{sec:outlook}
|
||
\begin{frame}{On the ``To Do'' List}
|
||
\begin{itemize}
|
||
\item verify/falsify weak coupling results in the literature
|
||
(engines)
|
||
\item three-level systems: there is an experimental paper ;)
|
||
\item parameter scan of two qubit model
|
||
\item filter modes
|
||
\item \ldots
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{Lessons Learned}
|
||
\begin{itemize}
|
||
\item careful convergence checks pay off
|
||
\item surveying literature is important
|
||
\item properly documenting observations is a great help
|
||
and should be done as early as possible
|
||
\item applications should be carefully chosen to answer interesting
|
||
questions
|
||
\item numerics are helpful, but physical insights are important
|
||
\item comparison with some experiments would have been nice
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\appendix
|
||
\begin{frame}[allowframebreaks]{References}
|
||
\printbibliography
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\begin{block}{Situation (Longer)}
|
||
Consider an open quantum system
|
||
\begin{equation}
|
||
\label{eq:openthermo}
|
||
H=\underbrace{H_\sys}_{\text{"small"}}\; +\;
|
||
\underbrace{H_\inter}_{\mathrm{?}} \;+\;
|
||
\underbrace{H_\bath}_{\text{"big", simple}}
|
||
\end{equation}
|
||
with \([H_S, H_B] = 0\).
|
||
\end{block}
|
||
\pause
|
||
\begin{itemize}[<+->]
|
||
\item weak coupling \(H_\inter\approx 0\)
|
||
thermodynamics\footnote{even in strong coupling equilibrium\ldots{}} of open systems
|
||
are somewhat understood \footcite{Rivas2019Oct,Talkner2020Oct}
|
||
\item strong coupling: \(\ev{H_\inter} \sim \ev{H_\sys}\)
|
||
\(\implies\) we can't neglect the interaction \(\implies\)
|
||
thermodynamics?
|
||
\item we do quantum mechanics \(\implies\) can't separate bath
|
||
and system\uncover<+->{, especially not dynamics!}
|
||
\item no consensus about strong coupling thermodynamics:
|
||
\note[item]{won't resolve within scope of masters thesis}
|
||
\item but what is clear: \emph{need to get access to exact dynamics
|
||
of \(H_\inter,H_\bath\)}
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{More Papers on Thermo}
|
||
\cite{Rivas2019Oct,Talkner2020Oct,Motz2018Nov,Wiedmann2020Mar,Senior2020Feb,Kato2015Aug,Kato2016Dec,Strasberg2021Aug,Talkner2016Aug,Bera2021Feb,Bera2021Jun,Esposito2015Dec}
|
||
\end{frame}
|
||
|
||
\begin{frame}{Ohmic SD BCF}
|
||
\begin{columns}
|
||
\begin{column}{.49\textwidth}
|
||
\plot{01/ohmic_sd}
|
||
\end{column}
|
||
\begin{column}{.49\textwidth}
|
||
\plot{01/ohmic_bcf}
|
||
\end{column}
|
||
\end{columns}
|
||
\end{frame}
|
||
\begin{frame}{NMQSD (Zero Temperature)}
|
||
Expanding in a Bargmann (unnormalized) coherent state basis~\cite{klauder1968fundamentals}
|
||
\(\qty{\ket{\vb{z}^{(1)},\vb{z}^{(2)},\ldots}=
|
||
\ket{\underline{\vb{z}}}}\)
|
||
|
||
\begin{equation}
|
||
\label{eq:projected}
|
||
\ket{ψ(t)} = ∫∏_{n=1}^N{\qty(\frac{\dd{\vb{z}\nth}}{π^{N_n}}\eu^{-\abs{\vb{z}}^2})}\ket{ψ(t,\underline{\vb{z}}^\ast)}\ket{\underline{\vb{z}}},
|
||
\end{equation}
|
||
we obtain
|
||
\note[item]{interpret the integral in a montecarlo sense}
|
||
\begin{equation}
|
||
\label{eq:multinmqsd}
|
||
∂_tψ_t(\vb{η}^\ast_t) = -\iu H ψ_t(\vb{η}^\ast_t) +
|
||
\vb{L}\cdot\vb{η}^\ast_tψ_t(\vb{η}^\ast_t) - ∑_{n=1}^N L_n^†∫_0^t\dd{s}α_n(t-s)\fdv{ψ_t(\vb{η}^\ast_t)}{η^\ast_n(s)},
|
||
\end{equation}
|
||
with
|
||
\begin{equation}
|
||
\label{eq:processescorr}
|
||
\begin{aligned}
|
||
\mathcal{M}(η^\ast_n(t)) &=0, & \mathcal{M}(η_n(t)η_m(s)) &= 0,
|
||
& \mathcal{M}(η_n(t)η_m(s)^\ast) &= δ_{nm}α_n(t-s),
|
||
\end{aligned}
|
||
\end{equation}
|
||
where
|
||
\(α_n(t-s)=∑_λ\abs{g_λ\nth}^2\eu^{-\iu
|
||
ω_λ\nth(t-s)}=\ev{B(t)B(s)}_{I,ρ(0)}\) \cite{Strunz2001Habil}
|
||
(fourier transf. of spectral density
|
||
\(J(ω) = π ∑_λ\abs{g_λ}^2δ(ω-ω_λ)\)).
|
||
\end{frame}
|
||
\begin{frame}{Fock-Space Embedding}
|
||
As in \refcite{Gao2021Sep} we can define
|
||
\begin{equation}
|
||
\label{eq:fockpsi}
|
||
\ket{Ψ} = \sum_\kmat\ket{\psi^\kmat}\otimes \ket{\kmat}
|
||
\end{equation}
|
||
where
|
||
\(\ket{\kmat}=\bigotimes_{n=1}^N\bigotimes_{μ=1}^{N_n}\ket{\kmat_{n,μ}}\)
|
||
are bosonic Fock-states.
|
||
|
||
Now \cref{eq:multihops} becomes
|
||
\begin{equation}
|
||
\label{eq:fockhops}
|
||
∂_t\ket{Ψ} = \qty[-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast -
|
||
∑_{n=1}^N∑_{μ=1}^{M_n}b_{n,μ}^\dag b_{n,μ} W\nth_μ +
|
||
\iu ∑_{n=1}^N∑_{μ=1}^{M_n} \sqrt{G_{n,μ}} \qty(b^†_{n,μ}L_n + b_{n,μ}L^†_n)] \ket{Ψ}.
|
||
\end{equation}
|
||
\pause
|
||
\(\implies\) possible to derive an upper bound for the norm of
|
||
\(\ket{\psi^\kmat}\) \pause \(\implies\) new truncation scheme
|
||
\note[item]{looks somewhat like a non-hermitian hamiltonian}
|
||
\end{frame}
|
||
\end{document}
|
||
|
||
\begin{frame}{Multiple Baths}
|
||
\begin{itemize}
|
||
\item theory generalizes easily to \(N\) baths
|
||
\item generalized our HOPS code to \(N\) baths
|
||
\note[item]{added docstrings and typehints to (almost) everything}
|
||
\note[item]{added (some tests) and CI that ensures they aren't broken}
|
||
\item solving a model with two coupled HOs is now possible
|
||
\note[item]{but with much anguish, root precision, bcf fit!!}
|
||
\begin{equation}
|
||
\label{eq:hamiltonian}
|
||
\begin{aligned}
|
||
H &= ∑_{i\in\qty{1,2}} \qty[H^{(i)}_O + q_iB^{(i)} + H_B^{(i)}] + \frac{γ}{4}(q_1-q_2)^2,
|
||
\end{aligned}
|
||
\end{equation}
|
||
where \(H_O^{(i)}= \frac{Ω_i}{4}\qty(p_i^2+q_i^2)\), \(B^{(i)}=\sum_λ\qty(g^{(i),\ast}_λb^{(i)}_λ + g^{(i)}_λ
|
||
b^{(i),†}_λ)\) and \(H_B^{(i)}=\sum_λ\omega_λ b^{(i),†}_λ b^{(i)}_λ\).
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
\section{Other Projects}
|
||
\begin{frame}[allowframebreaks]
|
||
\begin{itemize}
|
||
\item stabilized normalization in nonlinear HOPS
|
||
\begin{figure}
|
||
\centering
|
||
\plot{05/norm}
|
||
\end{figure}
|
||
\item stochastic process sampling via Cholesky decomposition
|
||
\note[item]{allows to generate processes that match up to a point in
|
||
time}
|
||
\note[item]{allows to extend stochastic processes to arbitrary times}
|
||
\note[item]{could be useful for steady state studies and long-time
|
||
propagation}
|
||
\begin{figure}
|
||
\centering
|
||
\plot{05/cholesky}
|
||
\end{figure}
|
||
\item norm based truncation scheme
|
||
\begin{itemize}
|
||
\note[item]{needs further investigation}
|
||
\item promising at ``friendly'' coupling strengths
|
||
\end{itemize}
|
||
\begin{figure}
|
||
\centering
|
||
\plot{06/convergence}
|
||
\end{figure}
|
||
\end{itemize}
|
||
\end{frame}
|
||
|
||
|
||
%%% Local Variables:
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||
%%% mode: latex
|
||
%%% TeX-master: t
|
||
%%% TeX-output-dir: "output"
|
||
%%% TeX-engine: luatex
|
||
%%% End:
|