diff --git a/index.tex b/index.tex index 78b2d1b..d205894 100644 --- a/index.tex +++ b/index.tex @@ -1,14 +1,36 @@ -\documentclass[reprint,aps,prl,superscriptaddress]{revtex4-2} +\documentclass[reprint,aps,superscriptaddress]{revtex4-2} \usepackage[unicode=true,bookmarks=true,bookmarksnumbered=false,bookmarksopen=false,breaklinks=false,pdfborder={0 0 1}, backref=false,colorlinks=true]{hyperref} \usepackage{orcidlink} \usepackage{microtype} \usepackage{mathtools} \usepackage{graphicx} \usepackage{physics} -\usepackage[math,random]{blindtext} +\usepackage{cleveref} \usepackage{bm} \bibliographystyle{apsrev4-2} -\blindmathtrue + + +% HOPS/NMQSD +\def\sys{\ensuremath{\mathrm{S}}} +\def\bath{\ensuremath{\mathrm{B}}} +\def\inter{\ensuremath{\mathrm{I}}} +\def\nth{\ensuremath{^{(n)}}} + +% unicode math +\iftutex +\usepackage{unicode-math} +\else +\usepackage{amssymb} +\def\z"{} +\def\UnicodeMathSymbol#1#2#3#4{% + \ifnum#1>"A0 + \DeclareUnicodeCharacter{\z#1}{#2}% + \fi} +\input{unicode-math-table} +\let\muprho\rho +\def\BbbR{\mathbb{R}} +\fi + \begin{document} \preprint{APS/123-QED} @@ -37,24 +59,138 @@ + \begin{abstract} -\blindtext \end{abstract} \maketitle +\tableofcontents + \section{Introduction} \label{sec:introduction} -\blindtext +The field of quantum thermodynamics has attracted much interest +recently~\cite{Talkner2020Oct,Rivas2019Oct,Riechers2021Apr,Vinjanampathy2016Oct,Binder2018,Kurizki2021Dec,Mukherjee2020Jan,Xu2022Mar}. +Quantum thermodynamics is, among other issues, concerned with +extending the standard phenomenological thermodynamic notions to +microscopic open systems. -\blindtext +The general type of model that is being investigated in this field is +given by the Hamiltonian +\begin{equation} + \label{eq:4} + H = H_{\sys} + ∑_{n} \qty[H_{\inter}^{(n)} + H_{\bath}^{(n)}], +\end{equation} +where \(H_{\sys}\) models a ``small'' system (from here on called +simply the \emph{system}) of arbitrary structure and the +\(H_{\bath}^{(n)}\) model the ``large'' bath systems with simple +structure but a large number of degrees of freedom. The +\(H_{I}^{(n)}\) acts on system and bath, mediating their interaction. -\blindtext +In this setting may make be possible to formulate rigorous microscopic +definitions of thermodynamic quantities such as internal energy, heat +and work that are consistent with the well-known laws of +thermodynamics. Currently, there is no consensus on this matter yet, +as is demonstrated by the plethora of proposals and discussions in +\cite{Rivas2019Oct,Talkner2020Oct,Motz2018Nov,Wiedmann2020Mar,Senior2020Feb,Kato2015Aug,Kato2016Dec,Strasberg2021Aug,Talkner2016Aug,Bera2021Feb,Bera2021Jun,Esposito2015Dec,Elouard2022Jul}. +This is particularly true for the general case where the coupling to +the baths may be arbitrarily strong. In this case the weak coupling +treatment that allows separate system and bath dynamics is not +applicable. Even the simple seeming question of how internal energy is +to be defined becomes non-trivial~\cite{Rivas2012,Binder2018} due to +the fact that \(\ev{H_{\inter}}\neq 0\). + +In this way the bath degrees of freedom interesting in themselves, +which necessitates a treatment of the exact global unitary dynamics of +system and bath. + +If no analytical solution for these dynamics is available, numerical +methods have to be relied upon. Notably there are perturbative methods +such as the Redfield equations for non-Markovian weak coupling +dynamics~\cite{Davidovic2020Sep} and also exact methods like the +Hierarchical Equations of Motion +HEOM~\cite{Tanimura1990Jun,Tang2015Dec}, multilayer +MCTDH~\cite{Wang2010May}, TEMPO~\cite{Strathearn2018Aug} and the +Hierarchy of Pure States HOPS~\cite{Suess2014Oct}\footnote{See + \cite{RichardDiss} for a detailed account.}. Although the focus of +these methods is on the reduced system dynamics, exact treatments of +open systems can provide access to the global unitary evolution of the +system and the baths. + +In this work we will focus on the framework of the ``Non-Markovian +Quantum State Diffusion'' (NMQSD)~\cite{Diosi1998Mar}, which is +briefly reviewed in~\cref{sec:nmqsd}. We will show in \cref{chap:flow} +that the NMQSD allows access to interaction and bath related +quantities. This novel application of the formalism constitutes the +main result of this work. + +Based on the NMQSD and inspired by the ideas behind HEOM, a numerical +method, the ``Hierarchy of Pure States'' +(HOPS)~\cite{RichardDiss,Hartmann2017Dec}, can be formulated. A brief +account of the method is given in \cref{sec:hops}. + +The results of \cref{sec:flow}, most importantly the calculation of +bath and interaction energy expectation values, can be easily +implemented within this numerical framework. By doing so we will +elucidate the role of certain features inherent to the method. The +most general case we will be able to handle is a system coupled to +multiple baths of differing temperatures under arbitrary time +dependent modulation. As HOPS on its own is already a method with a +very broad range of applicability~\cite{RichardDiss}, we will find it +to be suitable for the exploration of thermodynamical settings. + +In \cref{sec:applications} we apply this result to two simple systems. +As an elementary application, a brief study of the characteristics of +the energy flow out of a qubit into a zero temperature bath is +presented in \cref{sec:qubit-relax-char}. To demonstrate the current +capabilities of our method to the fullest we will turn to the +simulation of a quantum Otto-like +cycle~\cite{cite:Geva1992Feb,cite:Wiedmann2020Mar,cite:Wiedmann2021Jun} +in \cref{sec:quantum-otto-cycle}, which features a simultaneous time +dependence in both \(H_{\inter}\) and \(H_{\sys}\). \section{Energy Flow with HOPS} -\label{sec:method} +\label{sec:flow} + +\subsection{The NMQSD} +\label{sec:nmqsd} + +\subsection{The HOPS} +\label{sec:hops} + +\subsection{Bath Observables} +\label{sec:bath-observables} + +\subsubsection{Bath Energy Change} +\label{sec:bath-energy-change} + +\subsubsection{General Collective Bath Observables} +\label{sec:gener-coll-bath} +\section{Applications} +\label{sec:applications} + +\subsection{Qubit Relaxation Characteristics} +\label{sec:qubit-relax-char} + +\subsection{A Quantum Otto Cycle} +\label{sec:quantum-otto-cycle} + + + +\begin{itemize} +\item see the chapter in my thesis +\item \textbf{Ask richard about phase transitions in spin boson} +\end{itemize} + + +\section{Outlook and Open Questions} +\label{sec:outl-open-quest} +\begin{itemize} +\item steady state methods +\item energy flow for portions of the bath -> adaptive method? +\end{itemize} \bibliography{index} \end{document} diff --git a/latexmkrc b/latexmkrc index e49d144..429b68a 100644 --- a/latexmkrc +++ b/latexmkrc @@ -1,5 +1,6 @@ $pdf_mode = 1; @default_files = ('index.tex'); $out_dir = 'output'; +$pdflatex = 'pdflatex -synctex=1'; $pdf_previewer = "zathura %O %S";