mirror of
https://github.com/vale981/master-thesis
synced 2025-03-04 09:31:43 -05:00
remove old tex
This commit is contained in:
parent
d882ef9343
commit
f6a65271fc
9 changed files with 0 additions and 2138 deletions
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@ -1,2 +0,0 @@
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use_flake
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eval "$shellHook"
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42
tex/energy_transfer/flake.lock
generated
42
tex/energy_transfer/flake.lock
generated
|
|
@ -1,42 +0,0 @@
|
|||
{
|
||||
"nodes": {
|
||||
"flake-utils": {
|
||||
"locked": {
|
||||
"lastModified": 1653893745,
|
||||
"narHash": "sha256-0jntwV3Z8//YwuOjzhV2sgJJPt+HY6KhU7VZUL0fKZQ=",
|
||||
"owner": "numtide",
|
||||
"repo": "flake-utils",
|
||||
"rev": "1ed9fb1935d260de5fe1c2f7ee0ebaae17ed2fa1",
|
||||
"type": "github"
|
||||
},
|
||||
"original": {
|
||||
"owner": "numtide",
|
||||
"repo": "flake-utils",
|
||||
"type": "github"
|
||||
}
|
||||
},
|
||||
"nixpkgs": {
|
||||
"locked": {
|
||||
"lastModified": 1655567057,
|
||||
"narHash": "sha256-Cc5hQSMsTzOHmZnYm8OSJ5RNUp22bd5NADWLHorULWQ=",
|
||||
"owner": "NixOS",
|
||||
"repo": "nixpkgs",
|
||||
"rev": "e0a42267f73ea52adc061a64650fddc59906fc99",
|
||||
"type": "github"
|
||||
},
|
||||
"original": {
|
||||
"id": "nixpkgs",
|
||||
"ref": "nixos-unstable",
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||||
"type": "indirect"
|
||||
}
|
||||
},
|
||||
"root": {
|
||||
"inputs": {
|
||||
"flake-utils": "flake-utils",
|
||||
"nixpkgs": "nixpkgs"
|
||||
}
|
||||
}
|
||||
},
|
||||
"root": "root",
|
||||
"version": 7
|
||||
}
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||||
|
|
@ -1,42 +0,0 @@
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|||
{
|
||||
description = "Typesetting for Valentin's Masters Thesis";
|
||||
inputs = {
|
||||
nixpkgs.url = "nixpkgs/nixos-unstable";
|
||||
flake-utils.url = "github:numtide/flake-utils";
|
||||
flake-utils.inputs.nixpkgs.follows = "nixpkgs";
|
||||
};
|
||||
|
||||
outputs = { self, nixpkgs, flake-utils }:
|
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with flake-utils.lib; eachSystem allSystems (system:
|
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let
|
||||
pkgs = nixpkgs.legacyPackages.${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;
|
||||
};
|
||||
in rec {
|
||||
packages = {
|
||||
document = pkgs.stdenvNoCC.mkDerivation rec {
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||||
name = "masters-thesis";
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||||
src = self;
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||||
buildInputs = [ pkgs.coreutils tex pkgs.biber];
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||||
phases = ["unpackPhase" "buildPhase" "installPhase"];
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||||
buildPhase = ''
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export PATH="${pkgs.lib.makeBinPath buildInputs}";
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mkdir -p .cache/texmf-var
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||||
env TEXMFHOME=.cache TEXMFVAR=.cache/texmf-var \
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||||
OSFONTDIR=${pkgs.gyre-fonts}/share/fonts \
|
||||
latexmk -interaction=nonstopmode \
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./index.tex
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'';
|
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installPhase = ''
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mkdir -p $out
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cp document.pdf $out/
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'';
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||||
};
|
||||
};
|
||||
defaultPackage = packages.document;
|
||||
});
|
||||
}
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||||
|
|
@ -1,108 +0,0 @@
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|||
\ProvidesPackage{hiromacros}
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||||
|
||||
% Macros
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||||
|
||||
%% qqgg
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||||
\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
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||||
|
||||
%% ppgg
|
||||
\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma}
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||||
|
||||
%% Momenta and Polarization Vectors convenience
|
||||
\DeclareMathOperator{\ps}{\slashed{p}}
|
||||
|
||||
\DeclareMathOperator{\pe}{\varepsilon}
|
||||
\DeclareMathOperator{\pes}{\slashed{\pe}}
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||||
|
||||
\DeclareMathOperator{\pse}{\varepsilon^{*}}
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||||
\DeclareMathOperator{\pses}{\slashed{\pe}^{*}}
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||||
|
||||
%% Spinor convenience
|
||||
\DeclareMathOperator{\us}{u}
|
||||
\DeclareMathOperator{\usb}{\bar{u}}
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||||
|
||||
\DeclareMathOperator{\vs}{v}
|
||||
\DeclareMathOperator*{\vsb}{\overline{v}}
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||||
|
||||
%% Center of Mass energy
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||||
\DeclareMathOperator{\ecm}{E_{\text{CM}}}
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||||
|
||||
%% area hyperbolicus
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||||
\DeclareMathOperator{\artanh}{artanh}
|
||||
\DeclareMathOperator{\arcosh}{arcosh}
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|
||||
%% Fast Slash
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||||
\let\sl\slashed
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%% hermitian/complex conjugate
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||||
\DeclareMathOperator{\hc}{h.c.}
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\DeclareMathOperator{\cc}{c.c.}
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||||
%% eulers number
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||||
\def\eu{\ensuremath{\mathrm{e}}}
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||||
|
||||
%% Notes on Equations
|
||||
\newcommand{\shorteqnote}[1]{ & & \text{\small\llap{#1}}}
|
||||
|
||||
%% Typewriter Macros
|
||||
\newcommand{\sherpa}{\texttt{Sherpa}}
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||||
\newcommand{\rivet}{\texttt{Rivet}}
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||||
\newcommand{\vegas}{\texttt{VEGAS}}
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\newcommand{\lhapdf}{\texttt{LHAPDF6}}
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||||
\newcommand{\scipy}{\texttt{scipy}}
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|
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%% Sherpa Versions
|
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\newcommand{\oldsherpa}{\texttt{2.2.10}}
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\newcommand{\newsherpa}{\texttt{3.0.0} (unreleased)}
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||||
%% Special Names
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||||
\newcommand{\lhc}{\emph{LHC}}
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%% Expected Value and Variance
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\newcommand{\EX}[1]{\operatorname{E}\qty[#1]}
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\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]}
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||||
|
||||
%% Uppercase Rho
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\newcommand{\Rho}{P}
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||||
|
||||
%% Transverse Momentum
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\newcommand{\pt}[0]{p_\mathrm{T}}
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||||
|
||||
%% Sign Function
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\DeclareMathOperator{\sign}{sgn}
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|
||||
%% Stages
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||||
\newcommand{\stone}{\texttt{LO}}
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\newcommand{\sttwo}{\texttt{LO+PS}}
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\newcommand{\stthree}{\texttt{LO+PS+pT}}
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\newcommand{\stfour}{\texttt{LO+PS+pT+Hadr.}}
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\newcommand{\stfive}{\texttt{LO+PS+pT+Hadr.+MI}}
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%% GeV
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\newcommand{\gev}[1]{\SI{#1}{\giga\electronvolt}}
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\def\iu{\ensuremath{\mathrm{i}}}
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\def\i{\iu}
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\def\id{\ensuremath{\mathbb{1}}}
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\def\NN{\ensuremath{\mathbb{N}}}
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\def\RR{\ensuremath{\mathbb{R}}}
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\def\CC{\ensuremath{\mathbb{C}}}
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\def\dim{\ensuremath{\mathrm{dim}}}
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\def\hilb{\ensuremath{\mathcal{H}}}
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% fixme
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\newcommand{\fixme}[1]{\textbf{\textcolor{red}{FIXME:~#1}}}
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% HOPS/NMQSD
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\def\sys{\ensuremath{\mathrm{S}}}
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\def\bath{\ensuremath{\mathrm{B}}}
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\def\inter{\ensuremath{\mathrm{I}}}
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\def\nth{\ensuremath{^{(n)}}}
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\newcommand{\mat}[1]{\ensuremath{{\underline{\vb{#1}}}}}
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\def\kmat{{\mat{k}}}
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% Thermo
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\newcommand{\ergo}[1]{\ensuremath{\mathcal{W}\qty[#1]}}
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\newcommand{\qrelent}[2]{\ensuremath{S\qty(#1\,\middle|\middle|\,#2)}}
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@ -1,82 +0,0 @@
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\ProvidesPackage{hirostyle}
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\usepackage[utf8]{inputenc} % load early
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\usepackage[T1]{fontenc}
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% load early
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\usepackage[english]{babel}
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\usepackage{physics}
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\usepackage{graphicx, booktabs, float}
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\usepackage[tbtags]{mathtools}
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\usepackage{amssymb}
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\usepackage[backend=biber, language=english, style=phys]{biblatex}
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\usepackage{siunitx}
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\usepackage{caption}
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\usepackage[list=true, font=small,
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labelformat=brace, position=top]{subcaption}
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\usepackage{tikz}
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\usepackage{pgfplots}
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\usepackage{ifdraft}
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\usepackage[protrusion=true,expansion=true,tracking=true]{microtype}
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\usepackage{fancyvrb}
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\usepackage[autostyle=true]{csquotes}
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\usepackage{setspace}
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\usepackage{newunicodechar}
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\usepackage[pdfencoding=auto,hidelinks,colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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citecolor = black,
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urlcolor=cyan,]{hyperref} % load late
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\usepackage[capitalize]{cleveref}
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\usepackage{multirow,tabularx}
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\usepackage{bbold}
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\usepackage{scrhack}
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\usepackage{fontspec}
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\usepackage{unicode-math}
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\setmainfont{texgyrepagella}[
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Extension = .otf,
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UprightFont = *-regular,
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BoldFont = *-bold,
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ItalicFont = *-italic,
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BoldItalicFont = *-bolditalic,
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]
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\setmathfont{texgyrepagella-math.otf}
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\KOMAoptions{DIV=last}
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\usepackage[autooneside]{scrlayer-scrpage}
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%% use the current pgfplots
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\pgfplotsset{compat=1.16}
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%% Tikz
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\usetikzlibrary{arrows,shapes,angles,quotes,arrows.meta,external}
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\tikzexternalize[prefix=tikz/]
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%% Including plots
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\newcommand{\plot}[1]{%
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\ifdraft{\includegraphics[draft=false]{./figs/#1.pdf}}{\input{./figs/#1.pgf}}}
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\newcommand{\rivethist}[2][,]{%
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\includegraphics[draft=false,width=\textwidth,#1]{./figs/rivet/#2.pdf}}
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%% Including Results
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||||
\newcommand{\result}[1]{\input{./results/#1}\!}
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||||
%% SI units
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\sisetup{separate-uncertainty = true}
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%% Captions
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\captionsetup{justification=centering}
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%% Labels
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% \labelformat{chapter}{chapter~#1}
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% \labelformat{section}{section~#1}
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% \labelformat{figure}{figure~#1}
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% \labelformat{table}{table~#1}
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%% Cleverref
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\crefname{equation}{}{}
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\creflabelformat{equation}{(#2#1#3)}
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||||
%% Font for headings
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\addtokomafont{disposition}{\rmfamily}
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%% Minus Sign for Matplotlib
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\newunicodechar{−}{-}
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@ -1,31 +0,0 @@
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\documentclass[fontsize=10pt,paper=b5,open=any,
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,twoside=true,toc=listof,toc=bibliography,headings=optiontohead,
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captions=nooneline,captions=tableabove,english,DIV=calc,numbers=noenddot,final,parskip=half,
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headinclude=true,footinclude=false,BCOR=1cm]{scrbook}
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\usepackage{hirostyle}
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\usepackage{hiromacros}
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\addbibresource{references.bib}
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\synctex=1
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\title{Calculating heat flows with HOPS}
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\author{Valentin Boettcher}
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\date{\today}
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\begin{document}
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\maketitle
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\tableofcontents
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% Chapters
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\include{src/index.tex}
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\printbibliography{}
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\end{document}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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@ -1,2 +0,0 @@
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$pdf_mode = 4;
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@default_files = ('index.tex');
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File diff suppressed because it is too large
Load diff
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@ -1,800 +0,0 @@
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\section{Linear NMQSD, Zero Temperature}
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As in~\cite{Hartmann2017Dec} we choose
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\begin{equation}
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\label{eq:totalH}
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H = H_\sys + \underbrace{LB^† + L^† B}_{H_\inter} + H_\bath
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\end{equation}
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with the system hamiltonian \(H_\sys\), the bath hamiltonian
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||||
\begin{equation}
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\label{eq:bathh}
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||||
H_\bath = ∑_\lambda ω_\lambda a^† a,
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||||
\end{equation}
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the bath coupling system operator \(L\) and the bath coupling bath
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operator
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\begin{equation}
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\label{eq:bop}
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B=∑_{\lambda} g_{\lambda} a_{\lambda}
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||||
\end{equation}
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which define the interaction hamiltonian \(H_\inter\).
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||||
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We define the heat flow out of the system as in~\cite{Kato2015Aug}
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through
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\begin{equation}
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\label{eq:heatflowdef}
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J = - \dv{\ev{H_\bath}}{t}.
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\end{equation}
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Working, for now, in the Schr\"odinger picture the Ehrenfest theorem
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can be employed to find
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\begin{equation}
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\label{eq:ehrenfest}
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\i∂_t\ev{H_\bath} = \ev{[H_\bath,H]} = \ev{[H_\bath,H_\inter]}.
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||||
\end{equation}
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Thus, we need to calculate
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\begin{eqnarray}
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\label{eq:calccomm}
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\begin{aligned}
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[H_\bath,H_\inter] &= [H_\bath, LB^† + L^† B] \\
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&= L[H_\bath, B^† ] + L^† [H_\bath, B] \\
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&= L[H_\bath, B^† ] - \hc.
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||||
\end{aligned}
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||||
\end{eqnarray}
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||||
This checks out as the commutator has to be anti-hermitian due to
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\cref{eq:ehrenfest}.
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||||
Using \([H_\bath, B^† ]=∑_\lambda ω_\lambda g^\ast_\lambda
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a^†_\lambda\) it follows that
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\begin{equation}
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||||
\label{eq:expcomm}
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||||
\begin{aligned}
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||||
\ev{[H_\bath,H_\inter]} &= ∑_\lambda ω_\lambda g^\ast_\lambda
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\ev{La^†_\lambda} - \cc
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= ∑_\lambda ω_\lambda g^\ast_\lambda
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||||
\ev{La^†_\lambda \eu^{\i ω t}}_\inter - \cc\\
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||||
&= \frac{1}{\i}\ev{L∂_t{∑_\lambda
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g^\ast_\lambda a^†_\lambda \eu^{\i ω t}}}_\inter - \cc
|
||||
=\frac{1}{\i}\qty(\ev{L\dot{B}^†}_\inter + \cc)
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where we switched to the interaction picture with respect to \(H_\bath\)
|
||||
in keeping with the standard NMQSD formalism.
|
||||
In essence this is just the Heisenberg equation for \(H_\inter\). The
|
||||
expression for \(J\) follows
|
||||
\begin{equation}
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||||
\label{eq:final_flow}
|
||||
J(t) = \ev{L^†∂_t B(t) + L∂_t B^†(t)}_\inter.
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||||
\end{equation}
|
||||
|
||||
From this point on, we will assume the interaction picture and drop the
|
||||
\(I\) subscript.
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||||
|
||||
The two summands yield different expressions in terms of the NMQSD.
|
||||
For use with HOPS with the final goal of utilizing the auxiliary
|
||||
states the expression \(\ev{L^†∂_t B(t)}\) should be evaluated. When
|
||||
considering the complex conjugate of this expression, we find a
|
||||
formula involving the derivative of the driving stochastic
|
||||
process. This is undesirable as it does not exist for all bath
|
||||
correlation functions\footnote{Only for BCFs that are smooth at
|
||||
\(τ=0\).} and expressions involving the process directly are alleged
|
||||
to converge slower. The last fact may be explained by the fact, that
|
||||
one needs quite a lot of sample paths of the process for the mean of
|
||||
those sample paths to converge to zero. On the other hand, the first
|
||||
hierarchy states do contain an integral of-sorts of the sample paths
|
||||
and are not as sensitive to fluctuations.
|
||||
|
||||
We calculate
|
||||
\begin{equation}
|
||||
\label{eq:interactev}
|
||||
\ev{L^†∂_t B(t)}=\ev{L^†∂_t B(t)}{\psi(t)} =
|
||||
∫ \braket{\psi(t)}{z}\mel{z}{L^†∂_tB(t)}{\psi(t)}\frac{\dd[2]{z}}{\pi^N},
|
||||
\end{equation}
|
||||
where \(N\) is the total number of environment oscillators and
|
||||
\(z=\qty(z_{\lambda_1}, z_{\lambda_2}, \ldots)\).
|
||||
To that end,
|
||||
\begin{equation}
|
||||
\label{eq:nmqsdficate}
|
||||
\begin{aligned}
|
||||
\mel{z}{∂_tB(t)}{\psi(t)} &= ∑_\lambda g_\lambda
|
||||
\qty(∂_t \eu^{-\iω_\lambda
|
||||
t})∂_{z^\ast_\lambda}\ket{\psi(z^\ast,t)} \\
|
||||
&= ∫_0^t ∑_\lambda g_\lambda
|
||||
\qty(∂_t \eu^{-\iω_\lambda
|
||||
t})\pdv{η_s^\ast}{z^\ast_\lambda}\fdv{\ket{\psi(z^\ast,t)}}{η^\ast_s}\dd{s}\\
|
||||
&= -\i∫_0^t\dot{\alpha}(t-s)\fdv{\ket{\psi(z^\ast,t)}}{η^\ast_s}\dd{s},
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(η^\ast_t\equiv -\i ∑_\lambda g^\ast_\lambda
|
||||
z^\ast_\lambda \eu^{\iω_\lambda t}\).
|
||||
With this we can write
|
||||
\begin{equation}
|
||||
\label{eq:steptoproc}
|
||||
\ev{L^†∂_t B(t)} = -\i \mathcal{M}_{η^\ast}\bra{\psi(η,
|
||||
t)}L^†∫_0^t\dd{s} \dot{\alpha}(t-s)\fdv{η^\ast_s} \ket{\psi(η^\ast,t)}.
|
||||
\end{equation}
|
||||
Defining
|
||||
\begin{equation}
|
||||
\label{eq:defdop}
|
||||
D_t = ∫_0^t\dd{s} \alpha(t-s)\fdv{η^\ast_s}
|
||||
\end{equation}
|
||||
as in~\cite{Suess2014Oct} we can write
|
||||
\begin{equation}
|
||||
\label{eq:final_flow_nmqsd}
|
||||
J(t) = -\i \mathcal{M}_{η^\ast}\bra{\psi(η,
|
||||
t)}L^†\dot{D}_t\ket{\psi(η^\ast,t)} + \cc,
|
||||
\end{equation}
|
||||
where we've used that the integral in \(D_t\) can be expanded over the
|
||||
whole real axis. If we assume \(\alpha = \exp(-w t)\) then
|
||||
\begin{equation}
|
||||
\label{eq:hopsj}
|
||||
J(t) = \i \mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,
|
||||
t)}wL^†\ket{\psi^{(1)}(η^\ast,t)} + \cc.,
|
||||
\end{equation}
|
||||
where \(\ket{\psi^{(1)}(η^\ast,t)}\) is the first HOPS hierarchy
|
||||
state. This can be generalized to any BCF that is a sum of exponentials.
|
||||
|
||||
Interestingly one finds that
|
||||
\begin{equation}
|
||||
\label{eq:alternative}
|
||||
\ev{L∂_t B^†(t)} = \i∫\frac{\dd[2]{z}}{\pi^N}
|
||||
\dot{η}_t^\ast \mel{\psi(η,t)}{L}{\psi(η^\ast,t)}.
|
||||
\end{equation}
|
||||
However, this approach becomes more complicated in the nonlinear
|
||||
method.
|
||||
The previous expression has the advantage
|
||||
that we utilize the first hierarchy states that are already being
|
||||
calculated as a byproduct.
|
||||
|
||||
In the language of~\cite{Hartmann2021Aug} we can generalize to
|
||||
\(\alpha(t) = ∑_i G_i \eu^{-W_i t}\) and thus
|
||||
\begin{equation}
|
||||
\label{eq:hopsflowrich}
|
||||
J(t) = ∑_\mu\frac{G_\mu W_\mu}{\bar{g}_\mu} \i\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,
|
||||
t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc,
|
||||
\end{equation}
|
||||
where \(\psi^{\vb{e}_\mu}\) is the \(\mu\)-th state of the first
|
||||
hierarchy and \(\bar{g}_\mu\) is an arbitrary scaling introduced in
|
||||
the definition of the hierarchy in~\cite{Hartmann2021Aug} to help with
|
||||
the scaling of the norm.
|
||||
|
||||
With the new ``fock-space'' normalization however the expression
|
||||
becomes
|
||||
\begin{equation}
|
||||
\label{eq:hopsflowfock}
|
||||
J(t) = - ∑_\mu\sqrt{G_\mu}W_\mu
|
||||
\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,
|
||||
t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc.
|
||||
\end{equation}
|
||||
|
||||
|
||||
|
||||
|
||||
\section{Nonlinear NMQSD, Zero Temperature}
|
||||
\label{sec:nonlin}
|
||||
In the spirit of the usual derivation of the nonlinear NMQSD we write
|
||||
\begin{equation}
|
||||
\label{eq:newb}
|
||||
\begin{aligned}
|
||||
\ev{L^†\dot{B}(t)} &= ∫ \frac{\dd[2]{z}}{\pi^N} \eu^{-\abs{z}^2}
|
||||
\braket{\psi}{z}\!\braket{z}{\psi}
|
||||
\frac{\braket{\psi(t)}{z}\!\mel{z}{L^†\dot{B}(t)}{\psi(t)}}{\braket{\psi}{z}\!\braket{z}{\psi}}
|
||||
\\
|
||||
&= ∫ \frac{\dd[2]{z}}{\pi^N} \eu^{-\abs{z}^2}
|
||||
\frac{\mel{z(t)}{L^†\dot{B}(t)}{\psi(t)}}{\braket{\psi}{z(t)}\!\braket{z(t)}{\psi}},
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(z_{\lambda}^{*}(t)=z_{\lambda}^{*}+\i g_{\lambda} ∫_{0}^{t}
|
||||
\dd{s} \eu^{-\i ω_{\lambda} s}\ev{L^†}_{s}\).
|
||||
We find that next steps are the same as in \cref{sec:nonlin} by noting
|
||||
\begin{equation}
|
||||
\label{eq:deriv_trick}
|
||||
\eval{∂_{z^\ast_\lambda}}_{z^\ast=z_\lambda^\ast(t)} =
|
||||
∫_0^t\dd{s}\eval{\pdv{η^\ast_s}{z^\ast_\lambda}}_{z^\ast=z^\ast_\lambda(t)}
|
||||
\fdv{}{η^\ast_s(z^\ast=z^\ast(t))} =
|
||||
∫_0^t\dd{s}\eval{\pdv{η^\ast_s}{z^\ast_\lambda}}_{z^\ast=z^\ast(0)}
|
||||
\fdv{}{η^\ast_s(z^\ast=z^\ast(t))},
|
||||
\end{equation}
|
||||
which does alter the definition of \(D_t\) but results in the same
|
||||
HOPS equations.
|
||||
The shifted process \(\tilde{η}^\ast=
|
||||
η^\ast(z^\ast(t),t)=η^\ast(t) +
|
||||
∫_0^t\dd{s}\alpha^\ast(t-s)\ev{L^†}_{\psi_s}\) appears directly
|
||||
in the NMQSD equation but results only in a slight change in the
|
||||
functional derivative. Note however that
|
||||
\begin{equation}
|
||||
\label{eq:fdvclarification}
|
||||
\fdv{}{η^\ast_s(z^\ast=z^\ast(t))} \neq \fdv{}{\tilde{η}^\ast_s}
|
||||
\end{equation}
|
||||
which is not problematic as we have (implicit in~\cite{Diosi1998Mar})
|
||||
\begin{equation}
|
||||
\label{eq:fdvhops}
|
||||
\fdv{}{η^\ast_s(z^\ast=z^\ast(t))} \ket{\psi(z^\ast)} = \fdv{}{η^\ast_s}\ket{\psi(z^\ast(t, z^\ast_0), t)}
|
||||
\end{equation}
|
||||
so that the usual HOPS hierarchy follows. Note \(z^\ast_0 = z^\ast(0)\).
|
||||
|
||||
Therefore,
|
||||
\begin{equation}
|
||||
\label{eq:newbcontin}
|
||||
J(t) =
|
||||
-\i
|
||||
\mathcal{M}_{\tilde{η}^\ast}\frac{\mel{\psi(\tilde{η},t)}{L^†\dot{\tilde{D}}_t}{\psi(\tilde{η}^\ast,t)}}{\braket{\psi(\tilde{η},t)}{\psi(\tilde{η}^\ast,t)}}
|
||||
+ \cc,
|
||||
\end{equation}
|
||||
where the dependence on \(\tilde{η}\) is symbolic and to be
|
||||
understood in the context of \cref{eq:fdvhops}.
|
||||
|
||||
Again we express the result in the language of~\cite{Hartmann2021Aug}
|
||||
to obtain
|
||||
\begin{equation}
|
||||
\label{eq:nonlinhopsflowrich}
|
||||
J(t) = ∑_\mu\frac{G_\mu W_\mu}{\bar{g}_\mu}
|
||||
\i\mathcal{M}_{η^\ast}\frac{\bra{\psi^{(0)}(η,
|
||||
t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)}}{\bra{\psi^{(0)}(η,
|
||||
t)}\ket{\psi^{0}(η^\ast,t)}} + \cc.
|
||||
\end{equation}
|
||||
|
||||
With the new ``fock-space'' normalization however the expression
|
||||
becomes
|
||||
\begin{equation}
|
||||
\label{eq:nonlinhopsflowfock}
|
||||
J(t) = - ∑_\mu\sqrt{G_\mu}W_\mu
|
||||
\mathcal{M}_{η^\ast}\frac{\bra{\psi^{(0)}(η,
|
||||
t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)}}{\bra{\psi^{(0)}(η,
|
||||
t)}\ket{\psi^{0}(η^\ast,t)}} + \cc.
|
||||
\end{equation}
|
||||
|
||||
\section{Linear Theory, Finite Temperature}
|
||||
The finite temperature case needs some additional considerations as
|
||||
the previous sections dealt explicitly with mean values in a pure
|
||||
state. The Ehrenfest theorem still holds in mixed states, but we would
|
||||
like to recover the usual pure state zero temperature formalism. There
|
||||
are multiple methods for dealing with a thermal initial such as the
|
||||
thermofield method (see~\cite{Diosi1998Mar}), but because the results
|
||||
discussed here are to be applied with the HOPS method we shall use the
|
||||
method described in~\cite{Hartmann2017Dec}.
|
||||
|
||||
The shift operator
|
||||
\begin{equation}
|
||||
\label{eq:shiftop}
|
||||
\vb{D}(y) = \bigotimes_\lambda \eu^{y_\lambda a_\lambda^†-y^\ast_\lambda a_\lambda}
|
||||
\end{equation}
|
||||
the ground state of the environment into an arbitrary
|
||||
coherent state
|
||||
\begin{equation}
|
||||
\label{eq:shiftwork}
|
||||
\vb{D}(y)\ket{0} = \ket{y}
|
||||
\end{equation}
|
||||
where \(y=(y_1,y_2,\ldots)\) as usual.
|
||||
|
||||
This allows us to write the density matrix of the system with a
|
||||
thermal initial bath as
|
||||
\begin{equation}
|
||||
\label{eq:shiftbath}
|
||||
\rho =
|
||||
\prod_\lambda\qty(∫\dd[2]{y_\lambda}
|
||||
\frac{\eu^{-\abs{y_\lambda}^2\bar{n}_\lambda}}{\pi\bar{n}_\lambda})
|
||||
U(t)\vb{D}(y)\ketbra{\psi}\otimes\ketbra{0}\vb{D}(y)^† U(t)^†.
|
||||
\end{equation}
|
||||
The usual step is now to insert \(\id =\vb{D}(y)\vb{D}^†(y)\) to
|
||||
arrive at a new time translation operator
|
||||
\begin{equation}
|
||||
\label{eq:utilde}
|
||||
\tilde{U}(t) = \vb{D}^†(y)U(t)\vb{D}(y)
|
||||
\end{equation}
|
||||
and to interpret the integral in \cref{eq:shiftbath} in a monte-carlo
|
||||
sense which leads to a stochastic contribution to the system Hamiltonian
|
||||
\begin{equation}
|
||||
\label{eq:thermalh}
|
||||
H_{\mathrm{sys}}^{\mathrm{shift}}=L ξ^{*}(t)+L^{†} ξ(t)
|
||||
\end{equation}
|
||||
with the stochastic process
|
||||
\begin{equation}
|
||||
\label{eq:xiproc}
|
||||
ξ(t):=∑_{\lambda} g_{\lambda} y_{\lambda} \eu^{-\mathrm{i} ω_{\lambda} t}
|
||||
\end{equation}
|
||||
with corresponding moments \(\mathcal{M}(ξ(t))=0=\mathcal{M}(ξ(t) ξ(s))\) and
|
||||
\[
|
||||
\mathcal{M}\left(ξ(t) ξ^{*}(s)\right)=\frac{1}{\pi} ∫_{0}^{∞} \mathrm{d} ω \bar{n}(\beta ω) J(ω) e^{-\mathrm{i} ω(t-s)}.
|
||||
\]
|
||||
Remember that we want to calculate
|
||||
\begin{equation}
|
||||
\label{eq:whatreallymatters}
|
||||
\begin{aligned}
|
||||
\ev{L^†\dot{B}(t)} &= \tr[L^†\dot{B}(t)\rho(t)] \\
|
||||
&=\prod_\lambda\qty(∫\dd[2]{y_\lambda}
|
||||
\frac{\eu^{-\abs{y_\lambda}^2\bar{n}_\lambda}}{\pi\bar{n}_\lambda})\tr[L^†\dot{B}(t)
|
||||
U(t)\vb{D}(y)\ketbra{\psi}\otimes\ketbra{0}\vb{D}(y)^† U(t)^†] .
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
To recover the zero temperature formulation of this expectation value we
|
||||
again insert a \(\id\), but have to commute \(\vb{D}(y)^†\) past
|
||||
\(\dot{B}(t)\). This leads to the expression
|
||||
\begin{equation}
|
||||
\label{eq:pureagain}
|
||||
\begin{aligned}
|
||||
\ev{L^†\dot{B}(t)} &=\prod_\lambda\qty(∫\dd[2]{y_\lambda}
|
||||
\frac{\eu^{-\abs{y_\lambda}^2\bar{n}_\lambda}}{\pi\bar{n}_\lambda})\\
|
||||
&\qquad\times\tr[L^†(\dot{B}(t) + \dot{ξ}(t))
|
||||
\vb{D}^†(y) U(t)\vb{D}(y)\ketbra{\psi}\otimes\ketbra{0}\vb{D}^†(y)U(t)^†\vb{D}(y)] \\
|
||||
&=\prod_\lambda\qty(∫\dd[2]{y_\lambda}
|
||||
\frac{\eu^{-\abs{y_\lambda}^2\bar{n}_\lambda}}{\pi\bar{n}_\lambda})\tr[L^†\qty{\dot{B}(t) + \dot{ξ}(t)}
|
||||
\tilde{U}(t)\ketbra{\psi}\otimes\ketbra{0} \tilde{U}(t)^†].
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
which returns us to the zero temperature formalism with a transformed
|
||||
Hamiltonian and the replacement
|
||||
\begin{eqnarray}
|
||||
\label{eq:breplacement}
|
||||
B(t) \rightarrow B(t) + ξ(t)
|
||||
\end{eqnarray}
|
||||
which plausibly corresponds to the \(L^†\) part of \(H_\inter + H_{\mathrm{sys}}^{\mathrm{shift}}\).
|
||||
|
||||
The appearance of \(\dot{ξ}(t)\) may cause concern. However, for
|
||||
twice differentiable \(\mathcal{M}(ξ(t)ξ^\ast(s))\) the sample
|
||||
trajectories are smooth.
|
||||
|
||||
Alternatively we can calculate
|
||||
\begin{equation}
|
||||
\label{eq:gettingarounddot}
|
||||
\begin{aligned}
|
||||
\ev{\dot{H}_{\mathrm{sys}}^{\mathrm{shift}}} &=
|
||||
\dv{\ev{H_{\mathrm{sys}}^{\mathrm{shift}}}}{t} -
|
||||
\frac{1}{\iu}\qty(\ev{H_{\mathrm{sys}}^{\mathrm{shift}}H} -\ev{H
|
||||
H_{\mathrm{sys}}^{\mathrm{shift}}}) \\
|
||||
&=\dv{\ev{H_{\mathrm{sys}}^{\mathrm{shift}}}}{t} -
|
||||
\frac{1}{\iu}\ev{[H_{\mathrm{sys}}^{\mathrm{shift}}, H]} \\
|
||||
&=\dv{\ev{H_{\mathrm{sys}}^{\mathrm{shift}}}}{t} -
|
||||
\frac{1}{\iu}\ev{[H_{\mathrm{sys}}^{\mathrm{shift}}, H_\inter]}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
|
||||
Now,
|
||||
\begin{equation}
|
||||
\label{eq:hshcomm}
|
||||
[H_{\mathrm{sys}}^{\mathrm{shift}}, H_\inter] = ξ(t) [L^†, L]
|
||||
B(t)^† + ξ^\ast(t) [L, L^†] B
|
||||
\end{equation}
|
||||
and therefore
|
||||
\begin{equation}
|
||||
\label{eq:finalex}
|
||||
\ev{[H_{\mathrm{sys}}^{\mathrm{shift}}, H_\inter]} = -i \mathcal{M}_{η^\ast}\mel{\psi}{ξ(t)^\ast[L,L^†]D_t}{\psi}.
|
||||
\end{equation}
|
||||
This is an expression that we can easily evaluate with the HOPS
|
||||
method.
|
||||
|
||||
\section{Interaction Energy}
|
||||
\label{sec:intener}
|
||||
|
||||
By replacing the \(B(t)\) operators in \(H_\inter\) with derivatives as in
|
||||
the above considerations we obtain an expression for the expectation
|
||||
value of the interaction energy.
|
||||
|
||||
We have to find an expression for \(\ev{L^†B(t)}\)
|
||||
or its complex conjugate which would lead to an expression involving
|
||||
the driving stochastic process which is undesirable as discussed above.
|
||||
This is easily done by following the arguments in the previous
|
||||
chapters but omitting the time derivative.
|
||||
|
||||
For the most general case at zero temperature and for the nonlinear
|
||||
method we arrive at
|
||||
\begin{equation}
|
||||
\label{eq:intexp}
|
||||
\ev{H_\inter} =
|
||||
-\i
|
||||
\mathcal{M}_{\tilde{η}^\ast}\frac{\mel{\psi(\tilde{η},t)}{L^†\tilde{D}_t}{\psi(\tilde{η}^\ast,t)}}{\braket{\psi(\tilde{η},t)}{\psi(\tilde{η}^\ast,t)}}
|
||||
+ \cc.
|
||||
\end{equation}
|
||||
See \cref{eq:newbcontin} for an explanation of the constituents of
|
||||
that equation. The expression for the linear method is obtained by
|
||||
simply leaving out the normalization.
|
||||
|
||||
For nonzero temperature an extra term
|
||||
\begin{equation}
|
||||
\label{eq:interexptherm}
|
||||
\mathcal{M}_{\tilde{η}^\ast}\frac{\mel{\psi(\tilde{η},t)}{L^†ξ(t)}{\psi(\tilde{η}^\ast,t)}}{\braket{\psi(\tilde{η},t)}{\psi(\tilde{η}^\ast,t)}}
|
||||
+ \cc
|
||||
\end{equation}
|
||||
has to be added to \cref{eq:intexp}, where \(ξ\) is the thermal
|
||||
stochastic process.
|
||||
|
||||
|
||||
\section{Multiple Baths}
|
||||
\label{sec:multibath}
|
||||
|
||||
For the models we consider in \fixme{citation,reference}, we have
|
||||
\([H_\bath^{(i)}, H_\bath^{(j)}] = 0\), where \(i,j\) are the bath
|
||||
indices. Therefore, we can apply the formalism of the previous
|
||||
sections almost unchanged, by just taking care that all quantities
|
||||
involved in the expression of \(J_n=-\dv{\ev{H_B^{(n)}}}{t}\) refer to
|
||||
the \(n\)th bath.
|
||||
|
||||
This essentially boils down to the replacement
|
||||
\begin{equation}
|
||||
\label{eq:replacements}
|
||||
\begin{aligned}
|
||||
D_t &\rightarrow D_t^{(n)} \equiv
|
||||
∫_0^t\dd{s}α_n(t-s)\fdv{η^\ast_n(s)} \\
|
||||
ξ(t) &\rightarrow ξ_n(t)\equiv∑_{\lambda} g^{(n)}_{\lambda}
|
||||
y_{\lambda} \eu^{-\mathrm{i} ω^{(n)}_{\lambda} t},
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where the quantities involved are as in \fixme{reference} and
|
||||
\cref{eq:xiproc}.
|
||||
|
||||
\section{Pure Dephasing: The initial Slip}
|
||||
\label{sec:pure_deph}
|
||||
As seen in \fixme{include plots}, the short time behavior of the bath
|
||||
energy flow is dominated by characteristic peak at short
|
||||
times. Because this peak occurs at very short time scales, it may in
|
||||
part be explained by a simple calculation which neglects the system
|
||||
dynamics, setting \(H_\sys=0\).
|
||||
|
||||
We solve the model with the Hamiltonian (Schr\"odinger picture)
|
||||
\begin{equation}
|
||||
\label{eq:puredeph}
|
||||
H = L^†(t) B + L(t) B^† + H_\bath
|
||||
\end{equation}
|
||||
with \(L(t)=L(t)^†\), \([L(t), L(s)] = 0\;\forall t,s\) (so that
|
||||
Heisenberg Hamiltonian matches \cref{eq:puredeph}) and \(B,H_\bath\)
|
||||
as in \cref{eq:bop}.
|
||||
|
||||
Because \([L,H]=0\) we can immediately solve \(L_H(t)=L_S(t)\), where
|
||||
the subscript signify the Heisenberg and Schr\"odinger pictures
|
||||
respectively. The Heisenberg equations for the \(a_λ\) yield
|
||||
\begin{equation}
|
||||
\label{eq:alapuredeph}
|
||||
a_λ(t) = a_λ(0) \eu^{-\iu ω_λ t} - \iu g_λ^\ast∫_0^t\dd{s} L(s)
|
||||
\eu^{-\iu ω_λ (t-s)}.
|
||||
\end{equation}
|
||||
|
||||
This allows us to calculate
|
||||
\begin{equation}
|
||||
\label{eq:pureflow}
|
||||
\dot{H}_\bath = - ∑_λ g_λ L(t) \qty[∂_t a_λ(0) \eu^{\iu ω_λ t} - \iu
|
||||
g_λ^\ast∫_0^t\dd{s} L(s) ∂_t \eu^{-\iu ω_λ (t-s)}] + \hc,
|
||||
\end{equation}
|
||||
which gives with a state of the form \(ρ=\ketbra{ψ} \otimes ρ_β\)
|
||||
(\(ρ_β\) being a thermal state)
|
||||
\begin{equation}
|
||||
\label{eq:pureflowexpectation}
|
||||
\ev{\dot{H}_\bath } = -2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[\dot{α}(t-s)].
|
||||
\end{equation}
|
||||
|
||||
For time independent \(L\) this becomes
|
||||
\begin{equation}
|
||||
\label{eq:pureflowtimeindep}
|
||||
\ev{\dot{H}_\bath } = 2 \ev{L^2} \Im[\dot{α}(t)].
|
||||
\end{equation}
|
||||
|
||||
The proportionality to the imaginary BCF \(α\) does explain the
|
||||
initial peak in the bath energy flow. The imaginary part of the BCF is
|
||||
zero for \(t=0\) and then usually features a peak at rather short
|
||||
times (assuming finite correlation times). For the ohmic BCF used
|
||||
here, this feature is very prominent.
|
||||
\fixme{insert graph}
|
||||
|
||||
Interestingly, \cref{eq:pureflowexpectation} does not contain any
|
||||
reference to the temperature of the bath. Therefore, the bath energy
|
||||
can only surpass its initial value in this model, as the dynamics
|
||||
match the zero temperature case in which the bath has minimal energy
|
||||
in the initial state. A thermodynamically useful model should
|
||||
therefore feature an significant system dynamics that do not commute
|
||||
with the interaction or fast modulation so that the Hamiltonian does
|
||||
not commute with itself at different times. The latter may induce
|
||||
deviations from the pure-dephasing behavior at very short time scales
|
||||
and thus be useful for finite power output. \fixme{here the plot with
|
||||
energy extraction would be good.} Coupling that is not self-adjoint
|
||||
\fixme{plot} may also have this effect, but may be harder to
|
||||
physically motivate. For the spin-boson system it is the result of the
|
||||
random wave approximation, which however does not imply weak
|
||||
coupling~\cite{Irish2007Oct}.
|
||||
|
||||
For completeness, the interaction energy is given by
|
||||
\begin{equation}
|
||||
\label{eq:pureinter}
|
||||
H_\inter = L(t)\qty[∑_λg_λ\qty(a_λ(0)\eu^{-\i ω_λ t} - \i
|
||||
g^\ast_λ∫_0^t\dd{s} L(s) \eu^{\i ω_λ (t-s)})] + \hc,
|
||||
\end{equation}
|
||||
yielding
|
||||
\begin{equation}
|
||||
\label{eq:pureinterexp}
|
||||
\ev{H_\inter} = 2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[α(t-s)].
|
||||
\end{equation}
|
||||
\fixme{plots}
|
||||
|
||||
It is useful to normalize the BCF based on \cref{eq:pureinterexp}, so
|
||||
that the pure interaction energy build-up in the initial slip is
|
||||
canceled. To make the normalization independent of \(L(t)\),
|
||||
we choose the normalization to be
|
||||
\begin{equation}
|
||||
\label{eq:bcfnorm}
|
||||
\begin{aligned}
|
||||
\mathcal{N} &= 2 \abs{\frac{\max_t\norm{L(t)L^\dag(t)+\hc}}{\max_t{\norm{H(t)}}} ∫_0^∞ \Im[α_u(τ)]\dd{τ}}\\
|
||||
α(τ) &= α_u(τ)/\mathcal{N},
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(α_u\) is some unnormalized BCF. This normalization has the
|
||||
useful property, that it neutralizes any scaling in \(L\). Note that
|
||||
here the convention in which \(α\) is dimensionless is used.
|
||||
|
||||
% this is not true
|
||||
% imaginary part becomes proportional to the Dirac delta in the limit
|
||||
% where typical cutoff frequency \(ω_c\rightarrow ∞\). The integral over
|
||||
% the real part of \(α\) always gives zero if the spectral density obeys
|
||||
% \(J(0) = 0\) and tends to exhibit fast oscillations and fast decay in
|
||||
% the large-cutoff limit. For weak coupling, it may therefore be
|
||||
% neglected. This constitutes the Markov limit mentioned in
|
||||
% \cite{Strunz2001Habil}.
|
||||
|
||||
The Ohmic-type BCF is
|
||||
\begin{equation}
|
||||
\label{eq:normohmic}
|
||||
α(τ)=\frac{ω_c s }{ (\max_t\norm{H})(1+\iu ω_c τ)^{s+1}},
|
||||
\end{equation}
|
||||
in this normalization. Note however, that the norm of the Hamiltonian
|
||||
is assumed to be unity in the simulations referred to in this
|
||||
thesis. \fixme{maybe change}
|
||||
|
||||
\section{Ergotropy and Basic Thermodynamics of Open Systems}
|
||||
The ergotropy of a \emph{} quantum system is defined
|
||||
as~\cite{Binder2018}
|
||||
\begin{equation}
|
||||
\label{eq:ergo_def}
|
||||
\ergo{ρ} = \max_{U\,\text{unitary}}\tr[\qty(ρ - UρU^\dag) H],
|
||||
\end{equation}
|
||||
which is the maximal energy that can be extracted from a system through
|
||||
cyclic modulation of the Hamiltonian \(H\). A state is called passive
|
||||
iff the maximizing \(U\) \cref{eq:ergo_def} is the identity \(\id\).
|
||||
|
||||
A passive state \(ρ_P\) is always diagonal in the eigenbasis of \(H\) and its
|
||||
eigenvalues satisfy the following ordering condition~\cite{Lenard1978Dec}
|
||||
\begin{equation}
|
||||
\label{eq:passive_diag}
|
||||
ρ_{p}=∑_{j=1}^{n} \lambda_{j}|j\rangle\langle j|, \quad E_{j} \leq E_{j+1}, \quad \lambda_{j+1} \leq \lambda_{j},
|
||||
\end{equation}
|
||||
where \(n<∞\) is the Hilbert space dimension. This condition is both
|
||||
necessary and sufficient. Examples of passive states are the state of
|
||||
the micro-canonical ensemble or a Gibbs state. Gibbs are further
|
||||
distinguished by additional features as described
|
||||
in~\cite{Lenard1978Dec}, which can be connected to formulations of the
|
||||
zeroth and second laws of thermodynamics.
|
||||
|
||||
One of these properties is complete passivity. Completely passive
|
||||
states remain passive under the transformation \(ρ\to\otimes^Nρ\) (and
|
||||
an \(N\)-fold sum of the Hamiltonian) for finite \(N\). Therefore no
|
||||
energy can be extracted from multiple identical systems at the same
|
||||
temperature. For finite dimensional systems, the complete passivity
|
||||
implies the form of the Gibbs state. The open-systems case differs as
|
||||
here a ``small'' system is coupled to a bath of infinite size. If the
|
||||
system state is not a Gibbs state, the whole system becomes
|
||||
non-passive, even if the system state is passive with respect to the
|
||||
system Hamiltonian\footnote{for example being the ground state}.
|
||||
|
||||
For systems of infinite size, states fulfilling the
|
||||
Kubo–Martin–Schwinger (KMS) condition have been proposed as the
|
||||
generalizations of Gibbs states, having similar properties as
|
||||
Gibbs states. Under some conditions passivity implies the KMS
|
||||
condition. These conditions are related to the fact that KMS states
|
||||
are not necessarily unique~\cite{Binder2018,Pusz1978Oct}.
|
||||
|
||||
The KMS condition is stated for two arbitrary observables \(A,B\) and
|
||||
\(F_{AB}(t)=\tr[ρ_βA(t)B(0)]\) (Heisenberg picture,
|
||||
\(A(t)=\eu^{\iu H t}H\eu^{-\iu H t}\)) as
|
||||
\begin{equation}
|
||||
\label{eq:kmscond}
|
||||
F_{AB}(-t) = F_{BA}(t-\iu β)
|
||||
\end{equation}
|
||||
by virtue of analytic continuation.
|
||||
|
||||
For two initially uncorrelated KMS states, of different
|
||||
temperature, the Carnot efficiency bound can be
|
||||
proven~\cite{Pusz1978Oct}.
|
||||
|
||||
A simple application of ergotropy is an explanation for quantum
|
||||
friction. The buildup of coherence\footnote{Meaning a state which is
|
||||
non-diagonal in the energy basis.} in a quantum system makes the
|
||||
state non-passive and thus requires additional energy which cannot be
|
||||
extracted by modulating of the energy level gaps of the
|
||||
system\footnote{This is the usual mechanism of energy extraction in a
|
||||
quantum Otto cycle~\cite{Geva1992Feb}.}~\cite{Kurizki2021Dec}. The
|
||||
reduction of efficiency in through quantum coherence general has been
|
||||
termed quantum friction. However, the occurrence of coherence does not
|
||||
have to lead to a reduction in efficiency\fixme{do more research on
|
||||
that.refer to simulations}, if a diagonal state is restored \footnote{Shortcuts to
|
||||
adiabaticity, see for example~\cite{Chen2010Feb}.}.
|
||||
|
||||
Let us consider models with the Hamiltonians
|
||||
\begin{equation}
|
||||
\label{eq:simple_bath_models}
|
||||
H = \id_\sys\otimes H_\bath + H_\sys\otimes \id_\bath,
|
||||
\end{equation}
|
||||
where the system \(\sys\) is finite dimensional and \(H_\bath\) may
|
||||
chosen arbitrarily. Let the initial state of the system be
|
||||
\begin{equation}
|
||||
\label{eq:simple_initial_state}
|
||||
ρ=ρ_\sys\otimes τ_β,
|
||||
\end{equation}
|
||||
where \(τ_β=\eu^{-β H_\bath}/Z\) and \(ρ_\sys\) is arbitrary.
|
||||
|
||||
An interesting question is whether the ergotropy of such a state is
|
||||
finite. This amounts to the formulation of the second law: ``No energy
|
||||
may be extracted from a single bath in a cyclical manner''.
|
||||
|
||||
For systems obeying GKSL dynamics connected to a KMS state heat bath,
|
||||
thermodynamic laws can be derived in certain situations\footnote{very
|
||||
slow or very fast modulation of the system
|
||||
hamiltonian}\cite{Binder2018}, which imply the answer ``yes'' for the
|
||||
above questions. In the non-Markovian case, those arguments do not
|
||||
hold anymore.
|
||||
|
||||
For finite dimensional baths, we always have finite ergotropies, as
|
||||
their Hamiltonians are bounded. In the infinite dimensional case, we
|
||||
may expect that the ergotropy is still finite for some models, as long
|
||||
as the energies of the thermal states for those models is finite. This
|
||||
assumption breaks down when we consider infinite baths, whose thermal
|
||||
energy is unbounded even for finite temperatures.
|
||||
|
||||
Nevertheless, \fixme{graphics} the ergotropy appears to be
|
||||
bounded. Further, the system as if it was in a passive state as soon
|
||||
as the limit cycle is reached. In fact, there is a simple and general
|
||||
argument that provides and upper bound on the ergotropy of states of
|
||||
the form~\cref{eq:simple_initial_state} based on the special form of
|
||||
Gibbs states and relative entropy. The latter quantity allows the
|
||||
application of quantum informational tools, even in the presence of
|
||||
infinite baths if we are careful in taking limits.
|
||||
|
||||
The following is adapted
|
||||
from~\cite{Biswas2022May,Alicki2013Apr,Lobejko2021Feb} and we limit
|
||||
ourselves to finite dimensional problems for now. As unitary
|
||||
transformations leave the entropy invariant
|
||||
(\(\tr[ρ\ln(ρ)] = \tr[ρ_P\ln(ρ_P)]\)), we have for an arbitrary
|
||||
\(β > 0\) and \(ρ_β=\exp(-βH)/Z\)
|
||||
\begin{equation}
|
||||
\label{eq:ergo_entro}
|
||||
\begin{aligned}
|
||||
\ergo{ρ} &= E(ρ) - E(ρ_P) = \tr[(ρ-ρ_P) H] = -\frac{1}{β}\tr[(ρ-ρ_P)
|
||||
\qty(\ln(ρ_β) + \ln(Z))] \\
|
||||
&= -\frac{1}{β}\tr[(ρ-ρ_P) \ln(ρ_β)] =
|
||||
-\frac{1}{β}\tr[(ρ-ρ_P) \qty(\ln(ρ_β))]\\
|
||||
&=\frac{1}{β}\qty[\tr[ρ(\ln(ρ) - \ln(ρ_β))] -
|
||||
\tr[ρ_P(\ln(ρ_p) - \ln(ρ_β))]]\\
|
||||
&\equiv\frac{1}{β}\qty[\qrelent{ρ}{ρ_β} - \qrelent{ρ_P}{ρ_β}],
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where we have used \(\tr[ρ]=\tr[ρ_P]=1\). The relative entropies
|
||||
appearing in \cref{eq:ergo_entro} are always finite, as \(ρ\) is
|
||||
finite-dimensional and \(ρ_β\) has full rank. As energy is minimized
|
||||
by a Gibbs state when keeping the entropy fixed, we find an upper
|
||||
bound on the ergotropy by replacing \(ρ_P\to ρ_{β^\ast}\) in
|
||||
\cref{eq:ergo_entro} where
|
||||
\(S(ρ_{β^\ast})=S(ρ)\)~\cite{Alicki2013Apr}.
|
||||
|
||||
By choosing the temperature in \cref{eq:ergo_entro} accordingly, we
|
||||
arrive at
|
||||
\begin{equation}
|
||||
\label{eq:ergo_bound_single}
|
||||
\ergo{ρ} \leq \frac{1}{β^\ast}\qrelent{ρ}{ρ_{β^\ast}}.
|
||||
\end{equation}
|
||||
This bound can be saturated for states which are a permutation of a
|
||||
thermal state, as their corresponding passive states is the thermal
|
||||
state.
|
||||
|
||||
For our setting in
|
||||
\cref{eq:simple_bath_models,eq:simple_initial_state} we find a still
|
||||
better way to bound the ergotropy and fix the
|
||||
temperature~\cite{Lobejko2021Feb}. Substituting \(ρ\to ρ \otimes τ_β\)
|
||||
in \cref{eq:ergo_entro} we obtain
|
||||
\begin{equation}
|
||||
\label{eq:thermo_ergo_bound}
|
||||
\begin{aligned}
|
||||
\ergo{ρ\otimes τ_β} &= \frac{1}{β}
|
||||
\qty[\qrelent{ρ\otimes τ_β}{ρ_β\otimes τ_β} - \qrelent{(ρ_β\otimes
|
||||
τ_β)_P}{ρ_β\otimes τ_β}]\\
|
||||
&=\frac{1}{β}
|
||||
\qty[\qrelent{ρ}{ρ_β} - \qrelent{(ρ_β\otimes τ_β)_P}{ρ_β\otimes
|
||||
τ_β}] \leq \frac{1}{β} \qrelent{ρ}{ρ_β}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
Remarkably, the bound \cref{eq:thermo_ergo_bound} only depends on the
|
||||
system state and ``inherits'' the temperature of the bath. For any
|
||||
\(\dim[τ_β] = N\gg 1\) the bound stays valid. It is therefore
|
||||
reasonable to expected that it is also valid for an infinite bath. On
|
||||
the basis of physical intuition, a very large but finitely sized bath
|
||||
may be an arbitrarily good substitute for a continuous one. One might
|
||||
even argue, that the continuous bath is a mathematically convenient
|
||||
construct and the finite bath is the physical one. The objection to
|
||||
taking the limit outright is that the state \(τ_β\) does not exist as
|
||||
trace class operator for an infinite bath.
|
||||
|
||||
Interestingly, a saturation of \cref{eq:thermo_ergo_bound} is achieved
|
||||
in~\cite{Skrzypczyk2014Jun} with a continuous qubit
|
||||
bath. In~\cite{Lobejko2021Feb} a more generic argument is made in a
|
||||
similar setting. Both propose concrete protocols within the bounds of
|
||||
thermal operations and by considering explicit work reservoirs.
|
||||
|
||||
A corollary of \cref{eq:thermo_ergo_bound} is the Clausius form of the
|
||||
second law. By setting the system Hamiltonian to \(α \id\) in the
|
||||
above discussion the ergotropy becomes the change of bath energy
|
||||
\begin{equation}
|
||||
\label{eq:ergo_bath_change}
|
||||
\begin{aligned}
|
||||
\ergo{ρ} &= \max_{U\,\text{unitary}}\tr[\qty(ρ - UρU^\dag)
|
||||
(α\id\otimes H_\bath)] =
|
||||
\max_{U\,\text{unitary}}\tr_\bath[\qty(\tr_\sys[ρ-UρU^\dag])
|
||||
H_B]\\
|
||||
&\equiv\max_{U\,\text{unitary}}ΔE_B\leq \frac{1}{β}\qrelent{ρ}{\frac{\id_N}{N}},
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(N\) is the system dimension.
|
||||
|
||||
|
||||
Requiring a periodically modulated Hamiltonian with \(H(t+τ) = H(t)\)
|
||||
for \(τ>0\) we denote by \(U_{m,n}\) the propagator from \(mτ\) to
|
||||
\(nτ\) with \(m<n\). Assuming the system's reduced state
|
||||
\(ρ_\sys=\tr_\bath[ρ]\) reaches a limit cycle so that
|
||||
\(ρ_\sys(t+τ)=ρ_\sys(t)\) for all \(t > n_0τ\), we
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\subsection{Explicit Ergotropy Caluclation for a Bath of Identical
|
||||
Oscillators}
|
||||
\label{sec:explicitergo}
|
||||
Here, we explicitly calculate the ergotropy of a finite dimensional
|
||||
system connected to a bath of identical oscillators. This doesn't
|
||||
|
||||
|
||||
Let us choose \(H_S=α\id_N\) for simplicity,
|
||||
where \(α\) is an arbitrary energy scale. The ergotropy is then equal
|
||||
to the maximal energy reduction of the bath under arbitrary cyclic
|
||||
modulation.
|
||||
|
||||
The bound \cref{eq:thermo_ergo_bound} further simplifies to
|
||||
\begin{equation}
|
||||
\label{eq:thermo_ergo_bound_specific}
|
||||
\ergo{ρ\otimes τ_β} \leq \frac{1}{β} \qty[\ln(N) - S(ρ)],
|
||||
\end{equation}
|
||||
where \(S(ρ)=-\tr[ρ\ln(ρ)]\).
|
||||
For a pure state \cref{eq:thermo_ergo_bound_specific} is maximal. We
|
||||
therefore choose \(ρ=\ketbra{0}\).
|
||||
|
||||
|
||||
\subsection{Multiple Baths}
|
||||
As in the single bath case, some statement about the amount of energy
|
||||
that can be expected to be extracted in a cyclic manner. An argument
|
||||
based on entropy may be made for the periodic steady state as was
|
||||
shown in~\cite{Kato2016Dec} and is reproduced here. We will find the
|
||||
Clausius form of the second law.
|
||||
|
||||
We consider the situation given by the Hamiltonian for a system
|
||||
coupled to multiple baths under periodic driving
|
||||
\begin{equation}
|
||||
\label{eq:katoineqsys}
|
||||
H(t) = H_\sys(t) + ∑_i \qty(H_\bath^i + H_\inter^i(t)).
|
||||
\end{equation}
|
||||
Here, \(H_\sys(t)\) is the system Hamiltonian, \(H_\bath^i\) is the
|
||||
Hamiltonian of the \(i\)-th bath and \(H_\inter^i(t)\) is the coupling
|
||||
to the same. We demand periodic driving, that is \(H(t+τ) = H(t)\) for
|
||||
some \(τ\geq 0\).
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "../index.tex"
|
||||
%%% End:
|
||||
|
||||
% LocalWords: ergotropy
|
||||
Loading…
Add table
Reference in a new issue