diff --git a/.envrc b/.envrc new file mode 100644 index 0000000..c7a853c --- /dev/null +++ b/.envrc @@ -0,0 +1,2 @@ +use_flake +eval "$shellHook" diff --git a/.gitignore b/.gitignore index 7be3df5..fccffa8 100644 --- a/.gitignore +++ b/.gitignore @@ -301,3 +301,4 @@ TSWLatexianTemp* #*Notes.bib .direnv/ +/index.pdf diff --git a/flake.lock b/flake.lock new file mode 100644 index 0000000..294cff3 --- /dev/null +++ b/flake.lock @@ -0,0 +1,42 @@ +{ + "nodes": { + "flake-utils": { + "locked": { + "lastModified": 1653893745, + "narHash": "sha256-0jntwV3Z8//YwuOjzhV2sgJJPt+HY6KhU7VZUL0fKZQ=", + "owner": "numtide", + "repo": "flake-utils", + "rev": "1ed9fb1935d260de5fe1c2f7ee0ebaae17ed2fa1", + "type": "github" + }, + "original": { + "owner": "numtide", + "repo": "flake-utils", + "type": "github" + } + }, + "nixpkgs": { + "locked": { + "lastModified": 1655567057, + "narHash": "sha256-Cc5hQSMsTzOHmZnYm8OSJ5RNUp22bd5NADWLHorULWQ=", + "owner": "NixOS", + "repo": "nixpkgs", + "rev": "e0a42267f73ea52adc061a64650fddc59906fc99", + "type": "github" + }, + "original": { + "id": "nixpkgs", + "ref": "nixos-unstable", + "type": "indirect" + } + }, + "root": { + "inputs": { + "flake-utils": "flake-utils", + "nixpkgs": "nixpkgs" + } + } + }, + "root": "root", + "version": 7 +} diff --git a/flake.nix b/flake.nix new file mode 100644 index 0000000..ac8d623 --- /dev/null +++ b/flake.nix @@ -0,0 +1,42 @@ +{ + description = "Typesetting for Valentin's Masters Thesis"; + inputs = { + nixpkgs.url = "nixpkgs/nixos-unstable"; + flake-utils.url = "github:numtide/flake-utils"; + flake-utils.inputs.nixpkgs.follows = "nixpkgs"; + }; + + outputs = { self, nixpkgs, flake-utils }: + with flake-utils.lib; eachSystem allSystems (system: + let + pkgs = nixpkgs.legacyPackages.${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 { + name = "masters-thesis"; + src = self; + buildInputs = [ pkgs.coreutils tex pkgs.biber]; + phases = ["unpackPhase" "buildPhase" "installPhase"]; + buildPhase = '' + export PATH="${pkgs.lib.makeBinPath buildInputs}"; + mkdir -p .cache/texmf-var + env TEXMFHOME=.cache TEXMFVAR=.cache/texmf-var \ + OSFONTDIR=${pkgs.gyre-fonts}/share/fonts \ + latexmk -interaction=nonstopmode \ + ./index.tex + ''; + installPhase = '' + mkdir -p $out + cp index.pdf $out/ + ''; + }; + }; + defaultPackage = packages.document; + }); +} diff --git a/hiromacros.sty b/hiromacros.sty new file mode 100644 index 0000000..fd87060 --- /dev/null +++ b/hiromacros.sty @@ -0,0 +1,108 @@ +\ProvidesPackage{hiromacros} + +% Macros + +%% qqgg +\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma} + +%% ppgg +\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma} + +%% Momenta and Polarization Vectors convenience +\DeclareMathOperator{\ps}{\slashed{p}} + +\DeclareMathOperator{\pe}{\varepsilon} +\DeclareMathOperator{\pes}{\slashed{\pe}} + +\DeclareMathOperator{\pse}{\varepsilon^{*}} +\DeclareMathOperator{\pses}{\slashed{\pe}^{*}} + +%% Spinor convenience +\DeclareMathOperator{\us}{u} +\DeclareMathOperator{\usb}{\bar{u}} + +\DeclareMathOperator{\vs}{v} +\DeclareMathOperator*{\vsb}{\overline{v}} + +%% Center of Mass energy +\DeclareMathOperator{\ecm}{E_{\text{CM}}} + +%% area hyperbolicus +\DeclareMathOperator{\artanh}{artanh} +\DeclareMathOperator{\arcosh}{arcosh} + +%% Fast Slash +\let\sl\slashed + +%% hermitian/complex conjugate +\DeclareMathOperator{\hc}{h.c.} +\DeclareMathOperator{\cc}{c.c.} + +%% eulers number +\def\eu{\ensuremath{\mathrm{e}}} + +%% Notes on Equations +\newcommand{\shorteqnote}[1]{ & & \text{\small\llap{#1}}} + +%% Typewriter Macros +\newcommand{\sherpa}{\texttt{Sherpa}} +\newcommand{\rivet}{\texttt{Rivet}} +\newcommand{\vegas}{\texttt{VEGAS}} +\newcommand{\lhapdf}{\texttt{LHAPDF6}} +\newcommand{\scipy}{\texttt{scipy}} + +%% Sherpa Versions +\newcommand{\oldsherpa}{\texttt{2.2.10}} +\newcommand{\newsherpa}{\texttt{3.0.0} (unreleased)} + +%% Special Names +\newcommand{\lhc}{\emph{LHC}} + +%% Expected Value and Variance +\newcommand{\EX}[1]{\operatorname{E}\qty[#1]} +\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]} + +%% Uppercase Rho +\newcommand{\Rho}{P} + +%% Transverse Momentum +\newcommand{\pt}[0]{p_\mathrm{T}} + +%% Sign Function +\DeclareMathOperator{\sign}{sgn} + +%% Stages +\newcommand{\stone}{\texttt{LO}} +\newcommand{\sttwo}{\texttt{LO+PS}} +\newcommand{\stthree}{\texttt{LO+PS+pT}} +\newcommand{\stfour}{\texttt{LO+PS+pT+Hadr.}} +\newcommand{\stfive}{\texttt{LO+PS+pT+Hadr.+MI}} + +%% GeV +\newcommand{\gev}[1]{\SI{#1}{\giga\electronvolt}} + +\def\iu{\ensuremath{\mathrm{i}}} +\def\i{\iu} +\def\id{\ensuremath{\mathbb{1}}} +\def\NN{\ensuremath{\mathbb{N}}} +\def\RR{\ensuremath{\mathbb{R}}} +\def\CC{\ensuremath{\mathbb{C}}} +\def\dim{\ensuremath{\mathrm{dim}}} +\def\hilb{\ensuremath{\mathcal{H}}} + +% fixme +\newcommand{\fixme}[1]{\textbf{\textcolor{red}{FIXME:~#1}}} + +% HOPS/NMQSD +\def\sys{\ensuremath{\mathrm{S}}} +\def\bath{\ensuremath{\mathrm{B}}} +\def\inter{\ensuremath{\mathrm{I}}} +\def\nth{\ensuremath{^{(n)}}} + +\newcommand{\mat}[1]{\ensuremath{{\underline{\vb{#1}}}}} +\def\kmat{{\mat{k}}} + + +% Thermo +\newcommand{\ergo}[1]{\ensuremath{\mathcal{W}\qty[#1]}} +\newcommand{\qrelent}[2]{\ensuremath{S\qty(#1\,\middle|\middle|\,#2)}} diff --git a/hirostyle.sty b/hirostyle.sty new file mode 100644 index 0000000..18c94f9 --- /dev/null +++ b/hirostyle.sty @@ -0,0 +1,82 @@ +\ProvidesPackage{hirostyle} +\usepackage[utf8]{inputenc} % load early +\usepackage[T1]{fontenc} +% load early +\usepackage[english]{babel} +\usepackage{physics} +\usepackage{graphicx, booktabs, float} +\usepackage[tbtags]{mathtools} +\usepackage{amssymb} +\usepackage[backend=biber, language=english, style=phys]{biblatex} +\usepackage{siunitx} +\usepackage{caption} +\usepackage[list=true, font=small, +labelformat=brace, position=top]{subcaption} +\usepackage{tikz} +\usepackage{pgfplots} +\usepackage{ifdraft} +\usepackage[protrusion=true,expansion=true,tracking=true]{microtype} +\usepackage{fancyvrb} +\usepackage[autostyle=true]{csquotes} +\usepackage{setspace} +\usepackage{newunicodechar} +\usepackage[pdfencoding=auto,hidelinks,colorlinks=true, +linkcolor=blue, + filecolor=blue, + citecolor = black, + urlcolor=cyan,]{hyperref} % load late +\usepackage[capitalize]{cleveref} +\usepackage{multirow,tabularx} +\usepackage{bbold} +\usepackage{scrhack} +\usepackage{fontspec} +\usepackage{unicode-math} +\setmainfont{texgyrepagella}[ + Extension = .otf, + UprightFont = *-regular, + BoldFont = *-bold, + ItalicFont = *-italic, + BoldItalicFont = *-bolditalic, + ] +\setmathfont{texgyrepagella-math.otf} +\KOMAoptions{DIV=last} +\usepackage[autooneside]{scrlayer-scrpage} + +%% use the current pgfplots +\pgfplotsset{compat=1.16} + + +%% Tikz +\usetikzlibrary{arrows,shapes,angles,quotes,arrows.meta,external} +\tikzexternalize[prefix=tikz/] + +%% Including plots +\newcommand{\plot}[1]{% + \ifdraft{\includegraphics[draft=false]{./figs/#1.pdf}}{\input{./figs/#1.pgf}}} +\newcommand{\rivethist}[2][,]{% + \includegraphics[draft=false,width=\textwidth,#1]{./figs/rivet/#2.pdf}} + +%% Including Results +\newcommand{\result}[1]{\input{./results/#1}\!} + +%% SI units +\sisetup{separate-uncertainty = true} + +%% Captions +\captionsetup{justification=centering} + +%% Labels +% \labelformat{chapter}{chapter~#1} +% \labelformat{section}{section~#1} +% \labelformat{figure}{figure~#1} +% \labelformat{table}{table~#1} + +%% Cleverref +\crefname{equation}{}{} +\creflabelformat{equation}{(#2#1#3)} + +%% Font for headings +\addtokomafont{disposition}{\rmfamily} + +%% Minus Sign for Matplotlib +\newunicodechar{−}{-} diff --git a/index.tex b/index.tex new file mode 100644 index 0000000..7e2e3c7 --- /dev/null +++ b/index.tex @@ -0,0 +1,32 @@ +\documentclass[fontsize=10pt,paper=b5,open=any, +,twoside=true,toc=listof,toc=bibliography,headings=optiontohead, +captions=nooneline,captions=tableabove,english,DIV=calc,numbers=noenddot,final,parskip=half, +headinclude=true,footinclude=false,BCOR=1cm]{scrbook} + +\pdfvariable suppressoptionalinfo 512\relax + +\usepackage{hirostyle} +\usepackage{hiromacros} + + + +\addbibresource{references.bib} +\synctex=1 +\title{Calculating heat flows with HOPS} +\author{Valentin Boettcher} +\date{\today} +\begin{document} +\maketitle +\tableofcontents + +% Chapters +\include{src/index.tex} + +\printbibliography{} +\end{document} +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/latexmkrc b/latexmkrc new file mode 100644 index 0000000..6f3c6c9 --- /dev/null +++ b/latexmkrc @@ -0,0 +1,2 @@ +$pdf_mode = 4; +@default_files = ('index.tex'); diff --git a/references.bib b/references.bib new file mode 100644 index 0000000..481f606 --- /dev/null +++ b/references.bib @@ -0,0 +1,1029 @@ +@misc{vzitkovic2010, + title={Introduction to stochastic processes-lecture notes}, + author={{\v{Z}}itkovi{\'c}, Gordan}, + year=2010, + publisher={Austin, University of Texas-Department of Mathematics}, + url={https://web.ma.utexas.edu/users/gordanz/notes/introduction_to_stochastic_processes.pdf} +} + +@book{jacobs2010stochastic, + title={Stochastic Processes for Physicists: Understanding Noisy Systems}, + author={Jacobs, K.}, + isbn=9780521765428, + lccn=2010278127, + url={https://books.google.de/books?id=5PEjmQEACAAJ}, + year=2010, + publisher={Cambridge University Press}, +} + +@article{cufar2020, +doi = {10.1007/978-3-030-48408-8}, +title = {Probability and Stochastic Processes for Physicists ||}, +author = {Cufaro Petroni, Nicola}, +publisher = {Springer International Publishing}, +journal = {UNITEXT for Physics vol. 10.1007/978-3-030-48408-8}, + +isbn = {9783030484071; 3030484076; 9783030484088; 3030484084}, +year = 2020, +volume = {10.1007/978-3-030-48408-8}, +page = {--}, +url = {libgen.li/file.php?md5=0ac1c3786767232ebdc10b716eff3b96}, +} + +@book{Rivas2012, + author = {Rivas, {\ifmmode\acute{A}\else\'{A}\fi}ngel and Huelga, Susana F.}, + title = {{Open Quantum Systems}}, + year = 2012, + isbn = {978-3-642-23353-1}, + publisher = {Springer-Verlag}, + address = {Berlin, Germany}, + doi = {10.1007/978-3-642-23354-8} +} + +@article{klauder1968fundamentals, + title={Fundamentals of Quantum Optics Benjamin}, + author={Klauder, JR and Sudarshan, ECG}, + journal={Inc., New York}, + year=1968 +} + +@article{Gisin1992Nov, + author = {Gisin, N. and Percival, I. C.}, + title = {{The quantum-state diffusion model applied to open systems}}, + journal = {J. Phys. A: Math. Gen.}, + volume = 25, + number = 21, + pages = {5677--5691}, + year = 1992, + month = {Nov}, + issn = {0305-4470}, + publisher = {IOP Publishing}, + doi = {10.1088/0305-4470/25/21/023} +} + +@article{Diosi1995Jan, + author = {Di{\ifmmode\acute{o}\else\'{o}\fi}si, Lajos and Gisin, Nicolas and Halliwell, Jonathan and Percival, Ian C.}, + title = {{Decoherent Histories and Quantum State Diffusion}}, + journal = {Phys. Rev. Lett.}, + volume = 74, + number = 2, + pages = {203--207}, + year = 1995, + month = {Jan}, + issn = {1079-7114}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.74.203} +} + +@article{Diosi1997, + author = {Di{\ifmmode\acute{o}\else\'{o}\fi}si, Lajos and Strunz, Walter T.}, + title = {{The non-Markovian stochastic Schr{\ifmmode\ddot{o}\else\"{o}\fi}dinger equation for open systems}}, + journal = {Phys. Lett. A}, + volume = 235, + number = 6, + pages = {569--573}, + year = 1997, + month = {Nov}, + issn = {0375-9601}, + publisher = {North-Holland}, + eprint = {quant-ph/9706050}, + doi = {10.1016/S0375-9601(97)00717-2} +} + +@article{Diosi1998Mar, + author = {Diosi, L. and Gisin, N. and Strunz, W. T.}, + title = {{Non-Markovian Quantum State Diffusion}}, + journal = {arXiv}, + year = 1998, + month = {Mar}, + eprint = {quant-ph/9803062}, + doi = {10.1103/PhysRevA.58.1699} +} + +@article{Strunz1999Mar, + author = {Strunz, Walter T. and Di{\ifmmode\acute{o}\else\'{o}\fi}si, Lajos and Gisin, Nicolas}, + title = {{Open System Dynamics with Non-Markovian Quantum Trajectories}}, + journal = {Phys. Rev. Lett.}, + volume = {82}, + number = {9}, + pages = {1801--1805}, + year = {1999}, + month = {Mar}, + issn = {1079-7114}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.82.1801} +} + +@article{Suess2014Oct, + author = {Suess, D. and Eisfeld, A. and Strunz, W. T.}, + title = {{Hierarchy of stochastic pure states for open quantum system dynamics}}, + journal = {Phys. Rev. Lett.}, + volume = 113, + number = 15, + pages = {150403.}, + year = 2014, + month = {Oct}, + issn = {1079-7114}, + publisher = {See full text options at American Physical Society}, + eprint = 25375694, + doi = {10.1103/PhysRevLett.113.150403} +} + +@article{Hartmann2017Dec, + author = {Hartmann, Richard and Strunz, Walter T.}, + title = {{Exact Open Quantum System Dynamics Using the Hierarchy of Pure States (HOPS)}}, + journal = {J. Chem. Theory Comput.}, + volume = 13, + number = 12, + pages = {5834--5845}, + year = 2017, + month = {Dec}, + issn = {1549-9626}, + publisher = {J Chem Theory Comput}, + eprint = 29016126, + doi = {10.1021/acs.jctc.7b00751} +} + +@article{Hartmann2021Aug, + author = {Hartmann, Richard and Strunz, Walter T.}, + title = {{Open Quantum System Response from the Hierarchy of Pure States}}, + journal = {J. Phys. Chem. A}, + volume = {125}, + number = {32}, + pages = {7066--7079}, + year = {2021}, + month = {Aug}, + issn = {1089-5639}, + publisher = {American Chemical Society}, + doi = {10.1021/acs.jpca.1c03339} +} + +@article{Strunz1996LinearQS, + title={Linear quantum state diffusion for non-Markovian open quantum systems}, + author={Walter T. Strunz}, + journal={Physics Letters A}, + year=1996, + volume=224, + pages={25-30}, + eprint={quant-ph/9610035} +} + +@book{Press2007Sep, + author = {Press, William H. and Teukolsky, Saul A. and Vetterling, William T. and Flannery, Brian P.}, + title = {{Numerical Recipes}}, + journal = {Cambridge University Press}, + year = {2007}, + month = {Sep}, + isbn = {978-0-52188068-8}, + publisher = {Cambridge University Press}, + address = {Cambridge, England, UK}, + url = {https://www.cambridge.org/us/academic/subjects/mathematics/numerical-recipes/numerical-recipes-art-scientific-computing-3rd-edition?format=HB} +} + +@thesis{Strunz2001Habil, + title={Stochastic Schrödinger equation approach to the dynamics of + non-Markovian open quantum systems}, + author={Walter T. Strunz}, + type=Habilitation, + institution={Fachbereich Physik der Universität Essen}, + year=2001 +} + +@book{Weiss2008Mar, + author = {Weiss, Ulrich}, + title = {{Quantum Dissipative Systems {$\vert$} Series in Modern Condensed Matter Physics}}, + volume = {13}, + year = {2008}, + month = {Mar}, + isbn = {978-981-279-162-7}, + publisher = {World Scientific Publishing Company}, + address = {Singapore}, + doi = {10.1142/6738} +} + +@article{Feynman1963Oct, + author = {Feynman, R. P. and Vernon, F. L.}, + title = {{The theory of a general quantum system interacting with a linear dissipative system}}, + journal = {Ann. Phys.}, + volume = {24}, + pages = {118--173}, + year = {1963}, + month = {Oct}, + issn = {0003-4916}, + publisher = {Academic Press}, + doi = {10.1016/0003-4916(63)90068-X} +} + +@article{Kato2015Aug, + author = {Kato, Akihito and Tanimura, Yoshitaka}, + title = {{Quantum heat transport of a two-qubit system: Interplay between system-bath coherence and qubit-qubit coherence}}, + journal = {J. Chem. Phys.}, + volume = {143}, + number = {6}, + pages = {064107}, + year = {2015}, + month = {Aug}, + issn = {0021-9606}, + publisher = {American Institute of Physics}, + doi = {10.1063/1.4928192} +} + +@article{Gaul2007Jul, + author = {Gaul, Christopher and B{\ifmmode\ddot{u}\else\"{u}\fi}ttner, Helmut}, + title = {{Quantum mechanical heat transport in disordered harmonic chains}}, + journal = {Phys. Rev. E}, + volume = {76}, + number = {1}, + pages = {011111}, + year = {2007}, + month = {Jul}, + issn = {2470-0053}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.76.011111} +} + +@article{Motz2017May, + author = {Motz, T. and Ankerhold, J. and Stockburger, J. T.}, + title = {{Currents and fluctuations of quantum heat transport in harmonic chains}}, + journal = {New J. Phys.}, + volume = {19}, + number = {5}, + pages = {053013}, + year = {2017}, + month = {May}, + issn = {1367-2630}, + publisher = {IOP Publishing}, + doi = {10.1088/1367-2630/aa68bd} +} + +@thesis{RichardDiss, + title={Stochastic Schrödinger equation approach to the dynamics of + non-Markovian open quantum systems}, + author={Richard Hartmann}, + type=Dissertation, + institution={Institut für Theoretische Physik, Technische Universität Dresden}, + year=2021 +} + +@incollection{Pan1999May, + author = {Pan, Victor Y. and Chen, Zhao Q.}, + title = {{The complexity of the matrix eigenproblem}}, + booktitle = {{STOC '99: Proceedings of the thirty-first annual ACM symposium on Theory of Computing}}, + pages = {507--516}, + year = 1999, + month = {May}, + isbn = {978-158113067}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA}, + doi = {10.1145/301250.301389} +} + +@article{Reyes-Lega2016Dec, + author = {Reyes-Lega, A. F.}, + title = {{Some Aspects of Operator Algebras in Quantum Physics}}, + journal = {arXiv}, + year = 2016, + month = {Dec}, + eprint = {1612.07718}, + doi = {10.1142/9789814730884_0001} +} + +@article{Rivas2019Oct, + author = {Rivas, {\ifmmode\acute{A}\else\'{A}\fi}ngel}, + title = {{Strong Coupling Thermodynamics of Open Quantum Systems}}, + journal = {arXiv}, + year = 2019, + month = {Oct}, + eprint = {1910.01246}, + doi = {10.1103/PhysRevLett.124.160601} +} + + +@book {Bratteli87, +author = { Bratteli, Ola AND Robinson, Derek W. }, +title = { Operator algebras and quantum statistical mechanics 1 C*- and W*-algebras, symmetry groups, decomposition of states }, +edition = { 2. ed. } , +publisher = {Springer}, +isbn = 0387170936, +isbn = 3540170936, +isbn = 9783540170938, +isbn = 9780387170930, +year = 1987, +abstract = {Literaturverz. S. 469 - 488}, +address = { New York, NY }, +url = { http://slubdd.de/katalog?TN_libero_mab2147995 } +} + +@book{Schlosshauer2007, + author = {Schlosshauer, Maximilian}, + title = {{Decoherence and the Quantum-To-Classical Transition}}, + year = {2007}, + issn = {1612-3018}, + publisher = {Springer}, + address = {Berlin, Germany}, + doi = {10.1007/978-3-540-35775-9} +} + +@book{NISTHandbook, + title = {NIST handbook of mathematical functions}, + author = {Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark}, + publisher = {Cambridge University Press}, + isbn = {0521140633,9780521140638,9780521192255,0521192250}, + year = {2010}, + series = {}, + edition = {1 Pap/Cdr}, + volume = {}, +} + +@article{Senior2020Feb, + author = {Senior, Jorden and Gubaydullin, Azat and Karimi, Bayan and Peltonen, Joonas T. and Ankerhold, Joachim and Pekola, Jukka P.}, + title = {{Heat rectification via a superconducting artificial atom - Communications Physics}}, + journal = {Commun. Phys.}, + volume = {3}, + number = {40}, + pages = {1--5}, + year = {2020}, + month = {Feb}, + issn = {2399-3650}, + publisher = {Nature Publishing Group}, + doi = {10.1038/s42005-020-0307-5} +} + +@article{Motz2018Nov, + author = {Motz, T. and Wiedmann, M. and Stockburger, J. T. and Ankerhold, J.}, + title = {{Rectification of heat currents across nonlinear quantum chains: a versatile approach beyond weak thermal contact}}, + journal = {New J. Phys.}, + volume = {20}, + number = {11}, + pages = {113020}, + year = {2018}, + month = {Nov}, + issn = {1367-2630}, + publisher = {IOP Publishing}, + doi = {10.1088/1367-2630/aaea90} +} + +@article{Wiedmann2020Mar, + author = {Wiedmann, M. and Stockburger, J. T. and Ankerhold, J.}, + title = {{Non-Markovian dynamics of a quantum heat engine: out-of-equilibrium operation and thermal coupling control}}, + journal = {New J. Phys.}, + volume = {22}, + number = {3}, + pages = {033007}, + year = {2020}, + month = {Mar}, + issn = {1367-2630}, + publisher = {IOP Publishing}, + doi = {10.1088/1367-2630/ab725a} +} + +@article{Caldeira1983Sep, + author = {Caldeira, A. O. and Leggett, A. J.}, + title = {{Quantum tunnelling in a dissipative system}}, + journal = {Ann. Phys.}, + volume = {149}, + number = {2}, + pages = {374--456}, + year = {1983}, + month = {Sep}, + issn = {0003-4916}, + publisher = {Academic Press}, + doi = {10.1016/0003-4916(83)90202-6} +} + +@article{Bera2017Dec, + author = {Bera, Manabendra N. and Riera, Arnau and Lewenstein, Maciej and Winter, Andreas}, + title = {{Generalized laws of thermodynamics in the presence of correlations - Nature Communications}}, + journal = {Nat. Commun.}, + volume = {8}, + number = {2180}, + pages = {1--6}, + year = {2017}, + month = {Dec}, + issn = {2041-1723}, + publisher = {Nature Publishing Group}, + doi = {10.1038/s41467-017-02370-x} +} + +@article{Talkner2016Aug, + author = {Talkner, Peter and H{\ifmmode\ddot{a}\else\"{a}\fi}nggi, Peter}, + title = {{Open system trajectories specify fluctuating work but not heat}}, + journal = {Phys. Rev. E}, + volume = {94}, + number = {2}, + pages = {022143}, + year = {2016}, + month = {Aug}, + issn = {2470-0053}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.94.022143} +} + +@article{Strasberg2021Aug, + author = {Strasberg, Philipp and Winter, Andreas}, + title = {{First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy}}, + journal = {PRX Quantum}, + volume = {2}, + number = {3}, + pages = {030202}, + year = {2021}, + month = {Aug}, + issn = {2691-3399}, + publisher = {American Physical Society}, + doi = {10.1103/PRXQuantum.2.030202} +} + +@article{Bera2021Jun, + author = {Bera, Mohit Lal and Juli{\ifmmode\grave{a}\else\`{a}\fi}-Farr{\ifmmode\acute{e}\else\'{e}\fi}, Sergi and Lewenstein, Maciej and Bera, Manabendra Nath}, + title = {{Quantum Heat Engines with Carnot Efficiency at Maximum Power}}, + journal = {arXiv}, + year = {2021}, + month = {Jun}, + eprint = {2106.01193}, + url = {https://arxiv.org/abs/2106.01193v1} +} + +@article{Bera2021Feb, + author = {Bera, Mohit Lal and Lewenstein, Maciej and Bera, Manabendra Nath}, + title = {{Attaining Carnot efficiency with quantum and nanoscale heat engines - npj Quantum Information}}, + journal = {npj Quantum Inf.}, + volume = {7}, + number = {31}, + pages = {1--7}, + year = {2021}, + month = {Feb}, + issn = {2056-6387}, + publisher = {Nature Publishing Group}, + doi = {10.1038/s41534-021-00366-6} +} + + +@article{Vaccaro2011Jun, + author = {Vaccaro, Joan A. and Barnett, Stephen M.}, + title = {{Information erasure without an energy cost}}, + journal = {Proc. 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Rev. E}, + volume = {87}, + number = {4}, + pages = {042123}, + year = {2013}, + month = apr, + issn = {2470-0053}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.87.042123} +} +@article{Lobejko2021Feb, + author = {{\L}obejko, Marcin}, + title = {{The tight Second Law inequality for coherent quantum systems and finite-size heat baths}}, + journal = {Nat. Commun.}, + volume = {12}, + number = {918}, + pages = {1--6}, + year = {2021}, + month = feb, + issn = {2041-1723}, + publisher = {Nature Publishing Group}, + doi = {10.1038/s41467-021-21140-4} +} +@article{Skrzypczyk2014Jun, + author = {Skrzypczyk, Paul and Short, Anthony J. and Popescu, Sandu}, + title = {{Work extraction and thermodynamics for individual quantum systems}}, + journal = {Nat. Commun.}, + volume = {5}, + number = {4185}, + pages = {1--8}, + year = {2014}, + month = jun, + issn = {2041-1723}, + publisher = {Nature Publishing Group}, + doi = {10.1038/ncomms5185} +} diff --git a/src/index.tex b/src/index.tex new file mode 100644 index 0000000..86946b9 --- /dev/null +++ b/src/index.tex @@ -0,0 +1,800 @@ +\section{Linear NMQSD, Zero Temperature} +As in~\cite{Hartmann2017Dec} we choose +\begin{equation} + \label{eq:totalH} + H = H_\sys + \underbrace{LB^† + L^† B}_{H_\inter} + H_\bath +\end{equation} +with the system hamiltonian \(H_\sys\), the bath hamiltonian +\begin{equation} + \label{eq:bathh} + H_\bath = ∑_\lambda ω_\lambda a^† a, +\end{equation} +the bath coupling system operator \(L\) and the bath coupling bath +operator +\begin{equation} + \label{eq:bop} + B=∑_{\lambda} g_{\lambda} a_{\lambda} +\end{equation} +which define the interaction hamiltonian \(H_\inter\). + +We define the heat flow out of the system as in~\cite{Kato2015Aug} +through +\begin{equation} + \label{eq:heatflowdef} + J = - \dv{\ev{H_\bath}}{t}. +\end{equation} +Working, for now, in the Schr\"odinger picture the Ehrenfest theorem +can be employed to find +\begin{equation} + \label{eq:ehrenfest} + \i∂_t\ev{H_\bath} = \ev{[H_\bath,H]} = \ev{[H_\bath,H_\inter]}. +\end{equation} +Thus, we need to calculate +\begin{eqnarray} + \label{eq:calccomm} + \begin{aligned} + [H_\bath,H_\inter] &= [H_\bath, LB^† + L^† B] \\ + &= L[H_\bath, B^† ] + L^† [H_\bath, B] \\ + &= L[H_\bath, B^† ] - \hc. + \end{aligned} +\end{eqnarray} +This checks out as the commutator has to be anti-hermitian due to +\cref{eq:ehrenfest}. +Using \([H_\bath, B^† ]=∑_\lambda ω_\lambda g^\ast_\lambda +a^†_\lambda\) it follows that +\begin{equation} + \label{eq:expcomm} + \begin{aligned} + \ev{[H_\bath,H_\inter]} &= ∑_\lambda ω_\lambda g^\ast_\lambda + \ev{La^†_\lambda} - \cc + = ∑_\lambda ω_\lambda g^\ast_\lambda + \ev{La^†_\lambda \eu^{\i ω t}}_\inter - \cc\\ + &= \frac{1}{\i}\ev{L∂_t{∑_\lambda + 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} + \label{eq:final_flow} + J(t) = \ev{L^†∂_t B(t) + L∂_t B^†(t)}_\inter. +\end{equation} + +From this point on, we will assume the interaction picture and drop the +\(I\) subscript. + +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 \(m n_0τ\), 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