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.envrc Normal file
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use_flake
eval "$shellHook"

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@ -301,3 +301,4 @@ TSWLatexianTemp*
#*Notes.bib
.direnv/
/index.pdf

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{
"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
}

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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 }:
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;
});
}

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\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)}}

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\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{−}{-}

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\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:

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$pdf_mode = 4;
@default_files = ('index.tex');

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\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 \(m<n\). 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 \(τ\geq 0\).
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