more typo

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Valentin Boettcher 2022-09-27 16:19:09 +02:00
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8 changed files with 42 additions and 43 deletions

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@ -93,7 +93,7 @@
\def\hilb{\ensuremath{\mathcal{H}}} \def\hilb{\ensuremath{\mathcal{H}}}
% fixme % fixme
\newcommand{\fixme}[1]{\marginpar{\tiny\textcolor{red}{#1}}} \newcommand{\fixme}[1]{} %{\marginpar{\tiny\textcolor{red}{#1}}}
% HOPS/NMQSD % HOPS/NMQSD
\def\sys{\ensuremath{\mathrm{S}}} \def\sys{\ensuremath{\mathrm{S}}}
@ -140,3 +140,28 @@
\newcommand{\plot}[1]{% \newcommand{\plot}[1]{%
\includegraphics[draft=false]{./figs/#1.pdf}} \includegraphics[draft=false]{./figs/#1.pdf}}
\newcommand{\tval}[1]{{\input{./values/#1.tex}}} \newcommand{\tval}[1]{{\input{./values/#1.tex}}}
%% citing "in ref"
\NewBibliographyString{refname}
\NewBibliographyString{refsname}
\DefineBibliographyStrings{english}{%
refname = {Ref\adddot},
refsname = {Refs\adddot}
}
\DeclareCiteCommand{\refcite}
{%
\ifnum\thecitetotal=1
\bibstring{refname}%
\else%
\bibstring{refsname}%
\fi%
\addspace\bibopenbracket%
\usebibmacro{cite:init}%
\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite:comp}}
{}
{\usebibmacro{cite:dump}%
\usebibmacro{postnote}%
\bibclosebracket}

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@ -82,30 +82,4 @@ linkcolor=blue,
% cursive bold in maths % cursive bold in maths
\unimathsetup{math-style=TeX,bold-style=ISO} \unimathsetup{math-style=TeX,bold-style=ISO}
%% citing "in ref"
\NewBibliographyString{refname}
\NewBibliographyString{refsname}
\DefineBibliographyStrings{english}{%
refname = {Ref\adddot},
refsname = {Refs\adddot}
}
\DeclareCiteCommand{\refcite}
{%
\ifnum\thecitetotal=1
\bibstring{refname}%
\else%
\bibstring{refsname}%
\fi%
\addspace\bibopenbracket%
\usebibmacro{cite:init}%
\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite:comp}}
{}
{\usebibmacro{cite:dump}%
\usebibmacro{postnote}%
\bibclosebracket}
\recalctypearea \recalctypearea

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@ -38,7 +38,7 @@ non-Markovian strongly coupled open systems. Without modification of
the core method, it is possible to calculate the interaction energy the core method, it is possible to calculate the interaction energy
and the bath energy change. This is due to HOPS' foundation on the and the bath energy change. This is due to HOPS' foundation on the
global dynamics of the system and the bath in contrast to global dynamics of the system and the bath in contrast to
master-equation methods. We extended the result in~\cite{Kato2016Dec} master-equation methods. We extended the result in \refcite{Kato2016Dec}
for the Hierarchical Equations Of Motion method to arbitrary for the Hierarchical Equations Of Motion method to arbitrary
modulations of system and coupling inheriting all the advantages of modulations of system and coupling inheriting all the advantages of
the HOPS method. the HOPS method.

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@ -136,7 +136,7 @@ Defining
\label{eq:defdop} \label{eq:defdop}
D_t = ∫_0^t\dd{s} \alpha(t-s)\fdv{η^\ast_s} D_t = ∫_0^t\dd{s} \alpha(t-s)\fdv{η^\ast_s}
\end{equation} \end{equation}
as in~\cite{Suess2014Oct} we find as in \refcite{Suess2014Oct} we find
\begin{equation} \begin{equation}
\label{eq:final_flow_nmqsd} \label{eq:final_flow_nmqsd}
J(t) = -\i \mathcal{M}_{η^\ast}\bra{\psi(η, J(t) = -\i \mathcal{M}_{η^\ast}\bra{\psi(η,
@ -511,7 +511,7 @@ general. We refer to \cref{sec:hops_multibath} for an review of the
NMQSD theory and HOPS method for multiple baths. NMQSD theory and HOPS method for multiple baths.
Because the bath energy change is being calculated directly and not Because the bath energy change is being calculated directly and not
through energy conservation as in~\cite{Kato2016Dec}, we find through energy conservation as in \refcite{Kato2016Dec}, we find
\begin{equation} \begin{equation}
\label{eq:general_n_flow} \label{eq:general_n_flow}
J_n=-\dv{\ev{H_\bath^{(n)}}}{t} = \iu\ev{[H_\bath^{(n)}, J_n=-\dv{\ev{H_\bath^{(n)}}}{t} = \iu\ev{[H_\bath^{(n)},

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@ -319,7 +319,7 @@ arrive at an equation for \(\ket{ψ(t,\vb{z}^{\ast})}\)
From this point on there are multiple avenues open to us. We choose From this point on there are multiple avenues open to us. We choose
the canonical one of \cite{Strunz2001Habil}, but there is also a the canonical one of \cite{Strunz2001Habil}, but there is also a
time-discrete derivation, that avoids functional derivatives, time-discrete derivation, that avoids functional derivatives,
in~\cite{Hartmann2021Aug}. in \refcite{Hartmann2021Aug}.
We shift the perspective and define~\cite{RichardDiss,Strunz2001Habil} We shift the perspective and define~\cite{RichardDiss,Strunz2001Habil}
\begin{equation} \begin{equation}
@ -486,7 +486,7 @@ We call \cref{eq:singlehops} the \emph{Hierarchy of Pure States}
because each hierarchy state couples only to the hierarchy states one because each hierarchy state couples only to the hierarchy states one
level above and one level below. This is similar to the level above and one level below. This is similar to the
\emph{Hierarchical Equations of Motion} (HEOM) approach used \emph{Hierarchical Equations of Motion} (HEOM) approach used
in~\cite{Kato2016Dec}, but with the advantage of reducing the in \refcite{Kato2016Dec}, but with the advantage of reducing the
dimensionality from \(\dim{\hilb_{\sys}}^{2}\) to dimensionality from \(\dim{\hilb_{\sys}}^{2}\) to
\(\dim{\hilb_{\sys}}\) by treating pure states instead of density \(\dim{\hilb_{\sys}}\) by treating pure states instead of density
matrices. matrices.

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@ -109,7 +109,7 @@ the ``Markovianity'' of the bath.
The ohmic spectral density models an environment with a physical The ohmic spectral density models an environment with a physical
energy spectrum that is bounded from below and allows the application energy spectrum that is bounded from below and allows the application
of the finite temperature method described in~\cite{RichardDiss} and of the finite temperature method described in \refcite{RichardDiss} and
\cref{sec:lin_finite}. Also, \(J(0) = 0\) ensures that there is a \cref{sec:lin_finite}. Also, \(J(0) = 0\) ensures that there is a
unique zero temperature state of the bath. In~\cite{Kolar2012Aug} it unique zero temperature state of the bath. In~\cite{Kolar2012Aug} it
is argued (under weak coupling assumptions), that \(J(ω)\approx ω^γ\) is argued (under weak coupling assumptions), that \(J(ω)\approx ω^γ\)
@ -122,7 +122,7 @@ It may be remarked, that~\cref{eq:ohmic_bcf} does not correspond to a
simple sum of exponentials. As such it exercises the HOPS method and simple sum of exponentials. As such it exercises the HOPS method and
serves as a model for a general bath correlation function. For use serves as a model for a general bath correlation function. For use
with HOPS, a sum of exponentials must be fitted to the BCF. This has with HOPS, a sum of exponentials must be fitted to the BCF. This has
been done in~\cite{RichardDiss,Hartmann2021Aug}. In been done in \refcite{RichardDiss,Hartmann2021Aug}. In
\cref{sec:hopsvsanalyt} we will see, that this is indeed a valid \cref{sec:hopsvsanalyt} we will see, that this is indeed a valid
strategy for the application of \cref{chap:flow}. strategy for the application of \cref{chap:flow}.
@ -753,7 +753,7 @@ the stochastic process we chose the cutoff \(\abs{\vb{k}} \leq 4\)
(simplex truncation\footnote{see \cref{sec:hops_basics}}), (simplex truncation\footnote{see \cref{sec:hops_basics}}),
\(N=4.5 \cdot 10^5\) trajectories and an Ohmic BCF with \(α(0)=1.6\) \(N=4.5 \cdot 10^5\) trajectories and an Ohmic BCF with \(α(0)=1.6\)
and \(ω_c=4\). The sampling method uses the ``Fast Fourier and \(ω_c=4\). The sampling method uses the ``Fast Fourier
Transform'' (FFT) as described in~\cite{RichardDiss}. As the system Transform'' (FFT) as described in \refcite{RichardDiss}. As the system
Hilbert space dimension is small, a BCF expansion with seven terms was Hilbert space dimension is small, a BCF expansion with seven terms was
employed~\cite{Hartmann2021Aug,RichardDiss}. employed~\cite{Hartmann2021Aug,RichardDiss}.
@ -855,7 +855,7 @@ convergence as is also demonstrated in
\subsection{Hierarchy Truncation} \subsection{Hierarchy Truncation}
\label{sec:trunc} \label{sec:trunc}
As the systematics of the truncation depth have already been studied As the systematics of the truncation depth have already been studied
thoroughly in~\cite{RichardDiss,Hartmann2021Aug}, we will keep the thoroughly in \refcite{RichardDiss,Hartmann2021Aug}, we will keep the
discussion short. We chose \(N=4.5 \cdot 10^5\) trajectories and an discussion short. We chose \(N=4.5 \cdot 10^5\) trajectories and an
Ohmic BCF with \(α(0)=0.8\) and \(ω_c=2\). Again, a BCF expansion with Ohmic BCF with \(α(0)=0.8\) and \(ω_c=2\). Again, a BCF expansion with
seven terms has been used. The coupling strength has been chosen with seven terms has been used. The coupling strength has been chosen with

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@ -73,7 +73,7 @@ change over one cycle.
In \cref{sec:operational_thermo} a Gibbs like inequality for an In \cref{sec:operational_thermo} a Gibbs like inequality for an
arbitrary number of baths is derived as a slight generalization of the arbitrary number of baths is derived as a slight generalization of the
derivation in~\cite{Kato2016Dec}. The left hand side of this derivation in \refcite{Kato2016Dec}. The left hand side of this
inequality can be associated with a thermodynamic cost that should be inequality can be associated with a thermodynamic cost that should be
minimized for optimal efficiency. minimized for optimal efficiency.
@ -112,7 +112,7 @@ where \(n<∞\) is the Hilbert space dimension. This condition is both
necessary and sufficient. Examples of passive states are the state of necessary and sufficient. Examples of passive states are the state of
the micro-canonical ensemble or Gibbs states. Gibbs states are further the micro-canonical ensemble or Gibbs states. Gibbs states are further
distinguished by additional features described distinguished by additional features described
in~\cite{Lenard1978Dec}, which can be connected to formulations of the in \refcite{Lenard1978Dec}, which can be connected to formulations of the
zeroth and second laws of thermodynamics. zeroth and second laws of thermodynamics.
One of these properties is complete passivity. Completely passive One of these properties is complete passivity. Completely passive
@ -289,7 +289,7 @@ bath properties except the temperature. It is therefore reasonable to
expected that it is also valid for an infinite bath. expected that it is also valid for an infinite bath.
Interestingly, a saturation of \cref{eq:thermo_ergo_bound} is achieved Interestingly, a saturation of \cref{eq:thermo_ergo_bound} is achieved
in~\cite{Skrzypczyk2014Jun} with a continuous qubit in \refcite{Skrzypczyk2014Jun} with a continuous qubit
bath. In~\cite{Lobejko2021Feb} a more generic argument is made in a bath. In~\cite{Lobejko2021Feb} a more generic argument is made in a
similar setting. Both propose concrete protocols within the bounds of similar setting. Both propose concrete protocols within the bounds of
thermal operations and by considering explicit work reservoirs. In thermal operations and by considering explicit work reservoirs. In
@ -672,7 +672,7 @@ energy that can be extracted out of the system in relation to the
energy that is simply transferred between the baths. energy that is simply transferred between the baths.
An argument based on entropy may be made for the periodic steady state An argument based on entropy may be made for the periodic steady state
as was shown in~\cite{Kato2016Dec} and is reproduced here with the as was shown in \refcite{Kato2016Dec} and is reproduced here with the
slight generalization of multiple baths and modulated coupling. We slight generalization of multiple baths and modulated coupling. We
will find a Clausius like form of the second law. The left hand side will find a Clausius like form of the second law. The left hand side
of this inequality can then be interpreted as thermodynamic cost of of this inequality can then be interpreted as thermodynamic cost of
@ -789,7 +789,7 @@ If one defines heat in this
way~\cite{Kato2016Dec,Riechers2021Apr,Strasberg2021Aug}, way~\cite{Kato2016Dec,Riechers2021Apr,Strasberg2021Aug},
\cref{eq:secondlaw_cyclic} amounts to the Clausius inequality. The \cref{eq:secondlaw_cyclic} amounts to the Clausius inequality. The
definition of heat as bath energy change is corroborated definition of heat as bath energy change is corroborated
in~\cite{Esposito2015Dec} where it is shown, ableit for fermionic in \refcite{Esposito2015Dec} where it is shown, ableit for fermionic
baths, that a definition of heat involving any nonzero fraction of the baths, that a definition of heat involving any nonzero fraction of the
interaction energy will lead to the internal energy (as defined by the interaction energy will lead to the internal energy (as defined by the
first law) not being an exact differential. first law) not being an exact differential.
@ -1391,7 +1391,7 @@ however, we find that also the modulation of the interaction, i.e. the
coupling and decoupling, figures into the total power and reduces the coupling and decoupling, figures into the total power and reduces the
energy output. In a weak coupling scheme, this contribution can be energy output. In a weak coupling scheme, this contribution can be
neglected. Not so however in the generic case presented here. A neglected. Not so however in the generic case presented here. A
similar result was found in~\cite{Wiedmann2021Jun}. similar result was found in \refcite{Wiedmann2021Jun}.
The mean power output of this cycle is The mean power output of this cycle is
\(\bar{P}=0.002468\pm 0.000021\) with an efficiency, as defined in \(\bar{P}=0.002468\pm 0.000021\) with an efficiency, as defined in

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@ -817,7 +817,7 @@ labelformat=brace, position=top]{subcaption}
\(J(ω) = π ∑_λ\abs{g_λ}^2δ(ω-ω_λ)\)). \(J(ω) = π ∑_λ\abs{g_λ}^2δ(ω-ω_λ)\)).
\end{frame} \end{frame}
\begin{frame}{Fock-Space Embedding} \begin{frame}{Fock-Space Embedding}
As in~\cite{Gao2021Sep} we can define As in \refcite{Gao2021Sep} we can define
\begin{equation} \begin{equation}
\label{eq:fockpsi} \label{eq:fockpsi}
\ket{Ψ} = \sum_\kmat\ket{\psi^\kmat}\otimes \ket{\kmat} \ket{Ψ} = \sum_\kmat\ket{\psi^\kmat}\otimes \ket{\kmat}