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more typo
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8 changed files with 42 additions and 43 deletions
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@ -93,7 +93,7 @@
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\def\hilb{\ensuremath{\mathcal{H}}}
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% fixme
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\newcommand{\fixme}[1]{\marginpar{\tiny\textcolor{red}{#1}}}
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\newcommand{\fixme}[1]{} %{\marginpar{\tiny\textcolor{red}{#1}}}
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% HOPS/NMQSD
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\def\sys{\ensuremath{\mathrm{S}}}
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@ -140,3 +140,28 @@
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\newcommand{\plot}[1]{%
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\includegraphics[draft=false]{./figs/#1.pdf}}
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\newcommand{\tval}[1]{{\input{./values/#1.tex}}}
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%% citing "in ref"
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\NewBibliographyString{refname}
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\NewBibliographyString{refsname}
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\DefineBibliographyStrings{english}{%
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refname = {Ref\adddot},
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refsname = {Refs\adddot}
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}
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\DeclareCiteCommand{\refcite}
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{%
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\ifnum\thecitetotal=1
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\bibstring{refname}%
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\else%
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\bibstring{refsname}%
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\fi%
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\addspace\bibopenbracket%
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\usebibmacro{cite:init}%
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\usebibmacro{prenote}}
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{\usebibmacro{citeindex}%
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\usebibmacro{cite:comp}}
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{}
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{\usebibmacro{cite:dump}%
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\usebibmacro{postnote}%
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\bibclosebracket}
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@ -82,30 +82,4 @@ linkcolor=blue,
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% cursive bold in maths
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\unimathsetup{math-style=TeX,bold-style=ISO}
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%% citing "in ref"
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\NewBibliographyString{refname}
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\NewBibliographyString{refsname}
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\DefineBibliographyStrings{english}{%
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refname = {Ref\adddot},
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refsname = {Refs\adddot}
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}
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\DeclareCiteCommand{\refcite}
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{%
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\ifnum\thecitetotal=1
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\bibstring{refname}%
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\else%
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\bibstring{refsname}%
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\fi%
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\addspace\bibopenbracket%
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\usebibmacro{cite:init}%
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\usebibmacro{prenote}}
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{\usebibmacro{citeindex}%
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\usebibmacro{cite:comp}}
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{}
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{\usebibmacro{cite:dump}%
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\usebibmacro{postnote}%
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\bibclosebracket}
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\recalctypearea
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@ -38,7 +38,7 @@ non-Markovian strongly coupled open systems. Without modification of
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the core method, it is possible to calculate the interaction energy
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and the bath energy change. This is due to HOPS' foundation on the
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global dynamics of the system and the bath in contrast to
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master-equation methods. We extended the result in~\cite{Kato2016Dec}
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master-equation methods. We extended the result in \refcite{Kato2016Dec}
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for the Hierarchical Equations Of Motion method to arbitrary
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modulations of system and coupling inheriting all the advantages of
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the HOPS method.
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@ -136,7 +136,7 @@ Defining
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\label{eq:defdop}
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D_t = ∫_0^t\dd{s} \alpha(t-s)\fdv{η^\ast_s}
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\end{equation}
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as in~\cite{Suess2014Oct} we find
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as in \refcite{Suess2014Oct} we find
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\begin{equation}
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\label{eq:final_flow_nmqsd}
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J(t) = -\i \mathcal{M}_{η^\ast}\bra{\psi(η,
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@ -511,7 +511,7 @@ general. We refer to \cref{sec:hops_multibath} for an review of the
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NMQSD theory and HOPS method for multiple baths.
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Because the bath energy change is being calculated directly and not
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through energy conservation as in~\cite{Kato2016Dec}, we find
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through energy conservation as in \refcite{Kato2016Dec}, we find
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\begin{equation}
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\label{eq:general_n_flow}
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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 \(\ket{ψ(t,\vb{z}^{\ast})}\)
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From this point on there are multiple avenues open to us. We choose
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the canonical one of \cite{Strunz2001Habil}, but there is also a
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time-discrete derivation, that avoids functional derivatives,
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in~\cite{Hartmann2021Aug}.
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in \refcite{Hartmann2021Aug}.
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We shift the perspective and define~\cite{RichardDiss,Strunz2001Habil}
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\begin{equation}
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@ -486,7 +486,7 @@ We call \cref{eq:singlehops} the \emph{Hierarchy of Pure States}
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because each hierarchy state couples only to the hierarchy states one
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level above and one level below. This is similar to the
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\emph{Hierarchical Equations of Motion} (HEOM) approach used
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in~\cite{Kato2016Dec}, but with the advantage of reducing the
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in \refcite{Kato2016Dec}, but with the advantage of reducing the
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dimensionality from \(\dim{\hilb_{\sys}}^{2}\) to
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\(\dim{\hilb_{\sys}}\) by treating pure states instead of density
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matrices.
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@ -109,7 +109,7 @@ the ``Markovianity'' of the bath.
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The ohmic spectral density models an environment with a physical
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energy spectrum that is bounded from below and allows the application
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of the finite temperature method described in~\cite{RichardDiss} and
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of the finite temperature method described in \refcite{RichardDiss} and
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\cref{sec:lin_finite}. Also, \(J(0) = 0\) ensures that there is a
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unique zero temperature state of the bath. In~\cite{Kolar2012Aug} it
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is argued (under weak coupling assumptions), that \(J(ω)\approx ω^γ\)
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@ -122,7 +122,7 @@ It may be remarked, that~\cref{eq:ohmic_bcf} does not correspond to a
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simple sum of exponentials. As such it exercises the HOPS method and
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serves as a model for a general bath correlation function. For use
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with HOPS, a sum of exponentials must be fitted to the BCF. This has
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been done in~\cite{RichardDiss,Hartmann2021Aug}. In
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been done in \refcite{RichardDiss,Hartmann2021Aug}. In
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\cref{sec:hopsvsanalyt} we will see, that this is indeed a valid
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strategy for the application of \cref{chap:flow}.
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@ -753,7 +753,7 @@ the stochastic process we chose the cutoff \(\abs{\vb{k}} \leq 4\)
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(simplex truncation\footnote{see \cref{sec:hops_basics}}),
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\(N=4.5 \cdot 10^5\) trajectories and an Ohmic BCF with \(α(0)=1.6\)
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and \(ω_c=4\). The sampling method uses the ``Fast Fourier
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Transform'' (FFT) as described in~\cite{RichardDiss}. As the system
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Transform'' (FFT) as described in \refcite{RichardDiss}. As the system
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Hilbert space dimension is small, a BCF expansion with seven terms was
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employed~\cite{Hartmann2021Aug,RichardDiss}.
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@ -855,7 +855,7 @@ convergence as is also demonstrated in
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\subsection{Hierarchy Truncation}
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\label{sec:trunc}
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As the systematics of the truncation depth have already been studied
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thoroughly in~\cite{RichardDiss,Hartmann2021Aug}, we will keep the
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thoroughly in \refcite{RichardDiss,Hartmann2021Aug}, we will keep the
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discussion short. We chose \(N=4.5 \cdot 10^5\) trajectories and an
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Ohmic BCF with \(α(0)=0.8\) and \(ω_c=2\). Again, a BCF expansion with
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seven terms has been used. The coupling strength has been chosen with
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@ -73,7 +73,7 @@ change over one cycle.
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In \cref{sec:operational_thermo} a Gibbs like inequality for an
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arbitrary number of baths is derived as a slight generalization of the
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derivation in~\cite{Kato2016Dec}. The left hand side of this
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derivation in \refcite{Kato2016Dec}. The left hand side of this
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inequality can be associated with a thermodynamic cost that should be
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minimized for optimal efficiency.
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@ -112,7 +112,7 @@ where \(n<∞\) is the Hilbert space dimension. This condition is both
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necessary and sufficient. Examples of passive states are the state of
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the micro-canonical ensemble or Gibbs states. Gibbs states are further
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distinguished by additional features described
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in~\cite{Lenard1978Dec}, which can be connected to formulations of the
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in \refcite{Lenard1978Dec}, which can be connected to formulations of the
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zeroth and second laws of thermodynamics.
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One of these properties is complete passivity. Completely passive
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@ -289,7 +289,7 @@ bath properties except the temperature. It is therefore reasonable to
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expected that it is also valid for an infinite bath.
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Interestingly, a saturation of \cref{eq:thermo_ergo_bound} is achieved
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in~\cite{Skrzypczyk2014Jun} with a continuous qubit
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in \refcite{Skrzypczyk2014Jun} with a continuous qubit
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bath. In~\cite{Lobejko2021Feb} a more generic argument is made in a
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similar setting. Both propose concrete protocols within the bounds of
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thermal operations and by considering explicit work reservoirs. In
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@ -672,7 +672,7 @@ energy that can be extracted out of the system in relation to the
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energy that is simply transferred between the baths.
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An argument based on entropy may be made for the periodic steady state
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as was shown in~\cite{Kato2016Dec} and is reproduced here with the
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as was shown in \refcite{Kato2016Dec} and is reproduced here with the
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slight generalization of multiple baths and modulated coupling. We
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will find a Clausius like form of the second law. The left hand side
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of this inequality can then be interpreted as thermodynamic cost of
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@ -789,7 +789,7 @@ If one defines heat in this
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way~\cite{Kato2016Dec,Riechers2021Apr,Strasberg2021Aug},
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\cref{eq:secondlaw_cyclic} amounts to the Clausius inequality. The
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definition of heat as bath energy change is corroborated
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in~\cite{Esposito2015Dec} where it is shown, ableit for fermionic
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in \refcite{Esposito2015Dec} where it is shown, ableit for fermionic
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baths, that a definition of heat involving any nonzero fraction of the
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interaction energy will lead to the internal energy (as defined by the
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first law) not being an exact differential.
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@ -1391,7 +1391,7 @@ however, we find that also the modulation of the interaction, i.e. the
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coupling and decoupling, figures into the total power and reduces the
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energy output. In a weak coupling scheme, this contribution can be
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neglected. Not so however in the generic case presented here. A
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similar result was found in~\cite{Wiedmann2021Jun}.
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similar result was found in \refcite{Wiedmann2021Jun}.
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The mean power output of this cycle is
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\(\bar{P}=0.002468\pm 0.000021\) with an efficiency, as defined in
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@ -817,7 +817,7 @@ labelformat=brace, position=top]{subcaption}
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\(J(ω) = π ∑_λ\abs{g_λ}^2δ(ω-ω_λ)\)).
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\end{frame}
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\begin{frame}{Fock-Space Embedding}
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As in~\cite{Gao2021Sep} we can define
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As in \refcite{Gao2021Sep} we can define
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\begin{equation}
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\label{eq:fockpsi}
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\ket{Ψ} = \sum_\kmat\ket{\psi^\kmat}\otimes \ket{\kmat}
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