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src/num_results.tex
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@ -1,5 +1,19 @@
\chapter{Numerical Results}
\label{chap:numres}
\begin{itemize}
\item all the theory developed in \cref{chap:flow,chap:analytsol}
wants some applications
\item roadmap: first verify numerics and systematics using
\cref{chap:analytsol} in \cref{sec:hopsvsanalyt} and then using
energy conservation \cref{sec:prec_sim}. Although not the focus:
also some discussion of observations and the role of stronger
coupling and non-markovianity
\item especially short time behaviour in \cref{sec:pure_deph}
\item after systematics interlude about basic quantum-thermo
\cref{sec:basic_thermo}
\item finally short applications: single bath \cref{sec:singlemod} and
a simple otto cycle \cref{sec:otto}
\end{itemize}
In this chapter some application of the results described in
\cref{chap:flow,chap:analytsol} are presented. In
\cref{sec:hopsvsanalyt}, we begin by considering the bath energy for
@ -8,6 +22,11 @@ with the results obtained by hops.
\section{Some Remarks on the Methods}
\label{sec:meth}
\begin{itemize}
\item lowest common denominator of all models
\item error estimation important to claim convergence, compatibility
\end{itemize}
The figures presented may feature error funnels whose origin is,
unless otherwise stated, estimated from the empirical standard
deviation of the calculated quantities due to the finite sample
@ -79,8 +98,13 @@ derived. Using this solution, we are able to verify the results of
HOPS can indeed reproduce the \emph{exact} open system dynamics of the
bath energy flow.
\subsection{One Oscillator, One Bath}
\subsection{A Harmonic Oscillator coupled to a single Bath}
\label{sec:oneosccomp}
\begin{itemize}
\item a numerically simple model (because of Hilbert space, one bath)
to begin our verification
\item complex implementation needs high level tests
\end{itemize}
For the simulations with HOPS the model \cref{eq:one_ho_hamiltonian}
was made dimensionless by choosing \(Ω=1\). Simulations were run for
both for zero temperature and a finite temperature with varying bath
@ -296,8 +320,13 @@ the probability space is sampled.
% used in the last simulation of \cref{fig:comp_zero_t}.}
% \end{figure}
\subsection{Two Oscillators, Two Baths}
\subsection{Two coupled Harmonic Oscillators coupled to two Baths}
\label{sec:twoosccomp}
\begin{itemize}
\item first ever application of HOPS to multiple baths
\item important to verify fitness for thermodynamics simulations
\item much more challenging
\end{itemize}
The model of \cref{sec:oneosc} was generalized to two oscillators
coupled to two separate baths in \cref{sec:twoosc} and
@ -394,7 +423,7 @@ Hamiltonians. Because the NMQSD and also HOPS are largely agnostic of
these factors, we may safely assume that the results of the comparison
will be similar to the ones presented here.
\subsection{Pure Dephasing: The initial Slip}
\section{Pure Dephasing and The Initial Slip}
\label{sec:pure_deph}
As seen in \cref{fig:comp_finite_t,fig:comp_zero_t,fig:comp_two_bath},
the short time behavior of the bath energy flow is dominated by
@ -515,9 +544,16 @@ is assumed to be unity in the simulations referred to in this
thesis. \fixme{maybe change}
\section{Precision Simulations for a System without Analytical
Solution}
\section{Precision Simulations of the Zero Temperature Spin-Boson Model}
\label{sec:prec_sim}
\begin{itemize}
\item often no analytical solution available: how to verify
convergence?
\item at the same time: simplest model to start with, numerically
simple
\item good starting point to get to know the system and find out if
results can be trusted
\end{itemize}
In this section, we will study the energy flow of a simple model
connected to zero temperature bath. Both the characteristics of the
flow mediated by the concrete form of the bath correlation function
@ -726,7 +762,14 @@ of convergence.
\subsection{Dependence on the Cutoff Frequency}
\label{sec:one_bath_cutoff}
We now consider precision simulations of the spin-boson like system
\begin{itemize}
\item what role does non-markovianity play in this simple model for
the energy transfer?
\item want to deepen indications seen in precision simulations
\item also verify initial slip dynamics \cref{sec:pure_deph} for this model
\end{itemize}
We now consider precision simulations of the spin-boson like
system\fixme{Float placement is abysmal in this section}
with varying cutoff frequency. To make the interaction energies
comparable to each other, the BCF normalization of section
\cref{sec:pure_deph} is being used. Because of the small size of the
@ -1047,6 +1090,10 @@ the flow into the bath energy change.
\subsection{Dependence on the Coupling Strength}
\label{sec:one_bathcoup_strength}
\begin{itemize}
\item does it pay off to couple more strongly?
\item very brief discussion
\end{itemize}
After having studied the dependence of the bath energy flow for
various cutoff frequencies of the BCF in \cref{sec:one_bath_cutoff},
we now consider the case with fixed cutoff \(ω_c=2\) but varying
@ -1182,6 +1229,9 @@ optimal efficiency.
\subsection{The Ergotropy of Open Quantum Systems with a single Bath}
\label{sec:ergo_general}
\begin{itemize}
\item introduce ergotropy and passivity as basis for the rest
\end{itemize}
The ergotropy of a quantum system is defined\fixme{mention paper that
uses ergo for heat}
as~\cite{Binder2018}
@ -1252,7 +1302,12 @@ have to lead to a reduction in efficiency\fixme{do more research on
\subsection{The Ergotropy of Finite Systems Coupled to a Thermal Bath}
\label{sec:ergoonebath}
\begin{itemize}
\item thermal states are somewhat special (as seen above)
\item for thermodynamic consistency: we want to find a general bound
on ergotropy that may be valid also in the infinite dimensional case
\item great news: model independent and gives meaning to temperature!
\end{itemize}
Let us consider models with the Hamiltonians
\begin{equation}
\label{eq:simple_bath_models}
@ -1394,6 +1449,13 @@ time.
\subsection{The Ergotropy of a Two
Level System and a Bath of Identical Oscillators}
\label{sec:explicitergo}
\begin{itemize}
\item as an illustrative example: calculate ergotropy for a concrete
model
\item can show us how good the bound derived earlier is and whether it
holds for these infinite dimensional systems
\end{itemize}
Here, we explicitly calculate the ergotropy of a finite dimensional
system connected to a bath of identical oscillators. Throughout we
will set the zero-point energy of the oscillators to zero, meaning
@ -1713,6 +1775,14 @@ tight as may be expected, because the error one makes in
\subsection{A bound on the Energy Change of Multiple Baths in the
Periodic Steady State}
\label{sec:operational_thermo}
\begin{itemize}
\item for multiple baths of differing temperatures: previous
discussions not applicable
\item nevertheless, there is a limit on the combined energy change of
the baths in the PSS
\item a gibbs like inequality is derived, the left-hand-side may be
interpreted as thermodynamic cost, maybe input for future optimizations
\end{itemize}
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
@ -1812,15 +1882,14 @@ motivated in \cite{Riechers2021Apr}, where heat is identified with
into account system and bath is being considered and brought into
connection with information-theoretic quantities.
If one defines heat as above and in
\cite{Kato2016Dec,Riechers2021Apr,Strasberg2021Aug}, i.e. as the
energy change of the baths and substantiated, based on a microscopic
definition of entropy, in, \cref{eq:secondlaw_cyclic} amounts to the
Clausius form of the second law. This definition of heat is
corroborated in~\cite{Esposito2015Dec} where it is shown\footnote{for
fermionic baths} that a definition of heat involving any nonzero
fraction of the interaction energy will lead to the internal energy
(as defined by the first law) not being an exact differential.
If one defines heat as is done in
\cite{Kato2016Dec,Riechers2021Apr,Strasberg2021Aug} as the change of
bath energy, \cref{eq:secondlaw_cyclic} amounts to the Clausius form
of the second law. This definition of heat is corroborated
in~\cite{Esposito2015Dec} where it is shown\footnote{for fermionic
baths} that a definition of heat involving any nonzero fraction of
the interaction energy will lead to the internal energy (as defined by
the first law) not being an exact differential.
In contrast to~\cite{Strasberg2021Aug}, no interpretation in terms of
thermodynamical quantities is required for \cref{eq:secondlaw_cyclic}
@ -1834,6 +1903,16 @@ with two baths \cref{eq:secondlaw_cyclic} implies the Carnot bound.
\section{Modulation of System and Interaction for a Single Bath}
\label{sec:singlemod}
\begin{itemize}
\item again a numerically simple model to apply our findings from
\cref{sec:basic_thermo}
\item not clear how tight the bound is and if it is still valid
\item apply also findings about resonance (modified for modulation)
from previous section
\item despite simplicity: a lot of parameters and we only look at a
shallow subset of all possibilities
\end{itemize}
Because the HOPS allows us to simulate a full dynamical picture of our
model, let us now turn to a situation where the dynamics are of
@ -2185,7 +2264,13 @@ coupling.
\section{Quantum Otto Cycle}
\label{sec:otto}
\begin{itemize}
\item multi bath simulations are resource intensive
\item we focus therefore on a simple demonstration of HOPSs usefulness
in multiple baths
\item an opportunity to test \cref{sec:operational_thermo}
\item also: very popular model and good starting point for future work
\end{itemize}
As a demonstration of a standard thermodynamic cycle that is a popular
model in the literature\footnote{See
\cite{Wiedmann2021Jun,Karimi2016Nov,Binder2018}.}
@ -2368,20 +2453,20 @@ would be interesting to see if the slight deviations from theory in
\newpage
\section{Anti Zeno Engine}
\label{sec:antizeno}
\section{Miscellaneous Demonstrations of the Capabilities of HOPS}
\label{sec:miscdemo}
Very short mention of some results from ``side projects'' if I have
the time to include them.
\begin{itemize}
\item two qubits coupled to each other -> steady state flow
\item otto cycle
\item rotating engine
\item mention concept
\item results not reliable in time for thesis
\item interesting because: non markovian QUANTUM advantage. a bit
sensational ;P
\end{itemize}
\section{Some Proposals for future Work}
\begin{itemize}
\item a list of ideas and some papers I've came across
\item projects for future theses or papers
\end{itemize}
\begin{itemize}
\item ... list all those nice papers ...
\item the third law