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some touchups on the outro

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In this work, we set out to find a way of accessing bath related In this work, we set out to find a way of accessing bath related
observables, such as the expected bath energy change and the observables, such as the expected bath energy change and the
interaction energy expectation value, using the interaction energy expectation value using the
NMQSD\footnote{Non-Markovian Quantum State NMQSD\footnote{Non-Markovian Quantum State
Diffusion}/HOPS\footnote{Hierarchy of Pure States} framework which Diffusion}/HOPS\footnote{Hierarchy of Pure States} framework which
we introduced in \cref{chap:intro}. This endeavor was indeed we introduced in \cref{chap:intro}.
successful as was laid out in \cref{chap:flow}.
In \cref{chap:flow} we presented a solution to a well known model for This endeavour was indeed successful, as we laid out in
quantum Brownian motion. Using this solution, we were able to derive \cref{chap:flow}. The crucial point is, that we still have access to
expressions for the bath energy change \(∂_{t}\ev{H_{\bath}}\). the bath degrees of freedom \emph{before} performing a stochastic
average over the NMQSD/HOPS trajectories. This allows us to compute
the expectation values of observables which contain the bosonic bath
operators collectively.
In \cref{chap:flow} we presented an analytic solution to a well known
model for quantum Brownian motion. Using this solution, we were able
to derive expressions for the bath energy change
\(∂_{t}\ev{H_{\bath}}\) for models with one and two baths.
This enabled us to verify the results of \cref{chap:flow} in This enabled us to verify the results of \cref{chap:flow} in
\cref{chap:numres} by solving the same model numerically using \cref{chap:numres} by solving the same model numerically using
HOPS. Excellent agreement was found in HOPS. Excellent agreement was found in \cref{sec:hopsvsanalyt}.
\cref{sec:hopsvsanalyt}.
Turning to the spin-boson model in \cref{sec:prec_sim}, we used energy Turning to the spin-boson model in \cref{sec:prec_sim}, we used energy
conservation to verify again, that we can consistently and efficiently conservation to verify again, that we can consistently and efficiently
compute bath related observables with HOPS. In the cases where the compute bath related observables with HOPS upon the example of the
consistency condition was not met, we nevertheless found that interaction energy expectation value. In the cases where the
qualitatively correct results had been reached. The direct calculation consistency condition was not met, we nevertheless found that the
of the interaction energy by the use of \cref{sec:intener} gives results were qualitatively correct. When choosing less precise
results that are more precise than the ones obtained through energy numerical parameters, the direct calculation of the interaction energy
conservation. by the use of \cref{sec:intener} yields results that are generally
much more precise than the ones obtained through energy conservation.
We continued to explore the energy transfer behavior of the zero We continued to explore the energy transfer behavior of the zero
temperature spin-boson model and found that energy transfer temperature spin-boson model and found that energy transfer
performance for strong coupling has a complicated dependence on the performance for strong coupling has a complicated dependence on the
spectral density of the bath. Energy transfer performance can be shape of the spectral density of the bath. Energy transfer performance
optimized longer bath memories and resonant baths when the interaction can be optimized through longer bath memories and resonant baths when
is turned off at the right time. the interaction is switched off at the right time. Switching the
interaction off in finite time leads to cooling of the system,
especially when the steady state had been reached.
The short time dynamics of the bath energy change can be explained by Having explained short time dynamics of the bath energy change in
neglecting the system Hamiltonian, which we verified for the \cref{sec:pure_deph} neglecting the system Hamiltonian, which we
spin-boson model. It was also found, that this short time behaviour is verified for the spin-boson model in
already present on the trajectory level so that there are no \cref{sec:initial-slip-sb,sec:moder-init-slip}. It was further found,
stochastic fluctuations for short times. During this initial period, that this short time behaviour is already present on the trajectory
the auxiliary states of the HOPS are being populated. level so that there are no stochastic fluctuations for short
times. Instead, the auxiliary states of the HOPS are being populated
during this initial period.
In \cref{sec:singlemod} we turned to issues of quantum In \cref{sec:singlemod} we turned to some issues of quantum
thermodynamics. We reviewed some general analytical results that thermodynamics. We reviewed general analytical results that bounded
bounded energy extraction from open systems in energy extraction from open systems in \cref{sec:basic_thermo}, both
\cref{sec:basic_thermo}, both for the single-bath and the multi-bath for the single-bath and the multi-bath case. We then turned to some
case. We then turned to some more challenging applications of the HOPS more challenging applications of the HOPS method. First, a driven
method. First, a driven spin-boson model was considered. We found that spin-boson model was considered. We found that a not insignificant
a not insignificant fraction of the theoretical maximum of energy can fraction of the upper bound on the ergotropy can be extracted by
be extracted by modulating the coupling and providing a bath with long modulating the coupling and providing a bath with long memory time. We
memory time. We also demonstrated quantum friction, a quantum speed also demonstrated quantum friction, a quantum speed limit and a bath
limit and a bath resonance phenomenon. resonance phenomenon.
Finally, we treated a model with multiple baths in \cref{sec:otto} and Finally, we treated a model with multiple baths and non-harmonic
non-harmonic smooth modulation. A cyclic modulation protocol was smooth modulation in \cref{sec:otto}. A cyclic modulation protocol was
implemented upon a two level system coupled to two baths in a implemented upon a two level system coupled to two baths in a
spin-boson like fashion. We achieved finite power with finite spin-boson like fashion. We achieved finite power with finite
efficiency and verified a Gibbs-like inequality efficiency and verified a Gibbs-like inequality
@ -66,19 +77,20 @@ optimization for optimal power which leads to the
Novikov–Curzon–Ahlborn efficiency \(η_{ca}=1-\sqrt{T_{c}/T_{h}}\) is Novikov–Curzon–Ahlborn efficiency \(η_{ca}=1-\sqrt{T_{c}/T_{h}}\) is
interesting in the case of stronger coupling. interesting in the case of stronger coupling.
Another cycle to study would be a Carnot-type cycle, where the Another cycle to study would be a Carnot-type process, where the
modulation of the system and the thermalization with the bath occur at modulation of the system and the thermalization with the bath occur at
the same time. Interpolating between Otto and Carnot, as well as the same time. Interpolating between Otto and Carnot, as well as
studying the effect of overlapping and shifting strokes is a studying the effect of overlapping and shifting strokes is a
fascinating avenue for future exploration. fascinating avenue for future exploration.
Also, more interesting working media, such as a three level system are Also, more interesting working media, beginning with a three level
of interest. In \refcite{Uzdin2015Sep} it is shown, that in certain system, are of interest. In \refcite{Uzdin2015Sep} it is shown that
regimes quantum coherence can lead to superior power output. In the in certain regimes quantum coherence can lead to superior power
same regime different types heat engines are equivalent. Both these output. In the same regime different types heat engines are
effects have been observed experimentally in \refcite{Klatzow2019Mar}. It equivalent. Both these effects have been observed experimentally in
would be interesting to see if the slight deviations from theory in \refcite{Klatzow2019Mar}. It would be interesting to see if the slight
\cite{Klatzow2019Mar} could be explained using HOPS. deviations from theory in \cite{Klatzow2019Mar} could be explained
using HOPS.
The so called Anti-Zeno Effect occurring in systems under fast The so called Anti-Zeno Effect occurring in systems under fast
modulation has recently received some attention modulation has recently received some attention
@ -86,8 +98,8 @@ modulation has recently received some attention
due to the broadening of the resonance criterion which we have due to the broadening of the resonance criterion which we have
observed in observed in
\cref{sec:one_bath_cutoff,sec:modcoup_reso,sec:otto}. Being a \cref{sec:one_bath_cutoff,sec:modcoup_reso,sec:otto}. Being a
consequence of the energy time uncertainty it is being argued, that consequence of the energy time uncertainty it is argued that the
the origin of this advantage is truly quantum. The tools for the origin of this advantage is truly quantum. The tools for the
exploitation of this effect and its verification are provided in this exploitation of this effect and its verification are provided in this
work. However, a strong coupling analysis has already been performed work. However, a strong coupling analysis has already been performed
using HEOM in \refcite{Xu2022Mar}. using HEOM in \refcite{Xu2022Mar}.
@ -97,13 +109,17 @@ of finite ergotropy by letting energy flow through the working medium
and then extracting this ergotropy in a separate stroke. This work and then extracting this ergotropy in a separate stroke. This work
could be verified and expanded to the non-Markovian regime. could be verified and expanded to the non-Markovian regime.
A useful improvement of the method would be the ability to snapshot A useful technical improvement of the method would be the ability to
the total state of system and bath and then propagate this state with snapshot the total state of system and bath and then propagate this
different modulation protocols. Also, exploring the thermofield method state with different modulation protocols. Also, exploring the
for finite temperature to avoid the slow convergence of the flow may thermofield method for finite temperature to avoid slow convergence
be worthwhile. However, at least for coupling that is not hermitian, would be helpful. However, at least for coupling that is not
this would only trade computational effort for memory, as the number hermitian, this would likely only trade computational effort for
of hierarchy states would increase. memory, as the number of hierarchy states would increase.
The splitting up of the stochastic process to calculate, for example, the
energy change of parts of the bath as discussed at the end of
\cref{sec:general_obs} is also very interesting.
Finally, in the spirit of~\cite{Esposito2015Dec} one could employ the Finally, in the spirit of~\cite{Esposito2015Dec} one could employ the
HOPS to verify whether a given definition of internal energy that HOPS to verify whether a given definition of internal energy that