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\chapter{Conclusion and Ideas for future Work}
\chapter{Conclusion and Outlook}
\label{cha:concl-ideas-future}
In this work, we set out to find a way of accessing bath related
observables, such as the expected bath energy change and the
interaction energy expectation value, using the
NMQSD\footnote{Non-Markovian Quantum State
Diffusion}/HOPS\footnote{Hierarchy of Pure States} framework which
we introduced in \cref{chap:intro}. This endeavor was indeed
successful as was laid out in \cref{chap:flow}.
In \cref{chap:flow} we presented a solution to a well known model for
quantum Brownian motion. Using this solution, we were able to derive
expressions for the bath energy change \(∂_{t}\ev{H_{\bath}}\).
This enabled us to verify the results of \cref{chap:flow} in
\cref{chap:numres} by solving the same model numerically using
HOPS. Excellent agreement was found in
\cref{sec:hopsvsanalyt}.
Turning to the spin-boson model in \cref{sec:prec_sim}, we used energy
conservation to verify again, that we can consistently and efficiently
compute bath related observables with HOPS. In the cases where the
consistency condition was not met, we nevertheless found that
qualitatively correct results had been reached. The direct calculation
of the interaction energy by the use of \cref{sec:intener} gives
results that are more precise than the ones obtained through energy
conservation.
We continued to explore the energy transfer behavior of the zero
temperature spin-boson model and found that energy transfer
performance for strong coupling has a complicated dependence on the
spectral density of the bath. Energy transfer performance can be
optimized longer bath memories and resonant baths when the interaction
is turned off at the right time.
The short time dynamics of the bath energy change can be explained by
neglecting the system Hamiltonian, which we verified for the
spin-boson model. It was also found, that this short time behaviour is
already present on the trajectory level so that there are no
stochastic fluctuations for short times. During this initial period,
the auxiliary states of the HOPS are being populated.
In \cref{sec:singlemod} we turned to issues of quantum
thermodynamics. We reviewed some general analytical results that
bounded energy extraction from open systems in
\cref{sec:basic_thermo}, both for the single-bath and the multi-bath
case. We then turned to some more challenging applications of the HOPS
method. First, a driven spin-boson model was considered. We found that
a not insignificant fraction of the theoretical maximum of energy can
be extracted by modulating the coupling and providing a bath with long
memory time. We also demonstrated quantum friction, a quantum speed
limit and a bath resonance phenomenon.
Finally, we treated a model with multiple baths in \cref{sec:otto} and
non-harmonic smooth modulation. A cyclic modulation protocol was
implemented upon a two level system coupled to two baths in a
spin-boson like fashion. We achieved finite power with finite
efficiency and verified a Gibbs-like inequality
\cref{sec:operational_thermo}. When disabling the coupling modulation,
the power and efficiency were much reduced.
A worthwhile task for future work would be to verify the results
summarized in \refcite{Binder2018} for the Otto cycle. Especially the
optimization for optimal power which leads to the