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some more reordering and tweaking of the introduction of chap 4

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Valentin Boettcher 2022-09-23 14:55:09 +02:00
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@ -14,22 +14,33 @@ feature of the short time behavior of the bath energy flow that is
visible in all simulations will be discussed and explained in
\cref{sec:pure_deph}.
In the generic case where no analytic solution we nevertheless are
able to obtain consistent results as is demonstrated upon the example
of the spin-boson model \cref{sec:prec_sim}. We will also look into
the characteristics of the energy flow between system and bath
depending on the shape of the spectral density in
\cref{sec:energy-transf-char}. As the interaction energy is not
insubstantial in the regime that will be studied, we turn off the
interaction smoothly at the end of the evolution to be able to
adequately discuss our observations.
In the generic case where no analytic solution is known we
nevertheless are able to obtain consistent results as is demonstrated
upon the example of the spin-boson model \cref{sec:prec_sim}.
\Cref{sec:stocproc} investigates influence the precision of the
stochastic process sampling on the accuracy of the results. The same
analysis is performed for the hierarchy truncation in
\cref{sec:trunc}.
Finally, in \cref{sec:initial-slip-sb} we will study the short time
behavior of the spin-boson model, which will prove to be similar to
that of the harmonic oscillator discussed earlier. We will relate this
feature of the dynamics to the discussion in \cref{sec:pure_deph} and
also to the systematics of HOPS. Further, the effects of slowly and
smoothly turning on the interaction will be studied.
Subsequently, we will demonstrate some results with high consistency
\cref{sec:one_bath_cutoff} for varying bath memories. Prompted by the
results of this section we will also look into the characteristics of
the energy flow between system and bath depending on the shape of the
spectral density in \cref{sec:energy-transf-char}. As the interaction
energy is not insubstantial in the regime that will be studied, we
turn off the interaction smoothly at the end of the evolution to be
able to adequately discuss our observations.
The feasibility of precision studies is further demonstrated for
varying coupling strengths in \cref{sec:one_bathcoup_strength}.
Finally, in \cref{sec:initial-slip-sb,sec:moder-init-slip} the short
time behaviour of the spin-boson model is studied and will prove to be
similar to that of the harmonic oscillator discussed earlier. We will
relate this feature of the dynamics to the discussion in
\cref{sec:pure_deph} and also to the systematics of HOPS. The effects
of slowly and smoothly turning on the interaction will be studied in
\cref{sec:moder-init-slip}.
% These results will strengthen the confidence in
% the method so that we can turn to more complicated applications.
@ -38,6 +49,11 @@ smoothly turning on the interaction will be studied.
% will turn to two applications to demonstrate these features in
% \cref{sec:singlemod,sec:otto}.
However, before we begin in earnest some technical disscusions are in
order. \Cref{sec:meth} will explain our measures for the consistency
of results, as well as the spectral density used throughout the rest
of this work.
\section{Some Remarks on the Methods}
\label{sec:meth}
Before we begin with the applications in earnest, let us review some
@ -1404,10 +1420,107 @@ the Redfield master equation does not require the secular
approximation, but only weak coupling and can therefore capture
non-Markovian dynamics.
Let us now complete our precision studies of the zero temperature
spin-boson model energy flow by looking at the effect of different
coupling strengths in \cref{sec:one_bathcoup_strength}.
\subsection{Varying the Coupling Strength}%
\label{sec:one_bathcoup_strength}
\begin{wrapfigure}[-2]{o}{0.3\textwidth}*
\centering
\includegraphics{figs/one_bath_syst/final_states_flows}
\caption{\label{fig:delta_fs_flow} The absolute value difference of
the state energies and the maximal flows for the simulations in
\cref{fig:delta_energy_overview} from their value at coupling
strength \(α(0)=0.40\) normalized by their value at
\(α(0)=1.12\).}
\end{wrapfigure}
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
coupling strength. The results presented here are mainly a
demonstration of the feasibility of high-consistency simulations for a
range of coupling strengths. We will therefore keep the discussion of
the physical implications relatively short.
The chosen simulation parameters are the same as in
\cref{sec:one_bath_cutoff} and again consistent results have been
obtained throughout the whole range of coupling strengths as can be
gathered from \cref{fig:delta_interaction_consistency}. The
interaction strength was chosen linearly spaced and the simulation
results are presented in \cref{fig:delta_energy_overview}.
As the shape of the BCF is not altered between the simulations, the
bath energy flows look very similar as do the interaction
energies. The main difference between the simulations is the magnitude
of the interaction energy. With increased coupling strength there is
an increased interaction energy and an increased flow which leads to
faster energy loss in the system and faster energy gain of the
bath. The stronger the coupling, the more pronounced is the
non-monotonicity in time of the interaction energy, which is reflected
in a non-monotonicity in the bath energy expectation value.
The bath energy reaches a maximum and falls slightly for the strongest
coupling simulations (violet and brown lines). If the interaction is
strong enough, ``backflow'' can occur despite finite bath correlation
times. Here, the back flow is only occurring between interaction and
bath energy. In \cref{fig:markov_analysis_steady} the bath memory is
long, additionally to a strong coupling so that multiple oscillations
can be seen.
\begin{figure}[htp]
\centering
\includegraphics{figs/one_bath_syst/δ_energy_overview}
\caption{\label{fig:delta_energy_overview} Energy overview for the
model \cref{eq:one_qubit_model} for various coupling
strengths. The curves are converged out, and the error funnels are
not visible.}
\end{figure}
As a task for future work it might be worthwhile to ascertain the
exact conditions under which system energy might flow back into the
system. Is resonance required and what role is played by the system
time scale? Does backflow always occur provided the coupling is strong
enough?\footnote{The author expects a negative answer to this
question.}
Despite these differences for finite times, \cref{fig:delta_fs_flow}
demonstrates that the approximate steady state\footnote{excluding the
\(α(0)=0.4\) case} interaction energies
\(\ev{H_{\inter}}_{\mathrm{ss}}\) (blue), maximal flows
\(\abs{J}_{\mathrm{max}}\) (orange), system energies
\(\ev{H_{\sys}}_{\mathrm{ss}}\) (red) and bath energies
\(\ev{H_{\bath}}_{\mathrm{ss}}\) (green) are almost linearly dependent
on the coupling strength \(α(0)\) in the range of coupling strengths
studied.
We find that we can control the speed of the energy transfer between
bath and system with the coupling strength at the cost of greater
steady state interaction energy. Were we to turn off the interaction
diabatically, we would have to expend this energy in the worst
case. On the other hand, more adiabatic protocol as the one used in
\cref{sec:energy-transf-char} would likely be a remedy to this
drawback.
The cooling performance for a coupling that is being turned off at the
end would depend on the concrete protocol as we've seen in
\cref{sec:one_bath_cutoff} and a more detailed study is left to future
work. The interplay between the interaction time-scale mediated by the
coupling strength, the bath memory time and the system dynamics allows
for intricate tuning.
Both the final system and bath energies are increasing with the
coupling strength, compensating for the interaction energy which is
the main mechanism that leads to residual system energy in the steady
state which is further and further away from the ground state, which
would be the steady state of weak coupling dynamics.
We discussed the short term dynamics for a general model in
\cref{sec:pure_deph}. Now, in \cref{sec:initial-slip-sb} we will
apply them the spin-boson model.
\fixme{iftime: re-run with same coupling strength, more cutoff freqs,
less samples, see, longer times for coupling strengths, more coup}
\subsection{Initial Slip}%
\label{sec:initial-slip-sb}
\begin{figure}[htp]
@ -1496,7 +1609,7 @@ upon the bath energy change due to the initial slip, is the subject of
\subsection{Moderating the Inital Slip with Modulated Coupling}%
\label{sec:moder-init-slip}
\begin{wrapfigure}[-1]{O}{0.4\textwidth}*
\begin{wrapfigure}[-1]{o}{0.4\textwidth}*
\centering
\includegraphics{figs/one_bath_mod/modulation_protocols_init.pdf}
\caption{\label{fig:L_mod_init} The interaction is being switched on
@ -1583,105 +1696,6 @@ flow.
\end{figure}
Let us now complete our precision studies of the zero temperature
spin-boson model energy flow by looking at the effect of different
coupling strengths in \cref{sec:one_bathcoup_strength}.
\newpage
\subsection{Varying the Coupling Strength}%
\label{sec:one_bathcoup_strength}
\begin{wrapfigure}[-5]{O}{0.3\textwidth}*
\centering
\includegraphics{figs/one_bath_syst/final_states_flows}
\caption{\label{fig:delta_fs_flow} The absolute value difference of
the state energies and the maximal flows for the simulations in
\cref{fig:delta_energy_overview} from their value at coupling
strength \(α(0)=0.40\) normalized by their value at
\(α(0)=1.12\).}
\end{wrapfigure}
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
coupling strength. The results presented here are mainly a
demonstration of the feasibility of high-consistency simulations for a
range of coupling strengths. We will therefore keep the discussion of
the physical implications relatively short.
The chosen simulation parameters are the same as in
\cref{sec:one_bath_cutoff} and again consistent results have been
obtained throughout the whole range of coupling strengths as can be
gathered from \cref{fig:delta_interaction_consistency}. The
interaction strength was chosen linearly spaced and the simulation
results are presented in \cref{fig:delta_energy_overview}.
As the shape of the BCF is not altered between the simulations, the
bath energy flows look very similar as do the interaction
energies. The main difference between the simulations is the magnitude
of the interaction energy. With increased coupling strength there is
an increased interaction energy and an increased flow which leads to
faster energy loss in the system and faster energy gain of the
bath. The stronger the coupling, the more pronounced is the
non-monotonicity in time of the interaction energy, which is reflected
in a non-monotonicity in the bath energy expectation value.
The bath energy reaches a maximum and falls slightly for the strongest
coupling simulations (violet and brown lines). If the interaction is
strong enough, ``backflow'' can occur despite finite bath correlation
times. Here, the back flow is only occurring between interaction and
bath energy. In \cref{fig:markov_analysis_steady} the bath memory is
long, additionally to a strong coupling so that multiple oscillations
can be seen.
\begin{figure}[htp]
\centering
\includegraphics{figs/one_bath_syst/δ_energy_overview}
\caption{\label{fig:delta_energy_overview} Energy overview for the
model \cref{eq:one_qubit_model} for various coupling
strengths. The curves are converged out, and the error funnels are
not visible.}
\end{figure}
As a task for future work it might be worthwhile to ascertain the
exact conditions under which system energy might flow back into the
system. Is resonance required and what role is played by the system
time scale? Does backflow always occur provided the coupling is strong
enough?\footnote{The author expects a negative answer to this
question.}
Despite these differences for finite times, \cref{fig:delta_fs_flow}
demonstrates that the approximate steady state\footnote{excluding the
\(α(0)=0.4\) case} interaction energies
\(\ev{H_{\inter}}_{\mathrm{ss}}\) (blue), maximal flows
\(\abs{J}_{\mathrm{max}}\) (orange), system energies
\(\ev{H_{\sys}}_{\mathrm{ss}}\) (red) and bath energies
\(\ev{H_{\bath}}_{\mathrm{ss}}\) (green) are almost linearly dependent
on the coupling strength \(α(0)\) in the range of coupling strengths
studied.
We find that we can control the speed of the energy transfer between
bath and system with the coupling strength at the cost of greater
steady state interaction energy. Were we to turn off the interaction
diabatically, we would have to expend this energy in the worst
case. On the other hand, more adiabatic protocol as the one used in
\cref{sec:energy-transf-char} would likely be a remedy to this
drawback.
The cooling performance for a coupling that is being turned off at the
end would depend on the concrete protocol as we've seen in
\cref{sec:one_bath_cutoff} and a more detailed study is left to future
work. The interplay between the interaction time-scale mediated by the
coupling strength, the bath memory time and the system dynamics allows
for intricate tuning.
Both the final system and bath energies are increasing with the
coupling strength, compensating for the interaction energy which is
the main mechanism that leads to residual system energy in the steady
state which is further and further away from the ground state, which
would be the steady state of weak coupling dynamics.
\fixme{iftime: re-run with same coupling strength, more cutoff freqs,
less samples, see, longer times for coupling strengths, more coup}
\section{Conclusion}%
\label{sec:conclusion-1}