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7
src/further_graphics.tex
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@ -5,13 +5,6 @@
\subsection{Interaction Consisency Plots}
\label{sec:intercons}
\begin{figure}[H]
\centering
\includegraphics{figs/one_bath_syst/omega_interaction_consistency}
\caption{\label{fig:omega_interaction_consistency}Interaction
consistency plot for \cref{sec:one_bath_cutoff}, similar to
\cref{fig:stocproc_systematics}.}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics{figs/one_bath_syst/delta_interaction_consistency}

380
src/num_results.tex
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@ -10,7 +10,7 @@ The roadmap is the following. Using \cref{chap:analytsol} we will
verify the results of \cref{chap:flow} in
\cref{sec:hopsvsanalyt}. Excellent consistency of the analytical and
numeric solutions for the QBM models will be demonstrated. A common
feature of the short time behavior of the bath energy flow that is
feature of the short time behaviour of the bath energy flow that is
visible in all simulations will be discussed and explained in
\cref{sec:pure_deph}.
@ -550,17 +550,17 @@ 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.
Let us now turn to the short time behavior of the flow and the initial
slip in \cref{sec:pure_deph} .
Let us now turn to the short time behaviour of the flow and the
initial slip in \cref{sec:pure_deph} .
\section{Pure Dephasing and the Initial Slip}
\label{sec:pure_deph}
As evidenced 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 a characteristic
peak. Because this peak occurs at very short time scales, it may in
part be explained by a simple calculation which neglects the system
dynamics by setting \(H_\sys=0\).
time behaviour of the bath energy flow is dominated by a
characteristic peak. Because this peak occurs at very short time
scales, it may in part be explained by a simple calculation which
neglects the system dynamics by setting \(H_\sys=0\).
We solve the model with the Hamiltonian (Schr\"odinger picture)
\begin{equation}
@ -921,9 +921,9 @@ less focus on the systematics.
\subsection{Varying the Cutoff Frequency}
\label{sec:one_bath_cutoff}
The lessons learned in \cref{sec:stocproc,sec:trunc} will now be
applied to simulate \cref{eq:one_qubit_model} with high consistency
for various parameter choices of the cutoff \(ω_{c}\) of the Ohminc
BCF (see \cref{sec:meth})
applied to simulate the spin-boson model \cref{eq:one_qubit_model}
with high consistency for various parameter choices of the cutoff
frequency \(ω_{c}\in\{1,2,3,4\}\) of the Ohminc BCF (see \cref{sec:meth})
\begin{equation}
\label{eq:ohmic_bcf_repeat}
α(τ) =
@ -931,18 +931,35 @@ BCF (see \cref{sec:meth})
\end{equation}
Guided by these demonstrations, we'll turn to a more detailed analysis
of the role of non-Markovianity in the energy transfer characteristics
of our model. The quantification of the initial slip dynamics in
of the model. The quantification of the initial slip dynamics in
\cref{sec:pure_deph} will also be verified.
To make the interaction energies comparable to each other, the BCF
normalization of \cref{sec:pure_deph} is being used. Because
of the small size of the Hilbert space, we were able to choose a HOPS
normalization of \cref{sec:pure_deph} is being used. Because of the
small size of the Hilbert space, we were able to choose a HOPS
configuration\footnote{\(\norm{\vb{k}}\leq 7\), seven BCF terms,
\(\varsigma = 10^{-6}\)} that yields high-accuracy
results\footnote{A detailed account of the consistency is given in
\cref{fig:omega_interaction_consistency}.}, based on the results of
the previous section. The only problematic result is the one for cutoff
\(ω_c=1\), but there is good qualitative consistency in this case.
\(\varsigma = 10^{-6}\)} that yields high-accuracy results, based on
the results of the previous section. The only problematic result is
the one for the cutoff \(ω_c=1\), but there is good qualitative
consistency in this case. The failing of the consistency check may be
due to insufficient time resolution, or simply because the bath memory
is so long, that the hierarchy depth was insufficient despite the
rather weak coupling. The normalized difference between the two
interaction energies for this simulation (blue line, right panel of
\cref{fig:omega_interaction_consistency}) is of the order of
\(10^{-3}\) and thus quite acceptable. The consistency test is a very
high bar to cross.
\begin{figure}[htp]
\centering
\includegraphics{figs/one_bath_syst/omega_interaction_consistency}
\caption{\label{fig:omega_interaction_consistency}Interaction
consistency plot for \cref{sec:one_bath_cutoff}, similar to
\cref{fig:stocproc_systematics}. Good consistency is being
achieved. The \(ω_{c}=1\) case (blue) does not pass the
consistency test. Nevertheless, the qualitative agreement of the
direct and indirect interaction energies (left panel) is quite
good.}
\end{figure}
For all simulations \(N=5\cdot 10^{5}\) trajectories were integrated
to produce the results that are summarized in
@ -950,72 +967,72 @@ to produce the results that are summarized in
\begin{figure}[htp]
\centering
\includegraphics{figs/one_bath_syst/omega_energy_overview}
\caption{\label{fig:omega_systematics_system} Energy overview for the
model \cref{eq:one_qubit_model} for various coupling strengths and
cutoff frequencies. The curves are converged, and the error
funnels are not visible.}
\caption{\label{fig:omega_systematics_system} Energy overview for
the spin-boson model \cref{eq:one_qubit_model} for various
coupling strengths and cutoff frequencies. The curves are
converged, and the error funnels are not visible. The \(ω_{c}=1\)
stands apart from the others due to its long bath memory. The
system energy loss is the fastest of all cases despite the weaker
coupling as gauged by the magnitude of the interaction energy.}
\end{figure}
The interaction energy expectation values \(\ev{H_{\inter}}\)
(\cref{fig:omega_systematics_system}, upper left panel) differ
significantly, despite being roughly of the same order of magnitude
and not in the weak coupling regime. This illustrates the limitation
of the estimate in \cref{sec:pure_deph} and exemplifies the
nontriviality of the open system dynamics. Better estimates of the
interaction energy and thus interaction strength may be derived from
ideas similar to the ones discussed in \cref{sec:normest}. Besides
their magnitude, the qualitative time dependence of the interaction
energies varies, especially between the configuration with the cuoff
frequency \(ω_c=1\) (blue) and the others.
Let us preface the following discussion with a note of caution. All
the discussed phenomena are specific to the minimal model
\cref{eq:one_qubit_model}, although sometimes similarities to
phenomena observed in \cref{sec:hopsvsanalyt} can be seen. Whether
there is some universality to the results obtained is an interesting
question for a more detailed future detailed study.\fixme{Remove this?}
The interaction energy expectation values, despite being in the same
order of magnitude and not in the weak coupling regime, differ
significantly. This illustrates the limitation of the estimate in
\cref{sec:pure_deph} and exemplifies the nontriviality of the open
system dynamics. Better estimates of the interaction energy and thus
interaction strength may be derived from ideas similar to the ones
discussed in \cref{sec:normest}. Besides the magnitude, the
qualitative time dependence of the interaction energies varies,
especially between the configuration with the cuoff frequency
\(ω_c=1\) (blue) and the others.
The curve with the cuoff frequency \(ω_c=1\) (blue) exhibits two
pronounced turning points in contrast to the other simulations. This
behaviour is a symptom of the very long bath memory, as we will
discuss below. Further, despite having the weakest overall coupling
strength \(α(0)=0.4\) the system energy falls off the fastest after a
short period where it is above the other configurations. This
The interaction energy curve of the simulation with the cutoff
frequency \(ω_c=1\) (blue) in \cref{fig:omega_systematics_system}
exhibits two pronounced turning points in contrast to the other
simulations. This behaviour is a symptom of the very long bath memory,
as we will discuss below. Further, despite having the weakest overall
coupling strength \(α(0)=0.4\) the system energy falls off the fastest
after a short period where it is above the other configurations. This
behaviour is also visible in the bath energy expectation value, where
this simulation almost reaches the same levels as the \(ω_c=2\)
configuration (orange) for \(τ\gtrsim 5\).
The flow is generally negative (the bath gains energy) and decays
after an initial peak. All the flow graphs appear to be crossing in
about the same point \(τ\approx 7.7\) after which their ordering by
magnitude is reversed. The inset shows that the crossing is not
precisely at the same point, nevertheless further investigation of
this phenomenon may of interest in the future. Due to the pure
dephasing behavior discussed in \cref{sec:pure_deph}, a greater cutoff
coupled with a greater coupling strengths leads to a higher peak in
the flow for short times. After the peak the flow decays the faster,
the higher the cutoff. The situation is reversed after some time so
that the flow of higher cutoffs decays slower. This is due to the
fact that still more system energy is left to be transmitted to the
bath.
The flow \(J\) is generally negative (the bath gains energy) and
decays after an initial peak. All the flow graphs appear to be
crossing in about the same point \(τ\approx 7.7\) after which their
ordering by magnitude is reversed. The inset shows that this isn't
precisely the case, nevertheless further investigation of this
phenomenon may of interest in the future. Due to the pure dephasing
behaviour discussed in \cref{sec:pure_deph}, a greater cutoff coupled
with a greater coupling strengths leads to a higher peak in the flow
for short times. After the peak the flow decays the faster, the higher
the cutoff. The situation is reversed after some time. This is due to
the fact that still more system energy is left to be transmitted to
the bath.
The presentation in \cref{fig:omega_systematics_system} is not
conducive to comparing the actual performance of the energy transfer,
due to the variable shape of the spectral density and the chosen
coupling strengths. Because the maximum of the Ohmic spectral density
is located at \(ω_c\), the special observed energy transfer behaviour
for \(ω_c=1\) is likely due to a resonance effect (see above) as the
system energy level spacing is unity.
for \(ω_c=1\) is likely due to a resonance effect as the system energy
level spacing coincides with the maximum and the coupling is on the
weaker side.
The dissipator of the master equation for this two level system only
depends on the value of the spectral density at the level
depends on the value of the spectral density at the qubit level
spacing~\cite[p. 66]{Rivas2012}\footnote{There \(L=σ_{+}\), but this
has no bearing on the connection to \(ω_{0}\).} (\(ω_{0}=1\) here),
so that it is reasonable to expect, that there may be variations
whenever its magnitude at this point changes. On the other hand, a
strong dependence of the flow on the shape of the bath correlation
function beyond lamb-shift like influences, as we will find below, is
an indicator of the departure from the weak coupling limit.
\fixme{Here, some master equation comparison would be nice.}
has no bearing on the connection to \(ω_{0}\).} \(ω_{0}=1\), so
that it is reasonable to expect, that there may be variations whenever
its magnitude at this point changes. On the other hand, a strong
dependence of the flow on the shape of the bath correlation function
beyond lamb-shift like influences is an indicator of the departure
from the weak coupling limit. \fixme{Here, some master equation
comparison would be nice.}
For these reasons, we now perform a more detailed analysis in
\cref{sec:energy-transf-char}.
\subsection{Energy Transfer Characteristics}
\label{sec:energy-transf-char}
@ -1024,56 +1041,76 @@ in our simulations, we continue to shed some light on the role of bath
memory and resonance between system and bath in the behaviour of the
energy transfer between system and bath.
\begin{wrapfigure}{o}{0.3\textwidth}*
For a systematic study of resonance, we first compare the flow for
frequency shifted ohmic spectral densities\footnote{See
\cref{sec:shift_sp} for details.} with identical scaling \(α(0)\)
and cutoff frequency \(ω_c=2\). The shifted spectral density is given
by
\begin{equation}
\label{eq:shift_sd}
J_{\mathrm{s}}(ω_{s})= J(ω - ω_{s}),
\end{equation}
where \(ω_{s} > 0\).
\begin{wrapfigure}[-1]{O}{0.3\textwidth}*
\centering
\includegraphics{figs/one_bath_syst/L_mod}
\caption{\label{fig:L_mod} The smooth modulation of the coupling
operator \(L(τ)\).}
\end{wrapfigure}
For a systematic study of resonance, we first compare the flow for
shifted ohmic spectral densities\footnote{See \cref{sec:shift_sp} for
details.} with identical scaling \(α(0)\) and cutoff frequency
\(ω_c=2\). We turn off the interaction smoothly\footnote{A smoothstep
Also, we turn off the interaction smoothly\footnote{A smoothstep
function of order two with a transition period of two. See
\cref{sec:smoothstep}.} over two time units before the system has
reached its steady state and compare how much energy has been
transferred in terms of the final bath
\cref{sec:smoothstep}.} over two time units (see \cref{fig:L_mod})
before the system has reached its steady state and compare how much
energy has been transferred in terms of the final bath
\(\ev{H_{\bath}}\)\footnote{This is actually a slight misnomer, as we
consider the \emph{change} of bath energy. The bath energy itself is
infinite.} and system energies and the total energy change
\(Δ\ev{H}\) due to the modulated coupling.
The less energy the system has at the end of the process and the
faster the process concludes, the better the performance of
transfer. If the interaction energies are on the same scale, the
decoupling costs should be roughly the same in terms of total energy
change. Otherwise, they may lead to an additional change of system and
bath energies between the different simulations. Ideally the bath
energy changes to take up any energy introduced into the system by the
decoupling rather than the system energy.
This is the first time we make use of time dependent coupling. Despite
adding complexity to the method, this strategy simplifies our
discussion. Residual interaction energy is eliminated, making a
comparison of different simulations possible. Shifting the spectral
densities, another technical trick, allows us to study resonance
behaviour without altering the \emph{shape} of the spectral density,
again simplifying a comparison.
To make space for the shifted bath correlation functions in frequency
space, the system energy gap has been set to the value four so that
\(α(0)=8\) represents a quite reasonable coupling strength for our
purposes here.
space, the qubit energy gap has been set to the value four so that
\(α(0)=8\) represents a quite reasonable coupling strength for the
purpose of studying energy transfer out of the system on a suitable
timescale.
The results for three shifts are presented in
\cref{fig:resonance_analysis}. For all shifts the spectral density has
a finite value at the value of system level spacing, but only in the
case with the cutoff \(ω_s=2\) (orange), the resonance condition is
fulfilled. Indeed, the energy transfer out of the system is the best
for the resonant case (see \cref{fig:resonance_analysis}, middle
panel), as the final system energy is the lowest.
We now define our measure of energy transfer performance. The less
energy the system retains at the end of the protocol the better
performance of transfer. In other words, we want to transfer as much
energy out of the qubit as possible minimizing
\(\abs{\ev{H_{\sys}} + 2}\). If the interaction energies
\(\ev{H_{\inter}}\) are on the same scale, the decoupling costs should
be roughly the same in terms of total energy change. Otherwise, they
may lead to an additional change of system and bath energies between
the different simulations.
The change in total energy due to the decoupling of the bath is
moderately higher than in the resonant (orange) case than for cutoff
\(ω_s=1\) (blue) and seems to be somewhat proportional to the
interaction energy which is reasonable. The decoupling process appears
to affect the bath energy curves in this case. As can be gathered from
\cref{fig:resonance_analysis}, the bath energies generally decreases
when decoupling, compensating for the negative interaction energy. The
effect on the system energy is masked by the much greater energy loss
to the bath.
Simulations for three values of the shift \(ω_{s}\in\{1,2,3\}\) are
being presented in \cref{fig:resonance_analysis}. For all shifts the
spectral density has a finite value at the value of system level
spacing, but only in the case with the cutoff \(ω_s=2\) (orange), the
resonance condition is fulfilled. Indeed, the energy transfer out of
the system \(\abs{\ev{H_{\sys}} + 2}\) is the best for the resonant
case (see \cref{fig:resonance_analysis}, middle panel), as the final
system energy is the lowest.
The change in total energy \(Δ\ev{H}\) (\cref{fig:resonance_analysis},
middle panel) due to the decoupling of the bath is moderately higher
than in the resonant (orange) case than for cutoff \(ω_s=1\) (blue)
and seems to be somewhat proportional to the interaction energy, which
is reasonable. The decoupling process appears to affect mainly the
bath energy curves (dashed lines, left panel) in this case. Bath
energies \(\ev{H_{\bath}}\) generally decrease during decoupling,
compensating for the negative interaction energy. The effect on the
system energy is masked by the energy loss to the bath.
Were the interaction switched off abruptly, the system and bath
energies would remain untouched. Turning the interaction off in finite
@ -1086,40 +1123,40 @@ as we shall see below.
\begin{figure}[htp]
\centering
\includegraphics{figs/one_bath_syst/resonance_analysis}
\caption{\label{fig:resonance_analysis} Left
panel: The system, bath and interaction energies for various Ohmic
BFCs with \(α(0)=8,\,ω_c=2\) shifted by \(ω_s\). Mid panel: The
difference in total energy, the distance of the system energy to
the ground state energy and the distance of the bath energy to the
initial system energy. Right panel: The spectral density for the
three shift values. \(\norm{H_\sys}\) does mean in this case, that
the energy level spacing of the system is \(4\). There resonant
case has the best energy transfer behaviour as gauged by the final
system energy.}
\caption{\label{fig:resonance_analysis} Left panel: The system, bath
and interaction energies for various Ohmic BFCs with
\(α(0)=8,\,ω_c=2\) shifted by \(ω_s\). Middle panel: The
difference in total energy \(Δ\ev{H}\), the distance of the system
energy to the ground state energy \(\abs{\ev{H_{\sys}} + 2}\) and
the difference of the bath energy change and the initial system
energy \(\ev{H_{\bath}} - 4\). Right panel: The spectral density
for the three shift values \(ω_{s}\). \(\norm{H_\sys}\) does mean
in this case, that the energy level spacing of the system is
\(4\). The resonant case has the best energy transfer behaviour as
gauged by the final system energy.}
\end{figure}
The third simulation with cutoff \(ω_s=3\) (green) exhibits the worst
energy transfer with the highest residual system energy and the
highest change in total energy due the large magnitude of the
interaction energy. The lowering of the final bath is most pronounced
as the interaction energy is largest. As very much energy is shed into
interaction initially, the bath energy can't decay as much as in the
situation with cutoff \(ω_{s}=1\) (blue).
energy transfer performance with the highest residual system energy
and the highest change in total energy due the large magnitude of the
interaction energy. The lowering of the final bath energy is most
pronounced as the interaction energy is largest. As very much energy
is shed into to bath to build up the interaction, the bath energy
can't decay as much as in the simulation with cutoff \(ω_{s}=1\)
(blue).
Due to the asymmetry of the spectral density, the simulations for
\(ω_s=1\) and \(ω_s=3\) are not directly comparable. A repetition of
this investigation with a (pseudo~\cite{Mukherjee2020Jan}) Lorentzian
spectral density and for different interaction
time-frames\footnote{steady state vs. transient states} is left for
future work.
this investigation with a (pseudo) Lorentzian spectral density is left
to future work.
Interestingly, if we modulate the system periodically with angular
frequency \(Δ\), also bath frequencies of \(ω_{0} + n Δ\)
(\(n\in\NN\)) become important, so that shifting the spectral density
to higher frequencies is advantageous, albeit for the completely
different objective of extracting energy from system and bath, as we
will find out in \cref{sec:modcoup_reso}.
% Interestingly, also bath frequencies of \(ω_{0} + n Δ\) (\(n\in\NN\))
% become important if we modulate the system periodically with angular
% frequency \(Δ\), so that shifting the spectral density to higher
% frequencies is advantageous, albeit for the completely different
% objective of extracting energy from system and bath, as we will find
% out in \cref{sec:modcoup_reso}.
% Turning of the interaction leads mostly to a reduction of the final
% bath energy (see the middle panel). This effect will also appear in
@ -1153,42 +1190,32 @@ will find out in \cref{sec:modcoup_reso}.
frequency shifts of the spectral density lead to a higher
magnitude of the interaction energy and faster dynamics.}
\end{wrapfigure}
In general, the maximal absolute interaction energy is roughly
proportional to the shift \(ω_{s}\) of the spectral density, so that
the short term interaction strength as measured by the interaction
energy expectation value is not only dependent on the width and total
norm of the spectral density, but also on its absolute distribution in
frequency space. The shift adds a phase factor \(\eu^{-\i ω_{s} τ}\)
to the BCF which makes the imaginary part steeper for short
times. This in turn leads to faster initial slip dynamics and a speeds
up buildup of interaction energy as demonstrated in
\cref{fig:initial_slip_resonance}. The observed effect is sensible on
an intuitive level as higher frequency oscillators are being excited
in the bath leading to faster dynamics.
As a heuristic observation, the maximal absolute interaction energy
is roughly proportional to the shift \(ω_{s}\) of the spectral
density, so that the short term interaction strength as measured by
the interaction energy expectation value is not only dependent on the
width and total norm of the spectral density, but also on its absolute
distribution in frequency space. The shift adds a phase factor
\(\eu^{-\i ω_{s} τ}\) to the BCF which makes the imaginary part
steeper for short times. This in turn leads to faster initial slip
dynamics and a speeds up buildup of interaction energy as demonstrated
in \cref{fig:initial_slip_resonance}. The observed effect is sensible
on an intuitive level as higher frequency oscillators are being
excited in the bath leading to faster dynamics.
The longer term picture is being studied in
Turning the interaction off later leads to the results of
\cref{fig:resonance_analysis_steady}. We see broadly similar energy
transfer characteristics for the cutoffs \(ω_{s}=1\) (solid blue line)
and \(ω_{s}=2\) (solid orange line), where the off-resonant case
(blue) may be slightly advantageous. The left panel shows, that
although the system energy before the decoupling is lower in the
resonant case (orange), the situation is reversed during the
decoupling as now also the system energy is affected by the modulation
and the interaction energy of the off-resonant case is slightly larger
in magnitude. However these effects are quite marginal and should be
taken with care. The system energy difference amounts to
\(\Delta\langle H_\mathrm{S}\rangle=0.00397\pm 0.00010\), but only the
statistical error has been taken into account. The simulations were
run with \(N=10^{4}\) trajectories and the same HOPS settings as in
the discussion above, so that some confidence may be placed in them.
and \(ω_{s}=2\) (solid orange line).
For \(ω_{s}=3\) (green lines) the approximate steady state has not
been reached yet and the energy transfer is incomplete so that the
residual system energy is the highest. As remarked earlier, th fast
residual system energy is the highest. As remarked earlier, the fast
growth of the interaction energy leads to a slower initial loss of
system energy. On longer time scales we see a slow, almost linear
transfer of energy from the system (solid line) into the interaction
(dotted line). The system dynamics are catching up with the bath.
(dotted line) as there is still is system energy left to be
transferred.
\begin{figure}[htp]
\centering
\includegraphics{figs/one_bath_syst/resonance_analysis_steady}
@ -1201,24 +1228,29 @@ transfer into the bath. To study the effect of the bath memory, we
use Ohmic spectral densities with linearly spaced
\(τ_{\bath}\equiv ω_c^{-1}\) that have been shifted and scaled by
numerical optimization so that their peaks coincide and the resulting
maximal absolute interaction energies identical. The rightmost panel
of \Cref{fig:markov_analysis} shows plots of the spectral densities
obtained. We can see, that not only the magnitude at resonance point
enters, as the peak heights are quite different. We will encounter
this behavior again in \cref{sec:extr_mem}\footnote{Performing the
optimization in the weak coupling regime will actually yield
matching peak heights for the spectral densities.}.
maximal absolute interaction energies identical. This makes the
simulations more comparable, as otherwise very different energy
transfer speeds would be observed.
The results that can be obtained are very much dependent on the time
when the interaction is switched off. \Cref{fig:markov_analysis} has
been arrived at by tweaking the time point of decoupling so that an
extremum in the system energy of the long memory (\(τ_{B}=1\)) case
(green) is captured. This leads to an advantageous transfer
performance with a lower system energy and similar cost in terms of
total energy change, although residual system energy is still higher
than in \cref{fig:resonance_analysis_steady}. Despite the system
energy initially fall off the fastest for the short memory case (blue)
the situation is reversed after about \(τ=0.5\).
The rightmost panel of \Cref{fig:markov_analysis} shows plots of the
spectral densities obtained by the optimization. We can see, that not
only the magnitude at resonance point enters into the dynamics, as the
peak heights of the final spectral densities (right panel) are quite
different. We will encounter this behavior again in
\cref{sec:extr_mem}\footnote{Performing the optimization in the weak
coupling regime will actually yield matching peak heights for the
spectral densities.}.
The obtained results are very much dependent on the time when the
interaction is switched off. \Cref{fig:markov_analysis} has been
produced by tweaking the time point of decoupling so that an extremum
in the system energy of the long memory (\(τ_{B}=1\)) case (green) is
captured. This leads to an advantageous transfer performance with a
lower system energy and similar cost in terms of total energy change,
although residual system energy is still higher than in
\cref{fig:resonance_analysis_steady}. Despite the system energy
initially fall off the fastest for the short memory case (blue) the
situation is reversed after about \(τ=0.5\).
\cref{fig:resonance_analysis_steady}.
\begin{figure}[htp]
\centering