add spohn and finalize discussion

note: spohn has wrong y axis

src/num_results.tex

@@ -1016,7 +1016,7 @@ to higher frequencies is advantageous, as we will find out in
 % some part of the negative interaction energy. In this way, the removal
 % of the bath would
 
-\begin{wrapfigure}[16]{o}{0.4\textwidth}
+\begin{wrapfigure}[16]{O}{0.4\textwidth}
   \centering
   \includegraphics{figs/one_bath_syst/initial_slip_resonance}
   \caption{\label{fig:initial_slip_resonance} The interaction energies
@@ -1091,7 +1091,7 @@ 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}.
 Although initially the system energy falls fastest for the short
-memory case the situation is reversed after about \(τ=.5\).
+memory case the situation is reversed after about \(τ=0.5\).
 \cref{fig:resonance_analysis_steady}.
 \begin{figure}[htp]
   \centering
@@ -1139,10 +1139,12 @@ behavior can also be observed \cref{fig:markov_analysis}.
 For even longer times we find a picture similar to
 \cref{fig:markov_analysis_steady}. The short memory case shows hardly
 any backflow and performs best in terms of final system energy which
-is very close to zero. The medium memory case performs worse, but not
-by a large margin. The simulation with the longest bath memory stands
-out and having a very different final state as is exemplified by the
-final system energy and the interaction energy curve.
+is extremely close to the target value of negative two. The other two
+bath memories perform worse. We observe that simulation with the
+longest bath memory stands out and having a very different final state
+as is exemplified by the final system energy and the interaction
+energy curve which exhibits a greater magnitude and persistent
+oscillations.
 \begin{figure}[h]
   \centering
   \includegraphics{figs/one_bath_syst/markov_analysis_steady}
@@ -1159,6 +1161,46 @@ an effect that is strongest in the short memory case especially for
 the system energy. A slower, more adiabatic coupling modulation could
 likely further reduce the amount of energy introduced.
 
+As the steady state system energies\footnote{Before the interaction is
+  switched off.} are greater than the ground state energies in all
+simulations of \cref{fig:markov_analysis_steady} is a token of strong
+coupling. The ground state is not the steady state, as it would be
+with GKSL dynamics~\cite{Binder2018}.
+
+We have often alluded to the fact that oscillations in the system
+energy, the back-flow of energy into the system, are a token of
+departure from the Markovian regime. An explicit demonstration of this
+fact is given in \cref{fig:steady_relent}.
+
+Spohn's theorem~\cite{Breuer2002Jun} states that the negative time
+derivative of the relative entropy of system state and steady state
+must be positive if the dynamics are generated by GKSL dynamics.
+In mathematical terms Spohn's theorem can be formulated as
+\begin{equation}
+  \label{eq:spohn}
+  -\dv{\qrelent{ρ_{\sys}(t)}{ρ_{\sys}(∞)}}{t} \geq 0,
+\end{equation}
+where \(\qrelent{ρ}{σ}=\tr[ρ \log_{2} ρ - ρ \log_{2} σ]\) is the
+quantum relative entropy. The left hand side of \cref{eq:spohn} is
+called entropy production.
+\begin{figure}[htp]
+  \centering
+  \includegraphics{figs/one_bath_syst/steady_relent}
+  \caption{\label{fig:steady_relent} The negative time derivative of
+    the relative entropy of system state and the approximate steady
+    state (before the interaction is switched off) of the simulations shown in
+    \cref{fig:markov_analysis_steady}. The short memory case does not
+    violate Spohn's inequality, but the other two cases do. Note
+    however, that the \(τ_{\bath}=1\) case has not yet reached a
+    steady state and should therefore be treated with care.}
+\end{figure}
+
+\Cref{fig:steady_relent} demonstrates that the two simulations with
+longer bathe memories are inconsistent with Markovian dynamics, as
+their entropy production exhibits strong negativity. The
+\(τ_{\bath}=1\) must be taken with care, as the steady state hasn't
+been reached yet.
+
 In summary we find that the energy dynamics of system, interaction and
 bath depend strongly on the characteristics of the bath.  In the
 regime studied, optimizing for fast energy loss of the system favors
@@ -1245,7 +1287,7 @@ change.
 
 \subsection{Varying the Coupling Strength}
 \label{sec:one_bathcoup_strength}
-\begin{wrapfigure}[17]{o}{0.3\textwidth}
+\begin{wrapfigure}[17]{O}{0.3\textwidth}
   \centering
   \includegraphics{figs/one_bath_syst/final_states_flows}
   \caption{\label{fig:delta_fs_flow} The absolute difference of the state energies and the
@@ -1260,14 +1302,6 @@ 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.
 
-\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}
 The chosen simulation parameters are the same as in
 \cref{sec:one_bath_cutoff} and again consistent results have been
 obtained as can be gathered from
@@ -1276,6 +1310,14 @@ coupling strengths.
 The interaction strength was chosen linearly spaced and the simulation
 results are presented in \cref{fig:delta_energy_overview}.
 
+\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 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 is the magnitude of the interaction
@@ -1284,25 +1326,27 @@ 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 as the non-monotonicity in time of
 the interaction energy, which is reflected in a non-monotonicity in
-the bath energy expectation value, which reaches a maximum and falls
-slightly for the strongest coupling simulations. If the interaction is
-strong enough, ``backflow'' can occur despite finite bath correlation
-times. In \cref{fig:markov_analysis_steady} the bath memory is long,
-additionally to a strong coupling so that multiple oscillations can be
-seen.
+the bath energy expectation value. The bath energy reaches a maximum
+and falls slightly for the strongest coupling simulations. If the
+interaction is strong enough, ``backflow'' can occur despite finite
+bath correlation times. In \cref{fig:markov_analysis_steady} the bath
+memory is long, additionally to a strong coupling so that multiple
+oscillations can be seen.
 
 Despite these differences for finite times, the approximate steady
 state\footnote{excluding the \(α(0)=0.4\) cases} interaction energies,
 maximal flows, system energies and bath energies are almost linearly
-dependent on the coupling strength \(α(0)\) as is demonstrated in the
-log-log plot \cref{fig:delta_fs_flow}.
+dependent on the coupling strength \(α(0)\) in this regime as is
+demonstrated in the log-log plot \cref{fig:delta_fs_flow}.
 
 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
-very fast, we would have to expend this energy in the worst
-case. Also, both the final system and bath energies are increasing
-with the coupling strength, compensating for the interaction energy.
+very fast, we would have to expend this energy in the worst case. On
+the other hand, more adiabatic protocol as the one used in
+\cref{fig:markov_analysis_steady} 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
@@ -1310,6 +1354,12 @@ 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}