last fixes

src/thermo.tex

@@ -764,8 +764,8 @@ system enters a periodic steady state after the time \(n_0τ\) for some
 memory'' of the baths which implies an infinite bath\footnote{Or, as
   we have remarked earlier, a suitably large bath.}. Note that even in
 the absence of modulation, the steady state can exhibit periodic
-dynamics as can be seen in \cref{fig:markov_analysis_steady} on page
-\pageref{fig:markov_analysis_steady}. In the same spirit, we
+dynamics as can be seen in \cref{fig:markov_analysis_steady} on
+\cpageref{fig:markov_analysis_steady}. In the same spirit, we
 \emph{assume} that the energy change of each bath
 \(ΔE_{\bath^i}^\cyc =ΔE_{\bath^i}((n+1)τ)-ΔE_{\bath^i}(nτ) =
 E_{\bath^i}((n+1)τ)-E_{\bath^i}(nτ)\) is constant once the system
@@ -804,22 +804,22 @@ baths, that a definition of heat involving any nonzero fraction of the
 interaction energy will lead to the internal energy (as defined by the
 first law) not being an exact differential.
 
-In contrast to \refcite{Strasberg2021Aug}, no interpretation in terms of
-thermodynamical quantities is required for \cref{eq:secondlaw_cyclic}
-to be useful.  Assume that the interaction Hamiltonian in
-\cref{eq:katoineqsys} attains the same value periodically, so that
-system and bath energy expectation values can be cleanly separated. In
-the periodic steady state the system energy does not change after a
-cycle and the whole energy change amounts to the change in bath
-energies. The energy changes in the individual baths are then
-well-defined and useful (macroscopic) quantities, whether one calls
-them heat or not.  In a setting with two baths
-\cref{eq:secondlaw_cyclic} implies the Carnot bound for the efficiency
-given in \cref{eq:efficiency_definition}.
+In contrast to \refcite{Strasberg2021Aug}, no interpretation in terms
+of thermodynamical quantities is required for
+\cref{eq:secondlaw_cyclic} to be useful.  Assume that the interaction
+Hamiltonian in \cref{eq:katoineqsys} attains the same expectation
+value periodically, so that system and bath energy expectation values
+can be cleanly separated. In the periodic steady state the system
+energy does not change after a cycle and the whole energy change
+amounts to the change in bath energies. The energy changes in the
+individual baths are then well-defined and useful (macroscopic)
+quantities, whether one calls them heat or not.  In a setting with two
+baths \cref{eq:secondlaw_cyclic} implies the Carnot bound for the
+efficiency given in \cref{eq:efficiency_definition}.
 
 Generally, if one wishes to extract energy from hot baths (negative
 \(ΔE\)) and shed as little energy into cooler baths as possible,
-\cref{eq:secondlaw_cyclic} tells us, that it is advantageous to make
+\cref{eq:secondlaw_cyclic} tells us that it is advantageous to make
 the hot baths as hot as possible and the cold baths as cold as
 possible. In this way, the negative contributions are as small as
 possible and the positive contributions are maximized.
@@ -831,8 +831,9 @@ demonstration of the capabilities of the HOPS method.
 
 \section{Modulation of System and Interaction for a Single Bath}%
 \label{sec:singlemod}
-Because the HOPS method is best suited to situations where the actual
-finite time dynamics are of interest, let us now turn to such a
+The HOPS provide the full dynamical solution for the open system
+dynamics. It is therefore best suited to situations where the actual
+finite time dynamics are of interest. Let us now turn to such a
 problem.
 
 A classical dictum of thermodynamics is, that it is impossible to
@@ -857,18 +858,16 @@ with periodically modulated system and coupling
   a_k^†) + ∑_k ω_k a_k^\dag a_k,
 \end{equation}
 where \(λ,Δ\geq 0\) and \(f\in \{x, y, z\}\). The form of the system
-Hamiltonian has been chosen similar to \refcite{Mukherjee2020Jan} where
-Floquet theory was used, and it was shown that the relevant quantities
-are scaling with \(λ\). For \(λ=0\) the system Hamiltonian is positive
-semi-definite with the energies zero and one.  The modulation of the
-interaction has been chosen heuristically to always act in the same
-``direction'' and vanish periodically. The last fact is important, as
-we will be interested in the behaviour of the system before it has
-reached the steady state. The energy extracted from a system should be
-gauged in such a way, that an additional decoupling is not necessary.
-
-The goal is now to extract as much energy as possible from the total
-system.
+Hamiltonian has been chosen similar to \refcite{Mukherjee2020Jan}
+where Floquet theory was used, and it was shown that the relevant
+quantities scale with \(λ\). For \(λ=0\) the system Hamiltonian is
+positive semi-definite with the energies zero and one.  The modulation
+of the interaction has been chosen heuristically to always act in the
+same ``direction'' and vanish periodically. The last fact is
+important, as we will be interested in the behaviour of the system
+before it has reached the steady state. The energy extracted from a
+system should be gauged in such a way, that an additional decoupling
+is not necessary.
 
 We choose the ``down state'' with \(H(0)\ket{0}=0\) as initial state,
 as we want to extract energy from the bath and not the system. To
@@ -912,15 +911,6 @@ quantum friction in \cref{sec:quantum_friction}.
 
 \subsection{Quantum Friction}%
 \label{sec:quantum_friction}
-To justify the choice \(λ = 0\) for the model
-\cref{eq:one_qubit_model_driven}, we will briefly visit a phenomenon
-dubbed \emph{Quantum
-  Friction}~\cite{Binder2018,Mukherjee2020Jan}. This is creation
-coherences\footnote{Or more generally the creation of ergotropy.} in
-the system energy basis affects the performance of thermal quantum
-machines. These coherences raise the ergotropy of the system state,
-all the while consuming energy that could have been extracted by the
-external modulation.
 \begin{figure}[htp]
   \centering
   \includegraphics{figs/one_bath_mod/quantum_friction}
@@ -942,6 +932,15 @@ external modulation.
     dynamics. Quantum friction appears to be especially important for
     short times as it decays with time.}
 \end{figure}
+To justify the choice \(λ = 0\) for the model
+\cref{eq:one_qubit_model_driven}, we will briefly visit a phenomenon
+dubbed \emph{Quantum
+  Friction}~\cite{Binder2018,Mukherjee2020Jan}. This is creation
+coherences\footnote{Or more generally the creation of ergotropy.} in
+the system energy basis affects the performance of thermal quantum
+machines. These coherences raise the ergotropy of the system state,
+all the while consuming energy that could have been extracted by the
+external modulation.
 
 A simple demonstration of this fact can be observed in
 \cref{fig:quant_frict}. Here the simulation with nondiagonal
@@ -974,7 +973,7 @@ As it turns out, modulation of the system Hamiltonian leads to a
 deterioration of energy extraction performance for the model
 \cref{eq:one_qubit_model_driven} as is demonstrated in
 \cref{fig:quant_frict_sys_no_sys}. There, the total energy curves of a
-simulation with (orange) and without (blue) system modulation is
+simulation with (orange) and without (blue) system modulation are
 plotted.
 
 Again, the ``friction'' (system ergotropy) generated by the system
@@ -1025,7 +1024,8 @@ various cutoff frequencies \(ω_{c}\).
 The resulting interaction energies can be found in
 \cref{fig:omega_couplings_and_energies}. They roughly correspond to
 demanding \(α_{β}(0)=1.4\), where \(α_{β}\) is the finite temperature
-bath correlation function.  \Cref{fig:omega_couplings_and_energies}
+bath correlation function (see \cref{eq:finite_bcf} on
+\cpageref{eq:finite_bcf}).  \Cref{fig:omega_couplings_and_energies}
 also shows that for the simulations with small \(ω_{c}\) the positive
 parts of the interaction energy are especially large. This is a major
 departure from the pure dephasing regime where the interaction energy

src/title.tex

@@ -2,15 +2,15 @@
 \thispagestyle{empty}
 \begin{titlepage}
   \includegraphics[width=5cm]{figs/logo.pdf}
-  \hrule
   \vspace{1em}
+  \hrule
 
   {\centering
   \setstretch{1.1}
   \vspace*{5em}
   {\bfseries\Huge \@title\\}
-  \vspace*{2em}
-  {\Large\@subtitle}
+  % \vspace*{2em}
+  % {\Large\@subtitle}
     \vfill \vfill
     {\large {\bfseries Master-Arbeit} \\
       zur Erlangung des Hochschulgrades\\