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@ -764,8 +764,8 @@ system enters a periodic steady state after the time \(n_0τ\) for some
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memory'' of the baths which implies an infinite bath\footnote{Or, as
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we have remarked earlier, a suitably large bath.}. Note that even in
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the absence of modulation, the steady state can exhibit periodic
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dynamics as can be seen in \cref{fig:markov_analysis_steady} on page
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\pageref{fig:markov_analysis_steady}. In the same spirit, we
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dynamics as can be seen in \cref{fig:markov_analysis_steady} on
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\cpageref{fig:markov_analysis_steady}. In the same spirit, we
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\emph{assume} that the energy change of each bath
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\(ΔE_{\bath^i}^\cyc =ΔE_{\bath^i}((n+1)τ)-ΔE_{\bath^i}(nτ) =
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E_{\bath^i}((n+1)τ)-E_{\bath^i}(nτ)\) is constant once the system
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@ -804,22 +804,22 @@ baths, that a definition of heat involving any nonzero fraction of the
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interaction energy will lead to the internal energy (as defined by the
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first law) not being an exact differential.
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In contrast to \refcite{Strasberg2021Aug}, no interpretation in terms of
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thermodynamical quantities is required for \cref{eq:secondlaw_cyclic}
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to be useful. Assume that the interaction Hamiltonian in
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\cref{eq:katoineqsys} attains the same value periodically, so that
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system and bath energy expectation values can be cleanly separated. In
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the periodic steady state the system energy does not change after a
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cycle and the whole energy change amounts to the change in bath
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energies. The energy changes in the individual baths are then
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well-defined and useful (macroscopic) quantities, whether one calls
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them heat or not. In a setting with two baths
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\cref{eq:secondlaw_cyclic} implies the Carnot bound for the efficiency
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given in \cref{eq:efficiency_definition}.
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In contrast to \refcite{Strasberg2021Aug}, no interpretation in terms
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of thermodynamical quantities is required for
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\cref{eq:secondlaw_cyclic} to be useful. Assume that the interaction
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Hamiltonian in \cref{eq:katoineqsys} attains the same expectation
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value periodically, so that system and bath energy expectation values
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can be cleanly separated. In the periodic steady state the system
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energy does not change after a cycle and the whole energy change
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amounts to the change in bath energies. The energy changes in the
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individual baths are then well-defined and useful (macroscopic)
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quantities, whether one calls them heat or not. In a setting with two
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baths \cref{eq:secondlaw_cyclic} implies the Carnot bound for the
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efficiency given in \cref{eq:efficiency_definition}.
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Generally, if one wishes to extract energy from hot baths (negative
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\(ΔE\)) and shed as little energy into cooler baths as possible,
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\cref{eq:secondlaw_cyclic} tells us, that it is advantageous to make
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\cref{eq:secondlaw_cyclic} tells us that it is advantageous to make
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the hot baths as hot as possible and the cold baths as cold as
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possible. In this way, the negative contributions are as small as
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possible and the positive contributions are maximized.
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@ -831,8 +831,9 @@ demonstration of the capabilities of the HOPS method.
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\section{Modulation of System and Interaction for a Single Bath}%
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\label{sec:singlemod}
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Because the HOPS method is best suited to situations where the actual
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finite time dynamics are of interest, let us now turn to such a
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The HOPS provide the full dynamical solution for the open system
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dynamics. It is therefore best suited to situations where the actual
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finite time dynamics are of interest. Let us now turn to such a
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problem.
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A classical dictum of thermodynamics is, that it is impossible to
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@ -857,18 +858,16 @@ with periodically modulated system and coupling
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a_k^†) + ∑_k ω_k a_k^\dag a_k,
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\end{equation}
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where \(λ,Δ\geq 0\) and \(f\in \{x, y, z\}\). The form of the system
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Hamiltonian has been chosen similar to \refcite{Mukherjee2020Jan} where
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Floquet theory was used, and it was shown that the relevant quantities
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are scaling with \(λ\). For \(λ=0\) the system Hamiltonian is positive
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semi-definite with the energies zero and one. The modulation of the
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interaction has been chosen heuristically to always act in the same
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``direction'' and vanish periodically. The last fact is important, as
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we will be interested in the behaviour of the system before it has
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reached the steady state. The energy extracted from a system should be
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gauged in such a way, that an additional decoupling is not necessary.
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The goal is now to extract as much energy as possible from the total
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system.
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Hamiltonian has been chosen similar to \refcite{Mukherjee2020Jan}
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where Floquet theory was used, and it was shown that the relevant
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quantities scale with \(λ\). For \(λ=0\) the system Hamiltonian is
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positive semi-definite with the energies zero and one. The modulation
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of the interaction has been chosen heuristically to always act in the
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same ``direction'' and vanish periodically. The last fact is
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important, as we will be interested in the behaviour of the system
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before it has reached the steady state. The energy extracted from a
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system should be gauged in such a way, that an additional decoupling
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is not necessary.
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We choose the ``down state'' with \(H(0)\ket{0}=0\) as initial state,
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as we want to extract energy from the bath and not the system. To
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@ -912,15 +911,6 @@ quantum friction in \cref{sec:quantum_friction}.
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\subsection{Quantum Friction}%
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\label{sec:quantum_friction}
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To justify the choice \(λ = 0\) for the model
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\cref{eq:one_qubit_model_driven}, we will briefly visit a phenomenon
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dubbed \emph{Quantum
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Friction}~\cite{Binder2018,Mukherjee2020Jan}. This is creation
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coherences\footnote{Or more generally the creation of ergotropy.} in
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the system energy basis affects the performance of thermal quantum
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machines. These coherences raise the ergotropy of the system state,
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all the while consuming energy that could have been extracted by the
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external modulation.
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\begin{figure}[htp]
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\centering
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\includegraphics{figs/one_bath_mod/quantum_friction}
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@ -942,6 +932,15 @@ external modulation.
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dynamics. Quantum friction appears to be especially important for
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short times as it decays with time.}
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\end{figure}
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To justify the choice \(λ = 0\) for the model
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\cref{eq:one_qubit_model_driven}, we will briefly visit a phenomenon
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dubbed \emph{Quantum
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Friction}~\cite{Binder2018,Mukherjee2020Jan}. This is creation
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coherences\footnote{Or more generally the creation of ergotropy.} in
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the system energy basis affects the performance of thermal quantum
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machines. These coherences raise the ergotropy of the system state,
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all the while consuming energy that could have been extracted by the
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external modulation.
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A simple demonstration of this fact can be observed in
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\cref{fig:quant_frict}. Here the simulation with nondiagonal
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@ -974,7 +973,7 @@ As it turns out, modulation of the system Hamiltonian leads to a
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deterioration of energy extraction performance for the model
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\cref{eq:one_qubit_model_driven} as is demonstrated in
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\cref{fig:quant_frict_sys_no_sys}. There, the total energy curves of a
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simulation with (orange) and without (blue) system modulation is
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simulation with (orange) and without (blue) system modulation are
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plotted.
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Again, the ``friction'' (system ergotropy) generated by the system
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@ -1025,7 +1024,8 @@ various cutoff frequencies \(ω_{c}\).
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The resulting interaction energies can be found in
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\cref{fig:omega_couplings_and_energies}. They roughly correspond to
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demanding \(α_{β}(0)=1.4\), where \(α_{β}\) is the finite temperature
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bath correlation function. \Cref{fig:omega_couplings_and_energies}
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bath correlation function (see \cref{eq:finite_bcf} on
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\cpageref{eq:finite_bcf}). \Cref{fig:omega_couplings_and_energies}
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also shows that for the simulations with small \(ω_{c}\) the positive
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parts of the interaction energy are especially large. This is a major
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departure from the pure dephasing regime where the interaction energy
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@ -2,15 +2,15 @@
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\thispagestyle{empty}
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\begin{titlepage}
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\includegraphics[width=5cm]{figs/logo.pdf}
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\hrule
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\vspace{1em}
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\hrule
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{\centering
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\setstretch{1.1}
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\vspace*{5em}
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{\bfseries\Huge \@title\\}
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\vspace*{2em}
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{\Large\@subtitle}
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% \vspace*{2em}
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% {\Large\@subtitle}
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\vfill \vfill
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{\large {\bfseries Master-Arbeit} \\
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zur Erlangung des Hochschulgrades\\
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