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80
src/thermo.tex
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@ -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 memory'' of the baths which implies an infinite bath\footnote{Or, as
we have remarked earlier, a suitably large bath.}. Note that even in we have remarked earlier, a suitably large bath.}. Note that even in
the absence of modulation, the steady state can exhibit periodic the absence of modulation, the steady state can exhibit periodic
dynamics as can be seen in \cref{fig:markov_analysis_steady} on page dynamics as can be seen in \cref{fig:markov_analysis_steady} on
\pageref{fig:markov_analysis_steady}. In the same spirit, we \cpageref{fig:markov_analysis_steady}. In the same spirit, we
\emph{assume} that the energy change of each bath \emph{assume} that the energy change of each bath
\(ΔE_{\bath^i}^\cyc =ΔE_{\bath^i}((n+1)τ)-ΔE_{\bath^i}(nτ) = \(ΔE_{\bath^i}^\cyc =ΔE_{\bath^i}((n+1)τ)-ΔE_{\bath^i}(nτ) =
E_{\bath^i}((n+1)τ)-E_{\bath^i}(nτ)\) is constant once the system 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 interaction energy will lead to the internal energy (as defined by the
first law) not being an exact differential. first law) not being an exact differential.
In contrast to \refcite{Strasberg2021Aug}, no interpretation in terms of In contrast to \refcite{Strasberg2021Aug}, no interpretation in terms
thermodynamical quantities is required for \cref{eq:secondlaw_cyclic} of thermodynamical quantities is required for
to be useful. Assume that the interaction Hamiltonian in \cref{eq:secondlaw_cyclic} to be useful. Assume that the interaction
\cref{eq:katoineqsys} attains the same value periodically, so that Hamiltonian in \cref{eq:katoineqsys} attains the same expectation
system and bath energy expectation values can be cleanly separated. In value periodically, so that system and bath energy expectation values
the periodic steady state the system energy does not change after a can be cleanly separated. In the periodic steady state the system
cycle and the whole energy change amounts to the change in bath energy does not change after a cycle and the whole energy change
energies. The energy changes in the individual baths are then amounts to the change in bath energies. The energy changes in the
well-defined and useful (macroscopic) quantities, whether one calls individual baths are then well-defined and useful (macroscopic)
them heat or not. In a setting with two baths quantities, whether one calls them heat or not. In a setting with two
\cref{eq:secondlaw_cyclic} implies the Carnot bound for the efficiency baths \cref{eq:secondlaw_cyclic} implies the Carnot bound for the
given in \cref{eq:efficiency_definition}. efficiency given in \cref{eq:efficiency_definition}.
Generally, if one wishes to extract energy from hot baths (negative Generally, if one wishes to extract energy from hot baths (negative
\(ΔE\)) and shed as little energy into cooler baths as possible, \(ΔE\)) 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 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. In this way, the negative contributions are as small as
possible and the positive contributions are maximized. 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}% \section{Modulation of System and Interaction for a Single Bath}%
\label{sec:singlemod} \label{sec:singlemod}
Because the HOPS method is best suited to situations where the actual The HOPS provide the full dynamical solution for the open system
finite time dynamics are of interest, let us now turn to such a 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. problem.
A classical dictum of thermodynamics is, that it is impossible to 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, a_k^†) + ∑_k ω_k a_k^\dag a_k,
\end{equation} \end{equation}
where \(λ,Δ\geq 0\) and \(f\in \{x, y, z\}\). The form of the system 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 Hamiltonian has been chosen similar to \refcite{Mukherjee2020Jan}
Floquet theory was used, and it was shown that the relevant quantities where Floquet theory was used, and it was shown that the relevant
are scaling with \(λ\). For \(λ=0\) the system Hamiltonian is positive quantities scale with \(λ\). For \(λ=0\) the system Hamiltonian is
semi-definite with the energies zero and one. The modulation of the positive semi-definite with the energies zero and one. The modulation
interaction has been chosen heuristically to always act in the same of the interaction has been chosen heuristically to always act in the
``direction'' and vanish periodically. The last fact is important, as same ``direction'' and vanish periodically. The last fact is
we will be interested in the behaviour of the system before it has important, as we will be interested in the behaviour of the system
reached the steady state. The energy extracted from a system should be before it has reached the steady state. The energy extracted from a
gauged in such a way, that an additional decoupling is not necessary. 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.
We choose the ``down state'' with \(H(0)\ket{0}=0\) as initial state, 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 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}% \subsection{Quantum Friction}%
\label{sec: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] \begin{figure}[htp]
\centering \centering
\includegraphics{figs/one_bath_mod/quantum_friction} \includegraphics{figs/one_bath_mod/quantum_friction}
@ -942,6 +932,15 @@ external modulation.
dynamics. Quantum friction appears to be especially important for dynamics. Quantum friction appears to be especially important for
short times as it decays with time.} short times as it decays with time.}
\end{figure} \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 A simple demonstration of this fact can be observed in
\cref{fig:quant_frict}. Here the simulation with nondiagonal \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 deterioration of energy extraction performance for the model
\cref{eq:one_qubit_model_driven} as is demonstrated in \cref{eq:one_qubit_model_driven} as is demonstrated in
\cref{fig:quant_frict_sys_no_sys}. There, the total energy curves of a \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. plotted.
Again, the ``friction'' (system ergotropy) generated by the system 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 The resulting interaction energies can be found in
\cref{fig:omega_couplings_and_energies}. They roughly correspond to \cref{fig:omega_couplings_and_energies}. They roughly correspond to
demanding \(α_{β}(0)=1.4\), where \(α_{β}\) is the finite temperature 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 also shows that for the simulations with small \(ω_{c}\) the positive
parts of the interaction energy are especially large. This is a major parts of the interaction energy are especially large. This is a major
departure from the pure dephasing regime where the interaction energy departure from the pure dephasing regime where the interaction energy

6
src/title.tex
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@ -2,15 +2,15 @@
\thispagestyle{empty} \thispagestyle{empty}
\begin{titlepage} \begin{titlepage}
\includegraphics[width=5cm]{figs/logo.pdf} \includegraphics[width=5cm]{figs/logo.pdf}
\hrule
\vspace{1em} \vspace{1em}
\hrule
{\centering {\centering
\setstretch{1.1} \setstretch{1.1}
\vspace*{5em} \vspace*{5em}
{\bfseries\Huge \@title\\} {\bfseries\Huge \@title\\}
\vspace*{2em} % \vspace*{2em}
{\Large\@subtitle} % {\Large\@subtitle}
\vfill \vfill \vfill \vfill
{\large {\bfseries Master-Arbeit} \\ {\large {\bfseries Master-Arbeit} \\
zur Erlangung des Hochschulgrades\\ zur Erlangung des Hochschulgrades\\