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https://github.com/vale981/master-thesis-tex
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add first results about friction
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23 changed files with 88637 additions and 5 deletions
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figs/one_bath_mod/delta_dependence.pdf
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figs/one_bath_mod/delta_dependence.pdf
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figs/one_bath_mod/delta_dependence.pgf
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@ -0,0 +1,2 @@
|
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#! /usr/bin/env bash
|
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rsync -av ../../../masterarb/python/energy_flow_proper/08_dynamic_one_bath/figures/ .
|
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|
@ -1,6 +1,9 @@
|
|||
\chapter{Additional Plots\label{chap:plus_plots}}
|
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\chapter{Additional Plots and Parameters\label{chap:plus_plots_params}}
|
||||
|
||||
\section{Interaction Consisency Plots}
|
||||
\section{Plots}
|
||||
\label{sec:plus_plots}
|
||||
|
||||
\subsection{Interaction Consisency Plots}
|
||||
\label{sec:intercons}
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
|
|
@ -16,3 +19,39 @@
|
|||
consistency plot for \cref{sec:one_bathcoup_strength}, similar to
|
||||
\cref{fig:stocproc_systematics}.}
|
||||
\end{figure}
|
||||
|
||||
\section{Parameters}
|
||||
\label{sec:plus_params}
|
||||
|
||||
\subsection{Modulation with a single Bath}
|
||||
\label{sec:plus_mod_single}
|
||||
|
||||
\begin{table}
|
||||
\centering
|
||||
\begin{tabular}{lll}
|
||||
\hline
|
||||
$ω_c$ & $2$ & $2$ \\
|
||||
$α(0)$ & $0.7$ & $0.7$ \\
|
||||
$T$ & $5$ & $5$ \\
|
||||
$N$ & $10000$ & $2000$ \\
|
||||
$k_{\mathrm{max}}$ & $5$ & $5$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{\label{tab:plus_friction}Additional parameters for
|
||||
\cref{fig:quant_frict}.}
|
||||
\end{table}
|
||||
|
||||
\begin{table}
|
||||
\centering
|
||||
\begin{tabular}{lll}
|
||||
\hline
|
||||
$ω_c$ & $2$ & $2$ \\
|
||||
$α(0)$ & $0.7$ & $0.7$ \\
|
||||
$T$ & $5$ & $5$ \\
|
||||
$N$ & $1000$ & $1000$ \\
|
||||
$k_{\mathrm{max}}$ & $5$ & $5$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{\label{tab:plus_system}Additional parameters for
|
||||
\cref{fig:quant_frict_sys_no_sys}.}
|
||||
\end{table}
|
||||
|
|
|
|||
|
|
@ -1002,11 +1002,13 @@ interest.
|
|||
A classical dictum of thermodynamics is, that it is impossible to
|
||||
extract energy from a single bath in a cyclical manner. Indeed, we
|
||||
found in \cref{sec:ergoonebath} that this also holds for a finite
|
||||
quantum system coupled to a thermal bath. In \cref{sec:explicit ergo}
|
||||
quantum system coupled to a thermal bath. In \cref{sec:explicitergo}
|
||||
we found, that the bound given on the ergotropy in such a situation
|
||||
can be saturated for infinite baths, such as the ones used with
|
||||
NMQSD/HOPS. However, it is unclear if such a unitary transformation
|
||||
can be implemented without explicit construction of a model.
|
||||
can be implemented without explicit construction of a model. Our goal
|
||||
in this section is to extract as much energy from a one-bath system
|
||||
as is possible without extensive tuning.
|
||||
|
||||
In this section we will focus on the minimal dimensionless model
|
||||
\begin{equation}
|
||||
|
|
@ -1021,8 +1023,24 @@ 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.
|
||||
``direction'' and vanish periodically. 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 maximize energy flow, we will use
|
||||
resonant baths whose spectral densities have been shifted such that
|
||||
their maxima coincide with \(1 + Δ\).
|
||||
|
||||
For this model the ergotropy bound \cref{eq:ergo_bath_change}
|
||||
evaluates to
|
||||
\begin{equation}
|
||||
\label{eq:ergo_mod_model}
|
||||
\mathcal{W} \leq β^{-1} \ln(1+\eu^{-ωβ})=\mathcal{W}_{\mathrm{max}}.
|
||||
\end{equation}
|
||||
For \(ω\to ∞\) we find \(\mathcal{W} = 0\) as the temperature \(T\)
|
||||
must be of the order of the system energy gap \(ω\) for any
|
||||
energy lowering process to take place.
|
||||
|
||||
\subsection{Quantum Friction}
|
||||
\label{sec:quantum_friction}
|
||||
Before focusing on the \(λ = 0\) case, we will briefly visit a
|
||||
phenomenon coined ``Quantum Friction'', whereby the creation of
|
||||
coherences in the system energy basis hinders the performances of
|
||||
|
|
@ -1030,7 +1048,57 @@ thermal quantum machines. These coherences raise the ergotropy of the
|
|||
system without necessarily raising its energy and can thus not
|
||||
contribute to the operation of a machine.
|
||||
|
||||
A simple demonstration of this can be observed in
|
||||
\cref{fig:quant_frict}. Here the simulation with frictionless
|
||||
modulation \(σ_{f}=σ_{x}\) does extract less energy from the total
|
||||
system as \(σ_{f}=σ_{z}\) case. One reason for this is the not
|
||||
insignificant buildup of ergotropy in the system state which is being
|
||||
reduced to zero periodically, but does retard the energy extraction so
|
||||
that less energy is extracted before the periodic steady state is
|
||||
being reached which does only increase energy (see
|
||||
\cref{sec:operational_thermo}).
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics{figs/one_bath_mod/quantum_friction}
|
||||
\caption{\label{fig:quant_frict} Total energy change normalized by
|
||||
the maximal ergotropy in the left panel and the ergotropy of the system
|
||||
state \(ρ_{\sys}\) normalized by the maximal extracted energy
|
||||
(dashed lines in left panel) in the right panel over time. Here
|
||||
\(λ=0.1, Δ=5, T=5\) was chosen. In the ``With Friction'' case
|
||||
\(σ_{f}=σ_{x,y}\) and in the ``Frictionless'' case
|
||||
\(σ_{f}=σ_{z}\). The horizontal dashed lines mark the minimal
|
||||
total energy differences achieved. For more parameters see
|
||||
\cref{tab:plus_friction}.}
|
||||
\end{figure}
|
||||
|
||||
When choosing \(σ_{f}=σ_{y}\) the situation is even worse, because now
|
||||
the system modulation does not even commute with the interaction. The
|
||||
extracted energy is smallest and the relative system ergotropy buildup
|
||||
is maximal.
|
||||
|
||||
\subsection{System Modulation}
|
||||
\label{sec:sys_mod_v_no_sys_mod}
|
||||
As it turns out, the modulation generically 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}. Again, the ``friction'' (system
|
||||
ergotropy) generated by the system modulation is much greater than
|
||||
without system modulation. However, it is questionable whether this is
|
||||
the only reason for the performance advantage of the case without
|
||||
system modulation.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics{figs/one_bath_mod/system_vs_no_system}
|
||||
\caption{\label{fig:quant_frict_sys_no_sys} Similar to
|
||||
\cref{fig:quant_frict}, but for (frictionless) system modulation
|
||||
and no system modulation. The parameters were \(λ=0.1, Δ=5, T=5\)
|
||||
and further details can be found in \cref{tab:plus_system}.}
|
||||
\end{figure}
|
||||
|
||||
In light of this result, we will continue with \(λ=0\), which
|
||||
simplifies the situation.
|
||||
|
||||
\newpage
|
||||
\begin{itemize}
|
||||
\item quantum friction
|
||||
\item non-markovianity in the energy shovel
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue