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goodbye quantum friction

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@ -19,7 +19,7 @@ presentation of the subsequent results. A more comprehensive picture
can be obtained from can be obtained from
\cite{Binder2018,Kurizki2021Dec,Talkner2020Oct,Vinjanampathy2016Oct}. \cite{Binder2018,Kurizki2021Dec,Talkner2020Oct,Vinjanampathy2016Oct}.
Many central questions in thermodynamics are concerned with energy Many central questions in thermodynamics are concerned with work
extraction from macroscopic systems. These questions can be framed in extraction from macroscopic systems. These questions can be framed in
operational terms that don't require a specific definition of heat and operational terms that don't require a specific definition of heat and
just rely on the energy change in the total system or its just rely on the energy change in the total system or its
@ -42,8 +42,7 @@ studying bounds on the ergotropy of the system as is done in
\cref{sec:ergo_general}. Remarkably these bounds will turn out to be \cref{sec:ergo_general}. Remarkably these bounds will turn out to be
finite. We will review a general bound for single bath systems in finite. We will review a general bound for single bath systems in
\cref{sec:ergoonebath} and study an explicit calculation for a simple \cref{sec:ergoonebath} and study an explicit calculation for a simple
case in \cref{sec:explicitergo} after briefly discussing the subject case in \cref{sec:explicitergo}. The
of quantum friction in \cref{sec:quantum_friction_theory}. The
explicit ergotropy calculation will elucidate under which conditions explicit ergotropy calculation will elucidate under which conditions
the bound of \cref{sec:ergoonebath} may be expected to be tight. the bound of \cref{sec:ergoonebath} may be expected to be tight.
@ -118,7 +117,7 @@ zeroth and second laws of thermodynamics.
One of these properties is complete passivity. Completely passive One of these properties is complete passivity. Completely passive
states remain passive under the transformation \(ρ\to\otimes^Nρ\) (and states remain passive under the transformation \(ρ\to\otimes^Nρ\) (and
an \(N\)-fold sum of the Hamiltonian) for finite \(N\). Therefore no an \(N\)-fold sum of the Hamiltonian) for finite \(N\). Therefore, no
energy can be extracted from multiple identical systems in equilibrium energy can be extracted from multiple identical systems in equilibrium
at the same temperature. For finite dimensional systems, the complete at the same temperature. For finite dimensional systems, the complete
passivity even implies the form of the Gibbs state. passivity even implies the form of the Gibbs state.
@ -170,7 +169,7 @@ finite dimensional treatment in the following.
\begin{figure}[htp] \begin{figure}[htp]
\centering \centering
\includegraphics{figs/misc/bcf_approx} \includegraphics{figs/misc/bcf_approx}
\caption{\label{fig:bcf_approx} An ohmic BCF with \(ω_{c}=η=1\) \caption{\label{fig:bcf_approx} An Ohmic BCF with \(ω_{c}=η=1\)
approximated by the BCF of a finite number of linearly spaced approximated by the BCF of a finite number of linearly spaced
oscillators. The figure plots the relative difference between an oscillators. The figure plots the relative difference between an
approximation with \(N\) oscillators and the exact BCF over approximation with \(N\) oscillators and the exact BCF over
@ -179,42 +178,9 @@ finite dimensional treatment in the following.
\end{figure} \end{figure}
The Hamiltonian of a finite dimensional system is bounded and The Hamiltonian of a finite dimensional system is bounded and
therefore the ergotropy of such a system is finite. However, in the therefore the ergotropy of such a system is finite. However, in \cref{sec:ergoonebath} we will find that the ergotropy cannot even be made
following we will find that the ergotropy cannot even be made
arbitrarily large by enlarging the bath. arbitrarily large by enlarging the bath.
Now, we briefly introduce a simple application of quantum friction in
\cref{sec:quantum_friction_theory}.
\subsection{Quantum Friction}
\label{sec:quantum_friction_theory}
A simple application of the notion ergotropy is an explanation for so
called \emph{quantum
friction}~\cite{Binder2018,Mukherjee2020Jan}, a phenomenon with an
unfortunate name. From it one would expect that quantum friction has
some connection to dissipation. In fact, the reverse is true in most
cases where it is a concept applied to the reduced state of the
system.
Consider a modulated open quantum system. The buildup of energy basis
coherence in the system state makes it non-passive. Thus additional
energy which cannot be extracted by modulating of the energy level
gaps of the system\footnote{This is the usual mechanism of energy
extraction in a quantum Otto
cycle~\cite{Geva1992Feb}.}~\cite{Kurizki2021Dec} is tied up in the
system state, reducing power output. The reduction of power output
through quantum coherence in general has been termed quantum
friction. However, the occurrence of coherence is not necessarily
detrimental\fixme{do more research on that.refer to simulations}, if
the system is restored to a diagonal state\footnote{Shortcuts to
adiabaticity, see for example~\cite{Chen2010Feb}.}.
We will briefly demonstrate the effect of quantum friction in
\cref{sec:quantum_friction}. For now we will stay on a more general
track and turn to the ergotropy of an open quantum system coupled to a
thermal bath in \cref{sec:ergoonebath}.
\subsection{The Ergotropy of Finite Systems Coupled to a Thermal Bath} \subsection{The Ergotropy of Finite Systems Coupled to a Thermal Bath}
\label{sec:ergoonebath} \label{sec:ergoonebath}
We have argued above that Gibbs states play a special role. Here, we We have argued above that Gibbs states play a special role. Here, we
@ -928,13 +894,14 @@ 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 To justify the choice \(λ = 0\) for the model
\cref{eq:one_qubit_model_driven}, we will briefly revisit a phenomenon \cref{eq:one_qubit_model_driven}, we will briefly visit a phenomenon
introduced in \cref{sec:quantum_friction_theory}. The so called dubbed \emph{Quantum
\emph{Quantum Friction} is the creation of coherences\footnote{Or more Friction}~\cite{Binder2018,Mukherjee2020Jan}. This is creation
generally the creation of ergotropy.} in the system energy basis coherences\footnote{Or more generally the creation of ergotropy.} in
which affects the performances of thermal quantum machines. These the system energy basis affects the performance of thermal quantum
coherences raise the ergotropy of the system consuming energy that machines. These coherences raise the ergotropy of the system state,
could have been extracted by the external modulation. 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}