hiro/master-thesis-tex
1
0
Fork 0
mirror of https://github.com/vale981/master-thesis-tex synced 2025-03-05 09:31:39 -05:00
Code Issues Projects Releases Packages Wiki Activity Actions

some work on chap 5 (thx richard)

Browse source
This commit is contained in:
Valentin Boettcher 2022-09-25 19:53:29 +02:00
parent 967e2a1b17
commit bb875f08a3
No known key found for this signature in database
GPG key ID: E034E12B7AF56ACE
2 changed files with 208 additions and 147 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

30
references.bib
View file

@ -1790,3 +1790,33 @@
publisher = {American Physical Society}, publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.46.211} doi = {10.1103/PhysRevLett.46.211}
} }
@article{Allahverdyan2004Aug,
author = {Allahverdyan, A. E. and Balian, R. and
Nieuwenhuizen, {\relax Th}. M.},
title = {{Maximal work extraction from finite quantum
systems}},
journal = {Europhys. Lett.},
volume = 67,
number = 4,
pages = {565--571},
year = 2004,
month = aug,
issn = {0295-5075},
publisher = {IOP Publishing},
doi = {10.1209/epl/i2004-10101-2}
}
@article{Niedenzu2018Jan,
author = {Niedenzu, Wolfgang and Mukherjee, Victor and Ghosh, Arnab and Kofman, Abraham G. and Kurizki, Gershon},
title = {{Quantum engine efficiency bound beyond the second law of thermodynamics}},
journal = {Nat. Commun.},
volume = {9},
number = {165},
pages = {1--13},
year = {2018},
month = jan,
issn = {2041-1723},
publisher = {Nature Publishing Group},
doi = {10.1038/s41467-017-01991-6}
}

325
src/thermo.tex
View file

@ -2,21 +2,21 @@
\label{sec:therm_results} \label{sec:therm_results}
After the precision studies of \cref{sec:hopsvsanalyt,sec:prec_sim} we After the precision studies of \cref{sec:hopsvsanalyt,sec:prec_sim} we
turn to applications of the formalism developed in \cref{chap:flow} turn to applications of the formalism developed in \cref{chap:flow}
that are related to thermodynamic questions\fixme{add some that are related to the thermodynamic task of extracting energy from
citations}. Because the NMQSD HOPS provide access to the global systems connected to heat baths. Because the NMQSD and HOPS provide
unitary dynamics of system (working medium) and bath, it is access to the global unitary dynamics of system (working medium) and
predestined to be put to use in this field. bath, it is predestined to be put to use in this field.
We will begin with some theoretical notions in \cref{sec:basic_thermo} We will begin by reviewing some theoretical notions in
and then continue to apply them to some very simple cases in \cref{sec:basic_thermo} and then continue to apply them to some
\cref{sec:singlemod,sec:otto}. exemplary systems in \cref{sec:singlemod,sec:otto}.
\section{Some Theoretical Notions} \section{Some Theoretical Notions}
\label{sec:basic_thermo} \label{sec:basic_thermo}
The field of quantum thermodynamics is a complex one and we will The field of quantum thermodynamics is a complex one and we will
restrict the account given here to the minimum required for the restrict the account given here to the minimum required for the
presentation of the subsequent results. A comprehensive account can be presentation of the subsequent results. A more comprehensive picture
found in 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 energy
@ -24,36 +24,39 @@ 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
constituents. These quantities are now accessible to us in a rather constituents. These quantities are now accessible to us in a rather
general setting, making issues related energy extraction a prime general setting, making issues related to energy extraction a prime
application for our method. application for the HOPS method developed in
\cref{sec:hops_basics,chap:flow}.
Here, we will focus on two closely related problems. The first is We will focus on two closely related problems. The first is concerned
concerned with how much energy can be extracted through a unitary with how much energy can be extracted through a unitary transformation
transformation from a single infinite thermal bath coupled to an from a single infinite thermal bath coupled to an arbitrary finite
arbitrary finite dimensional working medium. We call this energy dimensional working medium. We call this energy
\emph{ergotropy}\footnote{see \cref{eq:ergo_def} for a more precise \emph{ergotropy}\footnote{see \cref{eq:ergo_def} for a more precise
definition}. definition}.
In Thermodynamics the second law tells us that in this setting no In thermodynamics the second law tells us that in this setting no
energy can be extracted in a periodic manner. It turns out that in energy can be extracted indefinitely in a periodic manner. It turns
the full quantum case such a result can be obtained by studying bounds out, that in the full quantum case such a result can be obtained by
on the ergotropy of the system as is done in 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}. The latter study will elucidate under case in \cref{sec:explicitergo} after briefly discussing the subject
which conditions we may expect the bound to be tight. of quantum friction in \cref{sec:quantum_friction_theory}. The
explicit ergotropy calculation will elucidate under which conditions
the bound of \cref{sec:ergoonebath} may be expected to be tight.
The second problem is a generalization of the above considerations to The second problem is a generalization of the above considerations to
systems coupled to multiple baths of different temperature. Here, systems coupled to multiple baths of different temperature. In this
naturally, the ergotropy is generally not bounded. However, there is a case the ergotropy is generally not bounded. However, there is a limit
limit to how efficiently energy can be extracted. To quantify to how efficiently energy can be extracted from the system. To
efficiency one has to introduce a notion of thermodynamic quantify efficiency one has to introduce a notion of thermodynamic
cost. Traditionally this is the entropy production or the amount of cost. Traditionally this is the entropy production or the amount of
``waste heat'' that is being shed into the cold reservoir instead of ``waste heat'' that is being shed into the cold reservoir instead of
being extracted as work. Both quantities require a proper definition, being extracted as work. Both quantities require a proper definition,
that can usually be given, but which is not unique due to the issue that can usually be given, but which is not unique due to the issue of
of finite interaction energy\footnote{see the discussion in finite interaction energy\footnote{see the discussion in
\cref{chap:intro}}. \cref{chap:intro}}.
Nevertheless, when the system is periodically modulated so that the Nevertheless, when the system is periodically modulated so that the
@ -70,15 +73,18 @@ over one cycle and \({Δ\ev{H}_{\mathrm{cyc}}}\) is the total energy
change over one cycle. change over one cycle.
In \cref{sec:operational_thermo} a Gibbs like inequality for an In \cref{sec:operational_thermo} a Gibbs like inequality for an
arbitrary number of baths is derived as a very slight generalization arbitrary number of baths is derived as a slight generalization of the
of the derivation in~\cite{Kato2016Dec}. The left hand side of this derivation in~\cite{Kato2016Dec}. The left hand side of this
inequality can be associated with a thermodynamic cost that should be inequality can be associated with a thermodynamic cost that should be
minimized for optimal efficiency. minimized for optimal efficiency.
Let us now continue to properly define ergotropy in
\cref{sec:ergo_general}.
\subsection{The Ergotropy of Open Quantum Systems} \subsection{The Ergotropy of Open Quantum Systems}
\label{sec:ergo_general} \label{sec:ergo_general}
The ergotropy of a quantum system is defined\fixme{mention paper that The ergotropy~\cite{Allahverdyan2004Aug} of a quantum system is
uses ergo for heat} defined\fixme{mention paper that uses ergo for heat}
as~\cite{Binder2018} as~\cite{Binder2018}
\begin{equation} \begin{equation}
\label{eq:ergo_def} \label{eq:ergo_def}
@ -87,11 +93,13 @@ as~\cite{Binder2018}
which is the maximal energy that can be extracted from a system which is the maximal energy that can be extracted from a system
through cyclic modulation of the Hamiltonian \(H\). A state is called through cyclic modulation of the Hamiltonian \(H\). A state is called
passive iff the maximizing \(U\) \cref{eq:ergo_def} is the identity passive iff the maximizing \(U\) \cref{eq:ergo_def} is the identity
\(\id\) and its ergotropy vanishes. In other words, a state is passive \(\id\). In other words, a state is passive if its energy can not be
if it's energy can not be reduced through unitary transformations. reduced through unitary transformations and its ergotropy vanishes.
The immediate appeal of this quantity for later applications is, that In \cite{Niedenzu2018Jan} the ergotropy of the system is employed for
it is formulated with respect to the full unitary dynamics which is the definition of heat to derive a tighter second law. The immediate
appeal of this quantity for the purposes of this work however is its
to the full unitary dynamics of system \emph{and} bath which is
accessible to us through HOPS. accessible to us through HOPS.
A passive state \(ρ_P\) is always diagonal in the eigenbasis of \(H\) A passive state \(ρ_P\) is always diagonal in the eigenbasis of \(H\)
@ -104,7 +112,7 @@ condition~\cite{Lenard1978Dec}
where \(n<∞\) is the Hilbert space dimension. This condition is both where \(n<∞\) is the Hilbert space dimension. This condition is both
necessary and sufficient. Examples of passive states are the state of necessary and sufficient. Examples of passive states are the state of
the micro-canonical ensemble or Gibbs states. Gibbs states are further the micro-canonical ensemble or Gibbs states. Gibbs states are further
distinguished by additional features as described distinguished by additional features described
in~\cite{Lenard1978Dec}, which can be connected to formulations of the in~\cite{Lenard1978Dec}, which can be connected to formulations of the
zeroth and second laws of thermodynamics. zeroth and second laws of thermodynamics.
@ -113,13 +121,13 @@ 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 the Gibbs state. passivity even implies the form of the Gibbs state.
The open-systems case differs as here a ``small'' system is coupled to The open-systems case differs as here a ``small'' system is coupled to
a bath of infinite size. If the system state is not a Gibbs state, the a bath of infinite size. If system and bath are not in Gibbs states
whole system becomes non-passive, even if the system state is passive of the same temperature, the whole system becomes non-passive, even if
with respect to the system Hamiltonian\footnote{for example being the the system state is passive with respect to the system
ground state}. Hamiltonian\footnote{for example being the ground state}.
For systems of infinite size, states fulfilling the For systems of infinite size, states fulfilling the
Kubo–Martin–Schwinger (KMS) condition have been proposed as the Kubo–Martin–Schwinger (KMS) condition have been proposed as the
@ -146,13 +154,13 @@ temperatures.
In the following we will restrict our discussions to finite In the following we will restrict our discussions to finite
dimensional systems, taking the thermodynamic limit when it is dimensional systems, taking the thermodynamic limit when it is
appropriate. KMS states only enter the NMQSD/HOPS formalism appropriate. KMS states only enter the NMQSD/HOPS formalism
indirectly, as they predict the expression for negative frequency part indirectly, as they predict the expression for the spectral density of
of the finite temperature spectral density. Due to the formulation of the finite temperature noise in \cref{sec:lin_finite}. Due to the
the NMQSD which only relies on bath correlation functions, the problem formulation of the NMQSD which only relies on bath correlation
of non-existing states is circumvented. functions, the problem of non-existing states is circumvented.
In fact, we see in \cref{fig:bcf_approx} that the BCF of an infinite In fact, we see in \cref{fig:bcf_approx} that the BCF of an infinite
bath can be approximate very well by a finite number of evenly spaced bath can be approximated very well by a finite number of evenly spaced
oscillators for finite times\footnote{Finite times include the oscillators for finite times\footnote{Finite times include the
lifetime of the universe.}. For such a bath the thermal state is lifetime of the universe.}. For such a bath the thermal state is
trace class, albeit not finite dimensional. However, for finite trace class, albeit not finite dimensional. However, for finite
@ -163,11 +171,11 @@ finite dimensional treatment in the following.
\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 linearly spaced oscillators. The figure approximated by the BCF of a finite number of linearly spaced
plots the relative difference between an approximation with \(N\) oscillators. The figure plots the relative difference between an
oscillators and the exact BCF over time. An order of magnitude approximation with \(N\) oscillators and the exact BCF over
more oscillators give an approximation which is valid for an order time. An order of magnitude more oscillators give an approximation
of longer dimensionless period.} which is valid for an order of magnitude longer.}
\end{figure} \end{figure}
The Hamiltonian of a finite dimensional system is bounded and The Hamiltonian of a finite dimensional system is bounded and
@ -175,35 +183,43 @@ therefore the ergotropy of such a system is finite. However, in the
following 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} \subsection{Quantum Friction}
\label{sec:quantum_friction_theory} \label{sec:quantum_friction_theory}
A simple application of the notion ergotropy is an explanation for so A simple application of the notion ergotropy is an explanation for so
called \emph{quantum called \emph{quantum
friction}~\cite{Binder2018,Mukherjee2020Jan}. This is an friction}~\cite{Binder2018,Mukherjee2020Jan}, a phenomenon with an
unfortunate term. From it one would expect that quantum friction has unfortunate name. From it one would expect that quantum friction has
some connection to dissipation. In fact the reverse is true in most 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 cases where it is a concept applied to the reduced state of the
system. system.
Consider a modulated open quantum system. The buildup of energy basis Consider a modulated open quantum system. The buildup of energy basis
coherence in the system state makes it non-passive. Thus additional coherence in the system state makes it non-passive. Thus additional
energy which cannot be extracted by modulating of the energy level energy which cannot be extracted by modulating of the energy level
gaps of the system\footnote{This is the usual mechanism of energy gaps of the system\footnote{This is the usual mechanism of energy
extraction in a quantum Otto extraction in a quantum Otto
cycle~\cite{Geva1992Feb}.}~\cite{Kurizki2021Dec} is tied up in the cycle~\cite{Geva1992Feb}.}~\cite{Kurizki2021Dec} is tied up in the
system state, reducing power output. The reduction of power output in system state, reducing power output. The reduction of power output
through quantum coherence general has been termed quantum through quantum coherence in general has been termed quantum
friction. However, the occurrence of coherence is not necessarily friction. However, the occurrence of coherence is not necessarily
detrimental\fixme{do more research on that.refer to simulations}, if detrimental\fixme{do more research on that.refer to simulations}, if
the system is restored to a diagonal state\footnote{Shortcuts to the system is restored to a diagonal state\footnote{Shortcuts to
adiabaticity, see for example~\cite{Chen2010Feb}.}. 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
explore the ergotropy of such a state to an arbitrary finite explore the ergotropy of such a state coupled to an arbitrary finite
dimensional systems. Our goal will be to ensure thermodynamic dimensional system. Our goal will be to ensure thermodynamic
consistency of the global unitary dynamics of such a system. consistency of the global unitary dynamics of such a system.
Let us consider models with the Hamiltonians Let us consider models with the Hamiltonians
@ -215,29 +231,29 @@ where the system \(\sys\) is finite dimensional and \(H_\bath\) may be
chosen arbitrarily. Let the initial state of the system be chosen arbitrarily. Let the initial state of the system be
\begin{equation} \begin{equation}
\label{eq:simple_initial_state} \label{eq:simple_initial_state}
ρ=ρ_\sys\otimes τ_β, ρ=ρ_\sys\otimes \varsigma_β,
\end{equation} \end{equation}
where \(τ_β=\eu^{-β H_\bath}/Z\) is a Gibbs state and \(ρ_\sys\) is where \(\varsigma_β=\eu^{-β H_\bath}/Z\) is a Gibbs state and \(ρ_\sys\) is
arbitrary. arbitrary.
An interesting question is whether the ergotropy of such a state is An interesting question is whether the ergotropy of such a state is
finite. This amounts to the formulation of the second law: ``No energy finite. This amounts to the formulation of the second law: ``No energy
may be extracted from a single bath in a cyclical manner''. may be extracted from a single bath in a cyclical manner''. For
For systems obeying GKSL dynamics connected to a KMS state heat bath, systems obeying weak coupling dynamics, thermodynamic laws can be
thermodynamic laws can be derived in certain situations\footnote{very derived in certain situations\footnote{very slow or very fast
slow or very fast modulation of the system modulation of the system hamiltonian}\cite{Binder2018}, which imply
hamiltonian}\cite{Binder2018}, which imply the answer ``yes'' for the the answer ``yes'' for the above questions. In the non-Markovian case
above questions. In the non-Markovian case, those arguments do not however, those arguments do not hold anymore.
hold anymore.
For finite dimensional baths, we always have finite ergotropies, as For finite dimensional baths, we always have finite ergotropies, as
their Hamiltonians are bounded. In the infinite dimensional case, we their Hamiltonians are bounded. In the infinite dimensional case, we
may expect that the ergotropy is still finite for some models, as long may expect that the ergotropy is still finite for some models, as long
as the energies of the thermal states for those models is finite. This as the energies of the thermal states for those models is finite. This
assumption breaks down when we consider infinite baths, whose thermal assumption breaks down when we consider infinite baths, whose thermal
energy is unbounded even for finite temperatures. In terms of finite energy is unbounded even for finite temperatures. We therefore
baths we may ask whether the ergotropy of the system can be made slightly reformulate our problem. The question is now, whether the
arbitrarily large by enlarging the bath. ergotropy can be made arbitrarily large by enlarging the bath. This is
consistent with approaching the infinite bath case as a limit.
There is a simple and general argument that provides and upper bound There is a simple and general argument that provides and upper bound
on the ergotropy of states of the form~\cref{eq:simple_initial_state} on the ergotropy of states of the form~\cref{eq:simple_initial_state}
@ -248,8 +264,9 @@ limits.
The derivation of the following bound is adapted The derivation of the following bound is adapted
from~\cite{Biswas2022May,Alicki2013Apr,Lobejko2021Feb}. We limit from~\cite{Biswas2022May,Alicki2013Apr,Lobejko2021Feb}. We limit
ourselves to finite dimensional problems for now. As unitary ourselves to finite dimensional problems for now. Let \(ρ\) be an
transformations leave the entropy invariant arbitrary state and \(ρ_{p}\) the corresponding passive state. As
unitary transformations leave the entropy invariant
(\(\tr[ρ\ln(ρ)] = \tr[ρ_P\ln(ρ_P)]\)), we have for an arbitrary (\(\tr[ρ\ln(ρ)] = \tr[ρ_P\ln(ρ_P)]\)), we have for an arbitrary
\(β > 0\) and \(ρ_β=\exp(-βH)/Z\), \(Z=\tr[\exp(-βH)]\) \(β > 0\) and \(ρ_β=\exp(-βH)/Z\), \(Z=\tr[\exp(-βH)]\)
\begin{align} \begin{align}
@ -284,24 +301,24 @@ state.
For our setting in For our setting in
\cref{eq:simple_bath_models,eq:simple_initial_state} we find a still \cref{eq:simple_bath_models,eq:simple_initial_state} we find a still
better way to bound the ergotropy and fix the better way to bound the ergotropy and fix the
temperature~\cite{Lobejko2021Feb}. Substituting \(ρ\to ρ \otimes τ_β\) temperature~\cite{Lobejko2021Feb}. Substituting \(ρ\to ρ \otimes \varsigma_β\)
in \cref{eq:ergo_entro} we obtain in \cref{eq:ergo_entro} we obtain
\begin{equation} \begin{equation}
\label{eq:thermo_ergo_bound} \label{eq:thermo_ergo_bound}
\begin{aligned} \begin{aligned}
\ergo{ρ\otimes τ_β} &= \frac{1}{β} \ergo{ρ\otimes \varsigma_β} &= \frac{1}{β}
\qty[\qrelent{ρ\otimes τ_β}{ρ_β\otimes τ_β} - \qrelent{(ρ\otimes \qty[\qrelent{ρ\otimes \varsigma_β}{ρ_β\otimes \varsigma_β} - \qrelent{(ρ\otimes
τ_β)_P}{ρ_β\otimes τ_β}]\\ \varsigma_β)_P}{ρ_β\otimes \varsigma_β}]\\
&=\frac{1}{β} &=\frac{1}{β}
\qty[\qrelent{ρ}{ρ_β} - \qrelent{(ρ\otimes τ_β)_P}{ρ_β\otimes \qty[\qrelent{ρ}{ρ_β} - \qrelent{(ρ\otimes \varsigma_β)_P}{ρ_β\otimes
τ_β}] \leq \frac{1}{β} \qrelent{ρ}{ρ_β}, \varsigma_β}] \leq \frac{1}{β} \qrelent{ρ}{ρ_β},
\end{aligned} \end{aligned}
\end{equation} \end{equation}
where the positivity of relative entropy has been used. where the positivity of relative entropy has been used.
Remarkably, the bound \cref{eq:thermo_ergo_bound} only depends on the Remarkably, the bound \cref{eq:thermo_ergo_bound} only depends on the
system state and ``inherits'' the temperature of the bath. For any system state and ``inherits'' the temperature of the bath. For any
\(\dim[τ_β] = N\gg 1\) the bound stays valid and independent of any \(\dim[\varsigma_β] = N\gg 1\) the bound stays valid and independent of any
bath properties except the temperature. It is therefore reasonable to bath properties except the temperature. It is therefore reasonable to
expected that it is also valid for an infinite bath. expected that it is also valid for an infinite bath.
@ -313,15 +330,16 @@ thermal operations and by considering explicit work reservoirs. In
\cref{sec:explicitergo} we will provide another example which \cref{sec:explicitergo} we will provide another example which
asymptotically saturates the bound. asymptotically saturates the bound.
For the term \(\qrelent{(ρ\otimes τ_β)_P}{ρ_β\otimes τ_β}\) to vanish In order for the term
in \cref{eq:thermo_ergo_bound}, the state bath and system should be as \(\qrelent{(ρ\otimes \varsigma_β)_P}{ρ_β\otimes \varsigma_β}\) to
close to the product thermal state as possible and so the bath state vanish in \cref{eq:thermo_ergo_bound}, the state of bath and system
should not change too much. This is achievable with a continuous should be as close to the product thermal state as possible and thus
infinite size bath. the bath state should not change too much during the time
evolution. This is achievable with a continuous infinite size bath.
A corollary of \cref{eq:thermo_ergo_bound} is the Clausius form of the A corollary of \cref{eq:thermo_ergo_bound} is a form of the second law
second law. By setting the system Hamiltonian to \(α \id\) in the of thermodynamics. By setting the system Hamiltonian to \(α \id\) in
above discussion the ergotropy becomes the change of bath energy the above discussion the ergotropy becomes the change of bath energy
\begin{equation} \begin{equation}
\label{eq:ergo_bath_change} \label{eq:ergo_bath_change}
\begin{aligned} \begin{aligned}
@ -340,6 +358,8 @@ periodic manner. If it were possible to extract a constant positive
amount of energy from the bath per cycle, \cref{eq:ergo_bath_change} amount of energy from the bath per cycle, \cref{eq:ergo_bath_change}
would be breached in finite time. would be breached in finite time.
Let us now illustrate the concept of ergotropy and the above bound by
an explicit calculation in \cref{sec:explicitergo}.
\subsection{The Ergotropy of a Two \subsection{The Ergotropy of a Two
Level System and a Bath of Identical Oscillators} Level System and a Bath of Identical Oscillators}
@ -357,18 +377,17 @@ harmonic oscillator with the usual annihilation operator \(a\).
Let us choose \(H_S=α\id_N\) as in \cref{eq:ergo_bath_change} for Let us choose \(H_S=α\id_N\) as in \cref{eq:ergo_bath_change} for
simplicity, where \(α\) is an arbitrary energy scale. The ergotropy is simplicity, where \(α\) is an arbitrary energy scale. The ergotropy is
then equal to the maximal energy reduction of the bath under arbitrary then equal to the maximal energy reduction of the bath under arbitrary
cyclic modulation. cyclic modulation. Then, the bound \cref{eq:thermo_ergo_bound}
further simplifies to
The bound \cref{eq:thermo_ergo_bound} further simplifies to
\begin{equation} \begin{equation}
\label{eq:thermo_ergo_bound_specific} \label{eq:thermo_ergo_bound_specific}
\ergo{ρ\otimes τ_β} \leq \frac{1}{β} \qty[\ln(N) - S(ρ)], \ergo{ρ\otimes \varsigma_β} \leq \frac{1}{β} \qty[\ln(N) - S(ρ)],
\end{equation} \end{equation}
where \(S(ρ)=-\tr[ρ\ln(ρ)]\). For a pure state where \(S(ρ)=-\tr[ρ\ln(ρ)]\). For a pure state
\cref{eq:thermo_ergo_bound_specific} is maximal and we therefore \cref{eq:thermo_ergo_bound_specific} is maximal and we therefore
choose \(ρ=\ketbra{0}\) as an arbitrary pure state basis state of the choose \(ρ=\ketbra{0}\) as an arbitrary pure state of the two level
two level system. The state \(\bra{1}\) is the second basis state, system. We call the second orthogonal state of the two level system
orthogonal to \(\ket{0}\). \(\ket{1}\).
If we take the system to be a qubit, the right hand side of If we take the system to be a qubit, the right hand side of
\cref{eq:thermo_ergo_bound_specific} is the Landauer bound \cref{eq:thermo_ergo_bound_specific} is the Landauer bound
@ -377,28 +396,31 @@ If we take the system to be a qubit, the right hand side of
the bath to erase one bit in a system of the same temperature as the the bath to erase one bit in a system of the same temperature as the
bath. Indeed, owing to \cref{eq:thermo_ergo_bound_specific} the closer bath. Indeed, owing to \cref{eq:thermo_ergo_bound_specific} the closer
the qubit state is to the infinite temperature (erased) state the more the qubit state is to the infinite temperature (erased) state the more
certain we are, that we have extracted the maximum energy out of the certain we can be, that we have extracted the maximum energy out of
bath. the bath.
\paragraph{One Oscillator} \paragraph{One Oscillator}
As a demonstration of the general program, let us first discuss the As a demonstration of the general program, let us first discuss the
ergotropy of a single harmonic oscillator with frequency \(ω\) as a ergotropy of a single harmonic oscillator with frequency \(ω\) coupled
bath. The initial state is given by to a qubit. Consider a state of qubit and oscillator given by
\begin{equation} \begin{equation}
\label{eq:onehoinit} \label{eq:onehoinit}
ρ_{0} = ∑_{n=0}^{∞} \underbrace{Z^{-1}\eu^{-βωn}}_{λ_{0,n}} \ketbra{{0,n}}, ρ_{0} = ∑_{n=0}^{∞} \underbrace{Z^{-1}\eu^{-βωn}}_{λ_{0,n}} \ketbra{{0,n}},
\end{equation} \end{equation}
where \(Z=Z_{1}=\frac{1}{1-\eu^{-βω}}\) is the partition sum of one where \(Z=Z_{1}=\frac{1}{1-\eu^{-βω}}\) is the partition sum of one
bosonic mode with frequency \(ω\). bosonic mode with frequency \(ω\). The \(a\) in
\(\ket{a,b}=\ket{a}\ket{b}\) labels the qubit state and \(b\) labels
the state the harmonic oscillator.
In contrast to the state characterized by \cref{eq:passive_diag} we In contrast to the passive state characterized by
only fill every second level with \cref{eq:onehoinit}. To construct a \cref{eq:passive_diag} we only fill every second level with
corresponding passive state\footnote{Due to the degeneracy of the \cref{eq:onehoinit}. To construct a corresponding passive
system Hamiltonian, the passive state is not unique.} we have to state\footnote{Due to the degeneracy of the system Hamiltonian, the
construct a sequence \(λ_{i},\, i \in \NN_{0}\) out of the weights passive state is not unique.} we have to construct a sequence
\(λ_{0,n}\) such that \(λ_{i}\geq λ_{j}\) for \(i\leq j\) and a \(λ_{i},\, i \in \NN_{0}\) out of the weights \(λ_{0,n}\) such that
sequence of states \(\ket{j}\) so that for \(H\ket{j} = E_{j}\ket{j}\) \(λ_{i}\geq λ_{j}\) for \(i\leq j\) and a sequence of states
we have \(E_{i}\leq E_{j}\) for \(i\leq j\). The passive state and its \(\ket{j}\) so that for \(H\ket{j} = E_{j}\ket{j}\) we have
\(E_{i}\leq E_{j}\) for \(i\leq j\). The passive state and its
corresponding energy is then corresponding energy is then
\begin{equation} \begin{equation}
\label{eq:specific_passive_state} \label{eq:specific_passive_state}
@ -418,7 +440,7 @@ and
\end{cases} \end{cases}
\end{equation} \end{equation}
Thus, we find a passive state Thus, we find the passive state
\begin{equation} \begin{equation}
\label{eq:one_ho_pass} \label{eq:one_ho_pass}
ρ_{p} = \frac{1}{Z} \qty[∑_{i} \eu^{-2i βω} \ketbra{0, i} + ρ_{p} = \frac{1}{Z} \qty[∑_{i} \eu^{-2i βω} \ketbra{0, i} +
@ -434,14 +456,14 @@ out to be
\rightarrow ∞} \frac{ω}{\eu^{βω} - 1}. \rightarrow ∞} \frac{ω}{\eu^{βω} - 1}.
\end{equation} \end{equation}
For low temperatures or large frequencies, the ergotropy of the For low temperatures or large frequencies, the ergotropy is just the
oscillator is just its mean thermal energy \(ω\bose(ωβ)\). In the mean thermal energy \(ω\bose(ωβ)\) of the oscillator. In the opposite
opposite limit we find \(\mathcal{W} = \frac{1}{7β}< β^{-1}\ln(2)\) limit we find \(\mathcal{W} = \frac{1}{7β}< β^{-1}\ln(2)\) verifying
verifying the bound derived in \cref{sec:ergoonebath}. the bound derived in \cref{sec:ergoonebath}.
\paragraph{Many Oscillators} \paragraph{Many Oscillators}
For the case of \(N>1\) oscillators with frequency \(ω\) the initial For the case of \(N>1\) oscillators with frequency \(ω\) we consider
state is the initial state
\begin{equation} \begin{equation}
\label{eq:manyhoinit} \label{eq:manyhoinit}
ρ_{0} = ∑_{\vb{n}\in\ZZ_{0}^{n}} \underbrace{Z^{-1}\eu^{-βω\abs{\vb{n}}}}_{λ_{0,\vb{n}}} \ketbra{{0,\vb{n}}}, ρ_{0} = ∑_{\vb{n}\in\ZZ_{0}^{n}} \underbrace{Z^{-1}\eu^{-βω\abs{\vb{n}}}}_{λ_{0,\vb{n}}} \ketbra{{0,\vb{n}}},
@ -449,16 +471,16 @@ state is
with the \(n_{i}=(\vb{n})_{i}\) labeling the state of the \(i\)th with the \(n_{i}=(\vb{n})_{i}\) labeling the state of the \(i\)th
oscillator and \(\abs{\vb{n}}=∑_{i=1}^{N}n_{i}\) and \(Z=Z_{1}^{N}\). oscillator and \(\abs{\vb{n}}=∑_{i=1}^{N}n_{i}\) and \(Z=Z_{1}^{N}\).
In this case we again call the ordered sequence of weights \(λ_{i}\), In this case we once again want to calculate the ordered sequence of
but its construction is somewhat more complicated and we will refrain weights \(λ_{i}\), but its construction is somewhat more complicated
from doing so here explicitly. Instead we take an enumeration of and we will refrain from doing so here explicitly. Instead we take an
state labels \(\{\vb{n}^{i}\}_{i\in\NN_{0}}\) such that enumeration of state labels \(\{\vb{n}^{i}\}_{i\in\NN_{0}}\) such that
\(i<j \implies m_{i} \leq m_{j}\) with \(i<j \implies m_{i} \leq m_{j}\) with
\(m_{i}\equiv\abs{\vb{n}^{i}}\). The energies of the states \(m_{i}\equiv\abs{\vb{n}^{i}}\). The energies of the states
\(\ket{k,\vb{n}^{i}}\) evaluate to \(E_{k, i} = ω m_{i}\) and the \(\ket{k,\vb{n}^{i}}\) evaluate to \(E_{k, i} = ω m_{i}\) and the
initial weights to \(λ_{0,i}=Z^{-1}\eu^{-βω m_{i}},\,λ_{1,i}=0\). initial weights to \(λ_{0,i}=Z^{-1}\eu^{-βω m_{i}},\,λ_{1,i}=0\).
The required enumeration of states and energies is then The required sequence of states and energies is then
\begin{equation} \begin{equation}
\label{eq:many_enum} \label{eq:many_enum}
\begin{aligned} \begin{aligned}
@ -470,7 +492,8 @@ The required enumeration of states and energies is then
& E_{i} &= ω m_{\lfloor{i/2}\rfloor} & E_{i} &= ω m_{\lfloor{i/2}\rfloor}
\end{aligned} \end{aligned}
\end{equation} \end{equation}
and the enumeration of weights is \(λ_{i} = λ_{0,i} =Z^{-1}\eu^{-βω m_{i}}\). and the sequence of weights is
\(λ_{i} = λ_{0,i} =Z^{-1}\eu^{-βω m_{i}}\).
The corresponding passive state energy will be The corresponding passive state energy will be
@ -487,9 +510,9 @@ until which \(m_{i}\) has the value \(m\). Likewise
\(\qty{y_{m}}_{m\in\NN_{0}}\) is defined so that \(\qty{y_{m}}_{m\in\NN_{0}}\) is defined so that
\(y_{m}=\big\lceil G^{N+1}_{m}/2\big\rceil\) denotes the index \(i\) \(y_{m}=\big\lceil G^{N+1}_{m}/2\big\rceil\) denotes the index \(i\)
until which \(m_{2i}\) retains the value \(m\). When using the floor until which \(m_{2i}\) retains the value \(m\). When using the floor
instead of ceil in the definition of \(y_{m}\) we find that this instead of ceiling in the definition of \(y_{m}\) we find that this
sequence \(\qty{z_{m}}_{m\in\NN_{0}}\) fulfills the same purpose but sequence \(\qty{z_{m}}_{m\in\NN_{0}}\) fulfills the same purpose for
for \(m_{2i+1}\). \(m_{2i+1}\).
Now, let Now, let
\begin{equation} \begin{equation}
@ -506,12 +529,11 @@ Now, let
\end{aligned} \end{aligned}
\end{equation} \end{equation}
The \(Δ^{e}_{m,k}\) denotes the number of indices \(i\) where The \(Δ^{e}_{m,k}\) denote the number of indices \(i\) where
\(m_{i}=m\) and \(m_{2i}=k\) and the \(Δ^{e}_{m,k}\) have the same \(m_{i}=m\) and \(m_{2i}=k\). The \(Δ^{e}_{m,k}\) have the same
function, but for \(m_{2i+1}\). function, but for \(m_{2i+1}\). The equations \cref{eq:deltas} list
The equations \cref{eq:deltas} lists explicit formulas only for the explicit formulas only for the case where the difference is at most
case where the difference is at most one, but this is sufficient for one, but this is sufficient for the estimate we will discuss now.
the estimate we will discuss now.
We find We find
\begin{equation} \begin{equation}
@ -628,23 +650,24 @@ findings as has been done in \cref{fig:numeric_n_ho_ergo}.
\includegraphics{figs/ergo_calc/ergo_numeric} \includegraphics{figs/ergo_calc/ergo_numeric}
\caption{\label{fig:numeric_n_ho_ergo} Numerical evaluation of \caption{\label{fig:numeric_n_ho_ergo} Numerical evaluation of
\cref{eq:many_ho_pass} and the estimates \cref{eq:many_ho_pass} and the estimates
\cref{eq:many_ho_ergo_bound,eq:ergo_esti_high_t} for \(N=110\) \cref{eq:many_ho_ergo_bound,eq:ergo_esti_high_t} for \(N=110\) and
and \(β=1\). Good agreement can be found and the bound \(β=1\). Good agreement can be found and the bound \(β^{-1}\ln2\)
\(β^{-1}\ln2\) is approximately saturated.} is approximately saturated. The general bound on the ergotropy
\(β^{-1}\ln2\) is valid.}
\end{figure} \end{figure}
In conclusion, we have found an upper bound In conclusion, we have found an upper bound
\cref{eq:many_ho_ergo_bound} for and a high temperature estimate \cref{eq:many_ho_ergo_bound} for and a high temperature estimate
\cref{eq:ergo_esti_high_t} of the ergotropy to \(N\) oscillators in a \cref{eq:ergo_esti_high_t} of the ergotropy to \(N\) oscillators
thermal state and a two level system in a pure state. The bound initially in a thermal state and a two level system initially in a
\cref{eq:thermo_ergo_bound_specific} on ergtropy is still valid and pure state. The bound \cref{eq:thermo_ergo_bound_specific} on
also tightest in the infinite bath case with vanishing level distance ergotropy is still valid and also tightest in the infinite bath case
\(ω\to 0\). These conditions are fulfilled for the baths usually with vanishing level distance \(ω\to 0\). These conditions are
considered in open quantum systems, where a continuum of frequencies fulfilled for the baths usually considered in open quantum systems,
are present instead of a single, very degenerate one. The energy level where a continuum of frequencies are present instead of a single, very
distance vanishes in this case. degenerate one. The energy level distance vanishes in this case.
Even for finitely many oscillators the ergtropy bound can be Even for finitely many oscillators the ergotropy bound can be
approached rather closely for finite \(ω\) as can be seen in approached rather closely for finite \(ω\) as can be seen in
\cref{fig:numeric_n_ho_ergo_nonmon}. Remarkably, the ergotropy becomes \cref{fig:numeric_n_ho_ergo_nonmon}. Remarkably, the ergotropy becomes
a non-monotonous function of the level spacing \(ω\) for larger \(N\) a non-monotonous function of the level spacing \(ω\) for larger \(N\)
@ -653,16 +676,24 @@ resonance phenomenon. It is always increasing with \(N\).
Also, for large \(N\) the transition into the region where the upper Also, for large \(N\) the transition into the region where the upper
bound \cref{eq:thermo_ergo_bound} is tight is very close to the bound \cref{eq:thermo_ergo_bound} is tight is very close to the
crossing point of this bound with the \(ω\ll 1\) estimate. This bound crossing point of this bound with the \(ω\ll 1\) estimate
is very tight as may be expected, because the error one makes in \cref{eq:ergo_esti_high_t}. This bound is very tight as may be
\cref{eq:manhoergoestimate} becomes negligible in the case \(N\gg 1\). expected, because the error one makes in \cref{eq:manhoergoestimate}
becomes negligible in the case \(N\gg 1\).
\begin{figure}[htp] \begin{figure}[htp]
\includegraphics{figs/ergo_calc/ergo_nonmonotonic} \includegraphics{figs/ergo_calc/ergo_nonmonotonic}
\caption{\label{fig:numeric_n_ho_ergo_nonmon} Numerical evaluation of \caption{\label{fig:numeric_n_ho_ergo_nonmon} Numerical evaluation
\cref{eq:many_ho_pass} for different \(N\) and \(β=1\).} of \cref{eq:many_ho_pass} for different \(N\) and \(β=1\). With a
rising number of oscillators \(N\) the ergotropy always increases
and becomes more non-monotonous, but never surpasses the bounds.}
\end{figure} \end{figure}
After validating the bound of \cref{sec:ergoonebath} for a concrete
example, we now return to a more generic setting in
\cref{sec:operational_thermo}.
\subsection{A bound on the Energy Change of Multiple Baths in the \subsection{A bound on the Energy Change of Multiple Baths in the
Periodic Steady State} Periodic Steady State}
\label{sec:operational_thermo} \label{sec:operational_thermo}
@ -798,7 +829,7 @@ thermodynamical quantities is required for \cref{eq:secondlaw_cyclic}
to be useful. Assume that the interaction Hamiltonian in to be useful. Assume that the interaction Hamiltonian in
\cref{eq:katoineqsys} vanishes periodically, so that system and bath \cref{eq:katoineqsys} vanishes periodically, so that system and bath
energy expectation values can be cleanly separated. In the periodic energy expectation values can be cleanly separated. In the periodic
steady state the system energy does not change during a cycle and the steady state the system energy does not change after a cycle and the
whole energy change amounts to the change in bath energy. In a setting whole energy change amounts to the change in bath energy. In a setting
with two baths \cref{eq:secondlaw_cyclic} implies the Carnot bound for with two baths \cref{eq:secondlaw_cyclic} implies the Carnot bound for
the efficiency given in \cref{eq:efficiency_definition}. the efficiency given in \cref{eq:efficiency_definition}.