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some work on chap 5 (thx richard)

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@ -1790,3 +1790,33 @@
publisher = {American Physical Society},
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
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@ -2,21 +2,21 @@
\label{sec:therm_results}
After the precision studies of \cref{sec:hopsvsanalyt,sec:prec_sim} we
turn to applications of the formalism developed in \cref{chap:flow}
that are related to thermodynamic questions\fixme{add some
citations}. Because the NMQSD HOPS provide access to the global
unitary dynamics of system (working medium) and bath, it is
predestined to be put to use in this field.
that are related to the thermodynamic task of extracting energy from
systems connected to heat baths. Because the NMQSD and HOPS provide
access to the global unitary dynamics of system (working medium) and
bath, it is predestined to be put to use in this field.
We will begin with some theoretical notions in \cref{sec:basic_thermo}
and then continue to apply them to some very simple cases in
\cref{sec:singlemod,sec:otto}.
We will begin by reviewing some theoretical notions in
\cref{sec:basic_thermo} and then continue to apply them to some
exemplary systems in \cref{sec:singlemod,sec:otto}.
\section{Some Theoretical Notions}
\label{sec:basic_thermo}
The field of quantum thermodynamics is a complex one and we will
restrict the account given here to the minimum required for the
presentation of the subsequent results. A comprehensive account can be
found in
presentation of the subsequent results. A more comprehensive picture
can be obtained from
\cite{Binder2018,Kurizki2021Dec,Talkner2020Oct,Vinjanampathy2016Oct}.
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
just rely on the energy change in the total system or its
constituents. These quantities are now accessible to us in a rather
general setting, making issues related energy extraction a prime
application for our method.
general setting, making issues related to energy extraction a prime
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
concerned with how much energy can be extracted through a unitary
transformation from a single infinite thermal bath coupled to an
arbitrary finite dimensional working medium. We call this energy
We will focus on two closely related problems. The first is concerned
with how much energy can be extracted through a unitary transformation
from a single infinite thermal bath coupled to an arbitrary finite
dimensional working medium. We call this energy
\emph{ergotropy}\footnote{see \cref{eq:ergo_def} for a more precise
definition}.
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
the full quantum case such a result can be obtained by studying bounds
on the ergotropy of the system as is done in
In thermodynamics the second law tells us that in this setting no
energy can be extracted indefinitely in a periodic manner. It turns
out, that in the full quantum case such a result can be obtained by
studying bounds on the ergotropy of the system as is done in
\cref{sec:ergo_general}. Remarkably these bounds will turn out to be
finite. We will review a general bound for single bath systems in
\cref{sec:ergoonebath} and study an explicit calculation for a simple
case in \cref{sec:explicitergo}. The latter study will elucidate under
which conditions we may expect the bound to be tight.
case in \cref{sec:explicitergo} after briefly discussing the subject
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
systems coupled to multiple baths of different temperature. Here,
naturally, the ergotropy is generally not bounded. However, there is a
limit to how efficiently energy can be extracted. To quantify
efficiency one has to introduce a notion of thermodynamic
systems coupled to multiple baths of different temperature. In this
case the ergotropy is generally not bounded. However, there is a limit
to how efficiently energy can be extracted from the system. To
quantify efficiency one has to introduce a notion of thermodynamic
cost. Traditionally this is the entropy production or the amount of
``waste heat'' that is being shed into the cold reservoir instead of
being extracted as work. Both quantities require a proper definition,
that can usually be given, but which is not unique due to the issue
of finite interaction energy\footnote{see the discussion in
that can usually be given, but which is not unique due to the issue of
finite interaction energy\footnote{see the discussion in
\cref{chap:intro}}.
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.
In \cref{sec:operational_thermo} a Gibbs like inequality for an
arbitrary number of baths is derived as a very slight generalization
of the derivation in~\cite{Kato2016Dec}. The left hand side of this
arbitrary number of baths is derived as a slight generalization of the
derivation in~\cite{Kato2016Dec}. The left hand side of this
inequality can be associated with a thermodynamic cost that should be
minimized for optimal efficiency.
Let us now continue to properly define ergotropy in
\cref{sec:ergo_general}.
\subsection{The Ergotropy of Open Quantum Systems}
\label{sec:ergo_general}
The ergotropy of a quantum system is defined\fixme{mention paper that
uses ergo for heat}
The ergotropy~\cite{Allahverdyan2004Aug} of a quantum system is
defined\fixme{mention paper that uses ergo for heat}
as~\cite{Binder2018}
\begin{equation}
\label{eq:ergo_def}
@ -87,11 +93,13 @@ as~\cite{Binder2018}
which is the maximal energy that can be extracted from a system
through cyclic modulation of the Hamiltonian \(H\). A state is called
passive iff the maximizing \(U\) \cref{eq:ergo_def} is the identity
\(\id\) and its ergotropy vanishes. In other words, a state is passive
if it's energy can not be reduced through unitary transformations.
\(\id\). In other words, a state is passive if its energy can not be
reduced through unitary transformations and its ergotropy vanishes.
The immediate appeal of this quantity for later applications is, that
it is formulated with respect to the full unitary dynamics which is
In \cite{Niedenzu2018Jan} the ergotropy of the system is employed for
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.
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
necessary and sufficient. Examples of passive states are the state of
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
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
energy can be extracted from multiple identical systems in equilibrium
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
a bath of infinite size. If the system state is not a Gibbs state, the
whole system becomes non-passive, even if the system state is passive
with respect to the system Hamiltonian\footnote{for example being the
ground state}.
a bath of infinite size. If system and bath are not in Gibbs states
of the same temperature, the whole system becomes non-passive, even if
the system state is passive with respect to the system
Hamiltonian\footnote{for example being the ground state}.
For systems of infinite size, states fulfilling 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
dimensional systems, taking the thermodynamic limit when it is
appropriate. KMS states only enter the NMQSD/HOPS formalism
indirectly, as they predict the expression for negative frequency part
of the finite temperature spectral density. Due to the formulation of
the NMQSD which only relies on bath correlation functions, the problem
of non-existing states is circumvented.
indirectly, as they predict the expression for the spectral density of
the finite temperature noise in \cref{sec:lin_finite}. Due to the
formulation of the NMQSD which only relies on bath correlation
functions, the problem of non-existing states is circumvented.
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
lifetime of the universe.}. For such a bath the thermal state is
trace class, albeit not finite dimensional. However, for finite
@ -163,11 +171,11 @@ finite dimensional treatment in the following.
\centering
\includegraphics{figs/misc/bcf_approx}
\caption{\label{fig:bcf_approx} An ohmic BCF with \(ω_{c}=η=1\)
approximated by the BCF of linearly spaced oscillators. The figure
plots the relative difference between an approximation with \(N\)
oscillators and the exact BCF over time. An order of magnitude
more oscillators give an approximation which is valid for an order
of longer dimensionless period.}
approximated by the BCF of a finite number of linearly spaced
oscillators. The figure plots the relative difference between an
approximation with \(N\) oscillators and the exact BCF over
time. An order of magnitude more oscillators give an approximation
which is valid for an order of magnitude longer.}
\end{figure}
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
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}. This is an
unfortunate term. From it one would expect that quantum friction has
some connection to dissipation. In fact the reverse is true in most
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 in
through quantum coherence general has been termed quantum
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}
\label{sec:ergoonebath}
We have argued above that Gibbs states play a special role. Here, we
explore the ergotropy of such a state to an arbitrary finite
dimensional systems. Our goal will be to ensure thermodynamic
explore the ergotropy of such a state coupled to an arbitrary finite
dimensional system. Our goal will be to ensure thermodynamic
consistency of the global unitary dynamics of such a system.
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
\begin{equation}
\label{eq:simple_initial_state}
ρ=ρ_\sys\otimes τ_β,
ρ=ρ_\sys\otimes \varsigma_β,
\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.
An interesting question is whether the ergotropy of such a state is
finite. This amounts to the formulation of the second law: ``No energy
may be extracted from a single bath in a cyclical manner''.
For systems obeying GKSL dynamics connected to a KMS state heat bath,
thermodynamic laws can be derived in certain situations\footnote{very
slow or very fast modulation of the system
hamiltonian}\cite{Binder2018}, which imply the answer ``yes'' for the
above questions. In the non-Markovian case, those arguments do not
hold anymore.
may be extracted from a single bath in a cyclical manner''. For
systems obeying weak coupling dynamics, thermodynamic laws can be
derived in certain situations\footnote{very slow or very fast
modulation of the system hamiltonian}\cite{Binder2018}, which imply
the answer ``yes'' for the above questions. In the non-Markovian case
however, those arguments do not hold anymore.
For finite dimensional baths, we always have finite ergotropies, as
their Hamiltonians are bounded. In the infinite dimensional case, we
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
assumption breaks down when we consider infinite baths, whose thermal
energy is unbounded even for finite temperatures. In terms of finite
baths we may ask whether the ergotropy of the system can be made
arbitrarily large by enlarging the bath.
energy is unbounded even for finite temperatures. We therefore
slightly reformulate our problem. The question is now, whether the
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
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
from~\cite{Biswas2022May,Alicki2013Apr,Lobejko2021Feb}. We limit
ourselves to finite dimensional problems for now. As unitary
transformations leave the entropy invariant
ourselves to finite dimensional problems for now. Let \(ρ\) be an
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
\(β > 0\) and \(ρ_β=\exp(-βH)/Z\), \(Z=\tr[\exp(-βH)]\)
\begin{align}
@ -284,24 +301,24 @@ state.
For our setting in
\cref{eq:simple_bath_models,eq:simple_initial_state} we find a still
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
\begin{equation}
\label{eq:thermo_ergo_bound}
\begin{aligned}
\ergo{ρ\otimes τ_β} &= \frac{1}{β}
\qty[\qrelent{ρ\otimes τ_β}{ρ_β\otimes τ_β} - \qrelent{(ρ\otimes
τ_β)_P}{ρ_β\otimes τ_β}]\\
\ergo{ρ\otimes \varsigma_β} &= \frac{1}{β}
\qty[\qrelent{ρ\otimes \varsigma_β}{ρ_β\otimes \varsigma_β} - \qrelent{(ρ\otimes
\varsigma_β)_P}{ρ_β\otimes \varsigma_β}]\\
&=\frac{1}{β}
\qty[\qrelent{ρ}{ρ_β} - \qrelent{(ρ\otimes τ_β)_P}{ρ_β\otimes
τ_β}] \leq \frac{1}{β} \qrelent{ρ}{ρ_β},
\qty[\qrelent{ρ}{ρ_β} - \qrelent{(ρ\otimes \varsigma_β)_P}{ρ_β\otimes
\varsigma_β}] \leq \frac{1}{β} \qrelent{ρ}{ρ_β},
\end{aligned}
\end{equation}
where the positivity of relative entropy has been used.
Remarkably, the bound \cref{eq:thermo_ergo_bound} only depends on the
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
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
asymptotically saturates the bound.
For the term \(\qrelent{(ρ\otimes τ_β)_P}{ρ_β\otimes τ_β}\) to vanish
in \cref{eq:thermo_ergo_bound}, the state bath and system should be as
close to the product thermal state as possible and so the bath state
should not change too much. This is achievable with a continuous
infinite size bath.
In order for the term
\(\qrelent{(ρ\otimes \varsigma_β)_P}{ρ_β\otimes \varsigma_β}\) to
vanish in \cref{eq:thermo_ergo_bound}, the state of bath and system
should be as close to the product thermal state as possible and thus
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
second law. By setting the system Hamiltonian to \(α \id\) in the
above discussion the ergotropy becomes the change of bath energy
A corollary of \cref{eq:thermo_ergo_bound} is a form of the second law
of thermodynamics. By setting the system Hamiltonian to \(α \id\) in
the above discussion the ergotropy becomes the change of bath energy
\begin{equation}
\label{eq:ergo_bath_change}
\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}
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
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
simplicity, where \(α\) is an arbitrary energy scale. The ergotropy is
then equal to the maximal energy reduction of the bath under arbitrary
cyclic modulation.
The bound \cref{eq:thermo_ergo_bound} further simplifies to
cyclic modulation. Then, the bound \cref{eq:thermo_ergo_bound}
further simplifies to
\begin{equation}
\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}
where \(S(ρ)=-\tr[ρ\ln(ρ)]\). For a pure state
\cref{eq:thermo_ergo_bound_specific} is maximal and we therefore
choose \(ρ=\ketbra{0}\) as an arbitrary pure state basis state of the
two level system. The state \(\bra{1}\) is the second basis state,
orthogonal to \(\ket{0}\).
choose \(ρ=\ketbra{0}\) as an arbitrary pure state of the two level
system. We call the second orthogonal state of the two level system
\(\ket{1}\).
If we take the system to be a qubit, the right hand side of
\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
bath. Indeed, owing to \cref{eq:thermo_ergo_bound_specific} the closer
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
bath.
certain we can be, that we have extracted the maximum energy out of
the bath.
\paragraph{One Oscillator}
As a demonstration of the general program, let us first discuss the
ergotropy of a single harmonic oscillator with frequency \(ω\) as a
bath. The initial state is given by
ergotropy of a single harmonic oscillator with frequency \(ω\) coupled
to a qubit. Consider a state of qubit and oscillator given by
\begin{equation}
\label{eq:onehoinit}
ρ_{0} = ∑_{n=0}^{∞} \underbrace{Z^{-1}\eu^{-βωn}}_{λ_{0,n}} \ketbra{{0,n}},
\end{equation}
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
only fill every second level with \cref{eq:onehoinit}. To construct a
corresponding passive state\footnote{Due to the degeneracy of the
system Hamiltonian, the passive state is not unique.} we have to
construct a sequence \(λ_{i},\, i \in \NN_{0}\) out of the weights
\(λ_{0,n}\) such that \(λ_{i}\geq λ_{j}\) for \(i\leq j\) and a
sequence of states \(\ket{j}\) 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
In contrast to the passive state characterized by
\cref{eq:passive_diag} we only fill every second level with
\cref{eq:onehoinit}. To construct a corresponding passive
state\footnote{Due to the degeneracy of the system Hamiltonian, the
passive state is not unique.} we have to construct a sequence
\(λ_{i},\, i \in \NN_{0}\) out of the weights \(λ_{0,n}\) such that
\(λ_{i}\geq λ_{j}\) for \(i\leq j\) and a sequence of states
\(\ket{j}\) 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
\begin{equation}
\label{eq:specific_passive_state}
@ -418,7 +440,7 @@ and
\end{cases}
\end{equation}
Thus, we find a passive state
Thus, we find the passive state
\begin{equation}
\label{eq:one_ho_pass}
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}.
\end{equation}
For low temperatures or large frequencies, the ergotropy of the
oscillator is just its mean thermal energy \(ω\bose(ωβ)\). In the
opposite limit we find \(\mathcal{W} = \frac{1}{7β}< β^{-1}\ln(2)\)
verifying the bound derived in \cref{sec:ergoonebath}.
For low temperatures or large frequencies, the ergotropy is just the
mean thermal energy \(ω\bose(ωβ)\) of the oscillator. In the opposite
limit we find \(\mathcal{W} = \frac{1}{7β}< β^{-1}\ln(2)\) verifying
the bound derived in \cref{sec:ergoonebath}.
\paragraph{Many Oscillators}
For the case of \(N>1\) oscillators with frequency \(ω\) the initial
state is
For the case of \(N>1\) oscillators with frequency \(ω\) we consider
the initial state
\begin{equation}
\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}}},
@ -449,16 +471,16 @@ state is
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 \(Z=Z_{1}^{N}\).
In this case we again call the ordered sequence of weights \(λ_{i}\),
but its construction is somewhat more complicated and we will refrain
from doing so here explicitly. Instead we take an enumeration of
state labels \(\{\vb{n}^{i}\}_{i\in\NN_{0}}\) such that
In this case we once again want to calculate the ordered sequence of
weights \(λ_{i}\), but its construction is somewhat more complicated
and we will refrain from doing so here explicitly. Instead we take an
enumeration of state labels \(\{\vb{n}^{i}\}_{i\in\NN_{0}}\) such that
\(i<j \implies m_{i} \leq m_{j}\) with
\(m_{i}\equiv\abs{\vb{n}^{i}}\). The energies of the states
\(\ket{k,\vb{n}^{i}}\) evaluate to \(E_{k, i} = ω m_{i}\) and the
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}
\label{eq:many_enum}
\begin{aligned}
@ -470,7 +492,8 @@ The required enumeration of states and energies is then
& E_{i} &= ω m_{\lfloor{i/2}\rfloor}
\end{aligned}
\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
@ -487,9 +510,9 @@ until which \(m_{i}\) has the value \(m\). Likewise
\(\qty{y_{m}}_{m\in\NN_{0}}\) is defined so that
\(y_{m}=\big\lceil G^{N+1}_{m}/2\big\rceil\) denotes the index \(i\)
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
sequence \(\qty{z_{m}}_{m\in\NN_{0}}\) fulfills the same purpose but
for \(m_{2i+1}\).
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 for
\(m_{2i+1}\).
Now, let
\begin{equation}
@ -506,12 +529,11 @@ Now, let
\end{aligned}
\end{equation}
The \(Δ^{e}_{m,k}\) denotes the number of indices \(i\) where
\(m_{i}=m\) and \(m_{2i}=k\) and the \(Δ^{e}_{m,k}\) have the same
function, but for \(m_{2i+1}\).
The equations \cref{eq:deltas} lists explicit formulas only for the
case where the difference is at most one, but this is sufficient for
the estimate we will discuss now.
The \(Δ^{e}_{m,k}\) denote the number of indices \(i\) where
\(m_{i}=m\) and \(m_{2i}=k\). The \(Δ^{e}_{m,k}\) have the same
function, but for \(m_{2i+1}\). The equations \cref{eq:deltas} list
explicit formulas only for the case where the difference is at most
one, but this is sufficient for the estimate we will discuss now.
We find
\begin{equation}
@ -628,23 +650,24 @@ findings as has been done in \cref{fig:numeric_n_ho_ergo}.
\includegraphics{figs/ergo_calc/ergo_numeric}
\caption{\label{fig:numeric_n_ho_ergo} Numerical evaluation of
\cref{eq:many_ho_pass} and the estimates
\cref{eq:many_ho_ergo_bound,eq:ergo_esti_high_t} for \(N=110\)
and \(β=1\). Good agreement can be found and the bound
\(β^{-1}\ln2\) is approximately saturated.}
\cref{eq:many_ho_ergo_bound,eq:ergo_esti_high_t} for \(N=110\) and
\(β=1\). Good agreement can be found and the bound \(β^{-1}\ln2\)
is approximately saturated. The general bound on the ergotropy
\(β^{-1}\ln2\) is valid.}
\end{figure}
In conclusion, we have found an upper bound
\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
thermal state and a two level system in a pure state. The bound
\cref{eq:thermo_ergo_bound_specific} on ergtropy is still valid and
also tightest in the infinite bath case with vanishing level distance
\(ω\to 0\). These conditions are fulfilled for the baths usually
considered in open quantum systems, where a continuum of frequencies
are present instead of a single, very degenerate one. The energy level
distance vanishes in this case.
\cref{eq:ergo_esti_high_t} of the ergotropy to \(N\) oscillators
initially in a thermal state and a two level system initially in a
pure state. The bound \cref{eq:thermo_ergo_bound_specific} on
ergotropy is still valid and also tightest in the infinite bath case
with vanishing level distance \(ω\to 0\). These conditions are
fulfilled for the baths usually considered in open quantum systems,
where a continuum of frequencies are present instead of a single, very
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
\cref{fig:numeric_n_ho_ergo_nonmon}. Remarkably, the ergotropy becomes
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
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
is very tight as may be expected, because the error one makes in
\cref{eq:manhoergoestimate} becomes negligible in the case \(N\gg 1\).
crossing point of this bound with the \(ω\ll 1\) estimate
\cref{eq:ergo_esti_high_t}. This bound is very tight as may be
expected, because the error one makes in \cref{eq:manhoergoestimate}
becomes negligible in the case \(N\gg 1\).
\begin{figure}[htp]
\includegraphics{figs/ergo_calc/ergo_nonmonotonic}
\caption{\label{fig:numeric_n_ho_ergo_nonmon} Numerical evaluation of
\cref{eq:many_ho_pass} for different \(N\) and \(β=1\).}
\caption{\label{fig:numeric_n_ho_ergo_nonmon} Numerical evaluation
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}
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
Periodic Steady State}
\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
\cref{eq:katoineqsys} vanishes periodically, so that system and bath
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
with two baths \cref{eq:secondlaw_cyclic} implies the Carnot bound for
the efficiency given in \cref{eq:efficiency_definition}.