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@ -1391,7 +1391,8 @@ 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
\emph{ergotropy}\footnote{See \cref{eq:ergo_def} for a more precise definition.}.
\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
@ -1434,11 +1435,8 @@ 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.
\subsection{The Ergotropy of Open Quantum Systems with a single Bath}
\subsection{The Ergotropy of Open Quantum Systems}
\label{sec:ergo_general}
\begin{itemize}
\item introduce ergotropy and passivity as basis for the rest
\end{itemize}
The ergotropy of a quantum system is defined\fixme{mention paper that
uses ergo for heat}
as~\cite{Binder2018}
@ -1448,17 +1446,23 @@ as~\cite{Binder2018}
\end{equation}
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 \(\id\).
iff the maximizing \(U\) \cref{eq:ergo_def} is the identity \(\id\)
and its ergotropy vanishes.
A passive state \(ρ_P\) is always diagonal in the eigenbasis of \(H\) and its
eigenvalues satisfy the following ordering condition~\cite{Lenard1978Dec}
The immediate appeal of this quantity for later applications is, that
it is formulated with respect to the full unitary dynamics which is
accessible to us through HOPS.
A passive state \(ρ_P\) is always diagonal in the eigenbasis of \(H\)
and its eigenvalues satisfy the following ordering
condition~\cite{Lenard1978Dec}
\begin{equation}
\label{eq:passive_diag}
ρ_{p}=∑_{j=1}^{n} \lambda_{j}|j\rangle\langle j|, \quad E_{j} \leq E_{j+1}, \quad \lambda_{j+1} \leq \lambda_{j},
\end{equation}
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 a Gibbs state. Gibbs are further
the micro-canonical ensemble or Gibbs states. Gibbs states are further
distinguished by additional features as described
in~\cite{Lenard1978Dec}, which can be connected to formulations of the
zeroth and second laws of thermodynamics.
@ -1466,20 +1470,22 @@ zeroth and second laws of thermodynamics.
One of these properties is complete passivity. Completely passive
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 at the same
temperature. For finite dimensional systems, the complete passivity
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}.
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.
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}.
For systems of infinite size, states fulfilling the
Kubo–Martin–Schwinger (KMS) condition have been proposed as the
generalizations of Gibbs states, having similar properties as
Gibbs states. Under some conditions passivity implies the KMS
condition. These conditions are related to the fact that KMS states
are not necessarily unique~\cite{Binder2018,Pusz1978Oct}.
generalizations of Gibbs states, having similar properties. Under some
conditions passivity implies the KMS condition. These conditions are
related to the fact that KMS states are not necessarily
unique~\cite{Binder2018,Pusz1978Oct}.
The KMS condition is stated for two arbitrary observables \(A,B\) and
\(F_{AB}(t)=\tr[ρ_βA(t)B(0)]\) (Heisenberg picture,
@ -1488,45 +1494,90 @@ The KMS condition is stated for two arbitrary observables \(A,B\) and
\label{eq:kmscond}
F_{AB}(-t) = F_{BA}(t-\iu β)
\end{equation}
by virtue of analytic continuation.
by virtue of analytic continuation and does not rely on a concrete
expression of the state \(ρ_{β}\) which may not exist in the
traditional sense.
For two initially uncorrelated KMS states, of different
temperature, the Carnot efficiency bound can be
proven~\cite{Pusz1978Oct}.
For example, the Carnot efficiency bound can be proven
rigorously~\cite{Pusz1978Oct} for two KMS states of different
temperatures.
A simple application of ergotropy is an explanation for quantum
friction. The buildup of coherence\footnote{Meaning a state which is
non-diagonal in the energy basis.} in a quantum system makes the
state non-passive and thus requires 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}. The
reduction of efficiency in through quantum coherence general has been
termed quantum friction. However, the occurrence of coherence does not
have to lead to a reduction in efficiency\fixme{do more research on
that.refer to simulations}, if a diagonal state is restored \footnote{Shortcuts to
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.
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
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
temperature the individual oscillator Hilbert spaces may be suitably
truncated making the whole bath finite dimensional, justifying our
finite dimensional treatment in the following.
\begin{figure}[h]
\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.}
\end{figure}
The Hamiltonian of a finite dimensional system is bounded and
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.
\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
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
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}.}.
\subsection{The Ergotropy of Finite Systems Coupled to a Thermal Bath}
\label{sec:ergoonebath}
\begin{itemize}
\item thermal states are somewhat special (as seen above)
\item for thermodynamic consistency: we want to find a general bound
on ergotropy that may be valid also in the infinite dimensional case
\item great news: model independent and gives meaning to temperature!
\end{itemize}
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
consistency of the global unitary dynamics of such a system.
Let us consider models with the Hamiltonians
\begin{equation}
\label{eq:simple_bath_models}
H = \id_\sys\otimes H_\bath + H_\sys\otimes \id_\bath,
\end{equation}
where the system \(\sys\) is finite dimensional and \(H_\bath\) may
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,
\end{equation}
where \(τ_β=\eu^{-β H_\bath}/Z\) and \(ρ_\sys\) is arbitrary.
where \(τ_β=\eu^{-β H_\bath}/Z\) 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
@ -1543,31 +1594,30 @@ 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.
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.
Nevertheless, \fixme{graphics} the ergotropy appears to be
bounded. Further, the system as if it was in a passive state as soon
as the limit cycle is reached. In fact, 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} based on the special form of
Gibbs states and relative entropy. The latter quantity allows the
application of quantum informational tools, even in the presence of
infinite baths if we are careful in taking limits.
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}
based on the special form of Gibbs states and relative entropy. The
latter quantity allows the application of quantum informational tools,
even in the presence of infinite baths if we are careful in taking
limits.
The following is adapted
from~\cite{Biswas2022May,Alicki2013Apr,Lobejko2021Feb} and we limit
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
(\(\tr[ρ\ln(ρ)] = \tr[ρ_P\ln(ρ_P)]\)), we have for an arbitrary
\(β > 0\) and \(ρ_β=\exp(-βH)/Z\)
\(β > 0\) and \(ρ_β=\exp(-βH)/Z\), \(Z=\tr[\exp(-βH)]\)
\begin{align}
\ergo{ρ} &= E(ρ) - E(ρ_P) = \tr[(ρ-ρ_P) H]\nonumber\\
&= -\frac{1}{β}\tr[(ρ-ρ_P)
\qty(\ln(ρ_β) + \ln(Z))] \nonumber\\
&= -\frac{1}{β}\tr[(ρ-ρ_P) \ln(ρ_β)] =
-\frac{1}{β}\tr[(ρ-ρ_P) \qty(\ln(ρ_β))]\nonumber\\
&=\frac{1}{β}\qty[\tr[ρ(\ln(ρ) - \ln(ρ_β))] -
\tr[ρ_P(\ln(ρ_p) - \ln(ρ_β))]]\nonumber\\
&= -\frac{1}{β}\tr[(ρ-ρ_P) \ln(ρ_β)]\nonumber\\
&=\frac{1}{β}\qty{\tr[ρ\qty(\ln(ρ) - \ln(ρ_β))] -
\tr[ρ_P\qty(\ln(ρ_p) - \ln(ρ_β))]}\nonumber\\
&\equiv\frac{1}{β}\qty[\qrelent{ρ}{ρ_β} - \qrelent{ρ_P}{ρ_β}]\label{eq:ergo_entro},
\end{align}
where we have used \(\tr[ρ]=\tr[ρ_P]=1\).
@ -1587,7 +1637,7 @@ arrive at
\ergo{ρ} \leq \frac{1}{β^\ast}\qrelent{ρ}{ρ_{β^\ast}}.
\end{equation}
This bound can be saturated for states which are a permutation of a
thermal state, as their corresponding passive states is the thermal
thermal state, as their corresponding passive state is the thermal
state.
For our setting in
@ -1603,35 +1653,30 @@ in \cref{eq:ergo_entro} we obtain
τ_β)_P}{ρ_β\otimes τ_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}.
τ_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. It is therefore
reasonable to expected that it is also valid for an infinite bath. On
the basis of physical intuition, a very large but finitely sized bath
may be an arbitrarily good substitute for a continuous one. One might
even argue, that the continuous bath is a mathematically convenient
construct and the finite bath is the physical one. The objection to
taking the limit outright is that the state \(τ_β\) does not exist as
trace class operator for an infinite bath.
\(\dim[τ_β] = N\gg 1\) 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.
Interestingly, a saturation of \cref{eq:thermo_ergo_bound} is achieved
in~\cite{Skrzypczyk2014Jun} with a continuous qubit
bath. In~\cite{Lobejko2021Feb} a more generic argument is made in a
similar setting. Both propose concrete protocols within the bounds of
thermal operations and by considering explicit work reservoirs.
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. As \(ρ_β\otimes τ_β\) is the steady state of GKSL
dynamics (without modulation) under mild
assumptions~\cite{Binder2018}, it can be said that in this case
ergotropy is being lost.
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
@ -1641,39 +1686,37 @@ above discussion the ergotropy becomes the change of bath energy
\begin{aligned}
\ergo{ρ} &= \max_{U\,\text{unitary}}\tr[\qty(ρ - UρU^\dag)
(α\id\otimes H_\bath)] \\
&=\max_{U\,\text{unitary}}\tr_\bath[\qty(\tr_\sys[ρ-UρU^\dag])
&=\max_{U\,\text{unitary}}\tr_\bath\qty[\qty(\tr_\sys[ρ-UρU^\dag])
H_\bath]\\
&\equiv\max_{U\,\text{unitary}}ΔE_\bath\leq \frac{1}{β}\qrelent{ρ}{\frac{\id_N}{N}},
&\equiv\max_{U\,\text{unitary}}ΔE_\bath\leq
β^{-1}\qrelent{ρ}{\frac{\id_N}{N}}=β^{-1}\pqty{\log(N) - S(ρ)}
\leq β^{-1}\log(N),
\end{aligned}
\end{equation}
where \(N\) is the system dimension. No finite amount of energy may
therefore be extracted from the bath in a 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.
where \(N\) is the system dimension and \(α\) is arbitrary. No finite
amount of energy may therefore be extracted from the bath in a
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.
\subsection{The Ergotropy of a Two
Level System and a Bath of Identical Oscillators}
\label{sec:explicitergo}
\begin{itemize}
\item as an illustrative example: calculate ergotropy for a concrete
model
\item can show us how good the bound derived earlier is and whether it
holds for these infinite dimensional systems
\end{itemize}
Before retreating to numerics to explore the bound from
\cref{sec:ergoonebath}, we will treat an illustrative example and
explicitly calculate the ergotropy for a simple model. This allows us
to see an indication how tight the bound can be.
Here, we explicitly calculate the ergotropy of a finite dimensional
system connected to a bath of identical oscillators. Throughout we
will set the zero-point energy of the oscillators to zero, meaning
that \(H=ωa^\dag a\) for a single harmonic oscillator with the usual
annihilation operator \(a\).
Consider a two dimensional system connected to a bath of identical
oscillators. Throughout we will set the zero-point energy of the
oscillators to zero, meaning that \(H=ωa^\dag a\) for a single
harmonic oscillator with the usual annihilation operator \(a\).
Let us choose \(H_S=α\id_N\) 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.
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
\begin{equation}
@ -1681,12 +1724,14 @@ The bound \cref{eq:thermo_ergo_bound} further simplifies to
\ergo{ρ\otimes τ_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.
\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 \(\ket{0}\).
If we take the system to be a qubit, the right hand side of
\cref{eq:thermo_ergo_bound_specific} is the Landauer bound
\(β^{-1}\ln2\). Therefore, up saturation of the bound
\(β^{-1}\ln2\). Therefore, upon saturation of the bound
\cref{eq:thermo_ergo_bound_specific} we can extract enough energy from
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
@ -1694,7 +1739,6 @@ 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.
\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
@ -2170,9 +2214,10 @@ energy lowering process to take place.
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
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.
thermal quantum machines (see
\cref{sec:quantum_friction_theory}). 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