# Quantum Heat Engines with Carnot Efficiency at Maximum Power

Mohit Lal Bera  ICFO – Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, ES-08860 Castelldefels, Spain  Sergi
Julià-Farré  ICFO – Institut de Ciències Fotòniques, The Barcelona Institute
of Science and Technology, ES-08860 Castelldefels, Spain  Maciej Lewenstein
ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and
Technology, ES-08860 Castelldefels, Spain  ICREA, Pg. Lluis Companys 23,
ES-08010 Barcelona, Spain  Manabendra Nath Bera<EMAIL_ADDRESS>Department
of Physical Sciences, Indian Institute of Science Education and Research
(IISER), Mohali, Punjab 140306, India

###### Abstract

Heat engines constitute the major building blocks of modern technologies.
However, conventional heat engines with higher power yield lesser efficiency
and vice versa and respect various power-efficiency trade-off relations. This
is also assumed to be true for the engines operating in the quantum regime.
Here we show that these relations are not fundamental. We introduce quantum
heat engines that deliver maximum power with Carnot efficiency in the _one-
shot finite-size_ regime. These engines are composed of working systems with a
finite number of quantum particles and are restricted to one-shot
measurements. The engines operate in a one-step cycle by letting the working
system simultaneously interact with hot and cold baths via semi-local thermal
operations. By allowing quantum entanglement between its constituents and,
thereby, a coherent transfer of heat from hot to cold baths, the engine
implements the fastest possible reversible state transformation in each cycle,
resulting in maximum power and Carnot efficiency. Finally, we propose a
physically realizable engine using quantum optical systems.

##  I Introduction

Since the beginning of the industrial revolution, heat engines have been
playing pivotal roles in shaping modern technologies. One of the central laws
of thermodynamics for heat engines in the classical regime, that is, the
second law, imposes a fundamental limit on the maximum heat-to-work conversion
efficiency in an engine, given by Carnot efficiency. This efficiency is only
achieved when the engine operates in a cycle using reversible transformations,
which requires it to run infinitely slowly. As a consequence, the engine’s
power - work extracted per unit time - becomes close to null. In general, the
realistic engines operate in finite time to deliver a non-vanishing power, and
then, the efficiency is compromised. The trade-off between efficiency and
power is studied extensively in the past decades; see, for example, Curzon and
Ahlborn (1975); Berry  _et al._ (2000); Salamon  _et al._ (2001), in the
context of finite-time classical engines.

In general, the laws of thermodynamics cannot be directly applied to the
engines that use working fluids made up of few particles. In that case, the
conventional (or statistical) notion of average quantities such as energy or
entropy becomes incomplete. The situation becomes further constrained for
engines operating in the quantum regime, where the working fluid is composed
of few quantum systems and the effects due to quantum fluctuations cannot be
ignored. There have been extensive studies to understand thermodynamics in
this regime, see for example Binder  _et al._ (2018); Jarzynski (1997); Crooks
(1999); Campisi  _et al._ (2011); Alhambra  _et al._ (2016); Åberg (2018);
Brandao  _et al._ (2013); Horodecki and Oppenheim (2013); Skrzypczyk  _et al._
(2014); Brandao  _et al._ (2015); Lostaglio  _et al._ (2015); Bera  _et al._
(2017); Gour  _et al._ (2018); Müller (2018); Sparaciari  _et al._ (2017);
Bera  _et al._ (2019); Uzdin and Rahav (2018); Baghali Khanian  _et al._
(2020), and it is revealed that, in general, a quantum system in contact with
a thermal bath delivers fluctuating work. Consequently, a quantum engine,
where a working fluid sequentially or incoherently interacts with two baths in
a cycle in the presence of highly fluctuating input and output energy fluxes,
is expected to have fluctuations in both efficiency and power, see for example
Kosloff and Levy (2014); Uzdin  _et al._ (2015); Klaers  _et al._ (2017);
Roßnagel  _et al._ (2014); Verley  _et al._ (2014); Funo and Ueda (2015); Ng
_et al._ (2017); Woods  _et al._ (2019); Manikandan  _et al._ (2019); Esposito
_et al._ (2010a); Scully  _et al._ (2011); Campisi and Fazio (2016); Holubec
and Ryabov (2016); Brandner  _et al._ (2017); Denzler and Lutz (2020); Abah
_et al._ (2012); Esposito  _et al._ (2010b); Guo  _et al._ (2013); Ma  _et
al._ (2018); Pietzonka and Seifert (2018); Holubec and Ryabov (2018); Dorfman
_et al._ (2018); Abiuso and Perarnau-Llobet (2020); Brandner and Saito (2020);
Miller and Mehboudi (2020); Singh (2020); Benenti  _et al._ (2020); Roßnagel
_et al._ (2016); Saryal and Agarwalla (2021); Saryal  _et al._ (2021). In
fact, finite size heat engines, in general, deliver fluctuating efficiency
Verley  _et al._ (2014); Funo and Ueda (2015); Manikandan  _et al._ (2019). It
is also true for power for engines operating with finite-time cycle Denzler
and Lutz (2020); Holubec and Ryabov (2018). Apart from that, there are power
losses due to energy coherence Brandner  _et al._ (2017). Also, there are
proposals that smartly exploit energy coherence to increase power, see for
example Scully  _et al._ (2011); Roßnagel  _et al._ (2014); Dorfman  _et al._
(2018). The overall performance of an engine considering inter-relations
between power and efficiency in the presence of quantum fluctuation are
studied in Esposito  _et al._ (2010b); Guo  _et al._ (2013); Pietzonka and
Seifert (2018); Holubec and Ryabov (2018); Dorfman  _et al._ (2018); Abiuso
and Perarnau-Llobet (2020); Brandner and Saito (2020); Miller and Mehboudi
(2020); Singh (2020); Benenti  _et al._ (2020). In Holubec and Ryabov (2016),
the authors derive the lower and upper bound on maximum efficiency at a given
power for the low dissipation heat engines. This bound generalizes the bound
on efficiency at maximum power given by Esposito  _et al._ (2010b). These
engines also exhibit universal constraints for efficiency and power Ma  _et
al._ (2018). For general cases, a universal trade-off between efficiency and
power is introduced in Pietzonka and Seifert (2018). A trade-off relation
based on geometric arguments is derived in Brandner and Saito (2020) for any
thermodynamically consistent microdynamics. Further studies based on the
geometry of work fluctuation and efficiency are made in Miller and Mehboudi
(2020) for microscopic heat engines.

As in classical engines, it is now commonly believed that yielding maximum
power at Carnot efficiency is impossible in a quantum engine. Earlier studies
have assumed engines with a working system that interacts with the hot and
cold baths at different stages of an engine cycle or with both baths
simultaneously but only enabling incoherent heat transfer. Furthermore, the
working system is either composed of a statistically large number of
particles, or a few particles, allowing a large number of measurements. The
role of quantum fluctuations in delimiting efficiency or power or both becomes
more prominent for the quantum engines operating in the one-shot finite-size
regime, i.e., engines with a finite number of quantum particles constituting
the working system and restricted to one-shot measurements or observations. So
far, there are no comprehensive studies on that.

Here we introduce quantum heat engines operating in the one-shot finite-size
regime and study the power and efficiency of heat-to-work conversion. We show
that these engines can simultaneously attain maximum efficiency, i.e., the
Carnot efficiency, and maximum power in the one-shot finite-size regime.
Therefore, there is no fundamental trade-off between power and efficiency that
an engine has to respect in the quantum regime. Our approach is fundamentally
different from the earlier ones in the sense that: (i) the engines are fully
quantum as they operate in the one-shot finite-size regime and allow genuine
entanglement between the baths and working system, (ii) the working system
simultaneously interacts with the hot and cold baths via semi-local thermal
operations, and (iii) the engines run in a one-step cycle. The framework
relies on the resource theory recently developed to establish the
thermodynamic laws in quantum heat engines Bera  _et al._ (2021). The engines
deliver maximum power, along with Carnot efficiency, purely because the
engines allow a coherent transfer of heat from hot to cold baths by
establishing quantum entanglement between the working system and the baths,
thereby attaining maximum quantum speed for the reversible state
transformation in each engine cycle. Finally, we also introduce a physically
realizable quantum heat engine based on a quantum-optical system.

The article is organized as follows. In Section II, we briefly describe the
thermodynamical transformation with which a quantum heat engine operates in a
one-step cycle. For this, we adhere to the resource theory quantum heat
engines, recently developed in Bera  _et al._ (2021). Then we present our main
results in Section III and IV. In Section III, we discuss how quantum heat
engines, equipped with the allowed (or resource-free) thermodynamical
operations, can deliver maximum power with Carnot efficiency. A physically
realizable model of such an engine is outlined in Section IV based on quantum
optical systems. Finally, we conclude with a discussion in Section V.

##  II Engine operating in one-step cycle

We consider an engine composed of two baths $B_{1}$ and $B_{2}$ with
corresponding Hamiltonians $H_{B_{1}}$ and $H_{B_{2}}$ and inverse
temperatures $\beta_{1}$ and $\beta_{2}$ respectively; a bipartite (working)
system $S_{12}$ with non-interacting subsystems $S_{1}$ and $S_{2}$ and
described by the Hamiltonian $H_{S_{12}}=H_{S_{1}}+H_{S_{2}}$; a bipartite
battery $S_{W_{S_{12}}}$ with non-interacting subsystems $S_{W_{1}}$ and
$S_{W_{2}}$ with the Hamiltonian $H_{S_{W_{12}}}=H_{S_{W_{1}}}+H_{S_{W_{2}}}$.
Here the battery plays the role of a piston in a traditional engine that takes
away work converted from heat. Throughout this work, we assume
$\beta_{1}<\beta_{2}$. All systems under consideration have Hamiltonians
bounded from below, with the lowest energy equal to zero. The baths are
considerably large compared to the systems, and the degeneracy in their
microcanonical ensembles scales exponentially with the change in energy. That
means the energies of the working systems and the battery are tiny compared to
the baths, while the latter have the highest energies close to infinity. The
properties of large baths are outlined in Appendix.

The engine lets the baths interact with the working system and the batteries
via a global unitary evolution ($U$) where the composite $S_{1}S_{W_{1}}$
semi-locally interacts with $B_{1}$ and $S_{2}S_{W_{2}}$ with $B_{2}$. As a
result, a semi-local thermal operation (SLTO) is implemented on the system-
battery composite $S_{12}S_{W_{12}}$ given by Bera  _et al._ (2021)

$\displaystyle\Lambda_{S_{12}S_{W_{12}}}$
$\displaystyle\left(\rho_{S_{12}}\otimes\rho_{S_{W_{12}}}\right)$
$\displaystyle={\operatorname{Tr\,}}_{B_{1}B_{2}}\left[U(\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes\rho_{S_{12}}\otimes\rho_{S_{W_{12}}})U^{\dagger}\right],$
(1)

where the global unitary $U$ satisfies

$\displaystyle\left[U,\
H_{B_{1}}+H_{S_{1}}+H_{S_{W_{1}}}+H_{B_{2}}+H_{S_{2}}+H_{S_{W_{2}}}\right]=0,$
(2) $\displaystyle\left[U,\ \beta_{1}\
(H_{B_{1}}+H_{S_{1}}+H_{S_{W_{1}}})+\beta_{2}\
(H_{B_{2}}+H_{S_{2}}+H_{S_{W_{2}}})\right]=0.$ (3)

Here the baths are in the equilibrium states denoted by
$\gamma_{B_{x}}=\frac{e^{-\beta_{x}H_{B_{x}}}}{{\operatorname{Tr\,}}[e^{-\beta_{x}H_{B_{x}}}]}$
for $x=1,2$, and $\rho_{S_{12}}$ is any state of $S_{12}$. The
$\rho_{S_{W_{12}}}$ is the state of the battery $S_{W_{12}}$, where the
subsystems $S_{W_{1}}$ and $S_{W_{2}}$ always remain in their energy
eigenstates and store or supply energy in the form of work. The commutation
relation (2) guarantees strict conservation of total energy of baths-system-
battery composite. Note, this ensures conservation of all moments of energy
and not just the average energy. The relation (2), in turn, represents the
quantum version of first law for engines.

The relation (3) ensures strict weighted energy conservation. We may place two
arguments to justify why quantum heat engines must satisfy this weighted
energy conservation. First, this relation is essential for the construction of
resource theory Bera  _et al._ (2021), ensuring that
$\Lambda_{S_{12}}\left(\gamma_{S_{1}}\otimes\gamma_{S_{2}}\right)=\gamma_{S_{1}}\otimes\gamma_{S_{2}}$
where
$\gamma_{S_{x}}=\frac{e^{-\beta_{x}H_{B_{x}}}}{{\operatorname{Tr\,}}[e^{-\beta_{x}H_{S_{x}}}]}$
for $x=1,2$. Here we ignore the batteries as they only store or supply without
directly interacting with the baths. This implies that if the subsystems
$S_{1}$ and $S_{2}$ are in (local) thermal equilibrium with the baths $B_{1}$
and $B_{2}$ respectively (also called semi-Gibbs state), then the SLTOs cannot
transform them out of the equilibrium. Note, this is a direct consequence of
the relation (3). Thus, these semi-Gibbs states are resource-free states from
the thermodynamic point of view. This also physically makes sense. Because if
the subsystems are thermalized to the baths (as mentioned above), nothing
interesting can happen in terms of energy (heat or work) exchange as
constrained by zeroth law. Thus, the local system will remain unaltered. This
is also true the traditional heat engines operating in a four-step Carnot
cycle. The second argument can be placed as follows. Here we implicitly assume
baths are considerably larger than the working system, and the working system
returns to its initial state at the end of each cycle. Let say, if the hot
bath $B_{1}$ releases $\Delta S$ amount of entropy then an associated energy
needs to flow out from the hot bath is given by $\Delta E_{1}$, where $\Delta
S=\beta_{1}\Delta E_{1}$. Since baths and systems together form an isolated
composite, the same amount of entropy $\Delta S$ has to be absorbed by the
cold bath because the entire process is strictly entropy conserving. This will
increase the energy by an amount $\Delta E_{2}$, where $\Delta
S=\beta_{2}\Delta E_{2}$. The entropy conservation implies $\beta_{1}\Delta
E_{1}+\beta_{2}\Delta E_{2}=0$, which is exactly the condition demanded by the
commutation relation (3). Note this argument above is based on the notion of
average entropy (Gibbs or von Neumann entropy), and it does not precisely
capture the notion of entropy in the one-shot finite-size regime. To
understand how the relation (3) ensures strict conservation of total entropy
in the one-shot regime, we need to turn to the microscopic picture based on
how degeneracy in the micro-canonical ensembles changes due to an energy
transfer from the hot to the cold composites. We discuss that in the Appendix.
The SLTOs satisfy several interesting properties and can be found in Bera  _et
al._ (2021).

Figure 1: One-step engine cycle. An engine consists of two baths $B_{1}$ and
$B_{2}$ at inverse temperatures $\beta_{1}$ and $\beta_{2}$
($\beta_{1}<\beta_{2}$), a working system $S_{12}\equiv S_{1}S_{2}$ and a
battery $S_{W_{12}}\equiv S_{W_{1}}S_{W_{2}}$. In each one-step engine cycle,
the composite $S_{1}S_{W_{1}}-S_{W_{2}}S_{2}$ semi-locally interact with the
baths $B_{1}-B_{2}$ and undergoes a transformation so that the working sub-
systems $S_{1}$ and $S_{2}$ swaps their states along with Hamiltonians (as
indicated by the arrows) and the battery sub-systems update their states. As a
result, there is an overall flow of heat from $B_{1}$ to $B_{2}$ and, in this
process, part of that heat is converted into work and stored in
$S_{W_{1}}S_{W_{2}}$. At the end of each cycle, the state of $S_{1}S_{2}$
becomes identical to its initial state upto a swap operation and is, again,
reused in the next cycles. See text for more details.

In an engine operating in cycles, the system $S_{12}$ mediates the heat
transfer from $B_{1}$ to $B_{2}$, while a part of that is converted into work
and stored in the battery $S_{W_{12}}$. At the end of each cycle, the $S_{12}$
should recover its initial state so it can be reused for the next cycle. But
the battery gets excited to a higher energy eigenstate to store work.
Interestingly, the engine executes this transformation in a one-step cycle
(see Figure 1) by implementing semi-local thermal operations on
$S_{12}S_{W_{12}}$, as

$\displaystyle\left(\rho_{S_{12}}\otimes\rho_{S_{W_{12}}}^{i},\
H_{S_{12}}+H_{S_{W_{12}}}\right)\rightarrow\left(\sigma_{S_{12}}^{\prime}\otimes\rho_{S_{W_{12}}}^{f},\
H_{S_{12}}^{\prime}+H_{S_{W_{12}}}\right).$

Consequently, the state of the working system transforms as
$\rho_{S_{12}}\to\sigma^{\prime}_{S_{12}}$ and at the same time the
Hamiltonian is modified as $H_{S_{12}}=H_{S_{1}}+H_{S_{2}}\to
H_{S_{12}}^{\prime}=H_{S_{1}}^{\prime}+H_{S_{2}}^{\prime}$. Further, it
satisfies the cyclicity conditions
$\sigma_{S_{12}}^{\prime}=U_{S_{1}\leftrightarrow
S_{2}}\left(\rho_{S_{12}}\right),\ \ H_{S_{1}}^{\prime}=H_{S_{2}},\ \
\mbox{and}\ \ H_{S_{2}}^{\prime}=H_{S_{1}}$, where the unitary
$U_{S_{1}\leftrightarrow S_{2}}$ swaps the states of the subsystems $S_{1}$
and $S_{2}$. The battery undergoes the transformation
$\rho_{S_{W_{12}}}^{i}\to\rho_{S_{W_{12}}}^{f}$ without updating its
Hamiltonian.

To understand how the above transformation executes the (four-step) Carnot
cycle in one-step, let us focus on the transformation happening in the system,
that is $(\rho_{S_{12}},H_{S_{12}})\to(\sigma_{S_{12}}^{\prime},\
H_{S_{12}}^{\prime})$. For this purpose, we ignore the battery as it only
changes states without updating its Hamiltonians and thereby stores or
releases work. Consider, $\rho_{S_{12}}=\rho\otimes\sigma$ and
$H_{S_{12}}=H+H^{\prime}$, where $H$ and $H^{\prime}$ are the Hamiltonians of
the subsystems $S_{1}$ and $S_{2}$ respectively. Then the (one-step) engine
operation leads to

$\displaystyle(\rho\otimes\sigma,\ H+H^{\prime})\to(\sigma\otimes\rho,\
H^{\prime}+H).$ (4)

This involves two simultaneous sub-transformations. One is $(\rho,\
H)\to(\sigma,\ H^{\prime})$ via a semi-local interaction with $B_{1}$, which
can be understood as the combination of an isothermal $(\rho,\ H)\to(\sigma,\
H)$ and then an adiabatic $(\sigma,\ H)\to(\sigma,\ H^{\prime})$
transformations. The other sub-transformation $(\sigma,\ H^{\prime})\to(\rho,\
H)$ takes place in semi-local interaction with the bath $B_{2}$, which again
can be understood as the combination of an isothermal $(\sigma,\
H^{\prime})\to(\rho,\ H^{\prime})$ and then an adiabatic $(\rho,\
H^{\prime})\to(\rho,\ H)$ transformations. Clearly, this mimics the situation
of a Carnot engine where one working system initially in $(\rho,H)$ undergoes
two isothermal (in interaction with two different baths) and two adiabatic
transformations, but in one-step (see Bera  _et al._ (2021) for more details).

##  III Maximum power with Carnot efficiency

The engines equipped with SLTOs can yield better performance than the
traditional heat engines. Not only can the engines execute the Carnot cycle in
one step, but they are also superior to conventional heat engines in
efficiency and power. Most importantly, these engines can deliver maximum
power with Carnot efficiency. Note, to attain maximum power and efficiency
simultaneously, the engine has to undergo the fastest possible
thermodynamically reversible transformation in each cycle, which we are going
to demonstrate below.

Without loss of generality, we consider the working subsystems $S_{1}$ and
$S_{2}$ are to be qubits with the Hamiltonians $H_{S_{1}}=a|1\rangle\\!\langle
1|_{S_{1}}$ and $H_{S_{2}}=a|1\rangle\\!\langle 1|_{S_{2}}$ respectively
having identical energy spacing. We also assume, without loss of generality,
that the battery subsystems $S_{W_{1}}$ and $S_{W_{2}}$ are qubits with the
Hamiltonians $H_{S_{W_{1}}}=E_{W_{1}}|1\rangle\\!\langle 1|_{S_{W_{1}}}$ and
$H_{S_{W_{2}}}=E_{W_{2}}|1\rangle\\!\langle 1|_{S_{W_{2}}}$ respectively. The
maximum heat-to-work conversion efficiency per (one-step) cycle is attained by
implementing a thermodynamically reversible state transformation in
$S_{1}S_{2}S_{W_{1}}S_{W_{2}}$ composite

$\displaystyle|0,1,0,0\rangle_{S_{1}S_{2}S_{W_{1}}S_{W_{2}}}\to|1,0,1,1\rangle_{S_{1}S_{2}S_{W_{1}}S_{W_{2}}},$
(5)

using a semi-local thermal operation Bera  _et al._ (2021), where the
subsystems $S_{1}$ and $S_{2}$ swap their states without changing the
Hamiltonians, and the batteries $S_{W_{1}}$ and $S_{W_{2}}$ get excited. Here
we denote
$|i,j,k,l\rangle_{S_{1}S_{2}S_{W_{1}}S_{W_{2}}}=|i\rangle_{S_{1}}\otimes|j\rangle_{S_{2}}\otimes|k\rangle_{S_{W_{1}}}\otimes|l\rangle_{S_{W_{2}}}$.

For simplicity, we may consider the working system and the battery to be the
parts of single system $S\equiv S_{1}S_{2}S_{W_{1}}S_{W_{2}}$ with the
Hamiltonian $H_{S}=a_{0}|0\rangle\\!\langle 0|_{S}+a_{1}|1\rangle\\!\langle
1|_{S}$, where $a_{0}=a$ and $a_{1}=a+E_{W_{1}}+E_{W_{2}}$ with the
corresponding energy eigenstates
$|0\rangle_{S}=|0,1,0,0\rangle_{S_{1}S_{2}S_{W_{1}}S_{W_{2}}}$ and
$|1\rangle_{S}=|1,0,1,1\rangle_{S_{1}S_{2}S_{W_{1}}S_{W_{2}}}$. Then, the
engine becomes compact and has three constituents; hot and cold baths ($B_{1}$
and $B_{2}$) and a two-level system ($S$). The engine cycle starts with the
initial state $|0\rangle_{S}$ and ends with the final state $|1\rangle_{S}$ of
$S$ (as shown in Figure 2). The corresponding global transformation, leading
to this one-step cycle, is

$\displaystyle\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes|0\rangle\\!\langle
0|_{S}\xrightarrow{U}\tau_{B_{1}B_{2}}\otimes|1\rangle\\!\langle 1|_{S},$ (6)

where $\tau_{B_{1}B_{2}}$ is final state of the baths. The unitary $U$
strictly conserves energy of $B_{1}B_{2}S$ composite and weighted-energy of
$B_{1}B_{2}$ composite, i.e.,

$\displaystyle\left[U,\ H_{B_{1}}+H_{B_{2}}+H_{S}\right]=0,$ (7)
$\displaystyle\left[U,\ \beta_{1}H_{B_{1}}+\beta_{2}H_{B_{2}}\right]=0.$ (8)

The commutation relation (7) ensures the strict conservation of total energy.
The relation (8) ensuring strict conservation of entropy is a special case of
the general condition (3) where the transformation is cyclic. Here the system
$S$ remains in the energy eigenstates before and after the cycle without
changing its energy and entropy. Because of that, the entropy conservation is
guaranteed by the strict weighted-energy conservation of the baths only. See
Appendix for more details. Note, the semi-local nature of the evolution is
clearly understood here as the system $S$ is simultaneously interacting with
both the baths via the unitary $U$.

Figure 2: A compact engine. An engine consists of two baths $B_{1}$ and
$B_{2}$ at inverse temperatures $\beta_{1}$ and $\beta_{2}$ respectively, and
a working system $S$. The system $S$ simultaneously interacts with both the
baths via a semi-local thermal operation. The engine operates in a one-step
cycle by exciting the system $S$ from lower energy to a higher energy
eigenstate. See text for more details.

Now we show that the unitary $U$, that satisfies relations (7) and (8),
leading to the transformation (6) indeed attains Carnot efficiency. Then we
consider constructing a driving Hamiltonian, corresponding to the unitary $U$,
that delivers maximum power with Carnot efficiency.

Since the initial (and final) state of $S$ is an energy eigenstate, the global
initial state of $B_{1}B_{2}S$ can be expressed in the block-diagonal form
with respect to total energies, as

$\displaystyle\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes|0\rangle\\!\langle
0|_{S}=\bigoplus_{E_{B_{12}}+a_{0}}[\gamma_{B_{1}}\otimes\gamma_{B_{2}}]_{E_{B_{12}}}\otimes|0\rangle\\!\langle
0|_{S},$ (9)

with $E_{B_{12}}=E_{B_{1}}+E_{B_{2}}$, where $E_{B_{1}}$ and $E_{B_{2}}$ are
the energies corresponding to the baths $B_{1}$ and $B_{2}$ respectively, and

$\displaystyle[\gamma_{B_{1}}\otimes\gamma_{B_{2}}]_{E_{B_{12}}}=p(E_{B_{12}})\sum_{i=1}^{d_{1}({E_{B_{1}}})d_{2}({E_{B_{2}}})}|E_{B_{12}}(i)\rangle\\!\langle
E_{B_{12}}(i)|,$

where $d_{1}({E_{B_{1}}})$ and $d_{2}({E_{B_{2}}})$ represent the degeneracies
corresponding to the bath energies $E_{B_{1}}$ and $E_{B_{2}}$ respectively
and
$p(E_{B_{12}})=e^{-\beta_{1}E_{B_{1}}-\beta_{2}E_{B_{2}}}/Z_{B_{1}}Z_{B_{2}}$.
A strictly total energy conserving unitary also takes a block-diagonal form,
$U=\bigoplus_{E_{B_{12}+a_{0}}}U_{E_{B_{12}+a_{0}}}$, where the unitary
$U_{E_{B_{12}+a_{0}}}$ operates only on the block with the total energy
$E_{B_{12}}+a_{0}$ and implements a transformation

$\displaystyle[\gamma_{B_{1}}\otimes\gamma_{B_{2}}]_{E_{B_{12}}}\otimes|0\rangle\\!\langle
0|_{S}\to[\tau_{B_{1}B_{2}}]_{E^{\prime}_{B_{12}}}\otimes|1\rangle\\!\langle
1|_{S},$ (10)

where $E_{B_{12}}^{\prime}=E_{B_{1}}^{\prime}+E_{B_{2}}^{\prime}$. Note,
$E_{B_{12}}+a_{0}=E_{B_{12}}^{\prime}+a_{1}$ as required by the total energy
conservation. The strict conservation of total weighted-energy of the baths
ensures

$\displaystyle\beta_{1}(E_{B_{1}}^{\prime}-E_{B_{1}})+\beta_{2}(E_{B_{2}}^{\prime}-E_{B_{2}})=\beta_{1}Q_{1}+\beta_{2}Q_{2}=0,$
(11)

where $Q_{1}$ and $Q_{2}$ are the heat flow out of the baths $B_{1}$ and
$B_{2}$ respectively. The (11) is nothing but the Clausius equality. This in
turn ensures the thermodynamic reversibility of the state transformation.
Note, this condition also implies
$d_{1}(E_{B_{1}})d_{2}(E_{B_{2}})=d_{1}(E_{B_{1}}^{\prime})d_{2}(E_{B_{2}}^{\prime})$,
where $d_{1}(E_{B_{1}}^{\prime})$ and $d_{2}(E_{B_{2}}^{\prime})$ are
degeneracies in the energies corresponding to $E_{B_{1}}^{\prime}$ and
$E_{B_{2}}^{\prime}$ of the baths $B_{1}$ and $B_{2}$ respectively (see
Appendix for more details). It is an essential requirement for a unitary
transformation where the rank and the spectra of the (un-normalized) state of
each total energy block remain unchanged.

Similar transformations, as in Eq. (10), also take place in all other total
energy blocks due to the evolution by the unitary $U$. As a result, the
desired state transformation, given in Eq. (6), is achieved and thereby
completes the one-step engine cycle. The extracted work per cycle is given by
$W_{ext}=a_{1}-a_{0}=Q_{1}+Q_{2}$ as a consequence of strict conservation of
total energy. Hence, the heat-to-work conversion efficiency becomes maximum in
the one-shot finite-size regime, given by

$\displaystyle\eta=\frac{W_{ext}}{Q_{1}}=1-\frac{\beta_{1}}{\beta_{2}},$ (12)

which is the Carnot efficiency, as expected for any reversible engine cycle.

Let us demonstrate how the global unitary $U$ can be implemented using an
interaction Hamiltonian

$\displaystyle H_{in}=\hbar
g\bigoplus_{E_{B_{12}}+a_{0}}\sum_{i=1}^{d_{1}(E_{B_{1}})d_{2}(E_{B_{2}})}|E_{B_{12}}^{\prime}(i),1\rangle\\!\langle
E_{B_{12}}(i),0|_{B_{1}B_{2}S}+h.c.,$

where, again, $|E_{B_{12}}(i)\rangle\equiv|E_{B_{1}}(i),E_{B_{2}}(i)\rangle$
and
$|E_{B_{12}}^{\prime}(i)\rangle\equiv|E_{B_{1}}^{\prime}(i),E_{B_{2}}^{\prime}(i)\rangle$,
and $g$ is a constant. The global unitary is then $U(t)=e^{-itH_{in}/\hbar}$
for any time $t$. Under this unitary, an initial state
$|E_{B_{1}}(i),E_{B_{2}}(i),0\rangle_{B_{1}B_{2}S}$ in the total energy block
evolves to
$|\psi(t)\rangle=U(t)|E_{B_{1}}(i),E_{B_{2}}(i),0\rangle_{B_{1}B_{2}S}$ at
time $t$, where

$\displaystyle|\psi(t)\rangle=\cos(gt)\ |$ $\displaystyle E_{B_{1}}(i),\
E_{B_{2}}(i),\ 0\rangle_{B_{1}B_{2}S}$ $\displaystyle-i\sin(gt)\
|E_{B_{1}}^{\prime}(i),\ E_{B_{2}}^{\prime}(i),\ 1\rangle_{B_{1}B_{2}S},$ (13)

which is a genuinely entangled state of $B_{1}$, $B_{2}$, and $S$ for $gt\neq
z\pi/2$ with $z\in\mathbb{Z}$. The desired final state is attained at time
$\tau=\pi/(2g)$, where all the constitutes become uncorrelated from each
other. It is important to highlight that the above engine evolution enables a
coherent heat transfer from $B_{1}$ to $B_{2}$, which happens due to
entanglement in the intermediate time and is fundamentally different from
conventional engines. Similar evolution takes place in every total energy
block, and the overall transformation (6) is attained at time $\tau$. With
this, the engine extracts $W_{ext}$ work in $\tau$ time. Thus, the power
delivered by the engine, i.e., work extraction per unit time, is

$\displaystyle P=\frac{W_{ext}}{\tau}=\frac{2gW_{ext}}{\pi}.$ (14)

Figure 3: Geodesic trajectory of evolution. Evolution of states on a quantum
state space (the gray area). An initial state $|\psi(0)\rangle$ at time $t=0$
is evolved to $|\psi(t)\rangle$ at time $t=t$. There are infinitely many paths
connecting the states. One with the shortest path is called the geodesic path
(solid (blue) line). Any other path (dotted (red) line) is longer than the
geodesic one.

Contrary to the traditional understanding, the reversible (one-step) engine
cycle via semi-local thermal operation requires finite time. Not only that, as
we argue below, the interaction Hamiltonian $H_{in}$ drives the evolution with
the maximum attainable speed to result in the shortest possible transformation
time. The speed of evolution is defined by the distance traversed by a system
per unit time in its quantum state space Anandan and Aharonov (1990). For pure
states, the distance is measured using Fubini-Study metric, given by
$s=\frac{1}{2}(1-|\langle\psi|\phi\rangle|^{2})$ for any two states
$|\psi\rangle$ and $|\phi\rangle$. The speed of evolution of the state
$|\psi(t)\rangle$ is

$\displaystyle v=\frac{ds}{dt}=\frac{\Delta H_{in}}{\hbar}=g,$ (15)

where $ds=\frac{1}{2}(1-|\langle\psi(t)|\psi(t+dt)\rangle|^{2})$ and energy
uncertainty $\Delta
H_{in}=\sqrt{\langle\psi(t)|H_{in}^{2}|\psi(t)\rangle-\langle\psi(t)|H_{in}|\psi(t)\rangle^{2}}$.
Note, the speed of evolution $v$ is same for every total energy block and,
hence, for the overall transformation. The uncertainty $\Delta H_{in}$ is
independent of time and has the maximum possible value, equals to $\hbar g$
for any driving Hamiltonian bounded by the operator norm $\hbar g$.
Furthermore, the interaction Hamiltonian $H_{in}$ drives the evolution
following a geodesic trajectory Anandan and Aharonov (1990) connecting the
initial and the final state which represents the shortest path (see Figure 3).
The evolution following shortest path with maximum speed results in the
minimum required time to complete the transformation in the one-step engine
cycle. As a consequence, the power $P$ in Eq. (14) is the maximum possible
one. Note, quantum effects such as entanglement are believed to degrade the
performance of engines. But, on the contrary, here we find that the engines
operating with semi-local thermal operations can exploit entanglement to
deliver maximum power with Carnot efficiency.

##  IV A quantum optics based heat engine

Here we discuss a physically realizable quantum heat engine transferring
maximum power with Carnot efficiency following the theoretical framework
presented above. We propose an engine composed of two thermal cavities and a
three-level working system (see Figure 4). The bath $B_{1}$ is a single-mode
optical cavity with a Hamiltonian
$H_{B_{1}}=\hbar\omega_{1}a_{1}^{\dagger}a_{1}=\sum_{n}n\omega_{1}|n\rangle\\!\langle
n|_{B_{1}}$, at inverse temperature $\beta_{1}$. Here $a_{1}^{\dagger}$ and
$a_{1}$ are the creation and annihilation operators of the mode in $B_{1}$
respectively, $\omega_{1}$ represents the mode frequency, and $n$ and
$|n\rangle$ are the number of excitation and the corresponding number state.
Similarly, the bath $B_{2}$ at inverse temperature $\beta_{2}$ is another
optical cavity with a Hamiltonian
$H_{B_{2}}=\hbar\omega_{2}a_{2}^{\dagger}a_{2}=\sum_{m}m\omega_{2}|m\rangle\\!\langle
m|_{B_{2}}$. The system $S$ is a three-level atom (in $\Lambda$-configuration)
with the Hamiltonian $H_{S}=\sum_{i=1}^{3}E_{i}|i\rangle\\!\langle i|_{S}$
with $E_{1}=0$. The overall Hamiltonian of the baths and the system composite
is then $H_{0}=H_{B_{1}}+H_{B_{2}}+H_{S}$.

A semi-local thermal operation, leading to a one-step cycle, is implemented by
introducing an intensity-dependent coupling between the bath modes and the
atom by the interaction Hamiltonian

$\displaystyle H_{I}=f_{1}(N_{1})+$ $\displaystyle f_{2}(N_{2})+\hbar
g_{1}\theta_{1}(N_{1})(a_{1}\sigma_{31}+h.c.)$ $\displaystyle+\hbar
g_{2}\theta_{2}(N_{2})(a_{2}\sigma_{32}+h.c.),$ (16)

with the number operator $N_{k}=a_{k}^{\dagger}a_{k}$ corresponds to the bath
$B_{k}$ for $k=1,2$, and $\sigma_{ij}=|i\rangle\\!\langle j|_{S}$ ($i\neq j$)
is the transition operator from $|j\rangle_{S}$ to $|i\rangle_{S}$ for
$i,j=1,2,3$. The $f_{1}(N_{1})$ and $f_{2}(N_{2})$ are some intensity-
dependent potentials in the cavity fields. The state $|3\rangle_{S}$ is
coupled with $|1\rangle_{S}$ and $|2\rangle_{S}$ via intensity-dependent
dipole-couplings $g_{1}\theta(N_{1})$ and $g_{2}\theta(N_{2})$ respectively,
where $\theta(N_{1})$ and $\theta(N_{2})$ are some functions of the number
operators, and $g_{1}$ and $g_{2}$ are some constants. There is no direct
coupling between $|1\rangle_{S}$ and $|2\rangle_{S}$. The technical details of
how the interaction Hamiltonian (16) may be realized are described in
Appendix.

With the choice of (identical) detuning
$\Delta=(E_{3}-E_{k})/\hbar-\omega_{k}$ and the couplings

$\displaystyle\frac{g_{k}^{2}}{\Delta}\theta_{k}^{2}(N_{k})=f_{k}(N_{k})=\frac{g_{k}^{2}}{\Delta}N_{k}^{-1},$
(17)

for $k=1,\ 2$. The above formula should be fulfilled possibly exactly for
large values of $N_{k}$, especially if we work at relatively high
temperatures. It has to be regularized, obviously, for $N_{k}=0$ to avoid the
singularity. Nevertheless, with these choices, the three-level problem can be
exactly reduced to a two-level problem irrespective of whether the detuning
$\Delta$ is small or large, similar to what is shown in Gerry and Eberly
(1990); Wu (1996); Greentree  _et al._ (2013). Then, the corresponding two-
level Hamiltonian becomes
$H_{S}^{\prime}=\frac{1}{2}\hbar\omega_{0}(|2\rangle\\!\langle
2|-|1\rangle\\!\langle 1|)$, and the interaction Hamiltonian, after rotating-
wave approximation, transforms to

$\displaystyle H_{I}^{\prime}=\hbar
g(A_{1}A_{2}^{\dagger}\sigma_{12}+A_{1}^{\dagger}A_{2}\sigma_{21}),$ (18)

where $A_{k}=a_{k}N_{k}^{-1/2}$ and $g=g_{1}g_{2}/\Delta$ is a constant. Here
$\omega_{0}\approx\omega_{1}-\omega_{2}$, i.e., the pump mode with
$\omega_{1}$ and Stokes mode with $\omega_{2}$ are in two-mode resonance with
the states $|1\rangle_{S}$ and $|2\rangle_{S}$. The $\omega_{1}$ and
$\omega_{2}$ are chosen so that $\beta_{1}\omega_{1}=\beta_{2}\omega_{2}$.
Then, the unitary $U(t)=\exp[-itH_{I}^{\prime}/\hbar]$ generated by
$H_{I}^{\prime}$ strictly conserves total energy of baths and system, and
total weighted-energy of the baths alone, as $[U(t),\
H_{S}^{\prime}+H_{B_{1}}+H_{B_{2}}]=0$ and $[U(t),\
\beta_{1}H_{B_{1}}+\beta_{2}H_{B_{2}}]=0$ respectively for all time $t$.

Figure 4: An optical-cavity based quantum heat engine. (a) A three-level
quantum system $S$ (i.e., working system) is placed in three overlapping
optical cavities. Thermal cavities with frequencies $\omega_{1}$ and
$\omega_{2}$ at inverse temperatures $\beta_{1}$ and $\beta_{2}$
($\beta_{1}<\beta_{2}$) represent the baths $B_{1}$ and $B_{2}$ respectively.
The cavity with frequency $\omega_{0}$ is in resonance with the transition
between the ground state and the first excited state of the system. (b) The
three-level system $S$ simultaneously interacts with baths $B_{1}$ and $B_{2}$
via two-mode amplitude-dependent coupling. In each engine cycle, it absorbs a
photon with energy $\hbar\omega_{1}$ from $B_{1}$ and emits a photon with
$\hbar\omega_{2}$ energy to the bath $B_{2}$ and excites itself from energy
$E_{1}$ to $E_{2}$. The system then emits a photon with energy
$\hbar\omega_{0}$ via stimulated emission in a cavity (green) in resonance
with the transition. See text for more details.

The initial state of the composite $B_{1}B_{2}S$ can now be expressed in
blocks classified by $(n,m)$, as

$\displaystyle\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes|1\rangle\\!\langle
1|_{S}=\bigoplus_{n,m}p_{nm}\ |n,m,1\rangle\\!\langle n,m,1|_{B_{1}B_{2}S},$

with
$p_{nm}=\exp[-\beta_{1}nE_{B_{1}}-\beta_{2}mE_{B_{2}}]/Z_{B_{1}}Z_{B_{2}}$,
where $Z_{B_{1}}$ and $Z_{B_{2}}$ are the partition functions of the baths
$B_{1}$ and $B_{2}$ respectively. Due to the constraints on strict total
energy conservation, the $U(t)$ operates on each block $(n,m)$ independently.
For a block $(n,m)$, the initial state $|n,m,1\rangle_{B_{1}B_{2}S}$ evolves
to $|\phi(t)\rangle=U(t)|n,m,1\rangle_{B_{1}B_{2}S}$ at some time $t$, and it
is given by

$\displaystyle|\phi(t)\rangle=\cos(gt)\ $
$\displaystyle|n,m,1\rangle_{B_{1}B_{2}S}$ $\displaystyle-i\sin(gt)\
|n-1,m+1,2\rangle_{B_{1}B_{2}S},$ (19)

which is an entangle state. This is true for all blocks $(n,m)$, except the
blocks $(0,m)$. Note, the time taken to evolve the initial state to the
desired final states are same for all blocks except $(0,\ m)$, and that is
$\tau=\pi/(2g)$. As a consequence, the joint initial state of $B_{1}B_{2}S$
evolves to
$\rho_{B_{1}B_{2}S}^{f}=U(t)(\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes|0\rangle\\!\langle
0|_{S})U(t)^{\dagger}$, and at time $\tau=\pi/(2g)$, the final state of the
system $S$ becomes,

$\displaystyle\rho_{S}^{f}={\operatorname{Tr\,}}_{B_{1}B_{2}}\
\rho_{B_{1}B_{2}S}^{f}=\frac{1}{Z_{B_{1}}}|1\rangle\\!\langle
1|_{S}+\left(1-\frac{1}{Z_{B_{1}}}\right)|2\rangle\\!\langle
2|_{S}\to|2\rangle\\!\langle 2|_{S}$

for $Z_{B_{1}}\to\infty$, which is true for low inverse temperature
$\beta_{1}$ of bath $B_{1}$, or $|B_{1}|\to\infty$. In each cycle, the the
system $S$ undergoes the transformation $|1\rangle_{S}\to|2\rangle_{S}$ and
thereby extracts $\hbar\omega_{0}$ amount of work with the Carnot efficiency
$\eta=1-\beta_{1}/\beta_{2}$. The transformation takes place with the maximum
quantum speed following a geodesic trajectory and time requires for that is
$\tau=\pi/(2g)$. Hence, the cycle delivers maximum power
$P=2g\hbar\omega_{0}/\pi$. The work is extracted in the form of photons at
$\omega_{0}$ by placing the atom in a resonant cavity and letting the
stimulated emission $|2\rangle_{S}\to|1\rangle_{S}$ (see Fig. 4).

It is worth mentioning that there have been several propositions of quantum
heat engines based on optical cavity or bosonic baths earlier, for example in
Scovil and Schulz-DuBois (1959); Ghosh  _et al._ (2018), where a quantum
system interacts with two bosonic thermal baths at different temperatures.
However, in contrast to the engines considered above, these engines only allow
incoherent heat transfer from hot to cold baths. They do not guarantee strict
conservation of total energy in order to characterize the energetics
correctly. Because of that, they cannot deliver maximum power with Carnot
efficiency.

##  V Discussion

For finite-time classical engines, it is known that the maximum power at
maximum heat-to-work conversion efficiency is impossible Curzon and Ahlborn
(1975). For quantum engines, where the working systems interacting with the
baths are quantum mechanical, the situation is quite different because the
quantum uncertainties present in the system further delimit the extractable
work in each cycle. For finite-time quantum heat engines considered earlier,
there are various trade-off relations between power and efficiency Pietzonka
and Seifert (2018); Brandner and Saito (2020), and both of these quantities
cannot be maximized simultaneously. So far, there are two kinds of quantum
engines that have been considered in the literature. In the first kind, the
working systems are often composed of a large number of quantum particles. In
the second kind, the working system comprises few quantum particles but is
allowed to be observed or measured repeatedly for an arbitrarily large number
of times. However, the working system interacts with the hot and cold baths in
different steps in both kinds. Hence the heat flow from the hot to the cold
baths is not continuous; rather, it occurs in different steps or in an
incoherent manner. And, possibly because of this feature of the engines, power
and efficiency satisfy trade-off relations and cannot be maximized
simultaneously.

The quantum engines considered here can deliver maximum power with maximum
efficiency and are fundamentally different from conventional ones studied
earlier. Firstly, the engine operates in the one-shot finite-size regime,
where the working system is genuinely quantum in the sense that it is made up
of a small number of quantum particles (i.e., of finite-size) and allows one
or few observations or measurements (i.e., one-shot measurement). Secondly,
the working system interacts with both hot and cold baths simultaneously via a
semi-local thermal operation. These operations are powerful compared to the
operations in traditional engines as they can implement a one-step engine
cycle and create entanglement between the baths and the working system.
Because of that, it enables a coherent flow of heat from hot to cold bath via
the working system and results in maximum power with maximum efficiency. With
this, our results have demonstrated that there, in principle, does not exist a
fundamental trade-off relation between power and efficiency. We have also put
forward an experimentally feasible quantum heat engine operating in the one-
shot finite-size regime with a three-level atom as a working system and two
thermal optical cavities as the baths. We have explicitly introduced an
intensity-dependent interaction between the atom and cavities that executes
the one-step engine cycle yielding maximum power at Carnot efficiency.

In summary:

  * •

We have introduced quantum heat engines that operate via one-step cycles in
the one-shot finite-size regime and enable a coherent heat transfer from hot
to cold baths by establishing genuine quantum entanglement between the working
system and the baths.

  * •

We have demonstrated that there is no fundamental trade-off relation between
power and efficiency.

  * •

We have shown a general protocol with which a quantum heat engine can deliver
maximum power with Carnot efficiency in the one-shot finite-size regime.

  * •

We have proposed a physically realizable model of such a quantum heat engine
based on an atom-cavity system.

  * •

Our work opens up avenues for an improved theoretical understanding of
thermodynamics in the quantum regime and new possibilities for quantum-enabled
technologies using heat engines.

Acknowledgments  – M.L.B., S.J.-F., and M.L. thankfully acknowledge support
from ERC AdG NOQIA, Agencia Estatal de Investigación (“Severo Ochoa” Center of
Excellence CEX2019-000910-S, Plan National FIDEUA
PID2019-106901GB-I00/10.13039/501100011033, FPI), Fundació Privada Cellex,
Fundació Mir-Puig, and from Generalitat de Catalunya (AGAUR Grant No. 2017 SGR
1341, CERCA program, QuantumCAT_U16-011424, co-funded by ERDF Operational
Program of Catalonia 2014-2020), MINECO-EU QUANTERA MAQS (funded by State
Research Agency (AEI) PCI2019-111828-2/10.13039/501100011033), EU Horizon 2020
FET-OPEN OPTOLogic (Grant No 899794), and the National Science Centre, Poland-
Symfonia Grant No. 2016/20/W/ST4/00314. M.N.B. gratefully acknowledges
financial supports from SERB-DST (CRG/2019/002199), Government of India.

##  Appendix A Properties of large baths

Our formalism exploits some useful properties of the baths that are
considerably large compared to the working systems under consideration. That
is why a bath always remains in thermal equilibrium at a fixed temperature
even after it interacts with a system. A heat engine has two baths and a
working body as components. Hence, a bath $B_{x}$ with Hamiltonian $H_{B_{x}}$
is expected to remain in the Gibb’s state
$\gamma_{B_{x}}=\frac{e^{-\beta_{x}H_{B_{x}}}}{{\operatorname{Tr\,}}[e^{-\beta_{x}H_{B_{x}}}]}$
always, with inverse temperature $\beta_{x}$. Now for two baths $B_{1}$ and
$B_{2}$, with corresponding Hamiltonians $H_{B_{1}}$ and $H_{B_{2}}$
respectively, the combined state can be expressed as

$\displaystyle\gamma_{B_{12}}=\gamma_{B_{1}}\otimes\gamma_{B_{2}}.$ (20)

In general, the baths and the working body have Hamiltonians bounded from
below, having the lowest energy to be zero. But, the baths may have
considerably large energy, i.e., $E^{max}_{B_{x}}\rightarrow\infty$. Although
the combined baths are probabilistic mixtures of all energy states, there
exists a set of total energies $\mathcal{E}_{B_{12}}$ where the baths can be
found with high probability. Mathematically, it means

$\displaystyle{\operatorname{Tr\,}}[P_{\mathcal{E}_{B_{12}}}\gamma_{B_{12}}]\geqslant
1-\delta,$ (21)

where $P_{\mathcal{E}_{B_{12}}}$ is a projector that spans over the space
corresponding to the set $\mathcal{E}_{B_{12}}$ and $\delta>0$. With this, the
properties of the combined baths are listed below (cf. Bera  _et al._ (2021)).

  * •

For any energy $E_{B_{12}}=E_{B_{1}}+E_{B_{2}}\in\mathcal{E}_{B_{12}}$
with$E_{B_{1}}$ and $E_{B_{2}}$ are the energies of the baths $B_{1}$ and
$B_{2}$ respectively, there is a peak around the mean value $\langle
E_{B_{12}}\rangle$, as $E_{B_{12}}\in\left\\{\langle
E_{B_{12}}\rangle-O(\sqrt{E_{B_{12}}}),\ldots,\langle
E_{B_{12}}\rangle+O(\sqrt{E_{B_{12}}})\right\\}$.

  * •

For any energy $E_{B_{12}}=E_{B_{1}}+E_{B_{2}}\in\mathcal{E}_{B_{12}}$,
$E_{S_{1}}\ll E_{B_{1}}$, and $E_{S_{1}}^{\prime}\ll E_{B_{1}}$, there exists
a
$E_{B_{12}}^{\prime}=E_{B_{1}}^{\prime}+E_{B_{2}}^{\prime}\in\mathcal{E}_{B_{12}}$
so that $E_{B_{1}}+E_{S_{1}}=E_{B_{1}}^{\prime}+E_{S_{1}}^{\prime}$ and
$E_{B_{2}}+E_{S_{2}}=E_{B_{2}}^{\prime}+E_{S_{2}}^{\prime}$, where
$E_{S_{2}}\ll E_{B_{2}}$ and $E_{S_{2}}^{\prime}\ll E_{B_{2}}$.

  * •

For any energy $E_{B_{1}}+E_{B_{2}}\in\mathcal{E}_{B_{12}}$, the degeneracy
$g_{B}(E_{B_{1}}+E_{B_{2}})$ scales exponentially with energy, and it
satisfies the relation $g_{B}(E_{B_{1}}^{\prime}+E_{B_{2}}^{\prime})\approx
g_{B}(E_{B_{1}}+E_{B_{2}})\ e^{\beta_{1}E_{S_{1}}+\beta_{2}E_{S_{2}}}$, where
$E_{B_{1}}^{\prime}=E_{B_{1}}+E_{S_{1}}$ and
$E_{B_{2}}^{\prime}=E_{B_{2}}+E_{S_{2}}$ with $E_{S_{1}}\ll E_{B_{1}}$ and
$E_{S_{2}}\ll E_{B_{2}}$.

##  Appendix B Reversible Engine Operation in a One-step Cycle

Here we reconsider the reversible engine operation, given in the main text
(see Eq. (5)), that yields maximum power with Carnot efficiency. We have
assumed a bipartite working system $S_{12}$ with the Hamiltonian
$H_{S_{12}}=H_{S_{1}}+H_{S_{2}}$ where $H_{S_{1}}=a|1\rangle\\!\langle
1|_{S_{1}}$ and $H_{S_{2}}=a|1\rangle\\!\langle 1|_{S_{2}}$. We have also
assumed a bipartite battery $S_{W_{12}}$ with the Hamiltonian
$H_{S_{W_{12}}}=H_{S_{W_{1}}}+H_{S_{W_{2}}}$, where
$H_{S_{W_{1}}}=E_{W_{1}}|1\rangle\\!\langle 1|_{S_{W_{1}}}$ and
$H_{S_{W_{2}}}=E_{W_{2}}|1\rangle\\!\langle 1|_{S_{W_{2}}}$. The one-step
cycle is executed by implementing a global unitary ($U$) operation on the
baths-system-battery composite leading to the transformation

$\displaystyle\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes\rho_{S_{12}}\otimes\rho^{i}_{S_{W_{12}}}\to\sigma_{B_{1}B_{2}}\otimes\sigma_{S_{12}}\otimes\rho^{f}_{S_{W_{12}}},$
(22)

where $\rho_{S_{12}}=|0\rangle\\!\langle 0|_{S_{1}}\otimes|1\rangle\\!\langle
1|_{S_{2}}$ and $\sigma_{S_{12}}=|1\rangle\\!\langle
1|_{S_{1}}\otimes|0\rangle\\!\langle 0|_{S_{2}}$ are the initial and final
states of the working system $S_{12}$, and
$\rho_{S_{W_{12}}}^{i}=|0\rangle\\!\langle
0|_{S_{W_{1}}}\otimes|0\rangle\\!\langle 0|_{S_{W_{2}}}$ and
$\rho_{S_{W_{12}}}^{f}=|1\rangle\\!\langle
1|_{S_{W_{1}}}\otimes|1\rangle\\!\langle 1|_{S_{W_{2}}}$ are the initial and
final states of the battery $S_{W_{12}}$. Recall, the global unitary $U$
respects strict conservation of total energy and total weighted-energy.
Therefore, we can study the transformation in each total energy block
separately. Consider a block of total energy $E_{1}+E_{2}$, where
$E_{1}=E_{S_{1}}+E_{B_{1}}$ is the sum of energies belonging to $S_{1}$ and
$B_{1}$, and similarly for $E_{2}=E_{S_{2}}+E_{B_{2}}$. In this total energy
block, the transformation becomes

$\displaystyle[\gamma_{B_{1}}\otimes\gamma_{B_{2}}\otimes\rho_{S_{12}}]_{E_{1}+E_{2}}\otimes\rho^{i}_{S_{W_{12}}}\to[\sigma_{B_{1}B_{2}}\otimes\sigma_{S_{12}}]_{E_{1}^{\prime}+E_{2}^{\prime}}\otimes\rho^{f}_{S_{W_{12}}},$
(23)

where $E_{1}^{\prime}=E_{S_{1}}^{\prime}+E_{B_{1}}^{\prime}$ and
$E_{2}^{\prime}=E_{S_{2}}^{\prime}+E_{B_{2}}^{\prime}$. The strict
conservation of the total weighted-energy and the total energy ensure that

$\displaystyle\beta_{1}E_{1}+\beta_{2}E_{2}$
$\displaystyle=\beta_{1}(E_{1}^{\prime}+E_{W_{1}})+\beta_{2}(E_{2}^{\prime}+E_{W_{2}}),$
(24) $\displaystyle E_{1}+E_{2}$
$\displaystyle=E_{1}^{\prime}+E_{2}^{\prime}+E_{W_{1}}+E_{W_{2}},$ (25)

where $E_{1}=E_{B_{1}}$, $E_{2}=E_{B_{2}}+a$,
$E_{1}^{\prime}=E_{B_{1}}^{\prime}+a$, and
$E_{2}^{\prime}=E_{B_{2}}^{\prime}$. Here we have assumed
$\beta_{1}<\beta_{2}$.

Note, similar transformations will follow in the other total weighted-energy
blocks with identical initial and final battery states. The reduced
transformation on the system $S_{12}$,
$\rho_{S_{12}}\rightarrow\sigma_{S_{12}}$ is reversible because all
$\alpha$-free-entropies for pure system and battery states considered here are
$\alpha$ independent Bera  _et al._ (2021). As a consequence, free-entropy
distances satisfy
$S_{d}(\rho_{S_{12}}\rightarrow\sigma_{S_{12}})=S_{d}(\rho_{S_{12}}\leftarrow\sigma_{S_{12}})$,
where

$\displaystyle
S_{d}(\rho_{S_{12}}\to\sigma_{S_{12}})=\beta_{2}a-\beta_{1}a=\beta_{1}E_{W_{1}}+\beta_{2}E_{W_{2}}>0.$
(26)

This relation guarantees that there is strict conservation of weighted-energy
of the working system and the battery together. Therefore, conservation of
total weighted-energy (24) is reduced down to the strict conservation of the
weighted-energy of the baths only, i.e.

$\displaystyle\beta_{1}(E_{B_{1}}-E_{B_{1}}^{\prime})+\beta_{2}(E_{B_{2}}-E_{B_{2}}^{\prime})$
$\displaystyle=\beta_{1}Q_{1}+\beta_{2}Q_{2}=0,$ (27)

where we have identified the heat as the change in energy of the bath $B_{1}$
given by $Q_{1}=E_{B_{1}}-E_{B_{1}}^{\prime}$ and similarly
$Q_{2}=E_{B_{2}}-E_{B_{2}}^{\prime}$ for bath $B_{2}$. This is true for all
energy blocks. The Eq. (27) represents the _Clausius equality_ for the cyclic
process. The other total energy blocks will result in identical Clausius
equality. The net extracted work in each (one-step) engine cycle is given by

$\displaystyle W_{ext}=E_{W_{1}}+E_{W_{2}}=Q_{1}+Q_{2}>0,$ (28)

Here we have used the strict total energy conservation (25). It is clear from
Eqs. (27) and (28) that the heat-to-work conversion is
$\eta_{C}=\frac{W_{ext}}{Q_{1}}=1-\frac{\beta_{1}}{\beta_{2}}$, which is
exactly the Carnot efficiency. Nevertheless for reversible engine
transformation, the global unitary evolution strictly ensures total energy
conservation of $B_{1}B_{2}S_{12}S_{W_{12}}$ and weighted-energy conservation
of $B_{1}B_{2}$, and that are mathematically expressed by the commutation
relations $\left[U,\ H_{B_{1}}+H_{B_{2}}+H_{S}\right]=0$ and $\left[U,\
\beta_{1}H_{B_{1}}+\beta_{2}H_{B_{2}}\right]=0$ respectively, where
$H_{S}=H_{S_{1}}+H_{S_{2}}+H_{S_{W_{1}}}+H_{S_{W_{2}}}$ is the Hamiltonian of
the system $S=S_{12}S_{W_{12}}$.

##  Appendix C Conservation of weighted-energy implies conservation of entropy

To understand the relationship between the conservation of entropy and the
conservation of weighted-energy, we analyze the engine process in terms of the
transformations happening in micro-canonical ensembles. The baths are assumed
to be considerably large compared to the systems and the batteries. Since the
global unitary implementation of the SLTOs strictly satisfy total energy
conservation, we may concentrate on the transformation happening in each total
energy block separately. For instance, consider the transformation (23). This
respects the total energy conservation, as given in Eq. (25). Again, the
overall process occurs unitarily in isolation, so total entropy must be
strictly conserved. The batteries only absorb or release work, and, by
definition, they cannot exchange entropy with the rest of the system. Thus the
entropy of system-bath composite ($B_{1}S_{1}S_{2}B_{2}$) must have to be
conserved. Given the initial total energy $E_{1}+E_{2}$, the strict entropy
conservation implies the conservation of degeneracy, i.e.,

$\displaystyle g_{B}(E_{1}+E_{2})$
$\displaystyle=g_{B}(E_{1}^{\prime}+E_{2}^{\prime})$
$\displaystyle=g_{B}(E_{1}+E_{2})e^{\beta_{1}(E_{1}^{\prime}-E_{1})+\beta_{1}(E_{2}^{\prime}-E_{2})}.$
(29)

Thus, the following must have to be satisfied
$\beta_{1}(E_{1}^{\prime}-E_{1})+\beta_{1}(E_{2}^{\prime}-E_{2})=0$. Here,
$E_{1}=E_{B_{1}}+E_{S_{1}}$ and $E_{2}=E_{B_{2}}+E_{S_{2}}$, and similarly for
$E_{1}^{\prime}$ and $E_{2}^{\prime}$. Thus, the above condition is reduced to

$\displaystyle\beta_{1}\left(\Delta E_{B_{1}}+\Delta
E_{S_{1}}\right)+\beta_{2}\left(\Delta E_{B_{2}}+\Delta E_{S_{2}}\right)=0,$
(30)

where $\Delta E_{x}=E_{x}^{\prime}-E_{x}$ is the change in energy for the
given total energy block. This is nothing but the condition for strict
weighted-energy conservation, as ensured by the commutation relation (3).

##  Appendix D Engineering intensity-dependence in Hamiltonian (16)

This section aims to show how the interaction Hamiltonian (16) can be realized
with designed cavities or ion traps. We will focus here on the case of one
cavity interacting with a two-level atom. Generalization to three-level
systems and two different cavities coupled to the two different transitions is
straightforward.

So, the starting point is a cavity (or trap) with a slight anharmonicity. That
is described by the Hamiltonian of a harmonic oscillator, with frequency
$\omega$ and a small controllable anharmonicity $V\left(x/x_{0}\right)$,

$H_{\rm
cav}=\frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}+V\left(x/x_{0}\right)=\hbar\omega\
a^{\dagger}a+V\left(\frac{a^{\dagger}+a}{\sqrt{2}}\right).$ (31)

where $x_{0}=\sqrt{\frac{\hbar}{m\omega}}$. Assuming that $\omega$ is much
larger than any other relevant frequency, it makes sense to go to interaction
picture with respect to the harmonic part of the Hamiltonian, and apply
rotating wave approximation, i.e. neglect all rapidly oscillating terms and
leave only diagonal terms in the Fock basis. The end result is

$H_{\rm cav,I}=f(N),$ (32)

where $N=a^{\dagger}a$ and $f(n)=\langle n|V(x)|n\rangle$ with
$N|n\rangle=n|n\rangle$.

Similarly, we assume that atom-cavity coupling originally has a general form

$H_{\rm c}=\hbar gb(x/x_{0})(\sigma^{\dagger}+\sigma),$ (33)

where $b(x/x_{0})$ is the cavity mode function, which we take to be odd, i.e.
$b(-y)=-b(y)$. Assuming that the atom is close to the bare cavity resonance
and, performing the same steps as before, we end up with the interaction
Hamiltonian

$H_{\rm c,I}=\hbar g(\theta(N)\sigma^{\dagger}a+h.c.),$ (34)

where $\langle n-1|b(x/x_{0})|n\rangle=\theta(n-1)\sqrt{n}$.

The intensity-dependent functions $f(N)$ and $\theta(N)$ are related to the
original functions $V\left(x/x_{0}\right)$ and $b\left(x/x_{0}\right)$. For
general $f(N)$ and $\theta(N)$, one needs to design the original functions.
This can be done using Monte Carlo (MC) optimization procedures. To this aim
one defines a cost function

$C[V,b]=||f_{\rm act}(\cdot)-f_{\rm tar}(\cdot)||+||\theta_{\rm
act}(\cdot)-\theta_{\rm tar}(\cdot)||,$ (35)

where $f,\theta_{\rm act,\rm tar}$ are the actual and target forms of the
functions $f(\cdot)$ and $\theta(\cdot)$, and $||\cdot||$ denotes any norm in
the space of the functions $f,\theta$. Judging from Eq. (17) it can be
$L^{2}$-norm for $f(\cdot)$ and $L^{q}$-norm with $q>2$ for $\theta(\cdot)$.
Now the MC procedure runs as follows: i) we choose actual form of $V_{\rm
act}(\cdot)$ and $b_{\rm act}(\cdot)$; ii) we calculate $f_{\rm act}(\cdot)$,
$\theta_{\rm act}(\cdot)$ and $C[V_{\rm act},b_{\rm act}]$; iii) we modify
slightly $V_{\rm act}(\cdot)$ and $b_{\rm act}$ and calculate the new value of
$C[V_{\rm act},b_{\rm act}]$; iv) we accept the modification, if the new value
of the error function is smaller than the previous one; v) we go to iii) and
repeat this steps until convergence is achieved. MC optimization maybe
modifies to allow small errors, if we treat the cost functions like energy and
minimize the corresponding free energy at some arbitrary auxiliary temperature
${\cal T}$.

The question of the convergence of the MC procedure, as well as the
sensitivity and the role of errors in the realization of our quantum engine is
very interesting but clearly goes beyond the scope of present work. We will
study it in a future publication.

##  Appendix E Effective Hamiltonian of the quantum optics based quantum heat
engine

We start by considering that our system $S$ is described by the Hamiltonian of
a $\Lambda$-system and that each of the two transitions is coupled to a
different bosonic mode. The Hamiltonian is divided into two parts
$H=H_{0}+H_{1}$, with

$\begin{split}H_{0}=\omega_{1}N_{1}&+\omega_{2}N_{2}+E_{1}|1\rangle\\!\langle
1|_{S}+E_{2}|2\rangle\\!\langle 2|_{S}+E_{3}|3\rangle\\!\langle 3|_{S},\\\
H_{1}=f_{1}(N_{1}&)+f_{2}(N_{2})+g_{1}\theta_{1}(N_{1})(\hat{a}_{1}\sigma_{31}+\text{h.c.})\\\
&+g_{2}\theta_{2}(N_{2})(\hat{a}_{2}\sigma_{32}+\text{h.c.}).\end{split}$

Here $\sigma_{ij}=|i\rangle\\!\langle j|_{S}$ is the transition operator, and
$N_{k}=a_{k}^{\dagger}a_{k}$ the number operator corresponding to the bath
$B_{k}$. The system and bath energies are given by $E_{i}$ and $\omega_{k}$
($\hbar=1$). The terms $f_{k}(N_{k})$ represent intensity-dependent energy
shifts of the baths, whereas $g_{k}(N_{k})$ also takes into account the
intensity-dependence of the dipole interaction between the system and the
baths. We now set $E_{1}=0$ and move to the interaction picture with respect
to $H_{0}^{\prime}=\omega_{1}N_{1}+\omega_{2}N_{2}+E_{2}|2\rangle\\!\langle
2|_{S}+\omega_{1}|3\rangle\\!\langle 3|_{S}$, imposing the resonant condition
$\omega_{1}=\omega_{2}+E_{2}$. This gives the interaction picture Hamiltonian

$\displaystyle H_{I}=\Delta|3\rangle\langle 3|$
$\displaystyle+f_{1}(N_{1})+f_{2}(N_{2})+g_{1}\theta_{1}(N_{1})(\hat{a}_{1}\sigma_{31}+\text{h.c.})$
$\displaystyle+g_{2}\theta_{2}(N_{2})(\hat{a}_{2}\sigma_{32}+\text{h.c.}),$
(36)

with $\Delta=E_{3}-\omega_{1}$. We express our quantum state in the
interaction picture as:

$|\Psi\rangle=|\tilde{\alpha}\rangle_{B}|1\rangle_{S}+|\tilde{\beta}\rangle_{B}|2\rangle_{S}+|\tilde{\gamma}\rangle_{B}|3\rangle_{S},$
(37)

where $|\tilde{\alpha}\rangle_{B},\ |\tilde{\beta}\rangle_{B}$, and
$|\tilde{\gamma}\rangle_{B}$ are unnormalized states of the $B_{1}B_{2}$
composite. This leads to the following form of the Schrödinger equation in
components

$i\frac{\text{d}}{\text{dt}}|\tilde{\alpha}\rangle_{B}=\left[f_{1}(N_{1})+f_{2}(N_{2})\right]|\tilde{\alpha}\rangle_{B}+g_{1}\theta_{1}(N_{1})\hat{a}_{1}^{\dagger}|\tilde{\gamma}\rangle_{B},$
(38)
$i\frac{\text{d}}{\text{dt}}|\tilde{\beta}\rangle_{B}=\left[f_{1}(N_{1})+f_{2}(N_{2})\right]|\tilde{\beta}\rangle_{B}+g_{2}\theta_{2}(N_{2})\hat{a}_{2}^{\dagger}|\tilde{\gamma}\rangle_{B},$
(39) $\displaystyle i\frac{\text{d}}{\text{dt}}|\tilde{\gamma}\rangle_{B}=$
$\displaystyle\left[f_{1}(N_{1})+f_{2}(N_{2})+\Delta\right]|\tilde{\gamma}\rangle_{B}$
$\displaystyle+g_{2}\theta_{2}(N_{2})\hat{a}_{2}|\tilde{\beta}\rangle_{B}+g_{1}\theta_{1}(N_{1})\hat{a}_{1}|\tilde{\alpha}\rangle_{B}.$
(40)

Now we consider a large detuning , i.e.,
$\left\|\frac{\text{d}}{\text{dt}}|\tilde{\gamma}\rangle_{B}\right\|\simeq 0$,
and $\Delta\gg\left\langle{f_{1}(N_{1})+f_{2}(N_{2})}\right\rangle$, which
allows us to express

$|\tilde{\gamma}\rangle\simeq-\frac{1}{\Delta}\left(g_{1}\theta_{1}(N_{1})\hat{a}_{1}|\tilde{\alpha}\rangle_{B}+g_{2}\theta_{2}(N_{2})\hat{a}_{2}|\tilde{\beta}\rangle_{B}\right).$
(41)

Introducing this result in the previous equations leads to

$\displaystyle i\frac{\text{d}}{\text{dt}}|\tilde{\alpha}\rangle_{B}=$
$\displaystyle\left[f_{1}(N_{1})+f_{2}(N_{2})-\frac{g_{1}^{2}}{\Delta}\theta_{1}^{2}(N_{1})\right]|\tilde{\alpha}\rangle_{B}$
$\displaystyle-\frac{g_{1}g_{2}}{\Delta}\theta_{1}(N_{1})\hat{a}_{1}^{\dagger}\hat{a}_{2}\theta_{2}(N_{2})|\tilde{\beta}\rangle_{B},$
(42) $\displaystyle i\frac{\text{d}}{\text{dt}}|\tilde{\beta}\rangle_{B}=$
$\displaystyle\left[f_{1}(N_{1})+f_{2}(N_{2})-\frac{g_{2}^{2}}{\Delta}\theta_{2}^{2}(N_{2})\right]|\tilde{\beta}\rangle_{B}$
$\displaystyle-\frac{g_{1}g_{2}}{\Delta}\theta_{2}(N_{2})\hat{a}_{2}^{\dagger}\hat{a}_{1}\theta_{1}(N_{1})|\tilde{\alpha}\rangle_{B}.$
(43)

The latter are the same equations of motion generated by an effective
interacting Hamiltonian given by

$\displaystyle H^{\prime}_{\text{eff}}=$ $\displaystyle
f_{1}(N_{1})+f_{2}(N_{2})-\frac{g_{1}^{2}}{\Delta}\theta_{1}^{2}(N_{1})-\frac{g_{2}^{2}}{\Delta}\theta_{2}^{2}(N_{2})$
$\displaystyle-\frac{g_{1}g_{2}}{\Delta}(\theta_{1}(N_{1})\hat{a}_{1}^{\dagger}\hat{a}_{2}\theta_{2}(N_{2})\sigma_{21}+\text{h.c.}).$
(44)

For suitable functions that satisfy

$f_{k}(N_{k})=\frac{g_{k}^{2}}{\Delta}\theta_{k}^{2}(N_{k}),$ (45)

we obtain the final effective Hamiltonian

$H_{\text{eff}}=-\frac{g_{1}g_{2}}{\Delta}\theta_{1}(N_{1})\hat{a}_{1}^{\dagger}\hat{a}_{2}\theta_{2}(N_{2})\sigma_{21}+\text{h.c.}.$
(46)

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