# Quantum Engines and Refrigerators

Loris Maria Cangemi 111Both authors contributed equally to this work
<EMAIL_ADDRESS>Chitrak Bhadra 222Both authors contributed
equally to this work<EMAIL_ADDRESS>Amikam Levy<EMAIL_ADDRESS>[ Department of Chemistry; Institute of Nanotechnology and Advanced Materials;
Center for Quantum Entanglement Science and Technology, Bar-Ilan University,
Ramat-Gan, 52900 Israel

###### Abstract

Engines are systems and devices that convert one form of energy into another,
typically into a more useful form that can perform work. In the classical
setup, physical, chemical, and biological engines largely involve the
conversion of heat into work. This energy conversion is at the core of
thermodynamic laws and principles and is codified in textbook material. In the
quantum regime, however, the principles of energy conversion become ambiguous,
since quantum phenomena come into play. As with classical thermodynamics,
fundamental principles can be explored through engines and refrigerators, but,
in the quantum case, these devices are miniaturized and their operations
involve uniquely quantum effects. Our work provides a broad overview of this
active field of quantum engines and refrigerators, reviewing the latest
theoretical proposals and experimental realizations. We cover myriad aspects
of these devices, starting with the basic concepts of quantum analogs to the
classical thermodynamic cycle and continuing with different quantum features
of energy conversion that span many branches of quantum mechanics. These
features include quantum fluctuations that become dominant in the microscale,
non-thermal resources that fuel the engines, and the possibility of scaling up
the working medium’s size, to account for collective phenomena in many-body
heat engines. Furthermore, we review studies of quantum engines operating in
the strong system-bath coupling regime and those that include non-Markovian
phenomena. Recent advances in thermoelectric devices and quantum information
perspectives, including quantum measurement and feedback in quantum engines,
are also presented.

###### keywords:

Quantum engines , Quantum refrigerators , Quantum thermodynamics , Open
quantum systems , Quantum fluctuations , Thermodynamic uncertainty relations ,
Non-thermal quantum sources , Quantum many-body , Non-Markovian quantum
dynamics , Strong coupling , Thermoelectrics , Quantum information , Maxwell’s
demon , Quantum measurement and feedback , Quantum transport

††journal: Physics Reports

https://levyresearch.com/]levyresearch.com

Quantum Engines provide a unique platform for studying the interplay of
quantum phenomena and nonequilibrium thermodynamics.

The role of quantum fluctuations, non-Markovianity, and strong coupling in
energy conversion can be investigated through Quantum Engines.

Many-body systems and non-thermal baths are building blocks of Quantum
Engines.

Recent developments in thermoelectric devices open new experimental
possibilities.

Quantum correlation measurements and feedback can serve as resources for work
extraction and cooling.

###### Contents

  1. 1 Introduction
  2. 2 Basic concepts
    1. 2.1 Quantum thermodynamic processes
    2. 2.2 Quantum engines and refrigeration cycles
      1. 2.2.1 Carnot cycle
      2. 2.2.2 Otto cycle
    3. 2.3 Continuous and autonomous thermal machines
      1. 2.3.1 The tree-level maser QHE: Static and dynamical pictures
      2. 2.3.2 Quantum absorption refrigerators
    4. 2.4 Quantum effects and signatures
  3. 3 Fluctuations and thermodynamic uncertainty relations in QHEs
    1. 3.1 The significance of thermodynamic fluctuations in small thermal machines
    2. 3.2 Fluctuations of QHEs: A Two-Point-Measurement-based study
    3. 3.3 Thermodynamic Uncertainty Relations in Quantum Heat Engines
  4. 4 Non-thermal quantum engines
    1. 4.1 The phaseonium and quantum coherence induced by dissipation
    2. 4.2 Squeezing as a thermodynamic resource
    3. 4.3 QHEs with negative temperature baths
    4. 4.4 QEs with correlated baths
    5. 4.5 Thermodynamics of non-thermal QE
  5. 5 Many-body Quantum Engines
    1. 5.1 Many-body engines and criticality
    2. 5.2 Long- and short-range interactions
  6. 6 Quantum heat engines beyond the weak coupling and Markovian paradigm
    1. 6.1 Quantum Thermodynamics in the strong coupling limit
      1. 6.1.1 Study of QHEs using the polaron transformation method
      2. 6.1.2 Reaction Coordinate Mapping in studying strongly coupled engines
      3. 6.1.3 Perturbative corrections to engine observables due to strong coupling
      4. 6.1.4 Strongly-coupled QHEs with Matrix Product States
    2. 6.2 Non-Markovian effects in quantum heat engines
      1. 6.2.1 QHEs with system-reservoir coupling control
      2. 6.2.2 Non-Markovian effects in slowly-driven QHEs
      3. 6.2.3 Correlation-induced non-Markovian QHEs
  7. 7 Thermoelectric devices
    1. 7.1 Models of nanoscale thermoelectric transport
      1. 7.1.1 Theoretical tools for studying thermoelectric devices
      2. 7.1.2 Linear response and performance
    2. 7.2 Multi-terminal devices
      1. 7.2.1 From the energy harvester to the thermal transistor
      2. 7.2.2 Hybrid thermal machines
    3. 7.3 Superconducting thermoelectric engines and coolers
      1. 7.3.1 Photonic superconducting engines
      2. 7.3.2 Nonlocal thermoelectric engines and coolers
      3. 7.3.3 Nonlinear thermoelectric engines
  8. 8 Quantum Entanglement, Information, and Measurement-driven QHEs
    1. 8.1 The classical Maxwell’s demon and the Szilárd engine
    2. 8.2 Quantum Maxwell’s demon heat engines
      1. 8.2.1 Quantum effects in MD engines
      2. 8.2.2 Experiments on quantum MD engines
    3. 8.3 Measurement-fuelled QHEs and refrigerators
    4. 8.4 Using QHEs to generate entanglement
  9. 9 Concluding Remarks

## Acronyms

2DEG

    Two-Dimensional Electron Gas
BCS

    Bardeen–Cooper–Schrieffer
CB

    Coulomb Blockade
CHSH

    Clauser-Horne-Shimony-Holt
COP

    Coefficient of Performance
CPS

    Cooper Pair Splitter
DFT

    Density Functional Theory
DOS

    Density of States
DP

    Differential Programming
EC

    Elastic Cotunneling
FTs

    Fluctuation Theorems
HO

    Harmonic Oscillator
KZ

    Kibble-Zurek
LDA

    Local Density Approximation
LGKS

    Lindblad-Gorini-Kossakowski-Sudarshan
LHS

    Local Hidden State
LMG

    Lipkin Meshkov Glick
LWI

    Lasing Without Inversion
MBL

    Many-body Localization
MD

    Maxwell’s Demon
MPS

    Matrix Product States
NEGF

    Nonequilibrium Green Function
NMR

    Nuclear Magnetic Resonance
NV

    Nitrogen Vacancy
POVM

    Positive Operator-Valued Measure
QAR

    Quantum Absorption Refrigerator
QD

    Quantum Dot
QE

    Quantum Engines
QED

    Quantum Electrodynamics
QHE

    Quantum Heat Engines
QPC

    Quantum Point Contact
QR

    Quantum Refrigerator
RC

    Reaction Coordinates
RL

    Reinforcement Learning
SLN

    Stochastic Liouville-von Neumann
STA

    Shortcut to Adiabaticity
TCL

    Time Convolution-Less
TLS

    Two-Level System
TN

    Tensor Networks
TPM

    Two-Point-Measurement
TUR

    Thermodynamics Uncertainty Relations

##  1 Introduction

The transformation of energy from one form into another is a fundamental
concept that is studied across many fields of research, ranging from physics
to chemistry and biology. Exploring the principles and mechanisms of energy
transformation is essential to understanding the behavior of both natural and
human-made systems. Such explorations are crucial, moreover, for designing and
improving technologies and finding alternative energy sources. Thermodynamics,
the field that investigates the conversion of work and heat on the macro
scale, was initiated by the theoretical work of Carnot on heat engines in
1824. In this work [1], “Reflection on Motive Power,”333Translation to English
by R. H. Thurston from 1890 can be found online. Carnot paved the way toward
the basic understanding that work can be produced using two thermal heat baths
at different temperatures, the common theoretical setting of heat engines even
today. Furthermore, the first step in formulating the second law of
thermodynamics was achieved thanks to his study on the maximal efficiency of
heat engines, which set an upper bound on the efficiency of heat conversion
into work in such cyclic operating devices.

Engines and refrigerators have provided a platform for exploring the basic
principles of energy conversion and thermodynamics for nearly 200 years. It
isn’t surprising, therefore, that not long after its invention, the maser
(i.e., microwave amplification via stimulated emission of radiation) was
analyzed as an engine operation. A pioneering work done by Scovil and Schulz-
DuBois [2] in 1959 is traditionally considered the first study of quantum heat
engines. Yet it provides only a quasistatic picture; its main quantum feature
is the discreet energy levels of the working medium, as will be discussed in
the next section. It was only much later that a dynamical picture of quantum
heat engines within an open quantum system formalism was established by Alicki
[3] and Kosloff [4]. Over the next several decades, the study of quantum
engines and refrigerators444Extensive work on Laser cooling took place during
those years, but its interpretation in terms of the quantum thermodynamic
framework was still missing. still took place within a relatively small
community. In the last decade, however, the field has seen tremendous growth,
thanks in large part to experimental advances in nanoscale devices,
superconducting circuits, circuit QED s, and quantum many-body devices. Both
theoretically and experimentally, this study of quantum engines and
refrigerators has expanded into many areas of quantum mechanics, including
quantum information, many-body systems, nonequilibrium quantum dynamics,
quantum fluctuations, quantum control, thermoelectric devices, and more.

After giving a short account of the most striking contributions to the field,
the majority of which were made in the last fifteen years, we focus on
theoretical proposals and experimental findings put forward over the last six
years, a period covering the emergence of quantum engines in several different
branches of quantum mechanics, as specified in the table of contents. A
previous review [5] from 2014 focused solely on continuous thermal devices,
and more recent ones [6, 7], published during the period in which we composed
the present work, offer a detailed list of promising platforms for
experimental studies of quantum heat engines (QHE s) and explore energy
dynamics, heat production and heat-to-work conversion with qubits. In each of
the main sections, we also provide references to reviews salient to the
specific topic considered. To keep our discussion self-contained, we had to
choose the research papers under our consideration quite selectively, and,
thus, covering all relevant studies was not possible.

The work is organized as follows: in Sec. 2, we summarize basic concepts and
models of driven and autonomous QHEs. We discuss both stroke engines that,
analogously to their classical counterparts, undergo thermodynamic cycles, and
continuous thermal machines that can provide thermodynamic insight into Lasers
and Masers. In these models, the working medium is a quantum object that is
driven out-of-equilibrium, by means of control fields and exchanges of work
and heat with a number of thermal reservoirs.

In Sec. 3, we discuss the significance of fluctuations in the operations of
QHEs. At the quantum scale, fluctuation may dominate the behavior of the
engine. We therefore introduce different protocols for characterizing the
fluctuations in the work, heat and efficiency of the devices, and examine
their possible pitfalls. Furthermore, we stress the connection between
thermodynamic uncertainty relations and bounds on the efficiency of QHEs.

In Sec. 4, we consider several proposals for prototypical non-thermal QHEs.
This class of engines is energized by means of non-thermal reservoirs, i.e.,
physical systems whose quantum states cannot be described with a thermal Gibbs
state. Engines belonging to this class harness additional resources with
respect to their thermal counterparts, e.g., coherent and squeezed baths. Due
to their special features that provide additional sources of work, the
efficiency of these engines has been shown to exceed, perhaps not
surprisingly, conventional heat engine bounds. These violations are only
apparent, and they can be avoided via a suitable theory presented here, which
takes into account additional thermodynamic resources linked to the non-
equilibrium nature of the bath. Moreover, this section also reviews the effect
of correlations between the QHE’s different baths.

In Sec. 5, we focus on recent theoretical proposals of QHEs, whose working
mediums consist of driven quantum many-body systems. This branch of research
is motivated by the search for genuine quantum effects that could provide an
advantage in the performance of QHEs, with respect to their classical analogs.
We take into account several features of many-body systems that could lead to
various advantages in the performance of heat engines. For instance,
entanglement and other forms of quantum correlations have been proven to boost
the output power and efficiency of QHEs, with respect to an ensemble of
noninteracting single-particle engines. Moreover, criticality due to
interactions among the constituents of the working medium can modify the
scaling of the power output with the engine size, such that the engine could
achieve finite output power with ideal efficiency. Critical QHEs working in
the presence of non-equilibrium phase transitions, ruled by the Kibble-Zurek
KZ mechanism, have also been studied, as have statistically indistinguishable
bosonic particle-based QHEs. We will review the main theoretical proposals
related to many-body QHEs, underlining the different definitions of quantum
advantage, as well as experimental platforms suitable for implementing such
engines.

Sec. 6 provides an overview of recent theoretical approaches and proposals
that address the impact of the working medium’s non-Markovian dynamics on the
performance of QHEs. Non-Markovianity in an open quantum system entails a wide
set of physical effects; these typically arise in the limit of strong system-
bath interactions and low-temperature regimes, where system-bath correlation
decay times are of the order of the system’s characteristic timescales. A non-
Markovian evolution can be described in terms of the occurrence of memory
effects of the open system dynamics. Such effects can lead, in turn, to
environment backaction and information backflows, which are absent in the
Markovian setting. As a result, from the theoretical perspective, these
dynamical effects cannot be caught within the framework of a quantum Markovian
semigroup. This phenomenon holds especially true for open quantum systems
subject to driving fields, as nonequilibrium dynamics are dominated by the
interplay between the driving fields and dissipation. We will thus focus on
recent proposals of QHEs undergoing arbitrary finite-time thermodynamic
cycles, where the coupling with thermal baths can also be suitably modulated
in time. Furthermore, we will present special proposals of QHEs whose working
medium is strongly coupled to heat baths, describing their features as
compared to more conventional settings.

In Sec. 7, we provide a short introduction to the recent theoretical and
experimental literature on nanoscale thermoelectric devices. This kind of QHE
belongs to the class of autonomous or continuous engines. Unlike driven QHEs,
such devices convert steady-state heat currents into charge currents against a
voltage bias, achieving heat to work conversion at the nanoscale. In addition
to heat-to-work conversion, steady-state machines can be employed as heat
rectificators, refrigerators, and on-chip coolers, working at sub-Kelvin
temperatures and employing superconducting circuit elements. In the first part
of the section, we outline the main theoretical approaches to modeling these
devices, referring to the excellent reviews available in the literature.
Subsequently, we review novel theoretical proposals and experimental
realizations, including the concepts of hybrid thermal machines and non-local
and non-linear superconducting thermoelectric devices.

In Sec. 8, we provide an overview of recent theoretical and experimental
advances that investigate the fundamental relationship between quantum thermal
devices and information theory. The cornerstone of this research in the
classical setting, Maxwell’s demon, has received great interest in recent
years, prompted by experimental advances in the control of nonequilibrium
processes. After summarizing the main steps taken during the two decades of
research on this topic, we then focus on recent studies that shed light on the
consequences of genuine quantum features in the measurement-feedback process,
which have been described using the framework of resource theories. We also
review novel theoretical proposals of quantum devices that harness quantum
measurement as an additional resource for achieving thermodynamic tasks, such
as heat-to-work conversion and refrigeration. Finally, we provide a brief
account of autonomous quantum heat engines that allow for markedly
nonclassical behavior, such as entanglement generation.

##  2 Basic concepts

###  2.1 Quantum thermodynamic processes

Classically, a thermodynamic process is defined as a change from an initial
equilibrium state to a final one, during which thermodynamic state variables
such as pressure, volume, and temperature may vary. In the quantum realm, the
system undergoing the thermodynamic process is on the scale of a single atom,
and among the relevant variables are the eigenenergies and the energy-level
occupations of the system. A quasistatic process, in which the system remains
at equilibrium at each instance of the process, can also be defined in the
quantum realm. In this case, the generator of the dynamics may depend on a
control parameter that varies along the process. For the quantum isothermal
process, the system is coupled to a heat bath and its energy levels are varied
slowly, such that it is kept at thermal equilibrium. When the thermalization
time scale is much shorter than the system’s internal one, the system remains
in a Gibbs state at a fixed temperature and with respect to the instantaneous
Hamiltonian. In such a process, the system is subject to both work and heat
changes.

On the other hand, the quantum isochoric analog process only involves changes
to the population in the energy basis. The eigenenergies are invariant and the
energy exchange with the thermal bath can be identified as heat, which is
accompanied by entropy variation. Another important process is the quantum
adiabatic evolution of a closed system. In this case, the control Hamiltonian
varies slowly, such that the state of the system follows the instantaneous
eigenenergies and the population remains stationary. These processes are the
building blocks that constitute quantum thermodynamic cycles, including the
Carnot and Otto cycles discussed below. In most cases, these basic processes
demonstrate a performance very similar to their classical counterparts. Much
of their quantum behavior, however, becomes conspicuous when finite-time
processes, non-equilibrium baths, and not fully thermalizing processes are
considered, all of which will be the subject of detailed discussion in this
review.

###  2.2 Quantum engines and refrigeration cycles

####  2.2.1 Carnot cycle

To understand the basic operation of an ideal quantum Carnot cycle, here we
consider the cycle of a heat engine with a two-level-system (TLS) working
medium, which operates quasistatically between two heat baths at temperatures
$T_{h}>T_{c}$. The cyclic operation of the engine consists of two isothermal
strokes and two adiabatic ones, as shown in Fig. 1a. In the first (isothermal)
stroke ($\rm{A}\rightarrow\rm{B}$), the TLS is coupled to a hot bath with
temperature $T_{h}$, and the energy gap of the TLS is reduced from
$\varepsilon_{A}$ to $\varepsilon_{B}$, while the temperature remains
constant. This condition is permissible given the energy level changes on a
time scale much larger than the system’s relaxation time. In this case, the
TLS remains in equilibrium with temperature $T_{h}$ at all times, such that
the population ratio of the exited and ground state is given by the Boltzmann
factor $P_{1}/P_{0}=\exp\left[-\beta_{h}\varepsilon(t)\right]$, with
$\beta_{h}=(k_{\rm B}T_{h})^{-1}$ and $k_{\rm B}$ the Boltzmann constant.
During this process heat is flowing into the TLS and work is done by the TLS.

In the subsequent second (adiabatic) stroke ($\rm{B}\rightarrow\rm{C}$), the
TLS is decoupled from the heat bath and follows a unitary evolution according
to some protocol that further reduces the gap from $\varepsilon_{B}$ to
$\varepsilon_{C}$. In this case, the entropy of the system is preserved. When
the system is driven quantum adiabatically, the population remains the same,
implying that no excitation or heat is being generated, and work is further
performed by the system. For the TLS, an effective temperature at the end of
the stroke can be expressed as $k_{\rm B}T_{\rm
eff}=\varepsilon_{C}\ln^{-1}(P_{1}/P_{0})$. This point is crucial as, in the
next stroke, we wish to couple the TLS to a cold bath with temperature
$T_{c}=T_{\rm eff}$, guaranteeing thermodynamic reversibility. We note in
passing that, for a multilevel system, an effective temperature is achieved if
all energy levels are shifted uniformly, such that the ratio between energy
gaps at the beginning and end of the stroke is constant and equal to
$T_{h}/T_{c}$ [8, 9].

In the third (isothermal) stroke ($\rm{C}\rightarrow\rm{D}$), the TLS is kept
in thermal equilibrium as the energy gap is enlarged to $\varepsilon_{D}$.
During this process, heat is removed from the system into a cold bath with
temperature $T_{c}$, while work is performed on the TLS. The cycle is
completed once in the fourth (adiabtic) stroke ($\rm{D}\rightarrow\rm{A}$) the
system is brought back to its initial condition at point $(\rm{A})$ via a
unitary evolution. In this stroke, work is performed on the system.

The Carnot theorem of maximal efficiency stems from the fact that, at the
beginning of strokes ($\rm{A}\rightarrow\rm{B}$) and
($\rm{C}\rightarrow\rm{D}$), the system’s temperature and that of the baths
are equal. Heat is exchanged with the baths only when there is no temperature
difference, leading to an efficient conversion of heat to work. The efficiency
is calculated simply via

$\eta_{\rm{C}}=\frac{|W|}{Q_{h}}=\frac{Q_{h}+Q_{c}}{Q_{h}}=1+\frac{T_{c}\Delta
S_{c}}{T_{h}\Delta S_{h}}=1-\frac{T_{c}}{T_{h}}.$ (1)

Here $W$ is the total average work gain over a cycle, which is equal to the
total energy exchanged with the baths. ${Q}_{h}$ and $Q_{c}$ are the heat
exchanged with the hot and cold bath respectively. Since the cycle is
thermodynamically reversible, the change of entropy $\Delta S_{c(h)}$ in the
two isothermal strokes is equal in magnitude but with the opposite sign. We
defined $\Delta S_{h}=S(\rho_{B})-S(\rho_{A})$ and $\Delta
S_{c}=S(\rho_{D})-S(\rho_{C})$ with
$S(\rho_{X})=-\mathrm{tr}[{\rho_{X}\ln(\rho_{X})}]$, the von-Neumann entropy
at point $X$. For the TLS considered above, the net average work over a cycle
is given by

$W=\eta_{\rm C}\left[k_{\rm
B}T_{h}\ln\left(\frac{Z(B)}{Z(A)}\right)+\varepsilon_{B}\tanh(\beta_{h}\varepsilon_{B})-\varepsilon_{A}\tanh(\beta_{h}\varepsilon_{A})\right],$
(2)

with partition function
$Z(X)=\exp(\beta_{h}\varepsilon_{X})+\exp(-\beta_{h}\varepsilon_{X})$. The
analysis of the quantum Carnot cycle follows reasoning similar to that of its
classical counterpart, and the cycle will exhibit the same efficiency. Yet
other characteristics of the quantum engine (QE) performance, such as the work
gained from the engine, will depend on the explicit details of the working
medium, e.g., whether it is a TLS, harmonic oscillator, or particle in a box
[10, 8, 9].

The Carnot cycle describes an ideal thermodynamic reversible cycle. In a
realistic cycle, however, i.e., finite-time thermodynamics, a dynamical
picture [11] of the engine evolution and its dissipation should be adopted.
When a weak coupling between the system and the bath is assumed, it is the
custom to apply the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) Master
equation approach to open quantum systems [12, 13, 14]. A dynamical treatment
gives rise to new questions and emphasizes the role of quantum phenomena in
the operation of the devices [15]. The finite-time operations in the Carnot-
type-cycles [16, 17, 18, 19] lead to processes of dephasing, loss of
correlations, and quantum friction [20], which become essential to optimizing
the power output of the QHE, the cooling rate of the Quantum refrigerator
(QR), and the efficiency of these devices. Optimization of Carnot cycle
performance was studied in different setups, including a single particle in a
potential well [21].

Figure 1: Schematic diagrams of the quantum Carnot cycle a) and the quantum
Otto cycle b) for a TLS. Arrows pointing towards the branches indicate work
performed on the TLS and heat flowing into the system from the bath. Arrows
pointing outwards indicate work performed by the TLS. a)
$\rm{A}\rightarrow\rm{B}$ and $\rm{C}\rightarrow\rm{D}$ are the isothermal
strokes, and $\rm{B}\rightarrow\rm{C}$ and $\rm{D}\rightarrow\rm{A}$ are the
adiabatic ones. b) $\rm{A}\rightarrow\rm{B}$ and $\rm{C}\rightarrow\rm{D}$ are
the isochoric strokes, and $\rm{B}\rightarrow\rm{C}$ and
$\rm{D}\rightarrow\rm{A}$ are the adiabatic ones. The plotted cycles
demonstrate the operation of heat engines when the cycle goes clockwise, via
reversal of the direction in which a refrigeration process can be realized.

####  2.2.2 Otto cycle

The quantum analog of the Otto cycle has become the canonical model for
exploring quantum engines and refrigerators, mainly due to its relatively
simple analysis. The separation of heat and work strokes in the cycle makes it
easier to define these magnitudes in the quantum regime, especially when
considering finite-time cycles.

In the Otto cycle, the two isothermal strokes are replaced by isochoric analog
strokes, as shown in Fig. 1b. In these strokes, only heat is exchanged between
the working medium and the bath. In stroke ($\rm{A}\rightarrow\rm{B}$), the
energy gap of the TLS, $\Delta_{h}$ remains invariant, while the exited state
occupation is increasing from $n_{c}$ to $n_{h}$, where
$n_{x}=(1+e^{\beta_{x}\Delta_{x}})^{-1}$ for $x=h,c$, and the TLS is
thermalized at the bath temperature $T_{h}$. This thermalization stage is
associated with heat absorbed by the TLS and equals
$Q_{h}=\Delta_{h}(n_{h}-n_{c})$. The stroke ($\rm{C}\rightarrow\rm{D}$)
describes the opposite population transition from $n_{h}$ to $n_{c}$. However,
since this transition accrues in a smaller energy gap $\Delta_{c}<\Delta_{h}$,
less heat is transferred to the cold bath, $Q_{c}=\Delta_{c}(n_{c}-n_{h})$.
Here the sign of the heat determines the direction of heat flow. Positive heat
implies that heat is absorbed by the working medium, whereas negative heat
implies that it is released. The work output over a closed cycle of the
quantum Otto engine can be easily calculated according to the first law of
thermodynamics

$W=-(Q_{h}+Q_{c})=-(\Delta_{h}-\Delta_{c})(n_{h}-n_{c}).$ (3)

Negative work implies that energy is extracted from of the working medium. The
efficiency of the Otto engine,

$\eta_{O}=\frac{|W|}{Q_{h}}=1-\frac{\Delta_{c}}{\Delta_{h}}\leq
1-\frac{T_{c}}{T_{h}}=\eta_{\rm C},$ (4)

is bounded by the Carnot efficiency, as can be immediately observed from the
condition for the cycle to operate as an engine, $n_{c}<n_{h}$, which boils
down to $\Delta_{h}/\Delta_{c}<T_{h}/T_{c}$.

When the working medium is considered to be a single Quantum Harmonic
Oscillator (HO), the efficiency of the Otto engine will be similar to that of
the TLS working medium; however, a difference is observed in the work output

$\eta_{O}^{\rm HO}=1-\frac{\omega_{c}}{\omega_{h}},\;\;\;\;\;W^{\rm
HO}=-(\omega_{h}-\omega_{c})(n_{h}^{\rm HO}-n_{c}^{\rm HO}).$ (5)

Here $\omega_{h}$ and $\omega_{c}$ are the HO frequency at the end of strokes
($\rm{D}\rightarrow\rm{A}$) and ($\rm{B}\rightarrow\rm{C}$) respectively.
Additionally, in this case, the “compression” ratio, i.e., the ratio between
the energy gaps, determines the efficiency. The work, on the other hand, is
governed by the thermal occupation, $n_{x}^{\rm
HO}=(e^{\beta_{x}\omega_{x}}-1)^{-1}$, which exceeds that of the TLS for a
given energy gap. This result is also manifested in other types of engines,
including those with finite-time operation, and becomes significant when
optimizing the efficiency at maximum power [5].

For a comparative study of the Carnot and Otto engines operating in the
quasistatic limit, see Ref.[9]. Much as in the Carnot QHE, in the Otto engine,
quantum phenomena emerge when finite-time processes are considered and a
dynamical picture is adopted [22, 23]. We note in passing that the cycles
described in Figs. 1 can be reversed and operate as a refrigerator, in which
heat is absorbed from the cold bath and is transferred to the hot one. This
reversal occurs when the cycle is completed anticlockwise.

###  2.3 Continuous and autonomous thermal machines

Thermodynamic analysis of heat engines is usually based on a cyclic
thermodynamic process, in which the system returns to its initial state at the
end of the last stroke. However, other engine types that are not stroke-based
can provide a thermodynamic description for many processes and devices in
biology, chemistry, and physics. These are sometimes referred to as continuous
engines, i.e. , a working medium simultaneously coupled to all heat baths and
work sources. Ref. [5] provides an extended review of the basic operation of
continuous quantum engines and refrigerators.

####  2.3.1 The tree-level maser QHE: Static and dynamical pictures

The earliest thermal QHE proposal, dating to 1959 [2], can be considered a
continuous operating engine and is based on the maser (i.e., microwave
amplification by stimulated emission of radiation) mechanism (see Fig. 2). A
three-level quantum system acts as a working fluid, while two sources of
thermal light set at different temperatures $T_{h}$ and $T_{c}$ play the role
of the heat baths. The working fluid interacts with the baths by means of
filter waveguides that only let photons of frequencies
$\omega_{h}=\omega_{3}-\omega_{1}$ and $\omega_{c}=\omega_{3}-\omega_{2}$
pass. As a consequence, the working fluid absorbs energy $\hbar\omega_{h}$
from the hot bath and releases part of it, namely $\hbar\omega_{c}$, to the
cold bath. Work can be produced via the stimulated emission of photons of
frequency $\omega_{s}=\omega_{h}-\omega_{c}$, if, as required by the maser
operating conditions, a population inversion between levels $\left|2\right>$
and $\left|1\right>$ is achieved. If the engine efficiency is assumed to be
$\eta_{\rm M}=\omega_{s}/\omega_{h}$ and $p_{i}$ denotes the equilibrium
population of the $i$-level, it follows from the population-inversion
condition $p_{2}/p_{1}>1$ that $\eta_{\rm M}<\eta_{\rm C}=1-T_{c}/T_{h}$,
i.e., the efficiency of the maser QHE is Carnot bounded [2, 24]. In fact, this
efficiency resembles that of the Otto cycle discussed in the previous section.

Figure 2: Schematic diagram of a three-level Maser QHE. Hot (cold) reservoirs
are modeled with thermal light sources at temperatures $T_{h}(T_{c})$. Filter
waveguides allow for the absorption of hot (cold) photons of frequency
$\omega_{h}$($\omega_{c}$). The population inversion takes place between
levels 1 and 2. Work is performed by stimulated emission of radiation of
frequency $\omega_{s}$.

In the original paper [2], no dynamical picture was associated with the engine
analysis. Here we provide a simple dynamical picture that will recover the
effect on the maser efficiency but will also provide further insight into and
information on the energy flows between the different baths. Our description
is based on the LGKS master equation for the reduced three-level system
interacting with three baths. As described above, the hot and cold baths with
temperature $T_{h}$ and $T_{c}$ couple the levels
$\left|1\right>\leftrightarrow\left|3\right>$ and
$\left|2\right>\leftrightarrow\left|3\right>$ respectively. The work bath is
assumed to have an infinite temperature, and thus no entropy flow is
associated with the energy flow from the system to the bath [24]. The LGKS
master equation inherently mimics the waveguide frequency filtering, by
assuming a weak system-bath coupling limit that emphasizes the resonant
frequency. The master equation for the three-level working medium reads,

$\frac{d\rho}{dt}=-i[H,\rho]+\mathcal{D}_{h}(\rho)+\mathcal{D}_{c}(\rho)+\mathcal{D}_{s}(\rho),$
(6)

with the Hamiltonian
$H=\sum_{i=1}^{3}\hbar\omega_{i}\left|i\right>\left<i\right|$ and the
thermelizing disspators,

$\mathcal{D}_{x}(\rho)=\gamma_{x}\left(A_{x}\rho A_{x}^{\dagger}\
-\frac{1}{2}\\{A^{\dagger}_{x}A_{x},\rho\\}\right)+\gamma_{x}e^{-\beta_{x}\hbar\omega_{x}}\left(A_{x}^{\dagger}\rho
A_{x}-\frac{1}{2}\\{A_{x}A_{x}^{\dagger},\rho\\}\right)$ (7)

for $x=h,c,w$, the hot, cold and work reservoirs. Here $\gamma_{x}$ are
relaxation rates given by the Fourier transform of the $x$-bath correlation
functions, and $\beta_{x}=(k_{\rm B}T_{x})^{-1}$ is the inverse temperature of
the $x$-bath, which implies $\beta_{w}=0$ for the work bath. The system
operators $A_{x}$ can be derived in the weak system-bath coupling limit and
are identified as $A_{h}=\left|1\right>\left<3\right|$,
$A_{c}=\left|2\right>\left<3\right|$, and
$A_{w}=\left|1\right>\left<2\right|$.

The energy flow between the system and each of the baths can be evaluated
according to
$J_{x}=\left<\mathcal{D}_{x}^{\dagger}(H)\right>\equiv\mathrm{tr}{[\rho\mathcal{D}^{\dagger}(H)]}$,
where $\mathcal{D}^{\dagger}$ is the adjoint of $\mathcal{D}$ [25, 26]. At a
steady state, these are simply given by

$\displaystyle J_{h}$ $\displaystyle=$
$\displaystyle\hbar\omega_{h}\gamma_{h}\left(e^{-\beta_{h}\hbar\omega_{h}}p_{1}-p_{3}\right),$
$\displaystyle J_{c}$ $\displaystyle=$
$\displaystyle\hbar\omega_{c}\gamma_{c}\left(e^{-\beta_{c}\hbar\omega_{c}}p_{2}-p_{3}\right),$
$\displaystyle J_{w}$ $\displaystyle\equiv$ $\displaystyle
P=\hbar(\omega_{c}-\omega_{h})\gamma_{w}\left(p_{2}-p_{1}\right).$ (8)

The steady-state populations $p_{i}$ can be solved analytically, and it
follows immediately that the efficiency is $\eta_{\rm
M}=-P/J_{h}=1-\omega_{c}/\omega_{h}$.

Exploring the thermodynamics of continuously operating quantum devices as
engines and refrigerators is very useful, as many quantum processes in
physics, biology, and chemistry fall into this category of mechanisms. Engines
and refrigerators provide a common language and set of tools and bounds on the
energetic, entropic, and efficiency of these quantum processes. The continuous
models can be extended far beyond the simple three-level QHE described above,
to include phenomena induced by many-body working media, time-dependent
external driving, multi-terminal machines, and more, all of which will be
reviewed in the following chapters.

####  2.3.2 Quantum absorption refrigerators

A further example of autonomous, multi-terminal devices, quantum absorption
refrigerators (QAR) have gained widespread attention over the last decade
[27]. Starting from early theoretical proposals [28, 29, 30, 26, 31, 32], a
great amount of theoretical work has been done to elucidate the working
mechanism of this engine [33, 34], as well as to investigate the possibility
of quantum advantage in its performance [35, 5] due to coherence and
entanglement. Since the majority of these developments were extensively
reviewed in a recent work [27], in what follows we limit ourselves to a brief
overview of this kind of engine’s working principles, focusing on more recent
theoretical and experimental proposals.

Figure 3: Sketch of a quantum absorption refrigerator’s basic setup, with a
three-body working medium. In the typical setting, quantum harmonic
oscillators as well as two-level systems can be used to model the working
medium. Reproduced from [27].

A quantum absorption refrigerator works with three thermal reservoirs. It
achieves cooling by removing heat from the cold ($c$) reservoir and dumping
part of it into the hot ($h$) one. While conventional refrigerator schemes
utilize work provided from the outside – by driving either the working medium
or its coupling to the reservoir (see Sec. 6) – here the energy required to
achieve cooling is provided by a third thermal reservoir, i.e., the work
reservoir ($w$).

In its simplest setting, the working medium is composed of a three-level
system, similar to a maser heat engine discussed in the previous section,
operated in reverse mode. However, more realistic schemes employ three qubits,
modeled as usual by means of two-level systems, each coupled to a different
reservoir. In addition, different settings have also been proposed that are
based on three quantum harmonic oscillators [5], as well as two quantum
harmonic oscillators [30] coupled via optomechanical interactions [36, 37,
38]. In what follows, we focus on the three-qubit setting as depicted in Fig.
3. The qubits are labeled as $h,c,w$, depending on the reservoir to which they
are coupled. Moreover, they interact with each other via a trilinear coupling
Hamiltonian, such that the whole Hamiltonian of the working medium takes the
form

$H_{\text{wm}}=H_{0}+H_{\text{int}}=\sum_{i=h,c,w}\omega_{i}\sigma^{+}_{i}\sigma^{-}_{i}+g(\sigma^{-}_{c}\sigma^{+}_{h}\sigma^{-}_{w}+\sigma^{+}_{c}\sigma^{-}_{h}\sigma^{+}_{w}).$
(9)

Although more recent schemes involve three qubits interacting with two-body
interactions [39], below we focus on the conventional, trilinear interaction
setting. Here, as in the maser engine, the bare qubit frequencies are chosen
such that $\omega_{w}=\omega_{h}-\omega_{c}$. With the ground and excited
states of each qubit denoted with $\\{\left|0\right>\left|1\right>\\}$
respectively, the interaction Hamiltonian of the form (9) couples only two
states belonging to the three qubits’ Hilbert space, namely
$\\{\left|0\right>_{c}\left|1\right>_{h}\left|0\right>_{w},\left|1\right>_{c}\left|0\right>_{h}\left|1\right>_{w}\\}$,
which are degenerate with respect to $H_{0}$. It follows that excitation in
the cold qubit can be coherently destroyed and created in the hot qubit,
provided that the work qubit is in its excited state. Thus, it is possible to
drive the cold qubit to its ground state by employing the energy coming from
the work qubit.

As each qubit is weakly coupled to its bath, cooling of the cold bath can, in
principle, be achieved. However, whether the present setup can work as a
cooler depends on the detailed study of its dynamics, when it is considered as
an open quantum system [5]. A rather intuitive argument for finding out the
temperature regime under which cooling is achieved can be found in Eq. (9), a
swap interaction between the cold qubit and a virtual two-level system defined
from states
$\\{\left|-\right>_{v}=\left|0\right>_{h}\left|1\right>_{w},\left|+\right>_{v}=\left|1\right>_{h}\left|0\right>_{w}\\}$
of energies $\omega^{-}_{v}=\omega_{h}-\omega_{c},$ and
$\omega^{+}_{v}=\omega_{h}$. In the absence of interaction among qubits, if
the hot and work qubits are found in thermal states with inverse temperatures
$\beta_{h},\beta_{w}$, the populations of the virtual states fulfill
$p_{+v}/p_{-v}=\exp(-\beta_{v}\omega_{c})$, where $\beta_{v}$ is the effective
inverse temperature of the thermal state that reads
$\beta_{v}=(1/\omega_{c})(\beta_{h}\omega_{h}-\beta_{w}\omega_{w})$. As a
consequence, if $T_{c}>1/\beta_{v}$, an excess of population in the excited
state of the cold qubit is present with respect to the virtual one. Once the
swap interaction is switched on, the excited state of the cold qubit will
start to deplete, i.e., the transition
$\left|1\right>_{c}\left|-\right>_{v}\rightarrow\left|0\right>_{c}\left|+\right>_{v}$
increasingly takes place. In the limit of weak qubit-bath coupling, thermal
excitations are progressively removed from the cold bath and transferred to
the hot bath. A nonequilibrium stationary state thus builds up, in which a net
heat current (the cooling power) flows between the cold and the hot bath.

Several theoretical proposals have been put forward that model quantum
absorption refrigerators in superconducting circuits [40], quantum dots [41,
42, 43], and cavity QED systems [44, 45]. In [46], a quantum absorption
refrigerator was experimentally realized via a trapped ion setup. An rf-driven
Paul trap is used to confine ${}^{+}\text{Yb}$ ions along a given spatial
direction. With the aid of Raman laser beams, optical dipole forces and spin-
motion coupling are applied to the ions, such that they are forced to follow
controlled motional states, i.e., two radial modes and one axial mode of
frequencies $(\omega_{x},\omega_{y},\omega_{z})$ of the order of KHz. The
three qubits in the Hamiltonian (9) are thus replaced by ionic normal modes of
oscillation. Thus, the platform simulates the quantum dynamics of a trilinear
Hamiltonian of the form
$H=\xi(a^{\dagger}_{c}a_{h}a^{\dagger}_{w}+a_{c}a^{\dagger}_{h}a_{w})$, where
$a_{i}$ are bosonic ladder operators. The coupling strength, $\xi$, is related
to the trap parameters and the equilibrium distance between ions aligned in
the trap.

###  2.4 Quantum effects and signatures

The study of quantum processes and devices in terms of engine operation
provides a means of relating quantum phenomena to energetic and entropic
currents and the efficiency of different energy and quantum resource
conversions (see also the book chapter [15] and references therein for a short
review on the manifestation of quantum phenomena in quantum thermal devices).
The operation of an engine in the quantum regime is affected by various
quantum features. The discrepancy between the classical and quantum operation
of engines can already be traced to the quantization of the energy levels and
to the uncertainty principle [47]. Other quantum effects such as coherence and
quantum correlations have far-reaching consequences on engine operation and
will be discussed at length in this review.

Beyond understanding the relationship between thermodynamics in the quantum
regime and quantum phenomena, one of the greatest challenges in this field is
identifying when these phenomena impact the process. The customary manner of
approaching this problem is revealing quantum signatures that witness the
presence of quantum effects. In the context of quantum engines, these
signatures can be expressed via classical bounds on measurable thermodynamic
quantities, such as heat and work, i.e., violation of these bounds is possible
only in the presence of quantum effects. Such a bound on the power output of
QHEs was introduced in [48] and was experimentally demonstrated in a nitrogen-
vacancy (NV) center in a diamond setup [49]. Other quantum thermodynamic
signatures have been revealed and demonstrated in heat exchange processes [50]
and work protocols [51].

##  3 Fluctuations and thermodynamic uncertainty relations in QHEs

###  3.1 The significance of thermodynamic fluctuations in small thermal
machines

The variation of heat and work in macroscopic heat engines becomes
insignificant from one cycle to another when such engines operate in the limit
cycle. Moreover, in the thermodynamic limit of large systems, fluctuations
vanish and the system can be fully characterized by thermodynamic averages
(excluding critical points and phase transitions). As the size of the thermal
machines is scaled down to mesoscopic and quantum length scales, due to the
finite nature of the working substance (single, few, or countably finite
n-particle systems), thermodynamic quantities like heat and work become truly
stochastic with a distribution characterized by the average, the variance and
higher order moments. Thus, evaluating the fluctuations of these thermodynamic
quantities becomes essential, as they affect the different efficiency and
output parameters that characterize a heat engine.

Fluctuation-dissipation relations for equilibrium [52, 53, 54] and
nonequilibrium steady-state [55, 56, 57] problems in classical and quantum
[58, 59] systems have been studied for quite some time. Over the last few
decades, such fluctuation relations have been extended to systems evolving
far-from-equilibrium and are known as Fluctuation Theorems (FTs). Broadly
speaking, FTs relate the probability distributions of the forward and reversed
nonequilibrium processes of some fluctuating quantity. Classically, these
theorems are relevant to biomolecules, molecular motors, colloidal particles,
etc. [60]. In the quantum regime, they are applicable to, and experimentally
observed in, a wide variety of quantum devices, such as trapped ions [61],
superconducting qubits [62], quantum dots [53, 63], NMR setups [64], and NV
centers in diamond [65, 66].

Fluctuation theorems reveal fundamental aspects of entropy production,
irreversible work, heat, and matter transport for both open and closed
systems, and thus bear important consequences for scaled-down heat engines.
The predominant relations include Crooks FT [67], which generalizes the
Jarzynski equality [68] and relates work fluctuations of nonequilibrium
processes to the difference in free energy between the initial and final
states at equilibrium; FT of heat exchange between thermal states [69]; and
steady-state FT of entropy production [70]. Many of the classical FTs have
been extended and derived for quantum systems [53, 63, 71]. Specifically, such
relations characterize the functioning of heat engines [72] and have important
implications for thermal devices functioning in the quantum domain [73, 74,
75].

###  3.2 Fluctuations of QHEs: A Two-Point-Measurement-based study

Unlike classical macroscopic heat engines, QHEs are crucially affected by the
tussle between non-zero quantum and thermal fluctuations, and, hence,
evaluating them is critical to understanding thermal devices at the mesoscopic
and quantum scale. To evaluate thermodynamic quantities like work and heat,
the standard phenomenological apparatus in Quantum Thermodynamics is the Two-
Point-Measurement (TPM) scheme [76]. In this scheme, two projective
measurements of an observable of interest are performed on the system, one at
the beginning and the other at the end of the dynamical process. This scheme
can provide, for example, the energy difference statistics of the process. Yet
one should keep in mind that the first measurement collapses the initial state
on an eigenstate of the measured observable, and may thus interfere with the
statistics.

The device-efficiency fluctuations can be evaluated by calculating the joint
probability distribution for heat and work on the different strokes [77]. Most
of the studies have focused on the quantum Otto cycle [78, 79], in order to
avoid ambiguity in distinguishing between heat and work. With the Otto engine,
for example, the work distribution of a certain work stroke is given by

$P(W)=\sum_{n,m}\delta\left[W-(E_{m}^{\tau}-E_{n}^{0})\right]P_{n\rightarrow
m}^{\tau}P_{n}^{0}(\beta),$ (10)

where $E_{m}^{0}$ and $E_{m}^{\tau}$ are the energy eigenvalues at the
beginning and at the end of a stroke of duration $\tau$ respectively.
$P_{n\rightarrow m}^{\tau}=|\langle n|U(\tau)|m\rangle|^{2}$ are the
transition probabilities, and $P_{n}^{0}(\beta)=\exp{(-\beta
E_{n}^{0})}/Z^{0}(\beta)$ are the initial thermal occupations. At the next
stroke, the conditional probability for a heat outcome $Q$ given the work $W$
is

$P(Q|W)=\sum_{k,l}\delta\left[Q-(E_{l}^{\tau}-E_{k}^{\tau})\right]P_{k\rightarrow
l}^{\tau^{\prime}}P_{k}^{\tau},$ (11)

with $p_{k}^{\tau}=\delta_{km}$ the occupation after the projective
measurement at time $\tau$ on the energy eigenstate $\left|m\right>$, and
$P_{k\rightarrow l}^{\tau^{\prime}}=p_{k}^{\tau}(\beta^{\prime})$ the thermal
occupation at the end of the stroke. In this case, the energy levels are those
projected at the beginning of the stroke at time $\tau$, and the inverse
temperature $\beta^{\prime}$ is that of the thermalizing stroke.

In a similar manner, one can construct the probability of a certain work/heat
stroke conditioned on previous measurements outcomes and apply the chain rule
to determine any joint probability. For example, an evaluation of the engine-
efficiency fluctuations $\eta=\frac{W_{\rm in}+W_{\rm out}}{Q_{h}}$ can be
carried out using the joint probability $P(W_{\rm in},Q_{h},W_{\rm
out})~{}=~{}P(W_{\rm out}|Q_{h},W_{\rm in})P(Q_{h}|W_{\rm in})P(W_{\rm in})$,
for the work done on the system $W_{\rm in}$, with the system $W_{\rm out}$
and the heat entering the system $Q_{h}$. The efficiency distribution then
reads

$P(\eta)=\int dW_{\rm in}dQ_{h}dW_{\rm out}P(W_{\rm in},Q_{h},W_{\rm
out})\delta\left(\eta-\frac{W_{\rm in}+W_{\rm out}}{Q_{h}}\right).$ (12)

Experimental studies have recovered the above distributions in several setups
of QHEs. In [80], the joint distribution of heat and work and the efficiency
distribution were reported in an NMR-based quantum Otto engine. On the other
hand, in [81], the joint probability for the energy exchanged in a two-stroke
engine was measured on an IBM quantum computer platform, verifying the heat-
engine fluctuation relation.

Power fluctuations in miniaturized quantum engines have been studied in the
same framework, with an emphasis on exploring the finiteness of the working
medium’s Hilbert space and the strict degeneracy of energy levels [82]. In the
quasistatic limit, all the moments of work distribution $\langle
W^{n}\rangle~{}=~{}\int dWP(W)W^{n}$ can be calculated via a joint p.d.f,
$P(W)$, for the full cycle of a Quantum Carnot Engine, derived on similar
lines as Eq. (12). This eventually leads to the evaluation of the Fano factor
and variation of the work coefficient for the QHE, with the two quantities
encapsulating power output and work fluctuations respectively. Strikingly, the
analysis points to the supremacy in performance of a quantum machine over its
nondegenerate counterparts, via optimization of the thermodynamic parameters
with respect to energy spacings or level numbers.

##### Work and heat fluctuations: beyond the TPM paradigm

Work and heat are thermodynamic quantities that are defined over two times. As
such, the fluctuations of these quantities are described by a joint
probability for the energy at the beginning and end of a given process. The
TPM scheme discussed above provides such a probability distribution, but with
the clear drawback of destroying all coherence and correlations in the
measured energy eigenbasis of the initial state. To circumvent this issue,
different paths have been taken to study interference and correlation effects
in the work and heat distribution. These include probability distributions
that can turn negative, as in the full counting statistics method [83, 84,
85], consistent histories [86], and the Margenau-Hill quasiprobability
distribution [87, 71]. In addition to turning negative, the Kirkwood-Dirac
quasiprobability can also have imaginary parts [88]. Methods in which the work
distribution is not linear in the density operator, such as the Bohmian
framework [89] and Bayesian networks [90, 91], offer another possible
direction [92]. These approaches provide tools for studying energy and
efficiency fluctuations of QHEs beyond the TPM scheme limitations and are
motivated by a recent experiment recovering the Margenau-Hill quasiprobability
distribution for the work in a quantum process [51]. Applying these ideas to
stroke engines that encode multiple measurement points would require devising
proper measurement schemes [93, 94].

###  3.3 Thermodynamic Uncertainty Relations in Quantum Heat Engines

Thermodynamic Uncertainty Relations (TUR) have been an area of intense study
over the past few years, in the context of stochastic thermodynamics and
nonequilibrium processes in classical and quantum systems [60, 95, 96, 97].
These relations help in providing bounds in the form of mathematical
inequalities for averages, fluctuations, and higher moments of important
thermodynamic quantities, like currents and entropy production for out-of-
equilibrium systems. Within the ambit of heat engines, classical and quantum,
TURs provide insights into the efficiency and power output measures,
especially away from the quasi-static limit. TUR in the classical setting was
first proposed in the context of multiaffinity problems in biomolecular
processes [95] and can be expressed in the following compact form:

$\frac{{\rm Var}(j)}{\langle j\rangle^{2}}\geq\frac{2k_{B}}{\sigma},$ (13)

where $\langle j\rangle$ is the average current, ${\rm Var}(j)\equiv\langle
j^{2}\rangle-\langle j\rangle^{2}$ is the variance, and $\sigma$ is the
entropy production, all of which are evaluated in the steady state limit.
Physically, these relations express a trade-off between precision,
characterized by the signal-to-noise ratio of currents at a steady state, and
the amount of entropy produced in the process. Eq. (13) reveals that, even
away from equilibrium, dissipation continues to regulate small fluctuations.
Extensions of such relations to phenomena away from a steady state (as in
classically driven or nonequilibrium quantum systems) are an active field of
research [98, 99].

A lot of work has focused on the quantum implications of such TURs in heat
transport, continuous thermal devices, and, most recently, QHEs with discrete
strokes [100, 101, 102, 103]. In the case of the QHEs, the effect of TURs are
best analyzed when higher cumulants of thermodynamic quantities are evaluated.
A joint probability distribution of heat and work (TPM scheme) evaluated in
one complete Otto cycle, borrowing from the results of section 3.2, provides a
platform for calculating such deviations from the average. With
$W=W_{1}+W_{2}$ being the total work output and $Q_{2}$ being the input heat,
a ratio of the two fluctuations can be defined as
$\eta^{(2)}=\frac{\langle\langle W^{2}\rangle\rangle}{\langle\langle
Q_{2}^{2}\rangle\rangle}$, with the double-bracket denoting cumulants (in this
case, ${\rm Var}(X)\equiv\langle\langle X^{2}\rangle\rangle$). Similar
relative ratios of higher-order cumulants, $\eta^{(n)}:=\frac{\langle\langle
W^{n}\rangle\rangle}{\langle\langle Q_{2}^{n}\rangle\rangle}$, hold as well.
In the quasistatic limit, the $n$-th order ratio becomes equivalent to the
$n$-th power of the standard thermodynamic efficiency [103], i.e.,
$\eta^{(n)}_{QS}=\eta^{n}_{QS}$. Interestingly, in certain exactly solvable
non-adiabatic QHEs, such ratios are lower-bounded, e.g.,
$\eta^{(2)}\geq\eta^{2}$. Away from the steady state, these inequalities
result from the TURs, which determine that the relative fluctuation of work
always surpasses the corresponding relative fluctuation of heat absorbed from
the hot bath. Similar calculations have been extended to Quantum Otto Engines
and Refrigerators, with both finite-time isentropic and isochoric strokes [73,
104] and an ensemble of quantum thermal machines [105].

In [106], a tighter bound than the Carnot limit, $\eta\leq\eta_{\rm
C}\>=1-(T_{C}/T_{H})$, was found in classical Steady-State Heat Engines
(SSHE), as a consequence of TURs that rigorously identify the influence of the
time-averaged fluctuations, $\Delta P_{W}$, of the generated power, $P_{W}$,
during the engine’s working cycle,

$\eta\leq\frac{\eta_{\rm C}}{1+2T_{C}(P_{W}/\Delta P_{W})}:=\eta^{PS}.$ (14)

Such universal trade-off relations in the quantum regime remain an open
problem, and a recent attempt explores the question in periodically driven
heat engines [107]. In the quantum mechanical case, it is evident that quantum
fluctuations and non-commutativity of operators will play a major role in the
performance of the QHE, and, indeed, such corrections to the classical formula
have been found. This modifies the above equation to

$\eta\leq\frac{\eta_{\rm C}}{1+2T_{C}G(P_{W},P^{QS}_{W},\Delta
P_{W},\Delta\mathcal{I}_{W})}:=\eta^{Q},$ (15)

where $G(P_{W},P^{QS}_{W},\Delta P_{W},\Delta\mathcal{I}_{W})$ includes the
additional effects of non-adiabatic power $P^{QS}_{W}$ and corrections due to
quantum friction $\Delta\mathcal{I}_{W}$, akin to the ones found in the
fluctuation-dissipation relations of work in slowly-driven systems [108].

##  4 Non-thermal quantum engines

In the quantum domain, due to the inherently quantum nature of the working
medium and the baths, counterintuitive physical effects can set in, leading to
apparent violations of thermodynamic laws. These effects pose many challenges
to a consistent formulation of thermodynamics in the quantum regime [109] and
to the assessment of the performance of QHEs. Recently, non-thermal QHEs,
i.e., engines working with baths whose quantum state differs from the Gibbs
form, have been the subject of much debate. We will more appropriately refer
to them as Quantum Engines (QE s), omitting ”heat.”

A non-thermal bath is typically prepared in a non-equilibrium quantum state:
this set-up can exhibit quantum coherence and correlations (classical and
quantum), shared among constituent parts as well as with the working medium.
Moreover, a bath can be prepared in a purely non-equilibrium state, such as a
squeezed state in the quantum optics setting [110], or go through a population
inversion procedure that makes it behave like a negative effective temperature
bath [111]. Indeed, several instances of non-thermal QEs have been
theoretically devised [112, 113, 114], and experimental proofs-of-principle
have been proposed [111, 115, 116]. These systems’ apparently dramatic
implications for thermodynamic laws have been addressed in several theoretical
works, which in some cases developed generalizations aimed at describing the
energetics of the systems. Below we provide an overview of several non-thermal
QEs, deferring our discussion of non-Gibbsian states arising from strong
correlations between the working medium and the bath to Sec. 6.

###  4.1 The phaseonium and quantum coherence induced by dissipation

Quantum coherent baths are a prominent type of thermal bath. A QE operating on
this type of bath shows additional thermodynamic features that are impossible
to achieve by means of conventional heat engines. Non-thermal QEs were first
introduced as quantum optical devices. Quantum interference in a system
subject to radiation fields can break the detailed balance between photon
absorption and emission processes. As a consequence, the thermodynamic
features of these systems must be modified, leading to phenomena like work
extraction from a single reservoir and the breaking of the Carnot limit in
two-reservoir engines.

Figure 4: Schematic diagram of early prototypes of QEs. $a)$ A three-level
Maser QE. Hot (cold) reservoirs are modeled with thermal light sources at
temperatures $T_{h}(T_{c})$. Filter waveguides allow the absorption of hot
(cold) photons of frequency $\omega_{p}$($\omega_{i}$). The population
inversion takes place between levels 1 and 2. Work is performed by spontaneous
emission of radiation of frequency $\omega_{s}$. $b)$ The Phaseonium QE. A
radiation mode inside a cavity performs work on the mirrors by means of
radiation pressure, playing the role of the engine’s working medium. The hot
reservoir is an ensemble of three-level $(a,b,c)$ atoms, which are in contact
with a thermal bath at temperature $T_{h}$. The two lower levels are quasi-
degenerate, such that the state of the atom can show quantum coherence,
dependent on the atomic phase difference $\phi$. The atoms are injected into
the cavity and thermalize the radiation field at temperature $T_{\phi}$,
controlled by $\phi$. The state of the reservoir is thus equivalent to a
thermal state with effective temperature $T_{\phi}$ that, due to quantum
coherence, can be higher than $T_{h}$. A conventional entropy sink at
temperature $T_{c}$ is also used. Reproduced from [117].

In their important contribution to the study of such engines, Scully et al.
[117] propose a modified version of the QHE that works with a quantum coherent
bath, i.e., the _phaseonium_ heat engine. The idea builds on previous
breakthroughs in quantum optics, namely micromasers and Lasing Without
Inversion (LWI) [118]. Unlike in conventional QHEs, the working medium is a
mode of radiation inside a cavity, and the work is accomplished through
radiation pressure on the cavity mirrors. An ensemble of three-level atoms
replaces the thermal light and acts as the hot bath. Each three-level atom has
a level structure as in Fig. 4 b, where coherence is allowed between the two
lower, quasi-degenerate levels. An entropy sink is also present, working as
the cold bath. As the atoms are injected into the cavity, the radiation inside
the cavity is brought to a quantum thermal state that depends on the phase
difference $\phi$ of the two lower atomic levels. The mean number of photons
inside the cavity, $\bar{n}_{\phi}$, obeys the following rate equation

$\dot{\overline{n}}_{\phi}=\alpha(2\rho_{aa}(\overline{n}_{\phi}+1)-(\rho_{bb}+\rho_{cc}+\rho_{bc}+\rho_{cb})\overline{n}_{\phi}),$
(16)

where $\rho_{ij}$ are the elements of the density matrix of the atoms and
$\alpha$ is a rate factor. It is thus clear that however small, quantum
coherence in the bath affects the steady-state number of photons in the
cavity. From a microscopic treatment [119], it follows that the steady-state
number of photons reads

$\overline{n}_{\phi}\simeq n_{\rm{th}}(1-n_{\rm{th}}\varepsilon\cos\phi),$
(17)

where $n_{\rm{th}}=((P_{b}+P_{c})/2P_{a}-1)^{-1}$ is the photon population in
the absence of coherence and $P_{i}$ is the Boltzmann weight of the level $i$.
It is worth stressing that the quantum state of radiation is thermal and can
be characterized by a parameter of temperature

$T_{\phi}=T_{\rm{h}}(1-n_{\rm{th}}\varepsilon\cos\phi),$ (18)

where $\varepsilon$ is a parameter linked to quantum coherence that, in the
high-temperature limit, reads $\varepsilon=3|\rho_{bc}|$. It follows that the
proper control of phase difference $\phi$ can lead to a thermal state of
radiation with effective temperature $T_{\phi}>T_{h}$.

A remarkable consequence is that work can be extracted from the engine even in
the case of equal temperature, i.e., $T_{c}=T_{h}$, which is conceptually
analogous to extracting work from a single reservoir. However, this task does
not violate thermodynamic laws, as additional energy is required to prepare
the initial coherences in the bath state. Thus, quantum coherence acts as an
additional quantum resource [120], similar to a special kind of _fuel_ for the
QE [121]. Indeed, it permits energy extraction from the high-temperature bath
more efficiently than traditional engines allow.

Moreover, this setup devises a thermodynamic cycle, composed of an isothermal
expansion at $T=T_{\phi}$, followed by an adiabatic expansion, where the atoms
are removed from the cavity. Subsequently, the radiation cavity is coupled
with the entropy sink, undergoing an isothermal compression at $T=T_{c}$
followed by an adiabatic compression that brings it back to its initial state.
With small cavity volume variations, using Eq. (18) the engine efficiency can
be written as

$\eta\simeq
1-\frac{T_{c}}{T_{h}}(1+n_{\rm{th}}\varepsilon\cos\phi)=\eta_{\rm{C}}-3\frac{T_{c}}{T_{h}}n_{\rm{th}}|\rho_{bc}|\cos\phi.$
(19)

In the presence of quantum coherences, by modulating the phase difference of
the two low-lying atomic levels, the phaseonium efficiency can be tuned to
exceed the Carnot bound. A recent study [122] revisiting this model explored
the power-efficiency trade-off when considering finite-time processes.

The detailed balance can also break in a more conventional QE working between
two thermal baths, such as a laser or photocell QE [123].

Figure 5:  Schematic diagram of the photocell QE. An ensemble of three-level
atoms (orange dots) immersed in a cavity acts as the working medium. Hot
(cold) reservoirs are again modeled with thermal light sources at temperatures
$T_{h}(T_{c})$. Lasing transition is achieved between levels $a,\beta$, after
pumping from $b_{i}$ to $a$ ($\beta$ to $b_{i}$) by means of hot (cold)
photons. Due to the degeneracy of the lower levels ($b_{1},b_{2}$), different
atomic paths interfere, so that quantum coherence builds up as induced by the
baths. No additional resources are required. Reproduced from [123].

The latter device, depicted in Fig. 5, is more similar to conventional maser
QHEs described in Sec. 2.3.1 above. However, here the two baths are thermal
light sources at different temperatures $T_{h},T_{c}$. The working medium is
again a three-level quantum system. The baths provide hot and cold photons to
optically pump the system and achieve lasing conditions. Unlike in
conventional QHEs, the lower level is replaced by a couple of nearly
degenerate levels. This change has a crucial consequence, as stimulated
emission processes from upper levels can interfere, producing coherences
between the two lower states that modify the steady state of the working
medium. Moreover, the bath-induced coherences enhance the photon absorption
from the reservoirs, as they lead to a faster depletion of the lower levels in
favor of the upper ones. As a result, a net improvement in the power output of
the engine can be obtained. The power can be written as

$P_{l}=A(\overline{n}_{h}-\overline{n}_{c})\hbar\omega_{l},$ (20)

where $\omega_{l}$ is the laser frequency and $\overline{n}_{h/c}$ is the
average number of hot/cold photons. In the presence of bath-induced
coherences, it has been shown that the prefactor $A$ doubles when convenient
model parameters are chosen. If the photocell QHE is considered, better
current-voltage characteristics can be obtained. This proposal’s distinctive
trait is that quantum coherence is not provided by any form of external fuel
but is rather induced by the noise linked to the same two baths that sustain
the laser operating condition. As a consequence, in contrast to the phaseonium
engine, no additional costs to produce quantum coherence are required. Other
models and experiments [124, 125, 126] have studied the enhancement of the
power output due to a coherence reservoir. In these studies, the engine
operates between superradiant and thermal reservoirs.

###  4.2 Squeezing as a thermodynamic resource

Non-thermal QEs can also be powered by squeezing [118]. Physically, squeezing
is a technique applied to a quantized field of radiation, in order to reduce
the quantum fluctuations of a given quadrature of the field at the expense of
the conjugate one. A non-thermal squeezed QE is composed of a working medium,
e.g., a driven harmonic oscillator, which exchanges energy between two
reservoirs at inverse temperatures $\beta_{1}$ and $\beta_{2}$, such that
$\beta_{1}>\beta_{2}$. In contrast to conventional heat engine setups, one of
the two reservoirs, let us say the hot one, is prepared into a squeezed
thermal state [127]. In the case of reservoirs modeled with noninteracting
bosons, the non-equilibrium squeezed thermal state reads

$\rho_{s}=P\frac{e^{-\beta H_{B}}}{Z_{B}}P^{\dagger},$ (21)

where $H_{B}=\sum_{k}\hbar\omega_{k}a^{\dagger}_{k}a_{k}$, $P=\prod_{k}P_{k}$,
and $P_{k}=\exp[1/2(\xi^{*}_{k}a^{2}_{k}-\xi_{k}a^{\dagger 2}_{k})]$ are the
single-mode squeezing operator corresponding to the squeezing parameter
$\xi_{k}=|\xi_{k}|e^{i\theta_{k}}$. The squeezed thermal bath is thus properly
characterized by its inverse temperature $\beta$ and the squeezing parameter
$\xi_{k}$ of each mode. In a squeezed thermal bath, the fluctuations of the
position and momentum of each of the bath’s modes can exponentially decrease
(or increase) depending on $\xi_{k}$, while their product does not vary [118].
Moreover, the mean correlation functions are not uniform in time, as with a
conventional bath [128]. These non-equilibrium features of the bath have
severe consequences for the working medium’s thermalization properties.
Indeed, in the weak-coupling, Markovian limit, and in the absence of driving
fields, the stationary state of the working medium is

$\rho_{\rm{st}}=P\frac{e^{-\beta H_{S}}}{Z_{S}}P^{\dagger},$ (22)

and it follows that the variances in the position and momentum of the working
medium are squeezed as well.

The idea of a squeezed QHE was first proposed in [110]. An Otto cycle was
devised (see also Sec. 2.3), composed of four strokes (ABCD), as depicted in
Fig. 6.

Figure 6: Sketch of the Otto cycle with a squeezed reservoir and the energy
levels of the working medium. $\text{A}\to\text{B}$ is the isentropic
compression stroke, the working medium is decoupled from the thermal
reservoir, and the oscillation frequency is increased. $\text{B}\to\text{C}$
is the isochore stroke; the working medium interacts with the squeezed
reservoir and reaches the squeezed thermal state in Eq. (22), increasing its
energy. $\text{C}\to\text{D}$ is the isentropic expansion stroke, the working
medium is decoupled from the squeezed reservoir, and the oscillation frequency
is decreased to the initial value. $\text{D}\to\text{A}$ is the final isochore
stroke, the working medium thermalizes, and its energy is released in the cold
bath.

The working medium is a driven harmonic oscillator described by the
Hamiltonian $H_{S}(t)=\hbar\omega(t)b^{\dagger}b$. It starts in a thermal
state at inverse temperature $\beta_{1}$. An isentropic compression stroke
($\text{A}\rightarrow\text{B}$) is then performed, i.e., the frequency of the
oscillator is increased from a starting value $\omega_{\rm{1}}$ to
$\omega_{\rm{2}}$, while it is decoupled from the reservoir. Next the
oscillator is put in contact with the squeezed bath, while its frequency is
held constant ($\text{B}\rightarrow\text{C}$). Its energy, which is a function
of the inverse temperature $\beta_{2}$ and the squeezing parameter $r$, is
increased, as is the asymmetry in the variances of position and momentum
[129]. Furthermore, the oscillator is decoupled from the bath and undergoes an
isentropic expansion stroke ($\text{C}\rightarrow\text{D}$), where its
frequency is brought back to its initial value $\omega_{1}$. Eventually, the
cycle is closed by means of another isochoric stroke
($\text{D}\rightarrow\text{A}$), where the oscillator is coupled to the
thermal reservoir at inverse temperature $\beta_{1}$.

It is assumed that the isochoric strokes take place in a much shorter time
than the isentropic strokes, and both the time needed to reach the steady-
state and the work needed to turn on the interactions with the baths [130] are
neglected. In the weak coupling limit, the final state after thermalization is
known, so that the amount of energy exchanged along each stroke can be derived
analytically. It follows that the efficiency of the cycle reads [110]

$\eta^{*}=1-\frac{\omega_{1}}{\omega_{2}}\frac{\coth(\hbar\beta_{1}\omega_{1}/2)-Q^{*}_{2}\coth(\hbar\beta_{2}\omega_{2}/2)\Delta
H(r)}{Q^{*}_{1}\coth(\hbar\beta_{1}\omega_{1}/2)-\coth(\hbar\beta_{2}\omega_{2}/2)\Delta
H(r)}.$ (23)

Here $\Delta H(r)=1+(2+1/n_{\rm{th}})\sinh^{2}(r)$ is the ration between the
squeezed and the thermal occupation
$n_{\rm{th}}=(\exp(\hbar\beta_{2}\omega_{2})-1)^{-1}$, and the $Q_{i}^{*}$ are
functions that arise from the exact non-equilibrium dynamics of the oscillator
during the isoentropic strokes [131]. When $r=0$, we have $\Delta H(r)=1$, and
the usual thermal equilibrium populations are restored. From Eq. (23), under
conditions of adiabatic compression/expansion strokes, and in the high-
temperature limit, the efficiency at maximum power of the cycle is derived:

$\eta^{*}=1-\sqrt{\frac{\beta_{2}}{\beta_{1}(1+2\sinh^{2}(r))}}.$ (24)

Eq. (24) shows that the efficiency of the engine equals the Curzon-Ahlborn
bound for $r=0$. However, when the squeezing parameter is increased, the
efficiency can surpass the Carnot bound, approaching unity for $r\gg 1$.

The surprising features of squeezed QEs have spurred novel theoretical
research efforts aimed at achieving a solid understanding of the
thermodynamics involved. The entropy production in squeezed QHEs was first
addressed in [129]. Here a generalization of the Spohn inequality [25, 128,
132] for the entropy production is obtained, in order to deal with the
squeezed nature of the bath at inverse temperature $\beta$. More specifically,
a non-thermal contribution to the _heat_ exchange is considered. The time
derivative of the entropy production is rewritten as

$\dot{\Sigma}=-\frac{d}{dt}D(\rho_{s}||\rho_{\rm{st}})=\dot{S}-\dot{\Phi},$
(25)

where $S=-\mathrm{tr}[\rho_{s}\log\rho_{s}]$ is the Von-Neumann entropy of the
working medium, $D(||)$ denotes the quantum relative entropy [133, 134], and
$\dot{\Phi}=-\mathrm{tr}[\dot{\rho}_{s}\log\rho_{\rm{st}}]$. From the form of
the steady-state in Eq. (21), it follows that

$\dot{\Phi}=\beta(\cosh(2r)\dot{Q}-\sinh(2r)\dot{\mathcal{A}}),$ (26)

where, after defining the asymmetry operator
$A=(\hbar\omega/2)(p^{2}_{\theta/2}-x^{2}_{\theta/2})$,
$\dot{\mathcal{A}}=\mathrm{tr}[A\dot{\rho_{s}}]$. The main consequence of Eqs.
(25) and (26) is that, in a dissipative evolution, the total variation
$\Delta\Phi$ can become negative, thereby overcoming the total entropy
production $\Sigma$ that is nonnegative by definition. As a consequence, the
working medium’s Von-Neumann entropy variation can become negative.

If simple two-stroke cycling processes are considered, where the system is put
in contact with a single reservoir, it follows from the previous analysis that
a positive amount of work $W_{\rm{out}}$ can be extracted from the reservoir
that is bounded from above,

$W_{\rm{out}}\leq\tanh(2r)\Delta\mathcal{A}.$ (27)

As a consequence, the amount of work extracted is linked to the increase in
the asymmetry of the reduced state. The maximum work can be extracted when the
initial state is thermal, i.e., $\rho_{s}=e^{-\beta H_{S}/Z_{S}}$, and reads

$W_{\rm{max}}=\hbar\omega(2n_{\rm{th}}+1)\sinh^{2}(r).$ (28)

Even more striking results appear if a slightly modified Otto cycle is
considered [129], where an additional unitary transformation is introduced to
unsqueeze the state after decoupling from the squeezed bath. The surplus of
work coming from the squeezed reservoir leads to entirely new operating
regimes for the Otto cycle. It follows that the QHE can work as a refrigerator
and perform work extraction simultaneously. The controversial results in the
thermodynamic performance of the Otto cycle in Eq. (22) have been traced back
to the additional amount of work extractable from the squeezed reservoir,
which is absent in the case of thermal baths. Squeezing, therefore, can be
seen as an additional quantum resource to perform thermodynamic work. This
result implies that the Carnot bound for heat engines operating between two
heat baths is not relevant when squeezing is involved, and a generalized bound
depending on the squeezing parameter $r$ has been shown to hold [110].

The derivation of generalized bounds is not limited to driven, four-stroke
heat engines. In [135], a generalized bound was derived for a three-terminal
continuous heat engine, in which one of the terminals is a photonic squeezed
thermal bath. The working medium is a TLS that exchanges energy with a
photonic squeezed reservoir with a two-terminal electronic circuit. The latter
is made of two metal leads acting as thermal reservoirs, with fixed values of
inverse temperature and chemical potential, i.e., $\beta,\mu$. In this way,
conversion from photonic energy to electrical work is studied through a full
counting statistics approach and TPM protocol. A universal bound on the
efficiency is derived in terms of effective inverse temperatures, which holds
for a generic non-equilibrium reservoir. In the instance of the proposed
photoelectric device, with leads at the same temperature and within the high-
temperature limit, the computed bound reduces to Eq. (24).

These controversial theoretical results were later confirmed by an
experimental work [115], which employs a GaAs nanobeam as a working medium,
modeled by a quantum harmonic oscillator. The nanobeam is piezoelectrically
coupled to a source of squeezed electronic noise. Both work extraction from a
single reservoir and the breaking of the Carnot bound in an Otto cycle have
been observed through this method. However, these experimental findings cannot
be interpreted as a way to circumvent the second law of thermodynamics.

Different setups for quantum thermal machines were also considered as
sensitive probing devices for temperature estimation [136, 137]. It was shown
in [137] that small temperature variations between the engine baths can be
evaluated by measuring a flywheel that stores the engine’s energy output. In
this specific setup, the flywheel is growing macroscopic oscillations of a
trapped ion that can be resolved using fluorescent imaging on a camera.
Squeezing the working medium after the thermalizing strokes significantly
amplifies the flywheel energy and the sensitivity in detecting very small
temperature differences. Moreover, when the squeezing attains the quantum
regime, the sensitivity is amplified by one order of magnitude.

###  4.3 QHEs with negative temperature baths

Over the last two decades, reservoir engineering techniques have been employed
in several tasks, e.g., quantum state preparation, generation of entangled
quantum states, and decoherence reduction. Negative-temperature quantum
states, i.e., inverted-population states [138, 139], have been studied since
the sixties by employing nuclear spins and NMR techniques [140, 141]. More
recently, cold atoms in optical lattices have been employed [142, 143]. Over
the years, the quest for a consistent thermodynamic description of these
states has spurred many debates [144, 145], as they were believed to lead to a
reformulation of the second law of thermodynamics. Quite recently, alternative
theoretical schemes have been proposed to describe these states without the
need for modification of thermodynamic laws. Indeed, a negative-temperature
state can be recast as a temperature-unstable state [146]. In contrast to
equilibrium states, which are the focus of the second law, these states can
store work, and it can be extracted to perform various thermodynamic tasks, as
with a flywheel [147, 116]. Adopting this scheme, several experimental
proposals of QEs working with engineered negative temperatures have been put
forward [111, 148, 149]. An explicit derivation of a Master equation with an
effective negative temperature can be found in the appendix of [147].

In [111] a QE is described that operates between a thermal reservoir at
positive temperatures and a second, engineered reservoir prepared at effective
negative temperatures. The experimental platform consists of
${}^{13}\text{C}$-labeled $\text{CHCl}_{3}$ liquid and an NMR spectrometer. In
this setting, the nuclear spin $1/2$ of the ${}^{13}\text{C}$ plays the role
of the working medium, while the nuclear spin of the ${}^{1}H$ atom mimics the
reservoir. The two nuclei mutually interact and they can be driven by RF
pulses. Similar platforms have been employed to engineer Quantum Maxwell’s
Demon engines (see Sec. 8.2.2). A pivotal feature of the setup, the
application of the RF fields, allows for full control of the spin populations,
that is, the state of the working medium can be prepared so as to obey
different Boltzmann distributions. The inverse temperature for a thermal state
of the working medium described by the Hamiltonian
$H_{i}=\omega_{i}\sigma_{i}$ is related to the populations of the excited
states via the relation
$\beta_{i}=(1/\hbar\omega_{i})\ln[(1-p^{+}_{i})/p^{+}_{i}]$. It follows that
inverting the population, $0.50<p^{+}\leq 1$, permits simulating a negative-
effective temperature state.

The engine cycle runs through a cooling stroke, where the carbon nucleus is
prepared in a thermal state $\rho_{1}=e^{-\beta_{c}H_{c}}/Z_{c}$, with
$H_{c}=-(1/2)\omega_{c}\sigma^{C}_{x}$. Then a gap expansion stroke takes
place in a finite time $\tau$, which is of the order of $\mu s$, i.e., much
smaller than the decoherence time. In this stroke, the evolution is unitary,
the frequency is increased to $\omega_{h}$, and the Hamiltonian changes to
$H_{h}=-(1/2)\omega_{h}\sigma^{C}_{y}$, in a rotating frame with the Larmor
frequency of the working medium. Next, the working medium goes through the
heating stroke. This thermalization is simulated by employing a combination of
RF pulses and free evolution gates under the Hamiltonian $H_{J}=(1/4)\hbar
J\sigma^{C}_{z}\sigma^{H}_{z}$, and the final state is
$\rho_{3}=e^{-\beta_{h}H_{h}}/Z_{h}$. The final, unitary stroke takes the
frequency of the working medium back to the initial value.

The engine’s performance shows remarkable properties, namely the efficiency is
shown to exceed the Otto efficiency limit. Note that the Otto efficiency has
been fulfilled in similar experimental setups [150], where thermal baths are
used. Moreover, during the finite-time gap expansion/compression strokes,
transitions between the eigenstates of the working medium are shown to occur
that increase the average extracted work $\left<W\right>$. Consequently, the
engine’s performance improves with decreasing running time.

###  4.4 QEs with correlated baths

A different type of QEs allows the baths to share correlations, both classical
and quantum, among their constituents [151]. A recently devised model of QHE
working with correlated baths is based on the theoretical framework of
collisional models [152, 153, 154, 155]. In [156], a QHE is proposed whose
working medium is made of two XXZ-coupled qubits $S_{1}$ and $S_{2}$, as
depicted in Fig. 7. The qubits, described by means of Pauli operators
$\sigma_{i},i=1,2$, are also subject to local magnetic fields $B_{1}$ and
$B_{2}$. The reservoirs are modeled by two sets of noninteracting qubits,
described by a local Hamiltonian $H_{B}=B_{i}\tilde{\sigma}_{i}$, and they are
initially prepared at equilibrium in a product of thermal states
$\rho_{B}=\rho_{\rm th}(\beta_{1})\otimes\rho_{\rm th}(\beta_{2})$.

Figure 7: Schematic diagram of the basic mechanism behind the two-qubit
correlated engine. The reservoirs are modeled with noninteracting qubits. They
emerge from the reservoirs and correlate after the action of the unitary gate
$U$. The working medium can thus interact with the correlated reservoir.
Several operating regimes can be studied by changing the quantum and classical
correlations generated by means of $U$. Reproduced from [156].

In the repeated interactions scheme, $S_{1}$ and $S_{2}$ interact with the
environmental qubits by means of collisions, i.e., the interaction takes place
over a finite time $\tau$, during which the Hamiltonian is constant in time.
In the continuous limit ($\tau\to 0$), the effect of multiple collisions can
be described by means of a Markovian quantum master equation [155].

In contrast to conventional approaches, before colliding with the working
medium, a unitary $U$ is applied to the baths, so that each couple of the
qubits is described by the state $\rho^{\prime}_{B}=U\rho_{B}U^{\dagger}$. As
a consequence, classical and quantum correlations develop, which affect the
steady-state of the system after a great number of collisions, $\rho^{\rm
steady}(t+\tau)=\rho^{\rm steady}(t)$. Here the steady state is the state of
the working medium and the baths for which the variation of the internal
energy amounts to zero, $\Delta
E=\mathrm{tr}[H_{S}(\rho^{\prime}_{SB}-\rho_{SB})]=0$, with
$\rho_{SB}=\rho^{\rm steady}_{S}\otimes\rho^{\prime}_{B}\mbox{,
}\rho^{\prime}_{SB}=U_{\rm coll}\rho_{SB}U^{\dagger}_{\rm coll}$, where
$U_{\rm coll}=\exp(-iH_{SB}t)$.

In suitable parameter ranges, the system performs thermodynamics tasks, and it
can work as a heat engine and refrigerator. The energetics of this machine is
analyzed by studying the work $W$, heat $Q$, and the entropy production
$\Sigma$. However, as with the squeezed reservoir (see 4.2, 4.5), the cost of
maintaining the bath correlations, that is
$W_{U}=\mathrm{tr}[H_{B}(\rho^{\prime}_{B}-\rho_{B})]$, directly enters the
energetic balance of the machine, such that in the complete case the
refrigeration effects are ruled out. A wide range of unitaries are chosen to
correlate the two baths. In the instance of a swap gate [75],
$S(\phi)=\exp[-i\frac{\phi}{2}(\tilde{\sigma}_{x,1}\tilde{\sigma}_{y,2}-\tilde{\sigma}_{y,1}\tilde{\sigma}_{x,2})]$,
the populations of the two qubits are partially exchanged, while the reduced
states are kept thermal with temperatures $\beta_{1},\beta_{2}$.

It is found that the work output is generally amplified with a partial swap
and maximized when a complete swap of the bath qubits’ populations is achieved
for $\phi=m\pi/2$ with $m\in\mathbb{Z}$. The engine develops a great amount of
both classical and quantum correlations during its operation, and the mutual
information is used to quantify these correlations and the quantum discord
[157, 158] to study its quantumness. Correlations among the baths qubits are
absent when the work output is maximal, while the system-bath qubits
correlations, on the other hand, play an important role in maximizing the
performance of the QE.

###  4.5 Thermodynamics of non-thermal QE

Following the proposals of QEs operating with non-thermal working medium, and
with non-thermal heat baths (see [121] for useful classification), the quest
to understand the thermodynamics of these machines regained momentum. The
apparent violations of thermodynamic laws could be traced to the thermodynamic
inconsistency of the adopted definition of heat in these engines. Competing
theoretical pictures have been developed, which rely on different
generalizations of fundamental thermodynamical concepts, such as heat, second-
law inequality, and entropy production [132].

Quantum correlations induced by non-Gibbsian equilibrium states modify the
entropic cost of every quasi-static thermodynamic process, as well as the work
exchanged [159]. It follows that the thermodynamic cost of correlations, i.e.,
the _housekeeping heat_ , has to be taken into account. A useful strategy for
performing the task was proposed in [159], for a rather generic non-thermal
state $\sigma$. The von-Neumann entropy of $\sigma$ is given by

$\mathcal{S}=\beta(\mathcal{E}-\mathcal{F}),$ (29)

with $\mathcal{E}=\mathrm{tr}{[\sigma H]}$,
$\mathcal{F}=F+\frac{1}{\beta}D(\sigma||\rho_{\rm{th}})$ the information free
energy, and $F$ the thermodynamic free energy of the thermal state
$\rho_{\rm{th}}$. Note that the relation in Eq. (29) reduces to traditional
thermodynamics potential for $\sigma=\rho_{\rm{th}}$. Using Eq. (29), the
entropic cost of a quasistatic transformation can be written in terms of the
excess heat, $d\mathcal{S}=\beta\delta Q_{\rm{ex}}$, with

$\delta Q_{\rm{ex}}=\delta Q_{\rm{tot}}-\delta Q_{c}.$ (30)

Here $\delta Q_{\rm{tot}}=\mathrm{tr}[\delta\sigma H]$ is the total heat and
$\delta Q_{c}=d\mathcal{F}-\mathrm{tr}[\sigma\delta{H}]$ is the correlation
heat, which can be interpreted as the price to be paid to maintain non-
Gibbsian correlations. Analogous reasoning leads to the definition of the
excess work as $\delta W_{\rm{ex}}=\delta W+\delta Q_{c}$, so that the first
law reads $d\mathcal{E}=\delta W_{\rm{ex}}+\delta Q_{\rm{ex}}$. The previous
definition of cost linked to quantum correlations was included in the
evaluation of the QE’s efficiency in performing the Carnot cycle. It was found
that, when the cost of non-Gibbsian correlation is correctly accounted for in
the heat to work conversion balance, the Carnot bound is restored.

Similar information-theoretic approaches have been pursued in studying
generalized system-bath correlated systems [160]. The second law can be
suitably reformulated if heat is quantified by means of conditional entropy.
Moreover, work extraction from system-bath correlations is possible and, as a
result, a generalized form of the Helmholtz free energy can be defined. As a
consequence, for generic correlated systems where non-thermal states are
involved, the inclusion of work extractable from correlations permits
reconciling the physics of correlated systems with traditional thermodynamics
and the Carnot bound.

Owing to the non-equilibrium nature of bath states, _ergotropy_ [161, 162] has
been proposed as a key ingredient to explain the energy exchange between the
working medium and the bath. Ergotropy is defined as the maximum amount of
work that can be extracted from a given state using only unitary
transformation that performs a cyclic variation of the Hamiltonian. Non-
equilibrium bath states, such as the squeezed-thermal state, are non-passive,
i.e., by applying a unitary transformation, it is possible to reduce their
energy and extract a finite amount of work. These states are a mainstay in the
study of quantum batteries [163, 164, 165].

Using these concepts of passive states and ergotropy, thermodynamic
inequalities have been reformulated to account correctly for the effects
linked to non-equilibrium baths [166]. When restricted to the weak-coupling
regime, i.e., neglecting system-bath correlations, for a generic quantum
evolution $\rho_{0}\to\rho(t)$ corresponding to a time-dependent Hamiltonian
$H(t)$, the change in the energy of the system can be written as

$\Delta E(t)=\mathcal{E}_{d}(t)+W(t),$ (31)

where
$\mathcal{E}_{d}(t)=\int_{0}^{t}\mathrm{tr}[\dot{\rho}(t^{\prime})H(t^{\prime})]\mathrm{d}t^{\prime}$
and
$W(t)=\int_{0}^{t}\mathrm{tr}[\rho(t^{\prime})\dot{H}(t^{\prime})]\mathrm{d}t^{\prime}$.
In the absence of system-bath correlations, the two contributions in Eq. (31)
can be identified as the heat and work exchanged with the reservoir,
respectively. For thermal states, the work can be extracted by controlling the
time-dependent Hamiltonian. However, for non-thermal states,
$\mathcal{E}_{d}(t)$ can be recast in terms of the passive state $\pi(t)$,
obtained from $\rho(t)$ by means of unitary operations, as follows

$\mathcal{E}_{d}(t)=\Delta E_{{\rm pas},d}(t)+\Delta W_{d}(t),$ (32)

with $\Delta E_{{\rm
pas},d}(t)=\int_{0}^{t}\mathrm{tr}[\dot{\pi}(t^{\prime})H(t^{\prime})]\mathrm{d}t^{\prime}$
and $\Delta
W_{d}(t)=\int_{0}^{t}\mathrm{tr}[(\dot{\rho}(t^{\prime})-\dot{\pi}(t^{\prime}))H(t^{\prime})]\mathrm{d}t^{\prime}$.
Eq. (32) separates between different contributions to $\mathcal{E}_{d}(t)$.
Here $\Delta E_{{\rm pas},d}(t)$ is the contribution due to the change in the
passive state $\pi(t)$, while $\Delta W_{d}(t)$ quantifies the change in the
system ergotropy as a result of system-bath interaction.

The crucial observation related to Eq. (32) is that only the passive part can
be linked to a change in the Von-Neumann entropy of the system, i.e., a change
in the passive state is a necessary condition for obtaining a change in
entropy. It follows that dissipative evolutions of the working medium exist
along which the energy change $\mathcal{E}_{d}(t)$ is different from zero,
without a corresponding change in the Von-Neumann entropy,
$\Delta\mathcal{S}=0$. As a consequence, the Spohn inequality,
$\Sigma=\mathcal{S}(\rho_{0}||\rho_{\rm{st}})\geq 0$, with
$\mathcal{S}(\rho_{1}||\rho_{2})=k_{\rm{B}}\mathrm{tr}[\rho_{1}\log\rho_{1}-\rho_{1}\log\rho_{2}]$,
may provide too loose a bound for dissipative systems in which non-equilibrium
baths are involved, i.e., ergotropy can be transferred from the bath to the
system ($\Delta W_{d}\geq 0$). In the case of a time-independent Hamiltonian,
the Spohn bound is entailed by the following relation

$\Delta\mathcal{S}\geq\frac{1}{T}\Delta E_{{\rm pas},d}.$ (33)

It follows that a tighter bound on entropy production is needed to deal with
non-equilibrium bath states. The proposal put forward in [166] only deals with
passive states,

$\mathcal{S(\pi_{\rm{0}}||\pi_{\rm{st}})}\geq 0.$ (34)

When a slowly-varying driving field is present, the scenario is more involved,
as the time-dependence of the Hamiltonian changes passive states into non-
passive ones. However, ergotropy might be extracted from the initial state
$\rho_{0}$ by performing a suitable unitary transformation,
$\pi_{0}=U^{{\dagger}}\rho_{0}U$. In this case, the system is put in contact
with the bath, at which point it is possible to apply the conventional Spohn
inequality. A bound for the entropy variation can thus be obtained as follows

$\Delta\mathcal{S}\geq\mathcal{E}^{\prime}_{d}\;\;{\rm
with}\;\;\mathcal{E}^{\prime}_{d}=\int_{0}^{+\infty}\mathrm{tr}[\dot{\tilde{\rho}}(t)H(t)]\mathrm{d}t,$
(35)

where $\tilde{\rho}(0)=\pi_{0}$. One of the main consequences of this
reformulation is that QEs working with non-equilibrium baths obey efficiency
bounds that are different and greater than the Carnot bound. If the bath
provides only ergotropy, the QE behaves like a mechanical engine and
$\eta_{\rm{max}}\to 1$. In the opposite limit, if no ergotropy is present in
the bath state (thermal engine), $\eta_{\rm{max}}=\eta_{\rm{C}}$ as with
conventional QHEs. Alternative approaches to modeling energetic and entropic
costs of non-thermal QEs can utilize studies on the thermodynamics of
generalized Gibbs reservoirs [167, 168].

##  5 Many-body Quantum Engines

QHEs have been studied extensively with single-particle working medium in both
the quasistatic limit as well as in finite time cycles. Recently, many-body
systems have been investigated for their influence on the performance of QHEs.
Many-body systems (free or interacting) provide a rich arena for studying and
engineering a wide variety of physical phenomena in condensed-matter physics.
Inarguably, the real power of quantum mechanics in fundamental sciences and
technology, in general, has been unleashed through the understanding and
control of many-body systems. In the case of Quantum Thermodynamics, many-body
systems are a natural choice of working medium for exploring quantum effects
in meso- and nano-scale thermal devices.

Many-body cooperative phenomena can already be observed in the two-spin
Lipkin-Meshkov-Glick (LMG) model as the working medium [169]

$H=-\frac{J}{4}\Big{[}\sigma^{x}_{1}\sigma^{x}_{2}+\gamma\sigma^{y}_{1}\sigma^{y}_{2}\Big{]}-\frac{h}{2}\Big{[}\sigma^{z}_{1}+\sigma^{z}_{2}\Big{]}-\frac{J(1+\gamma)}{4}$
(36)

where $\sigma^{i}$ are the Pauli matrices. This Hamiltonian exhibits a wide
range of behaviors depending on the parameter $\gamma$. When $\gamma=1$, the
model is reduced to the isotropic $XY$ model, while, in the range
$\gamma\in[-1,1)$, it becomes anisotropic. In [169] a detailed analysis of all
the Hamiltonian parameters is introduced, showing that anisotropy
significantly enhances the work output, even towards high efficiency of the
operating regime.

Another approach exploring collective phenomenon was put forward in [170]. In
this scheme, coherence was extracted from one single-body engine and injected
into another. The collective work extraction from the engines exhibits an
enhancement that scales quadratically with the number of engines. Improvements
in efficiency and work extraction are not the only advantages of collective
effects. In [171] it was further shown that these effects may stabilize the
power output fluctuations of the engine, as measured by the constancy [172].
In the following, we discuss some of the many-body QE unique phenomena in
detail and present exactly solvable models.

###  5.1 Many-body engines and criticality

The Carnot bound is a fundamental thermodynamic constraint on the performance
of thermal devices harvesting work, as has been discussed earlier in the
review. An important caveat in attaining this bound is the infinite time
required to complete a cycle. Thus, the power delivered by the engine vanishes
at this limit [5]. Likewise, any realization of finite generation of power in
a realistic engine cycle therefore sacrifices efficiency of the engine,
$\Delta\eta=\eta_{\rm C}-\eta$. The performance rate
$\dot{\Pi}=\frac{\mathcal{P}}{\eta}$ is relevant to characterizing the
relationship between power output ($\mathcal{P}$) and efficiency ($\eta$) in
QHEs.

One possibility for increasing the power without affecting the efficiency is
scaling the size of the working substance : an array of $N$ identical engines
working in parallel provide an N-fold larger power than each of them, at the
same efficiency with which $\dot{\Pi}\approx N$. However, this linear increase
in performance rate does not represent any real gain, as it is achieved at the
cost of a corresponding linear increase of resources.

Figure 8: Schematic diagram of the many-body heat engine at criticality. Left
(Right) panel: scaling of the performance rate $\dot{\Pi}$ of the
noninteracting (interacting) many-body engine. Here $a=(\alpha-z\nu)/d\nu$.
Reproduced from [173].

The question “Can the scaling power of the performance rate of QHEs be
increased beyond linear?” was first answered in the affirmative in an
interacting many-body working medium [173] driving an Otto engine. The basic
idea behind this heat engine is depicted in Fig. 8. It was found that a
universal behaviour, with anomalous scaling of the performance rate is given
by

$\dot{\Pi}\approx N^{1+(\alpha-z\nu)/d\nu},$ (37)

when the working medium is on the verge of a second-order phase transition.
Here $\alpha$ is the specific heat, $\nu$ is the correlation length, $z$ is
dynamical critical exponent, and $d$ is the working medium dimensions. The
performance rate is influenced by two important parameters: the scaling of the
heat capacity and of the relaxation time. The working medium is characterized
by a Hamiltonian with a control parameter $\lambda(t)$

$\hat{H}=\lambda(t)\mathcal{\hat{K}},$ (38)

and the residual part $\mathcal{K}$ makes up the many-body operators for a
generic multi-particle system. The heat exchanged with the bath and the work
extracted during the different strokes of the QHE, $W$, can be evaluated,
leading to an efficiency less than Carnot efficiency in accordance with heat
engine fluctuation relations. The performance of the QHE can be written as

$\Pi\approx\frac{\partial W}{\partial\Delta\eta}|_{\Delta\eta=0}\approx
Nc_{k}(\lambda,T),$ (39)

where $c_{k}$ is the specific heat of the working substance. To obtain a
super-linear scaling of the performance, an anomalous scaling of the specific
heat needs to be achieved, $c_{k}\approx N^{a}$ with $a>0$, which can happen
on the verge of a phase transition. The physical reason is that, at a phase
transition, finite exchanges of heat are accompanied by infinitesimal changes
of temperature, leading to the divergence of the specific heat.

Figure 9: Sketch of the many-body Otto cycle. A chain of $N$ spins interacting
by means of XY interactions acts as the working medium. It is equivalent to
$N/2$ independent thermal machines in the Fourier space. The parameter
$\lambda(t)$ modulates the gap in the many-body states. $\text{A}\to\text{B}$
($\text{C}\to\text{D}$) is the energizing (relaxing) isochore stroke, where
the working medium is coupled to a hot (cold) thermal reservoir.
$\text{B}\to\text{C}$ ($\text{D}\to\text{A}$) is the isentropic stroke, where
the working medium is decoupled from the bath and $\lambda(t)$ is increased
(decreased) linearly in time, with finite rates. During the isentropic
strokes, the working medium can cross its critical point, depending on
$\lambda_{1},\lambda_{2}$. Reproduced from [174].

QHEs dictated by quantum and topological phase transitions and critical points
of the many-body working medium have been investigated in various settings
[175, 176, 177, 178]. Transitions between a superfluid and an insulating phase
of the quantum matter driving a quantum Otto engine have recently been studied
[179], showing improved performance in work extraction with a finite-time
dynamical protocol for the unitary strokes.

Quantum criticality in many-body QEs has also been studied in analytically
solvable models [174], accessible in trapped ion experiments. In this study,
the Ising and the XY models in a transverse field with a ring geometry were
considered in a Otto cycle configuration (see Fig. 9). The many-body
Hamiltonian reads

$H=-\sum_{j}M_{j}(c^{\dagger}_{j}c_{j+1}-c_{j+1}c^{\dagger}_{j+1})-\sum_{j}N_{j}(c^{\dagger}_{j}c^{\dagger}_{j+1}-c_{j}c_{j+1})-\sum_{j}R_{j}(c^{\dagger}_{j}c_{j}-c_{j}c^{\dagger}_{j+1}),$
(40)

where $c$ and $c^{\dagger}$ are Fermionic annihilation and creation operators
and the parameters $M,N,R$ are scalars. During the two strokes of the quantum
Otto engine, the unitary dynamics of the working medium, under the influence
of an external parameter (say $\lambda$ with the rate of quench $1/\tau$),
might lead to the crossing of the quantum critical point governed by the
celebrated KZ scaling law. The scaling law is characterized by the density of
defects (excitations) with respect to the ground state Hamiltonian,
$n_{exc}\sim\tau^{\frac{\nu d}{\nu z+1}}$, where $d$ is the dimensionality of
the system, and $\nu,z$ are the correlation length and dynamical critical
exponent respectively. The efficiency of the QHE depends directly on the KZ
scaling law and can be diagnosed via its influence on the work distribution
statistics in quantum thermodynamical settings [180, 181].

###  5.2 Long- and short-range interactions

Many-body solvable Hamiltonians offer a rich array of physics to be explored
in the realm of quantum thermodynamics and nonequilibrium dynamics [182]. A
classic example of such a model describes a many-particle (N) dynamics under a
time-dependent harmonic trap with frequency $\omega(t)$, along with a two-body
interaction potential [183]

$H(t)=\sum_{i=1}^{N}\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{1}{2}\omega^{2}(t)r^{2}(t)\right)+\sum_{i\neq
j}V(|{r}_{i}-{r}_{j}|).$ (41)

The interaction potential is assumed to take the form $V(r/b)=b^{2}V(r)$,
which insure a scale-invariant dynamics in the unitary expansion and
compression strokes. This model is then reduced to the 1-d Calogero-Sutherland
gas of interacting Bosons in a harmonic trap with a pairwise interaction
potential $V(x_{i}-x_{j})=\frac{\lambda(\lambda-1)}{(x_{i}-x_{j})^{2}}$ .

Figure 10: a) The thermodynamic cycle of a many-body engine mediated by the
time-dependent control parameter $\omega$ that tunes the potential well of the
working medium. b) A schematic of the wave-function of a many-body state
mediated by the coupling strength $\lambda$ in contrast to c) a single
particle trapped in a harmonic bath. Reproduced from [183].

Two important parameter values for the interaction strength $\lambda$ are i)
$\lambda=0$ where the model becomes a free Bosonic gas in a harmonic trap and
ii) $\lambda=1$, a Tonks-Girardeu gas of repulsive Hard-Core Bosons [184]. For
other values of $\lambda$, the model exhibits an inverse square law (long-
range) potential and is often reinterpreted as an ideal gas of generalized
exclusion statistics [185]. Thus, the Hamiltonian in Eq. (41) has a rich
basket of intrinsic many-body physics with an exact spectrum of eigenstates
available.

Focussing on the nonequilibrium properties of such a system under external
driving, we note that the non-adiabatic average energy of the system turns out
to be related to the adaiabtic form through

$\langle H(t)\rangle=Q^{*}(t)\langle H(t)\rangle_{ad},$ (42)

where $Q^{*}(t)$ is the adiabatic factor that accounts for the non-adiabatic
excitations over the adiabatic average. The adiabatic limit is readily
achieved when $Q^{*}=1$. The total work input per cycle – corresponding to the
adiabatic compression phase (AB) and expansion phase (CD) when decoupled from
the thermal reservoir – is $\langle W\rangle=\langle W_{1}\rangle+\langle
W_{3}\rangle$, Fig 10. Finally, the non-adiabatic efficiency of the QHE is
given by

$\eta=1-\frac{\omega_{1}}{\omega_{2}}\left(\frac{Q^{*}_{CD}\langle
H\rangle_{C}-\frac{\omega_{1}}{\omega_{2}}\langle
H\rangle_{A}}{\frac{\omega_{1}}{\omega_{2}}\langle
H\rangle_{C}-Q^{*}_{AB}\langle H\rangle_{A}}\right).$ (43)

In the adiabatic limit $Q^{*}=1$, the efficiency reduces to the Otto
efficiency $\eta_{O}=1-\omega_{1}/\omega_{2}$. The efficiency Eq. (43) is
bound by $\eta\leq\frac{1-\omega_{1}Q^{*}_{CD}}{1-\omega_{1}}\eta_{O}$, which
is independent of the number of particles and the inter-particle interactions.

To characterize the many-body effects, it is useful to look at the ratio of
the optimal power output for the N-Particle engine and that of N single-
particle engines, i.e., $P^{N,\lambda}/NP^{1,\lambda}$’ for a given coupling
$\lambda$. Similarly, the ratio between the efficiency at optimal power for
these two cases can be explored, i.e., $\eta^{N,\lambda}/\eta^{1,\lambda}$. It
is concluded in [183] that many-particle quantum effects and nonadiabtic
dynamics can boost both power output and efficiency. Shortcut to adiabaticity
methods [186, 187] were employed in this model [188] and others [189, 190],
showing further improvement in the performance of the many-particle engines
and refrigerators.

A QHE with a working substance made up of tightly confined potential has also
been studied in similar lines [191, 192]. The Hamiltonian studied in this case
is the Lieb-Liniger type Hamiltonian with a contact interaction portrayed by
Dirac $\delta$-distribution. Such models probe the universal nature of Bose
gases at low temperature and the role of quantum criticality. The influence of
many-body effects on quantum thermodynamic cycles has also been explored in
Bose-Einstein condensate with nonlinear interactions [193], and in the LMG
model mentioned above. In this study, however, their influence is considered
in the thermodynamic limit of many particles [194].

A variety of many-body condensed matter systems show the presence of long-
range inter-particle interactions. Ref. [195] considers the implications of
such a working medium’s on the physics of the devices and quantitatively
compares it with the influence of short-ranged interaction on the performance
of a QHE. One such Hamiltonian is given by the extension of the famous Kitaev
chain to include long-ranged forces:

$H_{LR}=-J\sum_{j}^{L}(c^{\dagger}_{j}c_{j+1}+c_{j+1}c^{\dagger}_{j+1})-\mu\sum_{j}^{L}(c^{\dagger}_{j}c_{j}-\frac{1}{2})+\frac{\Delta}{2}\sum_{j}^{L}\sum_{l}^{L-1}\frac{1}{d_{l}^{\alpha}}(c_{j}c_{j+l}+c^{\dagger}_{j}c^{\dagger}_{j+l}).$
(44)

Here $c$ and $c^{\dagger}$ are the fermionic ladder operators on the $j$th
site, and $L$ is the length of the chain. The parameter $J$ controls the
nearest neighbor hopping strength, $\Delta$ controls the fermion pairing
interaction, and $\alpha$ characterizes the range of interaction (the short-
range limit is achieved for $\alpha\rightarrow\infty$). Long-range interaction
can play opposing roles in the performance of QHEs, depending on the parameter
regime and the thermodynamic cycle. In [195], the long-range effects of both
the Otto and Stirling cycles were explored. While the Otto cycle exhibits
enhancement of the work output and efficiency due to long-range interactions,
in the high-temperature regime the Stirling cycle is negatively impacted by
them. Opposite behavior in the two cycles is also observed near the critical
point. The work and efficiency of the Otto cycle decrease with the interaction
length, whereas in the Stirling cycle, they increase.

An Otto engine with a many-body localization (MBL) phase was studied in [196].
In this setup, the work strokes consist of transitions between thermal states
(after thermalization with heat baths) and MBL phases. The QE work per cycle
is expressed in terms of the localization length, and it is shown that the MBL
spectrum distribution leads to a reduction in the fluctuations and permits a
smooth operation of parallel engines compromising the macro-scale engine.

Optimizing the performance of many-body QHE becomes a difficult task, as the
number of possible configurations of the interactions increases tremendously
with the number of particles. Machine learning [197] provides tools to
approach such problems and was recently implemented in optimizing the
interaction of many-body nanothermoelectric systems [198]. In particular, the
Reinforcement Learning (RL) approach has been applied to a network topology of
interacting thermal systems, utilizing a strategy of optimizing power-
efficiency trade-off. A thermal device made up of a many-body working medium
is mapped onto a network, i.e., a mathematical graph with nodes and vertices.
Each node represents a single-electron level that exchanges electrons with
reservoirs and each of the edges between two nodes indicate the presence or
absence of interaction between the corresponding level. Then a training set
for the network is fed into the algorithm by varying the interaction
parameters, for the purpose of finding the set of optimized QHEs with respect
to power and efficiency.

We note in passing that machine learning is becoming a tool for optimizing the
performance of few-body quantum thermal devices as well. A specific technique
in machine learning, called Differential Programming (DP), has recently been
introduced to optimize refrigeration via an Otto cycle in STA conditions
[199]. This method employs a reward and penalty scheme to a fictitious agent,
aiming to minimize the energetic cost of running the QHE within specific
constraints imposed by the STA scheme. The devised strategies are optimized
via deep learning of the training dataset from each engine cycle.

##  6 Quantum heat engines beyond the weak coupling and Markovian paradigm

###  6.1 Quantum Thermodynamics in the strong coupling limit

In the formulation of thermodynamics, it is often assumed that the interaction
between the system of interest and a thermal reservoir is weak. This
assumption has several significant thermodynamic implications, one of which is
that the reservoir will generally force, in the long time limit, the system to
a canonical Gibbs state. This equilibrium state should then grant the
possibility of exploring changes in thermodynamic variables defined only at
equilibrium. Moreover, the weak system-bath coupling assumption is crucial for
consistent energy accounting and for the division of energy into heat and
work. Heat can be understood, in this case, as the energy dissipated to the
reservoir, and entropy production can be associated with it. On the
macroscale, weak coupling usually provides a good description, as the system
interacts with the reservoir through the boundaries that constitute a small
portion of the system’s degrees of freedom and contribute very little to its
energy. On the other hand, in the single-particle scale of the system, this
assumption doesn’t necessarily hold true. The energy scale of the system
becomes comparable to that of the interaction, the boundary between the system
and the reservoir is obscured, and the system will generally relax to a non-
Gibbisian state. As a consequence, the definition of heat becomes ambiguous.

Many paths to reconciling the thermodynamics of open quantum systems in the
strong coupling regime have been explored. Describing thermodynamics with a
single reservoir was established using the Hamiltonian of mean force, which
overcomes the deviation from a Gibbs state by defining an effective
Hamiltonian [200, 201, 202] to construct thermodynamic potentials. Quantum
thermodynamics in the strong coupling regime was also explored using well-
established nonequilibrium quantum dynamics theories and methods. This
includes the Polaron transformation [203, 204], noninteracting blip
approximation [205], Reaction coordinates [206, 207, 208], non-equilibrium
Green’s function methods [209, 210, 211], hierarchical equations of motion
[212, 213, 214], the stochastic surrogate Hamiltonian [215], and alternative
approach of strong coupling via heat exchangers [216] . The list above is
limited and the subject accedes the scope of this review, and in the
following, we will focus on a few of these methods in the context of QHE and
refrigerators.

####  6.1.1 Study of QHEs using the polaron transformation method

To study the dynamics of an open quantum system, out of equilibrium, and
strongly coupled to a bath at finite temperature, one of the earliest methods
developed follows the celebrated polaron transformation technique [217]. The
spin-boson model has been investigated via this method [218], and later was
applied to the externally driven variation of the model [219, 220], and to
nonequilibrium studies that include more than one bath [221, 214]. To give a
brief overview of the method, we follow the model in reference [220] of an
exciton-phonon system that is driven by an external field. The Hamiltonian in
a rotating frame, in terms of the Pauli matrices, can be expressed as

$H_{EP}=\delta\sigma_{+}+\frac{\Omega(t)}{2}\sigma_{x}+\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}+\sigma_{+}\sum_{k}g_{k}b^{\dagger}_{k}+g^{*}_{k}b_{k}.$
(45)

Here $\Omega(t)$ is the Rabi frequency, $\delta$ is the detuning, and $g_{k}$
are the couplings to the phononic modes. The polaron transform is then carried
out, $H_{p}=e^{S}H_{EP}e^{-S}$, with

$S=\sigma_{+}\sum_{k}\alpha_{k}b^{\dagger}_{k}-\alpha^{*}_{k}b_{k},\;\;\;{\rm
and}\;\;\;\alpha_{k}=\frac{g_{k}}{\omega_{k}}.$ (46)

The final transformed Hamiltonian,

$H_{p}=\delta^{\prime}\sigma_{z}+\frac{\Omega_{r}(t)}{2}\sigma_{x}+\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}+\frac{\Omega(t)}{2}\big{(}\sigma_{x}B_{x}+\sigma_{y}B_{y}\big{)},$
(47)

displays several distinct effects: a) renormalization of the exciton
parameters due to the strong coupling, in particular, a shift in the detuning,
$\delta^{\prime}=\delta-\sum_{k}\omega_{k}|\alpha_{k}|^{2}$, and an effective
Rabi frequency, $\Omega_{r}(t)=\Omega(t)\langle B\rangle$; b) renormalized
bath interaction operators $B_{x}=(B_{+}+B_{-}+2\langle B\rangle)/2$,
$B_{y}=i(B_{+}-B_{-})/2$, and
$B_{\pm}=e^{\pm(\sum_{k}\alpha_{k}b^{\dagger}_{k}-\alpha^{*}_{k}b_{k})}$. For
a phononic bath in thermal equilibrium at inverse temperature $\beta$, the
expectation value of $B_{\pm}$ is simply $\langle
B\rangle=\exp\Big{[}-\frac{1}{2}\sum|\alpha_{k}|^{2}\coth(\frac{\beta\omega_{k}}{2})\Big{]}$.

The dynamics of the system generated by the polaron Hamiltonian in Eq. (47)
can then be solved using different methods and perturbative expansions. In
[220], a time-local master equation for the reduced system was derived.
Phonon-mediated electronic transport, with nonequilbrium and coupled quantum
dots connected to external leads, have also been explored using master
equation techniques for the polaron [222, 223, 224]. More recently, the
polaron transformation, together with hierarchical equations of motion, were
implemented to study quantum friction and work extraction in an electronic
system coupled to two leads and to a phonon mode, with and without damping
[214]. This work spans both the weak and strong system-bath coupling regimes
and the slow- and fast-driving limits.

The polaron transformation was also adapted to studying the light-harvesting
systems modeled as quantum heat engines [225, 203]. The light-harvesting
energy process is simplified and reduced to a three-level system, which works
as the antenna that captures energy from the sun and transfers it to the
reaction center. Transitions between the ground state and the second excited
state are provided by light from the sun, the first and second excited states
are mediated by phonons, strong coupling is captured by the polaron
transformation, and, finally, the reaction center is modeled as a reservoir
that couples the ground state with the first excited one to trap the energy.
The polaron transformation covers a wide range of system-phonon coupling
strengths. This coupling modifies the steady state to include coherence, which
significantly affects the efficiency and energy flux in the process.

A different QHE, with a two-level system as a working medium, was investigated
in [204]. The system Hamiltonian including the external driving field in a
rotating wave approximation reads
$H_{S}=\frac{\omega_{0}}{2}\sigma_{z}+\frac{\Omega}{2}\big{(}\sigma_{+}e^{-i\omega_{R}t}+\sigma_{-}e^{i\omega_{R}t}\big{)}$.
The system is also simultaneously coupled to two Bosonic heat baths at inverse
temperatures $\beta_{c}$ and $\beta_{h}$. The interaction Hamiltonian,

$H_{SB}=\lambda_{c}\sigma_{z}\sum_{k}g_{c,k}a^{\dagger}_{k}+g^{*}_{c,k}a_{k}+\lambda_{h}\sigma_{x}\sum_{k}g_{h,k}b^{\dagger}_{k}+g^{*}_{h,k}b_{k},$
(48)

is coupled differently to the hot and the cold baths. The cold bath induces
dephasing while the hot bath can lead to population decay. The transformation
can be carried out in a manner similar to the one presented above, only with
$S=\sigma_{z}\sum_{k}\alpha_{k}a_{k}^{\dagger}-\alpha^{*}a_{k}$. In contrast
to the single-bath cases, the polaron transformation in the multi-bath model
reveals energy exchange between the two baths through their coupling to the
system, which is directly observed in the polaron-transformed Hamiltonian. The
compromised system-bath partition makes the identification of heat flows from
the baths convoluted in a master equation for the reduced-state approach.
Furthermore, at the ultra-strong coupling to the baths, the power output is
suppressed as the coupling increases, indicating that QHE’s require some
degree of separability to operate.

####  6.1.2 Reaction Coordinate Mapping in studying strongly coupled engines

A method in both classical and quantum dynamics, the Reaction Coordinate (RC)
mapping has opened up an area of study in the physics of strongly coupled open
systems [206, 207, 208]. The spirit of the framework lies in finding a
suitable coordinate transformation that can eventually lead to a weak coupling
description of a dressed open system in the presence of a bath. This outcome
helps in applying the usual weak coupling tools to the case of certain
strongly coupled open systems. For the purpose of our discussion, we focus on
a specific model, the TLS, which also serves as the working medium for
investigating the thermodynamics of a strongly coupled QHE [226]. The
Hamiltonian in terms of Pauli matrices and the identity reads,

$H_{S}(t)=\frac{\mu(t)}{2}I+\frac{\epsilon(t)}{2}\sigma_{z}+\frac{\Delta(t)}{2}\sigma_{x},$
(49)

where $\epsilon(t)$ is the TLS bias, $\Delta(t)$ the tunneling term
coefficient, and the level-splitting of the eigenstates
$\mu(t)=\sqrt{\epsilon^{2}(t)+\Delta^{2}(t)}$. The TLS couples to two
reservoirs, a model invoked to represent the discrete stroke Otto cycle of a
heat engine, in a Caldeira-Leggett form

$H_{SB}=\sum_{k}\frac{p_{k}^{2}}{2m_{k}}+\frac{m_{k}\omega_{k}^{2}}{2}\Big{(}x_{k}-\frac{d_{k}\sigma_{z}}{m_{k}\omega_{k}^{2}}\Big{)}^{2}.$
(50)

The bath is characterized by a spectrum of Bosonic oscillators encapsulated in
the spectral density (in this case both a cold and hot reservoir)
$J(\omega)\equiv\sum_{k}f_{k}^{2}\delta(\omega-\omega_{k})$, where the
rescaled system-bath coupling is given by $f_{k}=d_{k}/\sqrt{m_{k}\omega_{k}}$
and can be arbitrarily strong.

Figure 11:  Sketch of the Otto cycle in the strong coupling regime. a)
Operating cycle of the engine. b) Schematic diagram of the RC mapping scheme.
The star geometry on the left (red circles), where the TLS is coupled to each
bath oscillator, is exactly mapped to the scheme on the right (violet). In the
new setting, the TLS only interacts with a single oscillator (RC) (natural
frequency $\Omega$) with coupling strength $\lambda$, and the RC interacts
with a transformed bath of harmonic oscillator modes with frequencies
$\nu_{k}$. Reproduced from [226, 208].

We are now in a position to define the RC mapping scheme. It maps the
Hamiltonian in Eqs. (49) and (50) to an enlarged system $S^{\prime}$ in terms
of a collective coordinate, which is in turn weakly coupled to a redefined
bath $B^{\prime}$ (see Fig. 11). This process leads to a transformed system-
bath Hamiltonian

$\tilde{H}=\underbrace{\Big{(}H_{S}(t)-\lambda\sigma_{z}(a^{\dagger}+a)+\Omega
a^{\dagger}a\Big{)}}_{\rm{S^{\prime}}}+\underbrace{\Big{(}\sum_{k}g_{k}(a^{\dagger}+a)(r_{k}^{\dagger}+r_{k})+\sum_{k}\nu_{k}r_{k}^{\dagger}r_{k}\Big{)}}_{\rm{B^{\prime}}}.$
(51)

The first parenthesis represents the TLS coupled arbitrarily and strongly (via
$\lambda$) to a reaction coordinate, denoted by the annihilation and creation
operators, $a$ and $a^{\dagger}$, with frequency $\Omega$. The second
parenthesis accounts for the coupling between the collective coordinate and
the residual bath modes with couplings $g_{k}$, created and annihilated at
natural frequencies $\nu_{k}$ by the operators $r^{\dagger}_{k}$ and $r_{k}$
respectively. The analysis of the Otto cycle with the Hamiltonian
$H_{S^{\prime}}(t)$, rather than the reduced system $H_{S}(t)$, is
accomplished by virtue of the RC mapping formalism, reduced to the standard
weak coupling form with product system-bath states. Moreover, the usual
expressions for work and heat along the isentropic and isochoric strokes are
exactly calculated. In the two limits of adiabatic and quenched isentropic
strokes, the main source of work costs is the coupling and decoupling of the
working medium and the thermal bath, a feature ignored in the weak coupling
case. Thus, strongly coupled QHEs compromise on the efficiency of work
extraction concomitant with the predictions in continuous QHEs [204]. The
dependence of the QHE performance on the system-reservoir decoupling procedure
(instantaneous or adiabatic) is univocally influenced by the strong coupling
regime and it can therefore act as a sensor for the presence of strong
couplings in engineered experiments involving such thermal machines.

In a three-level QAR (see also Sec. 2.3.2), RC mapping has been applied to
figure out the most efficient cooling window and provides a comparison to the
weak coupling case [227]. In this work, the fact (discussed above) that the RC
coordinate ultimately couples weakly to the residual bath is utilized to
introduce the usual perturbative quantum Master equation approach. The merging
of these two theoretical techniques leads to a remarkable result in thermal
devices with multiple strongly coupled baths: the emergence of a direct heat-
flow pathway between the thermal reservoirs, without any change in the
population of the QAR’s energy levels. Such inter-bath leakage – which is
fundamentally a heat transport phenomenon, studied earlier via these two
techniques in [228]) – leads to a decrease in the cooling window and
diminished performance at strong coupling.

Finally, we would like to mention that such a scheme is also useful in
characterizing the generation of quantum coherence due to strong system-bath
coupling in QHEs fuelled by TLS (as described above in Eqs. (49) and (50)),
leading to bounds on efficiency and power output [229] that cannot be
delineated by the Born-Markov master equation. These results are traced back
to the generation of coherence in the working medium, which is directly
related to the system-bath coupling and not originated from the driving field.

####  6.1.3 Perturbative corrections to engine observables due to strong
coupling

A common theme of the above studies is that strong coupling between the system
and the bath produces correlations that adversely affect the performance of
QHEs. Irreversibility in the dynamics of a nonequilibrium system in contact
with a heat bath leads to universal bounds on work extraction of small thermal
machines, as well as on heat dissipation and Carnot efficiency. Perturbative
methods can be employed to evaluate these corrections due to strong coupling
in a systematic and universal manner [230, 231], without recourse to a
specific model per se.

A system S and a bath B are assumed to be coupled through a potential $V$,
allowing for strong interactions. Let us consider an N-step process consisting
of three elementary operations: (a) S and B are brought into contact, with the
total Hamiltonian $H_{S}+H_{B}+V$, and the possibility of $V$ being turned on
and off on the $i$-th step. The average work gained during such a step is
denoted by $W^{(i)}_{V}=\mathrm{tr}[\rho^{(i)}V]$; (b) a fast quench
transformation of the system Hamiltonian $H_{S}$, in which the state
$\rho^{(i)}$ remains unchanged and an amount of work,
$W^{(i)}=\mathrm{tr}[\rho_{S}^{(i)}(H_{S}^{(i)}-H_{S}^{(i+1)})]$, is obtained.
And; (c) a thermalization process at the $(i+1)$-th step, which only includes
energy exchange between S and B via the interaction $V$ (which involves zero
work cost). Finally, at equilibrium (N-th step), the protocol assumes a
reduced version of the global Gibbs state, i.e.,
$\rho_{S}^{(i+1)}=\mathrm{tr}_{B}[\omega_{\beta}H^{(i+1)}]$ where,
$\omega_{\beta}(H)=e^{-\beta H}/\mathrm{tr}e^{-\beta H}$. The optimal protocol
for maximizing the work extraction is found for initial uncorrelated state
$\rho^{(0)}=\rho_{S}\otimes\omega_{\beta}(H_{B})$ and cyclic boundary
condition $H^{(N+1)}=H^{(0)}$. The total work at the end of the protocol, in
this case, can be divided into three contributions,

$W=W^{({\rm weak})}-\Delta F^{({\rm rev})}-\Delta F^{({\rm irrev})},\;\;\;{\rm
with}\;\;\;\Delta F^{({\rm rev})},\Delta F^{(\rm irrev)}\geq 0.$ (52)

Here $W^{({\rm weak})}$ is the work assuming a weak coupling regime given by
the difference between the nonequilibrium free energy of the system and that
of a thermal state with the system Hamiltonian. $\Delta F^{(\rm irrev)}$ is a
correction term generated due to irreversible energy dissipation to the bath
$B$, and $\Delta F^{({\rm rev})}$ is the extractable work left in the final
state.

It is immediately evident that strong coupling diminishes work output over and
above the weak coupling term $W^{({\rm weak})}$. Taking a perturbative route
by defining the interaction potential as $gV$ (with $g$ being a perturbative
parameter), a rigorous minimum bound on $\Delta F^{(\rm irrev)}$ can be
evaluated and, up to second order in $g$, the term fundamentally depends on
the generalized Kubo-Mori covariance matrix of the interaction (see details in
[231] for the perturbative calculations). With the essentials of the
nonequilibrium dynamics under strong coupling in place, QHEs employing
equilibration with two baths at inverse temperatures $\beta_{c}$ and
$\beta_{h}$ can be investigated in the minimal dissipation regime, leading to
a corrected maximum efficiency

$\eta=\eta_{\rm{C}}-g^{2}\frac{\beta_{h}}{\beta_{c}}A+O(g^{3}),$ (53)

where $A$ is a non-negative function of perturbation coefficients and heat
flows in the weak coupling regime and $\eta_{\rm C}$ is the Carnot efficiency.
Although dissipation and coupling strengths reduce the efficiency of the QHE,
the power output can be enhanced by shortening the thermalization time scale
($\tau$). The correction terms for the maximum power output again follow a
series expansion, in the powers of the perturbation parameter $g$, and depend
on the characterization of equilibration times for the system S. The latter is
a challenging problem in itself for a variety of many-body models.

####  6.1.4 Strongly-coupled QHEs with Matrix Product States

The strong system-bath coupling regime entails quantum correlations among the
working medium and the reservoirs. In this regime, the same definitions of
quantum thermodynamics concepts, such as heat and work, become challenging