# Coherent Heat Transfer Leads to Genuine Quantum Enhancement in Performances
of Continuous Engines

Brij Mohan<EMAIL_ADDRESS>Department of Physical Sciences, Indian
Institute of Science Education and Research (IISER), Mohali, Punjab 140306,
India  Rajeev Gangwar  Department of Physical Sciences, Indian Institute of
Science Education and Research (IISER), Mohali, Punjab 140306, India  Tanmoy
Pandit  Fritz Haber Research Center for Molecular Dynamics, Hebrew University
of Jerusalem, Jerusalem 9190401, Israel  Mohit Lal Bera  Departamento de
Física Teórica and IFIC, Universidad de Valencia-CSIC, 46100 Burjassot
(Valencia), Spain  ICFO - Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
Maciej Lewenstein  ICFO - Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
ICREA, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain  Manabendra Nath Bera
<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of Science
Education and Research (IISER), Mohali, Punjab 140306, India

###### Abstract

The conventional continuous quantum heat engines rely on incoherent heat
transfer with the baths and, thus, have limited capability to outperform their
classical counterparts. In this work, we introduce distinct continuous quantum
heat engines that utilize coherent heat transfer with baths, yielding
significant quantum enhancement in performance. These continuous engines,
termed as coherent engines, consist of one qutrit system and two photonic
baths and enable coherent heat transfer via two-photon transitions involving
three-body interactions between the system and hot and cold baths. The closest
quantum incoherent analogs are those that only allow incoherent heat transfer
between the qutrit and the baths via one-photon transitions relying on two-
body interactions between the system and hot or cold baths. We demonstrate
that coherent engines deliver much higher power output and a much lower
signal-to-noise ratio in power, where the latter signifies the reliability of
an engine, compared to incoherent engines. Coherent engines manifest more non-
classical features than incoherent engines because they violate the classical
thermodynamic uncertainty relation by a greater amount and for a wider range
of parameters. Importantly, coherent engines can operate close to or at the
fundamental lower limit on reliability given by the quantum version of the
thermodynamic uncertainty relation, making them highly reliable. These genuine
enhancements in performance by hundreds of folds over incoherent engines and
the saturation of the quantum limit by coherent engines are directly
attributed to its capacity to harness higher energetic coherence which is,
again, a consequence of coherent heat transfer. The experimental feasibility
of the coherent engines and the improved understanding of how quantum
properties may enhance performance are expected to have significant
implications in emerging quantum-enabled technologies.

##  I introduction

Quantum heat engines – microscopic thermal devices designed to convert heat
into quantum mechanical work – have become one of the focal points of research
considering the current quantum industrial revolution Binder  _et al._ (2018);
Auffèves (2022). This leads to studying thermodynamics in the microscopic and
quantum regime, both from foundational and applied aspects Jarzynski (1997);
Crooks (1999); Campisi  _et al._ (2011); Brandão  _et al._ (2013); Horodecki
and Oppenheim (2013); Skrzypczyk  _et al._ (2014); Brandão  _et al._ (2015);
Lostaglio  _et al._ (2015); Alhambra  _et al._ (2016); Bera  _et al._ (2017);
Sparaciari  _et al._ (2017); Binder  _et al._ (2018); Åberg (2018); Gour  _et
al._ (2018); Müller (2018); Uzdin and Rahav (2018); Bera  _et al._ (2019,
2021, 2022); Khanian  _et al._ (2023); Bera  _et al._ (2024). The earliest
model of a quantum heat engine was proposed by Scovil and Schulz-DuBois (SSD),
which is composed of a qutrit interacting with two thermal baths Scovil and
Schulz-DuBois (1959). Later, it was re-investigated in a full quantum setting
using open quantum system dynamics Boukobza and Tannor (2006a, b, 2007);
Kosloff and Levy (2014). In the last decades, many other models of quantum
heat engines have been proposed; see Refs. Kosloff and Levy (2014); Binder
_et al._ (2018); Myers  _et al._ (2022); Cangemi  _et al._ (2023) for a
comprehensive overview of historical and recent advancements. Optomechanical
systems Sheng  _et al._ (2021), nitrogen-vacancy centers in diamond Klatzow
_et al._ (2019), trapped ions Roßnagel  _et al._ (2016); Bouton  _et al._
(2021), nuclear magnetic resonance (NMR) Peterson  _et al._ (2019), and
superconducting circuits de Araujo  _et al._ (2023) have emerged as versatile
experimental platforms to realize quantum heat engines, bringing these
theoretical concepts into practical realizations.

The conventional continuous quantum heat engines operate in a steady-state
regime, by interacting continuously with hot and cold baths Kosloff and Levy
(2014); Binder  _et al._ (2018); Myers  _et al._ (2022); Cangemi  _et al._
(2023). These engines, in general, deliver low power with high fluctuation
Kosloff and Levy (2014); Rahav  _et al._ (2012); Kalaee  _et al._ (2021);
Bayona-Pena and Takahashi (2021); Singh  _et al._ (2023). As a result, the
reliability, i.e., the ratio between the variance and average of power (or
relative fluctuation in power), of these engines is considerably compromised.
Recent studies focus on improving the performance of quantum heat engines,
aiming for more power with higher reliability (less relative fluctuation in
power), by harnessing energetic coherence. It has been observed that
continuous quantum thermal devices, when energetic coherence is present, may
enhance power Scully  _et al._ (2011); Um  _et al._ (2022) and efficiency
Scully (2010); Dorfman  _et al._ (2018), suppress fluctuation in power Kalaee
_et al._ (2021); Singh  _et al._ (2023). and may lead to violation of
classical thermodynamic trade-off relations (classical thermodynamic
uncertainty relation (cTUR) Barato and Seifert (2015) and power-efficiency-
constancy trade-off relation Pietzonka and Seifert (2018)) Ptaszyński (2018);
Liu and Segal (2019); Pal  _et al._ (2020); Rignon-Bret  _et al._ (2021);
Bayona-Pena and Takahashi (2021); Kalaee  _et al._ (2021); Van Vu and Saito
(2022); Souza  _et al._ (2022); Singh  _et al._ (2023); Prech  _et al._
(2023); Manzano and López (2023). These violations indicate that these engines
can operate in the quantum regime. However, it does not necessarily imply that
quantum engines are operating close to their optimal capacity in terms of
reliability. Ideally, one would expect negligible relative fluctuation in
power from an ideal continuous engine. However, relative fluctuation cannot be
suppressed to zero due to the existence of a finite lower bound on it
determined by the quantum thermodynamic uncertainty relation (qTUR) Hasegawa
(2020). This lower bound represents a fundamental quantum limit, which is
derived from the celebrated quantum Cramér-Rao bound Braunstein and Caves
(1994), and is closely related to the so-called quantum speed limits Hasegawa
(2020, 2023).

The characteristic feature of traditional continuous quantum heat engines is
that they utilize incoherent heat transfers between the working system and the
baths. It implies that the transitions induced in the working system by the
hot and cold baths are independent (or uncorrelated), rendering them highly
stochastic in nature. This feature constitutes one of the reasons for these
engines to have limited ability to outperform their classical counterparts.
Therefore, we are required to reduce the stochastic nature of the transitions
in the working system induced by the baths to overcome these limitations. The
natural question is, thus, how to employ an operationally distinct heat
transfer mechanism, rather than the incoherent one, in continuous heat engines
that inherently involve less stochastic transitions and lead to significant
enhancement in performance.

In this article, we affirmatively address the above question by introducing
the concept of a coherent heat transfer mechanism in continuous heat engines
in which the baths induce correlated (or mutually dependent) transitions in
the working system, and, as a result, the stochastic nature of transition
decreases. The continuous engines operating with this mechanism are termed
coherent quantum heat engines (CQHEs). These engines can be physically
realized by considering a qutrit coherently interacting with hot and cold
baths through two-photon transitions (Raman interaction, i.e., three-body
interactions between system and baths) in the presence of periodic driving by
an external field. The analogous incoherent quantum heat engines (IQHEs) are
the standard SSD engines Boukobza and Tannor (2006a, b, 2007); Kosloff and
Levy (2014), where a qutrit interacts incoherently (independently, through
one-photon transitions) with the hot and cold baths. For the same set of
qutrit and bath parameters, the CQHEs deliver much higher power and much lower
relative fluctuation in power compared to IQHEs. In fact, the performance of
CQHEs can be enhanced by hundreds of folds of that of IQHEs. This enhancement
is directly attributed to the presence of a much higher amount of energetic
coherence in CQHEs, which is a consequence of coherent heat transfer.
Moreover, for the same reason, the CQHEs not only exhibit a more profound
violation of cTUR and power-efficiency-constancy trade-off relations compared
to IQHEs but also can suppress relative fluctuation in power to the quantum
limit imposed by qTUR. Hence, CQHEs manifest genuine quantum enhancement over
IQHEs and classical engines.

The rest of the article is organized as follows. In section II, we introduce
the generic models of continuous quantum coherent and incoherent engines
involving coherent and incoherent heat transfers, respectively. We demonstrate
the genuine quantum enhancements in performances by coherent engines over
incoherent engines in section III. Finally, our results are summarized in
section IV.

##  II Continuous coherent quantum heat engines

A continuous heat engine consists of a working system that weakly interacts
with two heat baths at different temperatures while, at the same time, being
periodically driven by an external field. The simplest model for such an
engine utilizes a qutrit system interacting with two baths, widely studied in
literature Scovil and Schulz-DuBois (1959); Boukobza and Tannor (2006a, b,
2007); Kosloff and Levy (2014); Myers  _et al._ (2022); Cangemi  _et al._
(2023). Explicitly, a qutrit with Hamiltonian
$H_{S}=\omega_{h}\outerproduct{2}{2}+(\omega_{h}-\omega_{c})\outerproduct{1}{1}$
is coupled to two thermal (photon) baths with respective inverse temperatures
$\beta_{c}$ and $\beta_{h}$, where $\omega_{h}>\omega_{c}$ and
$\beta_{c}>\beta_{h}$. In addition, the qutrit is driven by an external field
following the Hamiltonian $H_{d}(t)=\alpha(e^{-i\omega
t}\outerproduct{1}{0}+e^{i\omega t}\outerproduct{0}{1})$. The condition
$\beta_{h}\omega_{h}<\beta_{c}\omega_{c}$ needs to be ensured for this device
to operate as a heat engine (see Ref. Kosloff and Levy (2014) and Appendix B).
We assume $\hbar=k_{B}=1$ throughout this work. The total Hamiltonian of the
qutrit-baths composite is

$\displaystyle H=H_{S}(t)+H_{B_{h}}+H_{B_{c}}+H_{SB_{h}B_{c}}^{X},$

where $H_{S}(t)=H_{S}+H_{d}(t)$ is the total Hamiltonian of the qutrit,
$H_{B_{h}}=\sum_{k}\Omega_{k,h}\ a^{\dagger}_{k,h}a_{k,h}$ and
$H_{B_{c}}=\sum_{k^{\prime}}\Omega_{k^{\prime},c}a^{\dagger}_{k^{\prime},c}a_{k^{\prime},c}$
are the Hamiltonians of the hot and cold (photon) baths with mode frequencies
$\Omega_{k,h}$ and $\Omega_{k^{\prime},c}$ respectively, and
$H_{SB_{h}B_{c}}^{X}$ represents the interaction between the qutrit and the
baths.

Below, we consider two qualitatively different models of continuous heat
engines that differ in the interaction between the qutrit and the baths, i.e.,
$H_{SB_{h}B_{c}}^{X}$. In particular, our goal is to compare the performances
of engines with an interaction Hamiltonian ($H_{SB_{h}B_{c}}^{I}$) that only
allows ‘incoherent’ energy transfer with the performances of engines with an
interaction Hamiltonian ($H_{SB_{h}B_{c}}^{C}$) that enables ‘coherent’ energy
transfer between the baths and the qutrit.

Incoherent Quantum Heat Engines (IQHEs) – We start with an engine that
operates via incoherent energy transfers between the constituents. Most of the
traditional (continuous) quantum heat engines utilize incoherent energy
transfers Boukobza and Tannor (2006a, b, 2007) with the interaction
Hamiltonian

$\displaystyle
H_{SB_{h}B_{c}}^{I}=g_{h}\sum_{k}(a_{k,h}b^{\dagger}_{h}+a_{k,h}^{\dagger}b_{h})+g_{c}\sum_{k^{\prime}}(a_{k^{\prime},c}b_{c}^{\dagger}+a_{k^{\prime},c}^{\dagger}b_{c}),$
(1)

where $b_{h}=\outerproduct{0}{2}$ and $b_{c}=\outerproduct{1}{2}$ are the
ladder operator acting on the qutrit space. The coefficients $g_{h}$ and
$g_{c}$ are the interaction strength with the hot and cold baths,
respectively. The interaction drives incoherent energy (heat) transfer in the
sense that the energy exchange between the states $|0\rangle$ and $|2\rangle$
with the hot bath is independent of the energy exchange between the states
$|1\rangle$ and $|2\rangle$ with the cold bath. For $|g_{h}|,|g_{c}|\ll 1$,
the local dynamics of the qutrit is expressed by the Lindblad master equation
Breuer and Petruccione (2007); Boukobza and Tannor (2006a, b, 2007)

$\displaystyle\dot{\rho}=i\ [\rho,\
H_{S}(t)]+\mathcal{D}_{h}(\rho)+\mathcal{D}_{c}(\rho),$ (2)

where $\rho$ is the density matrix representing the state of the qutrit. The
dissipators $\mathcal{D}_{h}(\rho)$ and $\mathcal{D}_{c}(\rho)$ represent
dissipative dynamics due to the interactions with the hot and cold baths and
are given by (for $x$ = $h$, $c$):

$\displaystyle\mathcal{D}_{x}(\rho)=\gamma_{x}$
$\displaystyle(n_{x}+1)(b_{x}\rho
b^{{\dagger}}_{x}-\\{b^{{\dagger}}_{x}b_{x},\rho\\}/2)$
$\displaystyle+\gamma_{x}n_{x}(b^{{\dagger}}_{x}\rho
b_{x}-\\{b_{x}b^{{\dagger}}_{x},\rho\\}/2),$

where the anti-commutator $\\{Y,Z\\}=YZ+ZY$, the coefficient $\gamma^{x}$ is
the Weiskopf-Wigner decay constant, and $n_{x}=1/(e^{\beta_{x}\omega_{x}}-1)$
is the average number of photons in the bath with frequency $\omega_{x}$. The
appearance of two dissipators, $\mathcal{D}_{h}(\rho)$ and
$\mathcal{D}_{c}(\rho)$, in the master equation (2) reflects that the heat
exchange with the hot bath is independent (or uncorrelated) of the heat
exchange with the cold baths. Thus, the heat exchanges between the baths are
incoherent.

Figure 1:  Schematics of incoherent and coherent heat engines. The engine is
constituted by a three-level quantum system (qutrit), which weakly interacts
with hot and cold baths with the inverse temperatures $\beta_{h}$ and
$\beta_{c}$. In incoherent heat engine, the energy (heat) transfer takes place
via (independent) single photon transitions, i.e., energy levels $|0\rangle$
and $|2\rangle$ interact with the hot bath and levels $|1\rangle$ and
$|2\rangle$ interact with the cold bath, governed by the interaction
Hamiltonian (1). Solid (red and blue) arrows indicate these independent or
incoherent energy transfers. In coherent heat engines, the energy transfer
takes place via two-photon transitions, where effectively energy levels
$|0\rangle$ and $|1\rangle$ participate in the process, and absorption of a
photon from the hot bath is associated with the release of a photon top the
cold bath and vice versa. This coherent heat transfer is governed by the
interaction Hamiltonian (4) and indicated here by the dotted (green) arrow.
The wavy arrow (solid-green) between $|0\rangle$ and $|1\rangle$ indicates the
external driving utilizing which the work is extracted. See text for more
details.

To quantify the power, heat currents, and other relevant quantities of IQHEs,
we move to a rotating frame using a transformation
$B_{R}=e^{i\tilde{H}t}Be^{-i\tilde{H}t}$, where $B$ is an arbitrary operator
and $[H_{S},\tilde{H}]=0$ Boukobza and Tannor (2007); Bera  _et al._ (2024).
This transformation eliminates the time dependence of $H_{S}(t)$ and reduces
it to $H_{S}-\tilde{H}+H_{dR}$, where
$H_{dR}=\alpha(\outerproduct{1}{0}+\outerproduct{0}{1})$. The dissipators
remain unchanged in the rotating frame, and the dynamics leads to a steady
state $\sigma_{I}$ with $\dot{\sigma}_{I}=0$ (see Appendix A). Now the average
power $\langle P_{I}\rangle$ and the average heat currents
$\langle\dot{J}^{x}_{I}\rangle$ are given by

$\displaystyle\langle P_{I}\rangle=-i\Tr([H_{S},H_{dR}]\sigma_{I}),\
\mbox{and}\
\langle\dot{J}^{x}_{I}\rangle=\Tr(\mathcal{D}_{x}(\sigma_{I})H_{S}).$ (3)

Note, $\langle P_{I}\rangle\leq 0$ for a heat engine, and the heat-to-work
conversion efficiency is $\eta_{I}=-\langle
P_{I}\rangle/\langle\dot{J}^{h}_{I}\rangle\geq 0$. Other important quantities,
such as fluctuation in power ($\Delta P_{I}$) and fluctuation in heat currents
($\Delta J_{I}^{x}$), where power and heat currents are considered as random
variables, are computed using full counting statistics of the steady state
dynamics. See Appendix D for more details.

Coherent Quantum Heat Engines (CQHEs) – We consider an alternative engine that
involves energy transfer between the baths and the qutrit via a two-photon
process, driven by an interaction Hamiltonian Gerry and Eberly (1990); Gerry
and Huang (1992); Wu (1996)

$\displaystyle
H_{SB_{h}B_{c}}^{C}=g_{0}\sum_{k,k^{\prime}}(a_{k,h}a_{k^{\prime},c}^{\dagger}b_{hc}^{\dagger}+a_{k,h}^{\dagger}a_{k^{\prime},c}b_{hc}),$
(4)

where $b_{hc}=\outerproduct{0}{1}$ and $g_{0}$ is the coupling strength. Here,
the energy transfer between the baths and the system is coherent in the sense
that any photon absorbed from the hot bath is associated with a release of a
photon to the cold bath and the excitation $|0\rangle\to|1\rangle$, and vice
versa. For $|g_{0}|\ll 1$, the local dynamics of the qutrit reduces to

$\displaystyle\dot{\rho}=i\ [\rho,\ H_{S}(t)]+\mathcal{D}_{hc}(\rho)$ (5)

for a qutrit state $\rho$, where the only dissipator in the Lindblad master
equation is given by,

$\displaystyle\mathcal{D}_{hc}(\rho)$ $\displaystyle=\gamma_{1}(b_{hc}\rho
b_{hc}^{{\dagger}}-\\{b_{hc}^{{\dagger}}b_{hc},\rho\\}/2)$
$\displaystyle+\gamma_{2}(b_{hc}^{{\dagger}}\rho
b_{hc}-\\{b_{hc}b_{hc}^{{\dagger}},\rho\\}/2),$

with $\gamma_{1}=\gamma_{0}n_{c}(n_{h}+1)$,
$\gamma_{2}=\gamma_{0}n_{h}(n_{c}+1)$, and $\gamma_{0}$ is Weiskopf-Wigner
decay constant. The derivation of the above Lindblad master equation is
outlined in Appendix B. The dissipator $\mathcal{D}_{hc}$ involves the
parameters of both hot and cold baths and induces dissipation utilizing the
levels $|0\rangle$ and $|1\rangle$. The level $|2\rangle$ is never “engaged”
in the process. Due to the coherent nature of the interaction, the energy
(heat) transfer between the baths and the qutrit is less random (i.e.,
involves less stochastic transitions) due to correlated heat transfer than
that of the engines with incoherent heat transfer considered earlier.

To calculate the power, heat currents, and other relevant quantities, we move
to a rotating frame employing the transformation
$B_{R}=e^{i\tilde{H}t}Be^{-i\tilde{H}t}$, where $B$ is an operator satisfying
$[H_{S},\tilde{H}]=0$, similar to the case of IQHEs. With the resultant time-
independent qutrit Hamiltonian $H_{S}-\tilde{H}+H_{dR}$, where
$H_{dR}=\alpha(\outerproduct{1}{0}+\outerproduct{0}{1})$, the dynamics attains
a steady state in the rotating frame. For the steady state $\sigma_{C}$, with
$\dot{\sigma}_{C}=0$, the average power $\langle P_{C}\rangle$ is given by

$\displaystyle\langle P_{C}\rangle=-i\ \Tr([H_{S},H_{dR}]\sigma_{C})\leq 0.$
(6)

The average heat currents $\langle\dot{J}^{x}_{C}\rangle$ cannot be quantified
directly (like in the case of IQHEs) because there are no independent
dissipators corresponding to hot and cold baths. For that, we employ full
counting statistics of the steady state dynamics (see Appendix D). This
enables us to calculate the heat currents, the fluctuations in power ($\Delta
P_{C}$), and the fluctuation in heat currents ($\Delta J^{x}_{C}$). With heat
current from the hot bath $\langle\dot{J}^{h}_{C}\rangle$, we may compute the
heat-to-work conversion efficiency $\eta_{C}=-\langle
P_{C}\rangle/\langle\dot{J}^{h}_{C}\rangle$ of CQHEs.

Figure 2: Comparisons of energetic coherence and power outputs in coherent and
incoherent engines. The computations are carried out with the parameters
$\gamma_{0}=0.01$, $\omega_{h}=10$, $\omega_{c}=5$. (a) The figure on the left
illustrates the variation in energetic coherence
$\mathcal{C}_{C}=\mathcal{C}(\sigma_{C})$ and
$\mathcal{C}_{I}=\mathcal{C}(\sigma_{I})$ for both coherent and incoherent
heat engines, respectively with respect to the driving field strength
$\alpha$, for $\beta_{h}=0.01$ and $\beta_{c}=0.8$. The expressions of
energetic coherence are given Eqs. (7) and (8). The traces in solid-blue and
dashed-blue represent $\mathcal{C}_{C}$ and $\mathcal{C}_{I}$, respectively.
The corresponding power outputs $\mathcal{P}_{C}$ and $\mathcal{P}_{I}$, given
in Eq. (9), by coherent and incoherent engines, are presented with the solid-
red and dashed-red traces, respectively. (b) The figure of the right displays
the ratio of powers
$\mathcal{P}_{C}/\mathcal{P}_{I}=\mathcal{C}_{C}/\mathcal{C}_{I}$ of the
coherent and incoherent heat engine, with $\beta_{h}=0.001$, against $\alpha$
and $\beta_{c}$. In fact, for these parameters, the ratio can be
$\mathcal{P}_{C}/\mathcal{P}_{I}\geq 135$. See text for more details.

##  III Quantum enhancements in coherent engines

An evaluation of the performance of a continuous quantum heat engine requires
a comprehensive analysis of three metrics: (i) efficiency, which signifies how
efficiently heat is being converted into work; (ii) power, which is the rate
of work output; and (iii) noise-to-signal ratio (NSR) in power, which
signifies the relative fluctuation or inverse of precision in the power
output. We compare these metrics for coherent and incoherent heat engines and
demonstrate that the former have substantial quantum enhancements in
performance over the latter.

Our analysis reveals that the engine performance is related to the energetic
coherence present in the steady state $\sigma_{X}$ (for $X=I,C$) in the
rotating frame. Henceforth, a steady state refers to the steady state in the
rotating frame. The quantum enhancements in the performance of CQHEs over the
IQHEs are the direct consequence of the fact that the energetic coherence in
$\sigma_{C}$ is higher than that of $\sigma_{I}$, in general. Note that the
energetic coherence in the steady state results from a balance between two
opposing processes - the (periodic) driving that creates coherence and the
dissipation(s) that destroys coherence in the qutrit. Due to coherent heat
transfer, the dissipative ‘tendency’ in CQHEs is weaker compared to the
dissipative ‘tendency’ in IQHEs. As a result, we observe more energetic
coherence in the former.

We start our analysis by studying the coherence in the steady states. In what
follows, we set $\gamma_{h}=\gamma_{c}=\gamma_{0}$ and equal driving strength
$\alpha$ for fair comparisons. The energetic coherence is measured using the
$l$-1 norm of coherence Baumgratz  _et al._ (2014), given by
$\mathcal{C}(\sigma_{X})=\sum_{i\neq j}|\sigma_{X}^{(ij)}|$, where
$\sigma_{X}^{(ij)}=\langle i|\sigma_{X}|j\rangle$. For CQHEs and IQHEs,
$\sigma_{X}^{(ij)}=\sigma_{X}^{(ji)*}$, and the corresponding amount of
energetic coherence in the steady states are given by

$\displaystyle\mathcal{C}(\sigma_{C})$ $\displaystyle=\frac{4\alpha\
\gamma_{0}(n_{h}-n_{c})}{8\alpha^{2}+\gamma_{0}^{2}(n_{hc}+2n_{c}n_{h})^{2}},$
(7) $\displaystyle\mathcal{C}(\sigma_{I})$ $\displaystyle=\frac{4\alpha\
\gamma_{0}(n_{h}-n_{c})}{4\alpha^{2}(3n_{hc}+4)+\gamma_{0}^{2}n_{hc}(n_{hc}+3n_{h}n_{c})},$
(8)

where $n_{hc}=n_{h}+n_{c}$. We refer to Appendices A and B for detailed
derivation. For fixed $\gamma_{0}$, $n_{h}$, and $n_{c}$, the energetic
coherence is a function of the driving strength $\alpha$. As shown in Fig.
2(a), the energetic coherence $\mathcal{C}(\sigma_{C})$ for CQHEs are higher
than the energetic coherence $\mathcal{C}(\sigma_{I})$ of IQHEs in general.
Even for some reasonable values of system and bath parameters, the
$\mathcal{C}(\sigma_{C})$ becomes more than 135 times of
$\mathcal{C}(\sigma_{I})$, i.e., $\mathcal{C}(\sigma_{C})\geq 135\
\mathcal{C}(\sigma_{I})$. We also note that, for fixed $n_{h}$, $n_{c}$, and
$\gamma_{0}$, there is a critical value of the driving strength $\alpha_{cr}$
for which $\mathcal{C}(\sigma_{C})=\mathcal{C}(\sigma_{I})$. We calculate the
critical value (see Appendix C) and observe that
$\mathcal{C}(\sigma_{C})\leq\mathcal{C}(\sigma_{I})$ for
$\alpha\leq\alpha_{cr}$. However, the $\alpha_{cr}$ is generally very small,
representing extremely weak periodic driving, except for the case of the baths
with very high temperatures, i.e., $n_{h}\approx n_{c}\gg 1$. In all
reasonable physical situations, the engines operate with $\alpha>\alpha_{cr}$,
which we consider for evaluating engine performances below.

Figure 3: Comparisons of noise-to-signal ratios (NSRs) in coherent and
incoherent engines. The parameters $\gamma_{0}=0.01$, $\omega_{h}=10$, and
$\omega_{c}=5$ are considered for all the figures. (a) The figure on the left
displays the ratio $\mathcal{N}_{I}/\mathcal{N}_{C}$ of NSRs in power (see Eq.
(12)) corresponding to incoherent and coherent heat engines against
$\beta_{c}$ and $\alpha$, while $\beta_{h}=0.001$. Note,
$\mathcal{N}_{I}>\mathcal{N}_{C}$ signifies that the coherent engine produces
less NSR in power than the incoherent engine, and the ratio can reach up to
$\mathcal{N}_{I}/\mathcal{N}_{C}\geq 330$. (b) The figure in the middle shows
the difference between the NSR and its lower bound for CQHEs and IQHEs (i.e.,
degree of saturation of qTUR) in Eq. (12), involving NSRs and their quantum
bounds with respect to $\alpha$ for $\beta_{h}=0.01$ and $\beta_{c}=0.8$. The
traces in dark-blue and light-red represent $\mathcal{N}_{C}-f_{C}$ and
$\mathcal{N}_{I}-f_{I}$ for the coherent and incoherent engines, respectively.
The dashed-green trace corresponds to the zero value. (c) The figure on the
right represents the saturation of qTUR by CQHEs for the parameters
$\beta_{h}=0.01$ and $\beta_{c}=3$ with a large amount of energetic coherence.
Here, $\mathcal{C}_{C}=\mathcal{C}(\sigma_{C})$ represents the energetic
coherence in the steady state of CQHEs.

Power and efficiency – Now, we study power and efficiency. The power delivered
by a steady state engine has a monotonic relation with the energetic coherence
present in the steady state, and it is given by (see Appendices A and B)

$\displaystyle\mathcal{P}_{X}=|\langle P_{X}\rangle|=\alpha\
(\omega_{h}-\omega_{c})\ \mathcal{C}(\sigma_{X}),$ (9)

which is a non-linear function of $\alpha$. As shown in Fig. 2(a), it
increases with $\alpha$. The power is proportional to coherence for a given
$\alpha$. In fact, the ratio of the powers of CQHEs and IQHEs becomes equal to
the ratio of the energetic coherence present in their respective steady
states, i.e.,
$\mathcal{P}_{C}/\mathcal{P}_{I}=\mathcal{C}(\sigma_{C})/\mathcal{C}(\sigma_{I})$.
Given that $\mathcal{C}(\sigma_{C})>\mathcal{C}(\sigma_{I})$ in general, the
power of CQHEs is higher than the power delivered by IQHEs or
$\mathcal{P}_{C}/\mathcal{P}_{I}>1$. A numerical analysis of the power ratio
is presented in Fig. 2(b) with respect to the bath temperatures and the
driving strength, which displays that not only
$\mathcal{P}_{C}/\mathcal{P}_{I}$ is greater than one, but also the ratio may
reach more than 135. Clearly, CQHEs exhibit quantum enhancements over IQHEs in
power.

The heat current from the hot bath is given by

$\displaystyle\langle\dot{J}_{X}^{h}\rangle=\alpha\ \omega_{h}\
\mathcal{C}(\sigma_{X}),$ (10)

for both coherent and incoherent heat engines, and it has a monotonic relation
with energetic coherence in the steady states. Yet again, due to energetic
coherence, the heat current in CQHEs is higher than in IQHEs. In other words,
the CQHEs have a higher capacity to draw heat from the hot bath than the
IQHEs. However, the former also produces more power than the latter.
Consequently, the efficiency $\eta_{X}=-\langle
P_{X}\rangle/\langle\dot{J}_{X}^{h}\rangle$ remains same for both the engines,
i.e.,

$\displaystyle\eta_{I}=\eta_{C}=1-\omega_{c}/\omega_{h}.$ (11)

Thus, CQHEs perform as good as IQHEs as far as heat-to-work conversion
efficiency is concerned. See Appendices A and D.2 for the derivations.

Noise-to-signal ratio (NSR) in power – In microscopic heat engines, power
output often fluctuates. This, in turn, delimits the reliability or stability
of the engines. The fluctuation is usually expressed in terms of the variance
of power $\Delta P_{X}$, for $X=I,C$. For CQHEs and IQHEs, they are

$\displaystyle\Delta P_{X}=\lambda_{1}^{X}\langle
P_{X}\rangle-\lambda_{2}^{X}\langle P_{X}\rangle^{3},$

where coefficients $\lambda_{i}^{X}$s are functions of system and bath
parameters. See Appendix D for more details.

Ideally, a good engine is expected to deliver high power output and low power
output fluctuations. This quality is characterized by the NSR in power, i.e.,
the ratio between the fluctuation in power $\Delta P_{X}$, and the square of
the average power output $\langle P_{X}\rangle^{2}$, and it is lower bounded
by a quantum limit Hasegawa (2020) as

$\displaystyle\mathcal{N}_{X}=\frac{\rm\Delta P_{X}}{\langle
P_{X}\rangle^{2}}\geq f_{X},$ (12)

where the lower bound $f_{X}$ is determined by quantum dynamical activity and
coherent dynamical contribution. This relation is known as the quantum
thermodynamic uncertainty relation (qTUR), and it is derived using quantum
Cramér-Rao bound Hasegawa (2020). The bounds $f_{X}$ in Eq. (12) are different
for coherent and incoherent heat engines as they depend on the underlying
Markovian dynamics. We find that the $\mathcal{N}_{X}$ depends on the
energetic coherence present in the steady states and, for CQHEs and IQHEs,
they are (see Appendices D.1 and D.2)

$\displaystyle\mathcal{N}_{C}$ $\displaystyle=\frac{F_{p}}{\alpha\
\mathcal{C}(\sigma_{C})}\left(1-\frac{3}{2}\mathcal{C}(\sigma_{C})^{2}\right),$
(13) $\displaystyle\mathcal{N}_{I}$ $\displaystyle=\frac{F_{p}}{\alpha\
\mathcal{C}(\sigma_{I})}\left(1-\frac{k}{F_{p}}\mathcal{C}(\sigma_{I})^{2}\right),$
(14)

respectively, where

$\displaystyle F_{p}=\frac{2n_{h}n_{c}+n_{hc}}{n_{h}-n_{c}},\
k=\frac{4\alpha^{2}+\gamma_{0}^{2}(n_{hc}^{2}+2n_{hc}+3n_{h}n_{c})}{\gamma_{0}^{2}(n_{h}-n_{c})}.$
(15)

From the Eqs. (13) and (14), it is seen that the NSR in both coherent and
incoherent engines can be suppressed by accessing energetic coherence in the
steady state for fixed $n_{h}$ and $n_{c}$. We observe that the NSR for CQHEs
is, in general, much lower than that of IQHEs, which is the consequence of
$\mathcal{C}(\sigma_{C})\gg\mathcal{C}(\sigma_{I})$. As shown in Fig. 3(a),
the NSR in CQHEs can be as less as 330 times or lower than the NSR attained in
IQHEs. Clearly, CQHEs are more reliable or deliver more precision in power
than IQHEs.

The saturation of the relation (12), i.e., $\mathcal{N}_{X}=f_{X}$, implies
that the engine is producing the least possible NSR in power that is given by
its quantum bound. This is the best possible operating condition one would
desire from an engine. A numerical analysis presented in Fig. 3(b)
demonstrates that the CQHEs can operate in a regime where they yield very low
NSR in power close to the quantum bound. In contrast, the IQHE has more NSR in
power, which is far from its quantum bound. In addition, the CQHEs can
saturate the qTUR by harnessing a large amount of energetic coherence, as
shown in Fig. 3(c). Overall, the CQHEs are highly reliable and exhibit
substantial quantum enhancements over IQHEs.

Violations of cTUR – For classical heat engines, it is known that the rate of
entropy production and the noise-to-signal ratio (NSR) in power follow a
trade-off relation. This feature has been studied in terms of classical
thermodynamic uncertainty relation (cTUR) Barato and Seifert (2015), given by

$\displaystyle\mathcal{Q}=\dot{S}\mathcal{N}\geq 2,$ (16)

where
$\dot{S}=-\beta_{h}\langle\dot{J}_{h}\rangle-\beta_{c}\langle\dot{J}_{c}\rangle$
is the entropy production rate due to steady state dynamics and
$\mathcal{N}=\Delta P/\langle P\rangle^{2}$ is NSR in power. It implies that a
reduction in NSR can be achieved at the cost of increasing the entropy
production rate $\dot{S}$, particularly when the bound in (16) is saturated.
This, in turn, represents more degree of irreversibility in the engine
operation, leading to a reduced heat-to-work conversion efficiency. A similar
conclusion is also drawn from another relation, known as the power-efficiency-
constancy trade-off relation Pietzonka and Seifert (2018). Interestingly, it
coincides with cTUR for CQHEs and IQHEs (see Appendix E).

We have discussed earlier that, for both coherent and incoherent heat engines,
the NSR in power can be reduced while keeping the engine efficiency the same.
This is why we witness violations of cTUR by CQHEs and IQHEs for some values
of system-bath parameters, signifying that the engines can operate in the
quantum regime.

The left-hand side of relation (16) reduces to (for $X=I,C$)

$\displaystyle\mathcal{Q}_{X}=\ln{\left(\frac{n_{h}(n_{c}+1)}{n_{c}(n_{h}+1)}\right)}F_{X}.$
(17)

Here $F_{X}=\frac{\Delta\dot{N}_{X}}{\langle\dot{N}_{X}\rangle}$ is the Fano
factor, where ${\langle\dot{N}_{X}\rangle}={|\langle
P_{X}\rangle}|/(\omega_{h}-\omega_{c})$ is the photon current and
${\Delta\dot{N}_{X}}=\Delta{P}_{X}/(\omega_{h}-\omega_{c})^{2}$ is the
fluctuation in photon current. The violation of cTUR by CQHEs and IQHEs
implies the violation of $\mathcal{Q}_{C}\geq 2$ and $\mathcal{Q}_{I}\geq 2$,
respectively. Interestingly, the corresponding Fano factor can be expressed in
terms of energetic coherence as

$\displaystyle F_{C}$
$\displaystyle=F_{p}\left(1-\frac{3}{2}\mathcal{C}(\sigma_{C})^{2}\right),$
(18) $\displaystyle F_{I}$
$\displaystyle=F_{p}\left(1-\frac{k}{F_{p}}\mathcal{C}(\sigma_{I})^{2}\right),$
(19)

where $F_{p}$ and $k$ are given in Eq. (15). We refer to Appendix E for the
derivations. In the absence of energetic coherence,
$\mathcal{Q}=\ln{\left(\frac{n_{h}(n_{c}+1)}{n_{c}(n_{h}+1)}\right)}F_{p}$. In
that case, the cTUR is respected because
$\ln{\left(\frac{n_{h}(n_{c}+1)}{n_{c}(n_{h}+1)}\right)}F_{p}\geq 2$ Kalaee
_et al._ (2021). On the contrary, for quantum engines, the violations of cTUR
can necessarily be attributed to the presence of energetic coherence in the
steady states.

Figure 4: Violations of cTUR by CQHEs and IQHEs. (a) The figure on the left
displays the range of violations of cTUR by coherent and incoherent heat
engines with respect to $\alpha$, for the parameters $\gamma_{0}=0.01$,
$\omega_{h}=10$, $\omega_{c}=5$, $\beta_{h}=0.01$, and $\beta_{c}=0.1$. (b)
The figure on the right depicts the depth of violation of cTUR for the
parameters $\gamma_{0}=0.01$, $\omega_{h}=10$, $\omega_{c}=5$,
$\beta_{h}=0.003$, and $\beta_{c}=0.7$. The figures show that the CQHE
violates cTUR for a wider range of parameter $\alpha$. Further, the minimum
value of $\mathcal{Q}_{I}$ is $1.997$ while the minimum value of
$\mathcal{Q}_{C}$ can be $1.24$. See text for more details.

The important point we highlight here is that the CQHEs violate cTUR not only
for a wider range of parameters but also by a higher amount than IQHEs. This
is, yet again, due to the fact that
$\mathcal{C}(\sigma_{C})>\mathcal{C}(\sigma_{I})$ in general. A numerical
study is carried out to compare $\mathcal{Q}_{C}$ and $\mathcal{Q}_{I}$ and
presented in Fig. 4(a) and 4(b). We observe that $\mathcal{Q}_{C}$ can have
values as low as $1.24$, while the lowest value of $\mathcal{Q}_{I}$ remains
very close to $1.997$. Thus, IQHEs only marginally violate the classical
limit. Overall, the violations of cTUR for a wider range of parameters and
with a larger amount indicate that CQHEs possess more non-classical features
than IQHEs.

##  IV Summary

Recent studies have indicated that the performance of microscopic heat engines
can be enhanced by harnessing quantum mechanical features, like energetic
quantum coherence. To harness more energetic coherence, we have introduced
continuous quantum heat engines that utilize coherent energy (heat) transfers
between the working system and the baths via two-photon transitions (Raman
transitions). These coherent heat engines are analogous to the traditional
Scovil and Schulz-DuBois (SSD) engines, except that the latter only allow
incoherent heat transfers via one-photon transitions. The analysis and results
presented above clearly demonstrate that, due to coherent heat transfers,
coherent heat engines harness much more energetic coherence in the working
system than traditional (incoherent) quantum engines. Consequently, the power
and noise-to-signal ratio in power is enhanced by hundreds of folds compared
to their incoherent analogs. The noise-to-signal ratio in power has a
fundamental lower bound derived from the quantum Cramér-Rao bound, and the
inequality is termed the quantum thermodynamic uncertainty relation (qTUR). We
have shown that coherent engines can yield a substantially low noise-to-signal
ratio in power, which is very close to the lower bound (quantum limit). Even
the CQHEs can saturate this quantum bound by harnessing high energetic
coherence. This suggests that saturation of qTUR requires a high amount of
coherence. Thus, coherent engines are highly reliable. In addition, unlike
incoherent engines, coherent engines violate classical thermodynamic
uncertainty relation for a much wider range of parameters and by a much higher
amount. Altogether, the coherent engines possess more quantum features and
greatly outperform conventional quantum and classical heat engines,
manifesting genuine quantum enhancements.

Two-photon Raman transitions provide a very common and standard tool in
contemporary applications of quantum optics (cf. Cohen-Tannoudji  _et al._
(1998); Haroche and Raimond (2006); Meystre and Scully (2021); Larson and
Mavrogordatos (2021)). This paves the way for the realization of coherent
quantum heat engines on various experimental platforms. Raman transitions have
been easily demonstrated in various experimental setups, such as
superconducting circuits Aamir  _et al._ (2022); Zanner  _et al._ (2022),
atom-optical systems Gauguet  _et al._ (2008); Kristensen  _et al._ (2021),
and nitrogen-vacancy centers in diamond Böhm  _et al._ (2021), among others.
Thus, our present analysis and results not only improve the understanding of
quantum thermal devices, particularly how energetic coherence greatly enhances
engine performance, but also open up new avenues for quantum-enabled
technologies in the near future.

An executive summary of our main results is below.

  * •

A new model of continuous quantum heat engines is introduced that enables
coherent heat transfer between the working system (qutrit) and the baths via
two-photon transitions.

  * •

These coherent engines harness a much greater amount of energetic coherence in
the qutrit than the analogous incoherent engines, where the latter are the
traditional SSD engines.

  * •

The coherent engines yield much higher power and much less signal-to-noise
ratio in power compared to incoherent engines, while the efficiency remains
the same.

  * •

The coherent engines can operate at or very close to the quantum limit on the
noise-to-signal ratio in power imposed by quantum thermodynamic uncertainty
relation. Thus, the coherent engines are highly reliable.

  * •

The improvements in performance by coherent engines, exhibiting genuine
quantum enhancements, are attributed to the presence of high energetic
coherence, which is a consequence of coherent heat transfer.

  * •

The new model of engines with coherent heat transfer and the improved
understanding of the role of quantum properties in their performance are
expected to find important implications in emerging quantum-enabled
technologies.

## Acknowledgements

R.G. thanks the Council of Scientific and Industrial Research (CSIR),
Government of India, for financial support through a fellowship (File No.
09/947(0233)/2019-EMR-I). M.L.B. acknowledges financial support from the
Spanish MCIN/AEI/10.13039/501100011033 grant PID2020-113334GB-I00, Generalitat
Valenciana grant CIPROM/2022/66, the Ministry of Economic Affairs and Digital
Transformation of the Spanish Government through the QUANTUM ENIA project call
- QUANTUM SPAIN project, and by the European Union through the Recovery,
Transformation and Resilience Plan - NextGenerationEU within the framework of
the Digital Spain 2026 Agenda, and by the CSIC Interdisciplinary Thematic
Platform (PTI+) on Quantum Technologies (PTI-QTEP+). This project has also
received funding from the European Union’s Horizon 2020 research and
innovation program under grant agreement CaLIGOLA MSCA-2021-SE-01-101086123.
M.L. acknowledges financial supports from Europea Research Council AdG NOQIA;
MCIN/AEI (PGC2018-0910.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project
funded by MCIN/AEI/10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR” (PRTR-C17.I1), FPI); QUANTERA MAQS PCI2019-111828-2);
QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European
Union’s Horizon 2020 program under Grant Agreement No 101017733); Ministry for
Digital Transformation and of Civil Service of the Spanish Government through
the QUANTUM ENIA project call - Quantum Spain project, and by the European
Union through the Recovery, Transformation and Resilience Plan -
NextGenerationEU within the framework of the Digital Spain 2026 Agenda;
Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social
Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-1-0013); EU Quantum
Flagship PASQuanS2.1, 101113690, EU Horizon 2020 FET-OPEN OPTOlogic, Grant No
899794; EU Horizon Europe Program (This project has received funding from the
European Union’s Horizon Europe research and innovation program under grant
agreement No 101080086 NeQSTGrant Agreement 101080086 — NeQST); European
Union’s Horizon 2020 program under the Marie Sklodowska-Curie grant agreement
No 847648; ICFO Internal “QuantumGaudi” project; “La Caixa” Junior Leaders
fellowships, La Caixa” Foundation (ID 100010434): CF/BQ/PR23/11980043. Views
and opinions expressed are, however, those of the author(s) only and do not
necessarily reflect those of the European Union, European Commission, European
Climate, Infrastructure and Environment Executive Agency (CINEA), or any other
granting authority. Neither the European Union nor any granting authority can
be held responsible for them.

##  Appendix A Steady state solution of incoherent quantum heat engines in
rotating frame

For incoherent engines, the total Hamiltonian of the qutrit system and two
photonic (bosonic) thermal baths can be written as

$H=H_{S}+H_{B_{h}}+H_{B_{c}}+H_{SB_{h}B_{c}}^{I},$ (20)

where the Hamiltonian and of the qutrit system is given by

$H_{S}=\omega_{h}\outerproduct{2}{2}+(\omega_{h}-\omega_{c})\outerproduct{1}{1},$
(21)

with $\omega_{h}$ and $\omega_{h}-\omega_{c}$ being the frequencies
corresponding to the energy gaps. The Hamiltonians of the photonic thermal
baths $H_{B_{h}}$ and $H_{B_{c}}$ and the interaction $H_{SB_{h}B_{c}}^{I}$
given in the main text. The corresponding Lindblad master equation describing
the local dynamics of the qutrit is given in Eq. (2) in the main text. In a
rotating frame, given by $B_{R}=e^{i\tilde{H}t}Be^{-i\tilde{H}t}$ and any
operator $B$ and $[H_{0},\tilde{H}]=0$, the master equation becomes

$\displaystyle\dot{\rho_{R}}=$
$\displaystyle-i[H_{S}-\tilde{H}+H_{dR},\rho_{R}]+\mathcal{D}_{h}(\rho_{R})+\mathcal{D}_{c}(\rho_{R}).$
(22)

Without loss of generality, we consider $H_{S}=\tilde{H}$. Thus, the steady-
state solution of the above master equation can be obtained by solving
$\dot{\rho}_{R}=0$ (we denote the steady state by $\sigma_{I}$), and it is

$\displaystyle\sigma_{I}$
$\displaystyle=\frac{4\alpha^{2}(\gamma_{c}+\gamma_{h}+\gamma_{c}n_{c}+\gamma_{h}n_{h}+\gamma_{c}\gamma_{h}n_{c}(n_{h}+1)(\gamma_{c}n_{c}+\gamma_{h}n_{h})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}\outerproduct{0}{0}$
$\displaystyle-\frac{2i\alpha\gamma_{c}\gamma_{h}(n_{h}-n_{c})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}\outerproduct{0}{1}$
$\displaystyle+\frac{2i\alpha\gamma_{c}\gamma_{h}(n_{h}-n_{c})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}\outerproduct{1}{0}$
$\displaystyle+\frac{4\alpha^{2}(\gamma_{c}+\gamma_{h}+\gamma_{c}n_{c}+\gamma_{h}n_{h}+\gamma_{c}\gamma_{h}n_{h}(n_{c}+1)(\gamma_{c}n_{c}+\gamma_{h}n_{h})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}\outerproduct{1}{1}$
$\displaystyle+\frac{(4\alpha^{2}+\gamma_{c}\gamma_{h}n_{c}n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}\outerproduct{2}{2}.$
(23)

The $l$-1 norm of energetic coherence Baumgratz  _et al._ (2014) in the steady
state on the rotating frame is

$\displaystyle\mathcal{C}(\sigma_{I})=|\sigma_{I}^{(01)}|+|\sigma_{I}^{(10)}|=\frac{4\alpha\gamma_{c}\gamma_{h}(n_{h}-n_{c})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})},$
(24)

where $\sigma_{I}^{(ij)}=\langle i|\sigma_{I}|j\rangle$. Now, the average
power and the average heat currents in IQHEs corresponding to the hot and cold
baths are given by

$\displaystyle\langle
P_{I}\rangle=-i\tr([H_{S},H_{dR}]\sigma_{I})=-i\alpha(\omega_{h}-\omega_{c})(\sigma^{(01)}_{I}-\sigma^{(10)}_{I})=-\alpha(\omega_{h}-\omega_{c})\
\mathcal{C}(\sigma_{I}),$ (25)
$\displaystyle\langle\dot{J}^{h}_{I}\rangle=\Tr[\mathcal{D}_{h}(\sigma_{I})H_{S}]=\alpha\
\omega_{h}\ \mathcal{C}(\sigma_{I}),$ (26) and
$\displaystyle\langle\dot{J}^{c}_{I}\rangle=\Tr[\mathcal{D}_{c}(\sigma_{I})H_{S}]=-\alpha\
\omega_{c}\ \mathcal{C}(\sigma_{I}),$ (27)

respectively. Accordingly, the heat-to-work conversion ratio for IQHEs is

$\eta_{I}=-\frac{\langle
P_{I}\rangle}{\langle\dot{J}^{h}_{I}\rangle}=1-\frac{\omega_{c}}{\omega_{h}}.$
(28)

##  Appendix B Derivation of Lindblad master equation for coherent quantum
heat engines

In this section, we derive the Lindblad master equation for a three-level
quantum system coupled with the two photonic (bosonic) thermal baths (hot and
cold baths), where the system and baths interact via two-photon transitions
(Raman Interactions, i.e., three-body interactions). Our derivation follows
the standard textbook approach discussed in Refs. Breuer and Petruccione
(2007); Lidar (2020). The total Hamiltonian of the system and two photonic
thermal baths can be written as

$H=H_{S}+H_{B_{h}}+H_{B_{c}}+H_{SB_{h}B_{c}}^{C}.$ (29)

where suffixes $h$ and $c$ correspond to hot and cold baths, respectively. We
assume $\hbar=k_{B}=1$ throughout this work. In Eq. (29), the system
Hamiltonian $H_{S}$ describes a three-level system (qutrit), given by

$H_{S}=\omega_{h}\outerproduct{2}{2}+(\omega_{h}-\omega_{c})b_{hc}^{\dagger}b_{hc}=\omega_{h}\outerproduct{2}{2}+(\omega_{h}-\omega_{c})\outerproduct{1}{1},$
(30)

where $\omega_{h}$ and $\omega_{h}-\omega_{c}$ refers to the frequencies
corresponding to the energy gaps, and $b_{hc}^{\dagger}=|1\rangle\langle 0|$
and $b_{hc}=|0\rangle\langle 1|$. In Eq. (29), the photonic baths are a
collection of infinite dimensional systems whose total Hamiltonian is given as

$H_{B_{h}}+H_{B_{c}}=\sum_{k}\Omega_{k,h}a^{\dagger}_{k,h}a_{k,h}+\sum_{k^{\prime}}\Omega_{k^{\prime},c}a^{\dagger}_{k^{\prime},c}a_{k^{\prime},c}.$
(31)

Furthermore, in Eq. (29), the interaction Hamiltonian between the system and
the baths has the following form Gerry and Eberly (1990); Gerry and Huang
(1992); Wu (1996)

$H_{SB_{h}B_{c}}^{C}=g_{0}\sum_{kk^{\prime}}(a_{k,h}a^{\dagger}_{k^{\prime},c}b_{hc}^{\dagger}+a^{\dagger}_{k,h}a_{k^{\prime},c}b_{hc}),$
(32)

Here, we consider system-baths coupling to be very weak, i.e., $g_{0}\ll 1$.
The total Hamiltonian of the composite system (system + baths) in the
interaction picture can be written as

$\tilde{H}(t)=g_{0}\sum_{k,k^{\prime}}(a_{k,h}(t)a^{\dagger}_{k^{\prime},c}(t)b_{hc}^{\dagger}(t)+a^{\dagger}_{k,h}(t)a_{k^{\prime},c}(t)b_{hc}(t)),$
(33)

where $b_{hc}(t)=b_{hc}e^{-i(\omega_{h}-\omega_{c})t}$,
$b_{hc}^{\dagger}(t)=b_{hc}^{\dagger}e^{i(\omega_{h}-\omega_{c})t}$,
$a_{p}(t)=a_{p}e^{-i\omega_{p}t}$ and
$a_{p}^{\dagger}(t)=a_{p}^{\dagger}e^{i\omega_{p}t}$. For convenience, we can
write the above Hamiltonian

$\tilde{H}_{I}(t)=g_{0}\sum_{k,k^{\prime}}\sum_{\alpha=\\{1,2\\}}A_{\alpha}(t)\otimes
B_{\alpha,kk^{\prime}}(t),$ (34)

where $A_{1}(t)=b_{hc}^{\dagger}(t)$, $A_{2}(t)=b_{hc}(t)$,
$B_{kk^{\prime},1}(t)=a_{k,h}(t)a^{\dagger}_{k^{\prime},c}(t)$ and
$B_{kk^{\prime},2}(t)=a^{\dagger}_{k,h}(t)a_{k^{\prime},c}(t)$. In the
interaction picture, the dynamics of the composite system is given by the von
Neumann equation,

$\frac{d\tilde{\rho}(t)}{dt}=-i[\tilde{H}(t),\tilde{\rho}(t)].$ (35)

For the cases where the system and baths are initially in a product state and
very weakly coupled, using Born and Markov approximations, we obtain the
following dynamical equation of the system

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}=-g_{0}^{2}\sum_{\alpha\beta}\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau\\{\mathcal{B}_{\alpha\beta,kk^{\prime}ss^{\prime}}(\tau,0)[A_{\alpha}(t),A_{\beta}(t-\tau)\tilde{\rho}(t)]+\mathcal{B}_{\beta\alpha,ss^{\prime}kk^{\prime}}(0,\tau)[\tilde{\rho}(t)A_{\beta}(t-\tau),A_{\alpha}(t)]\\},$
(36)

where
$\mathcal{B}_{\alpha\beta,kk^{\prime}ss^{\prime}}(\tau,0)=\tr(e^{iH_{B}\tau}B_{\alpha,kk^{\prime}}e^{-iH_{B}\tau}B_{\beta,ss^{\prime}}\rho_{\beta_{h}}\otimes\rho_{\beta_{c}})$
and
$\mathcal{B}_{\beta\alpha,ss^{\prime}kk^{\prime}}(0,\tau)=\tr(B_{\beta,ss^{\prime}}e^{iH_{B}\tau}B_{\alpha,kk^{\prime}}e^{-iH_{B}\tau}\rho_{\beta_{h}}\otimes\rho_{\beta_{c}})$.
Here $H_{B}=H_{B_{h}}+H_{B_{c}}$ is total free Hamiltonian of the baths. The
states $\rho_{\beta_{h}}$ and $\rho_{\beta_{c}}$ are the thermal states of hot
and cold baths at inverse temperatures $\beta_{h}$ and $\beta_{c}$. The above
dynamical equation in the frequency domain can be written as

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}=-g_{0}^{2}\sum_{kk^{\prime}ss^{\prime}}$
$\displaystyle\big{[}\int_{0}^{\infty}d\tau\mathcal{B}_{12,kk^{\prime}ss^{\prime}}(\tau,0)e^{i\omega_{hc}\tau}[b_{hc}^{\dagger},b_{hc}\tilde{\rho}(t)]+\int_{0}^{\infty}d\tau\mathcal{B}_{21,kk^{\prime}ss^{\prime}}(\tau,0)e^{-i\omega_{hc}\tau}[b_{hc},b_{hc}^{\dagger}\tilde{\rho}(t)]$
$\displaystyle+\int_{0}^{\infty}d\tau\mathcal{B}_{12,kk^{\prime}ss^{\prime}}(0,\tau)e^{-i\omega_{hc}\tau}[\tilde{\rho}(t)b_{hc}^{\dagger},b_{hc}]+\int_{0}^{\infty}d\tau\mathcal{B}_{21,kk^{\prime}ss^{\prime}}(0,\tau)e^{i\omega_{hc}\tau}[\tilde{\rho}(t)b_{hc},b_{hc}^{\dagger}]\big{]},$
(37)

where $\omega_{hc}=\omega_{h}-\omega_{c}$. The bath correlation functions can
be simplified as

$\displaystyle\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau\mathcal{B}_{12}(\tau,0)e^{i\omega_{hc}\tau}=\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau\langle
a_{k,h}(\tau)a^{\dagger}_{k^{\prime},c}(\tau)a^{\dagger}_{s,h}a_{s^{\prime},c}\rangle
e^{i\omega_{hc}\tau}=\sum_{kk^{\prime}}(n_{k,h}(\Omega_{k,h})+1)n_{k^{\prime},c}(\Omega_{k^{\prime},c})\int_{0}^{\infty}d\tau
e^{-i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau},$
$\displaystyle\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau\mathcal{B}_{21}(\tau,0)e^{-i\omega_{hc}\tau}=\sum_{kk^{\prime}}(n_{k,c}(\Omega_{k,h})+1)n_{k^{\prime},h}(\Omega_{k^{\prime},c})\int_{0}^{\infty}d\tau
e^{i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau},$
$\displaystyle\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau\mathcal{B}_{12}(0,\tau)e^{-i\omega_{hc}\tau}=\sum_{kk^{\prime}}(n_{k,h}(\Omega_{k,h})+1)n_{k^{\prime},c}(\Omega_{k^{\prime},c})\int_{0}^{\infty}d\tau
e^{i(\Delta_{hc}-\omega_{kk^{\prime},hc})\tau},$
$\displaystyle\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau\mathcal{B}_{21}(\tau,0)e^{-i\omega_{hc}\tau}=\sum_{kk^{\prime}}(n_{k,c}(\Omega_{k,h})+1)n_{k^{\prime},c}(\Omega_{k^{\prime},h})\int_{0}^{\infty}d\tau
e^{-i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau},$
$\displaystyle\sum_{kk^{\prime}ss^{\prime}}\int_{0}^{\infty}d\tau e^{\pm
i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau}=\sum_{kk^{\prime}}\pi\delta(\Delta_{kk^{\prime},hc}-\omega_{hc})\pm
i\mathbb{P}(\frac{1}{\Delta_{kk^{\prime},hc}-\omega_{hc}}),$

where $\Delta_{kk^{\prime},hc}=\Omega_{k,h}-\Omega_{k^{\prime},c}$. Here we
have used the relations $\langle
a^{\dagger}_{p}a_{p^{\prime}}\rangle=n_{p}\delta_{pp^{\prime}}$, $\langle
a_{p}a^{\dagger}_{p^{\prime}}\rangle=(n_{p}+1)\delta_{pp^{\prime}}$ and
$\langle a_{p}a_{p^{\prime}}\rangle=\langle
a^{\dagger}_{p}a^{\dagger}_{p}\rangle=0$ to simplify the bath correlation
functions. To further simplify these functions, now we also convert
$\sum_{p}\sum_{p^{\prime}}=\int_{0}^{\infty}\int_{0}^{\infty}d\Omega
d\Omega^{\prime}D(\Omega)D(\Omega^{\prime})$, where $D(\Omega)$ is the photon
density of states, i.e. the number of photon modes in a small frequency
interval $[\Omega,\Omega+d\Omega]$. Ignoring the principal value part for the
moment, we then obtain

$\displaystyle\sum_{k,k^{\prime}}f$
$\displaystyle\left(n_{h}(\Omega_{k,h}),n_{c}(\Omega_{k^{\prime},c})\right)\int_{0}^{\infty}d\tau
e^{\pm i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau}$
$\displaystyle=\pi\int_{0}^{\infty}d\Omega_{h}D(\Omega_{h})\int_{0}^{\infty}d\Omega_{c}D(\Omega_{c})f\left(n_{h}(\Omega_{h}),n_{c}(\Omega_{c})\right)\delta\left((\Omega_{h}-\Omega_{c})-(\omega_{h}-\omega_{c})\right),$
(38)

where $f$ is a function of $n_{h}(\Omega_{k,h})$ and
$n_{c}(\Omega_{k^{\prime},c})$. The double integral on the right-hand side is
correlated. To match it with the incoherent quantum heat engines case, we
enforce the resonance condition $(\Omega_{c}=\omega_{c})$ and
$(\Omega_{h}=\omega_{h})$. As a consequence, the expression reduces to

$\sum_{k,k^{\prime}}f(n_{h}(\Omega_{k,h}),n_{c}(\Omega_{k^{\prime},c}))\int_{0}^{\infty}d\tau
e^{\pm
i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau}=\pi\int_{0}^{\infty}d\Omega_{h}D(\Omega_{h})\int_{0}^{\infty}d\Omega_{c}D(\Omega_{c})f(n_{h}(\Omega_{h}),n_{c}(\Omega_{c}))\delta(\Omega_{h}-\omega_{h})\delta(\Omega_{c}-\omega_{c}),$
(39)

and finally to

$\sum_{k,k^{\prime}}f(n_{h}(\Omega_{k,h}),n_{c}(\Omega_{k^{\prime},c}))\int_{0}^{\infty}d\tau
e^{\pm i(\Delta_{kk^{\prime},hc}-\omega_{hc})\tau}=\pi
f(n_{h}(\omega_{h}),n_{c}(\omega_{c}))D(\omega_{c})D(\omega_{h}).$ (40)

After substituting the expression of simplified bath correlation functions in
Eq. (B), we obtain the Lindblad master equation

$\displaystyle\frac{d\tilde{\rho}(t)}{dt}=\gamma_{1}\left(b_{hc}\tilde{\rho}(t)b_{hc}^{\dagger}-\frac{1}{2}\\{b_{hc}^{\dagger}b_{hc},\tilde{\rho}(t)\\}\right)+\gamma_{2}\left(b_{hc}^{\dagger}\tilde{\rho}(t)b_{hc}-\frac{1}{2}\\{b_{hc}b_{hc}^{\dagger},\tilde{\rho}(t)\\}\right),$

where $\gamma_{1}=\gamma_{0}n_{c}(n_{h}+1)$,
$\gamma_{2}=\gamma_{0}n_{h}(n_{c}+1)$, $\gamma_{0}=2g_{0}^{2}\pi
D(\omega_{c})D(\omega_{h})$ is Weiskopf-Wigner decay constant, and
$n_{x}=1/(e^{\beta_{x}\omega_{x}}-1)$ is average boson number of the bath
’$x$’ with inverse temperature $\beta_{x}$ ($x=h,c$). The Lindblad master
equation derived above is in the interaction picture. It can be expressed in
Schrodinger’s Picture as

$\displaystyle\frac{d{\rho}(t)}{dt}$
$\displaystyle=-i[H_{S},\rho(t)]+\gamma_{1}\left(b_{hc}{\rho}(t)b_{hc}^{\dagger}-\frac{1}{2}\\{b_{hc}^{\dagger}b_{hc},{\rho}(t)\\}\right)+\gamma_{2}\left(b_{hc}^{\dagger}{\rho}(t)b_{hc}-\frac{1}{2}\\{b_{hc}b_{hc}^{\dagger},{\rho}(t)\\}\right).$

This dynamics leads to a steady state $\rho_{ss}$, i.e., $\dot{\rho}_{ss}=0$,
given by

$\rho_{ss}=\frac{\text{$\gamma$}_{1}}{(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})}\outerproduct{0}{0}+\frac{\text{$\gamma$}_{2}}{(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})}\outerproduct{1}{1}.$
(41)

For the steady state, the ratio of populations of exited state $|1\rangle$ and
ground state $|0\rangle$ is given as

$\frac{\rho^{(11)}_{ss}}{\rho^{(00)}_{ss}}=\frac{\gamma_{2}}{\gamma_{1}}=e^{{-(\beta_{h}\omega_{h}-\beta_{c}\omega_{c})}}=e^{-\frac{(\beta_{h}\omega_{h}-\beta_{c}\omega_{c})}{(\omega_{h}-\omega_{c})}(\omega_{h}-\omega_{c})},$
(42)

where $\rho^{(ij)}_{ss}=\langle i|\rho_{ss}|j\rangle$. To have an engine
operation by utilizing two-photon transitions, we need population inversion,
i.e., $\frac{\rho^{(11)}_{ss}}{\rho^{(00)}_{ss}}>1$. For this, the required
condition is $\beta_{h}\omega_{h}-\beta_{c}\omega_{c}<0$. This also implies
$n_{h}>n_{c}$.

###  B.1 Steady state solution of coherent quantum heat engines in rotating
frame

With an external periodic driving on the qutrit $H_{d}(t)=\alpha(e^{-i\omega
t}|1\rangle\langle 0|+e^{i\omega t}|0\rangle\langle 1|)$, the Lindblad master
equation describing the dynamics of a coherent quantum heat engine becomes

$\displaystyle\dot{\rho}=$
$\displaystyle-i[H_{S}+H_{d}(t),\rho]+\mathcal{D}_{hc}(\rho),$ (43)

where the master equation involves single dissipator $\mathcal{D}_{hc}(\rho)$,
given by

$\displaystyle\mathcal{D}_{hc}(\rho)=\gamma_{1}(b_{hc}{\rho}(t)b_{hc}^{\dagger}-\frac{1}{2}\\{b_{hc}^{\dagger}b_{hc},{\rho}(t)\\})+\gamma_{2}(b_{hc}^{\dagger}{\rho}(t)b_{hc}-\frac{1}{2}\\{b_{hc}b_{hc}^{\dagger},{\rho}(t)\\}).$
(44)

We can transform the above Lindblad master equation to the rotating frame
using the transformation $B_{R}=e^{i\tilde{H}t}Be^{-i\tilde{H}t}$, where $B$
is an arbitrary operator and $[H_{S},\tilde{H}]=0$, as follows:

$\displaystyle\dot{\rho}_{R}=$
$\displaystyle-i[H_{S}-\tilde{H}+H_{dR},\rho_{R}]+\mathcal{D}_{hc}(\rho_{R}),$
(45)

where $H_{dR}$ corresponds the driving Hamiltonian $H_{d}(t)$ in rotating
frame. Without loss of generality, we consider $H_{S}=\tilde{H}$.

A steady-state solution of the above master equation can be obtained by
solving $\dot{\rho}_{R}=0$ (we denote the steady state by $\sigma_{C}$), which
yields

$\sigma_{C}=\frac{4\alpha^{2}+\gamma_{1}(\gamma_{1}+\gamma_{2})}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}\outerproduct{0}{0}+\frac{2i\alpha(\gamma_{1}-\gamma_{2})}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}\outerproduct{0}{1}\\\
-\frac{2i\alpha(\gamma_{1}-\gamma_{2})}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}\outerproduct{1}{0}+\frac{4\alpha^{2}+\gamma_{2}(\gamma_{1}+\gamma_{2})}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}\outerproduct{1}{1}.$
(46)

The $l$-1 norm of coherence Baumgratz  _et al._ (2014) of the steady state in
CQHEs can be expressed as

$\displaystyle\mathcal{C}(\sigma_{C})=|\sigma_{C}^{(01)}|+|\sigma_{C}^{(10)}|=\frac{4\gamma_{0}\alpha(n_{h}-n_{c})}{8\alpha^{2}+\gamma_{0}^{2}(2n_{h}n_{c}+n_{h}+n_{c})^{2}},$
(47)

where $\sigma_{C}^{(ij)}=\langle i|\sigma_{I}|j\rangle$. The average power is
directly related to energetic coherence as

$\displaystyle\langle
P_{C}\rangle=-i\tr([H_{S},H_{dR}]\sigma_{C})=-i\alpha(\omega_{h}-\omega_{c})(\sigma^{(01)}_{C}-\sigma^{(10)}_{C})=-\alpha(\omega_{h}-\omega_{c})\
\mathcal{C}(\sigma_{C}).$ (48)

The dynamics due to heat transfer with the baths is governed by single
dissipator $\mathcal{D}_{hc}$, unlike in IQHEs discussed in Appendix A, and it
takes into account the contributions from hot and cold baths together. Because
of that, we cannot directly calculate the heat currents from the hot and cold
baths with the dissipator. We overcome this limitation by employing the full
counting statistics (FCS) of the steady-state dynamics in the rotating frame
(see Appendix D).

##  Appendix C Comparison of energetic coherences in coherent and incoherent
heat engines

The energetic coherence in the steady state of the qutrit is non-linearly
dependent on the driving parameter $\alpha$ for both coherent and incoherent
heat engines. It is, in general, higher in the coherent heat engines compared
to the incoherent ones. However, for some values of $\alpha$, the energetic
coherence in the coherent heat engines can be lower than the incoherent ones.
The driving parameter has a critical value, given by $\alpha_{cr}$, below
which the energetic coherence is higher for incoherent heat engines. We
determine the $\alpha_{cr}$ by solving the condition

$\mathcal{C}(\sigma_{C})=\mathcal{C}(\sigma_{I}),$ (49)

and it is

$\displaystyle\alpha_{cr}=\gamma_{0}\sqrt{\frac{(n_{h}+n_{c})(n_{h}+n_{c}+3n_{h}n_{c})-\gamma_{0}^{2}(n_{c}+n_{h}+2n_{c}n_{h})^{2}}{8-4\left(3(n_{h}+n_{c})+4\right)}}.$
(50)

The $\mathcal{C}(\sigma_{C})>\mathcal{C}(\sigma_{I})$ for $\alpha>\alpha_{cr}$
and $\mathcal{C}(\sigma_{C})\leq\mathcal{C}(\sigma_{I})$ for
$\alpha\leq\alpha_{cr}$. Note that we need to satisfy the condition
$n_{h}>n_{c}$ for the continuous device to operate as a heat engine. However,
for reasonable values of the parameters $n_{h}$, $n_{c}$, and $\gamma_{0}$,
the critical value $\alpha_{cr}$ remains very small, corresponding to a very
weak external driving. Fig. 5 illustrates how $\alpha_{cr}$ varies with
respect to inverse temperatures of the baths. In the exceptional cases where
the baths are extremely hot, i.e., $n_{h}\approx n_{c}\gg 1$, the
$\alpha_{cr}$ becomes very high. Nevertheless, considering the usual
experimental situations, the engines operate with $\alpha>\alpha_{cr}$, and
the coherent engines yield more energetic coherence in their steady state than
the incoherent engines.

Figure 5:  The figure depicts the variation of the critical value of the
driving parameter against the inverse temperature of the cold bath. Here we
consider $\gamma_{0}$ = 0.01, $\omega_{h}$ = 10, $\omega_{c}$ = 5, and
$\beta_{h}$ = 0.001.

##  Appendix D Full Counting Statistics

Full Counting Statistics (FCS) provides an analytical approach to determine
the statistics of the quantity of interests $M$, such as power, currents
corresponding to each bath, and their fluctuations in an open quantum system
dynamics Esposito  _et al._ (2009). This approach incorporates counting fields
into the master equation. Suppose $\rho(\chi,t)$ represents the solution of
the dressed Lindblad master equation. In that case, we define the moment-
generating function $\mathcal{M}(\chi,t)$ and the cumulant-generating function
$\mathcal{F}(\chi,t)$ as follows:

$\displaystyle\mathcal{M}(\chi,t)=\mathrm{tr}{\\{\rho(\chi,t)\\}},\
\mbox{and}\ \mathcal{F}(\chi,t)=\ln\mathcal{M}(\chi,t).$ (51)

Sometimes, a description in terms of cumulants is more convenient. The
advantage lies in the fact that the dominant eigenvalue of the Liouvillian
usually determines the long-time evolution of the cumulant-generating
function:

$\mathcal{C}(\chi,t)\approx\lambda(\chi)t,$ (52)

where $\lambda(\chi)$is the eigenvalue of
$\mathcal{L}(\chi)=\mathcal{L}(\chi,0)$ with the largest real part (uniqueness
assumed) and it vanishes when $\chi=0$.

In the long-time limit, the cumulants of the quantity of interest $M$ in the
steady state can be obtained using the following formula:

$\langle\langle
M^{k}\rangle\rangle=\bigg{(}\frac{d}{d(i\chi)}\bigg{)}^{k}\lambda(\chi)\bigg{|}_{\chi=0}.$
(53)

The first and second cumulants correspond to the mean and variance of the
quantity of interest $M$, respectively:

$\displaystyle\langle
M\rangle=\bigg{(}\frac{d}{d(i\chi)}\bigg{)}\lambda(\chi)\bigg{|}_{\chi=0},\ \
\mbox{and}\ \ \Delta M=\langle\langle
M^{2}\rangle\rangle=\bigg{(}\frac{d}{d(i\chi)}\bigg{)}^{2}\lambda(\chi)\bigg{|}_{\chi=0}.$
(54)

A direct computation of $\lambda(\chi)$ is not straightforward. To
analytically determine the mean and variance from the derivatives, we follow
the method outlined in Refs. Bruderer  _et al._ (2014); Kalaee  _et al._
(2021); Prech  _et al._ (2023); Landi  _et al._ (2024). Consider the
characteristic polynomial of $\cal{L}(\chi)$

$\sum_{n}a_{n}\lambda(\chi)^{n}=0,$ (55)

where the terms $a_{n}$ are functions of $\chi$. Derivatives of $a_{n}$ are
defined as

$\displaystyle a^{\prime}_{n}=i\frac{d}{d\chi}a_{n}|_{\chi=0},\ \mbox{and}\
a^{\prime\prime}_{n}=\big{(}i\frac{d}{d\chi}\big{)}^{n}a_{n}|_{\chi=0}.$ (56)

With a little analysis, we can express mean and variance as (for more details,
see appendices of Refs. Kalaee  _et al._ (2021); Prech  _et al._ (2023); Landi
_et al._ (2024)):

$\displaystyle\langle M\rangle=-\frac{a^{\prime}_{0}}{a_{1}},\ \mbox{and}\
\Delta
M=\bigg{(}\frac{a^{\prime\prime}_{0}}{a^{\prime}_{0}}-\frac{2a^{\prime}_{1}}{a_{1}}\bigg{)}\langle
M\rangle-\frac{2a_{2}}{a_{1}}\langle M\rangle^{2}.$ (57)

Note that the above expressions of mean and variance hold for all systems with
Lindblad dynamics with a unique steady state.

###  D.1 Counting field statistics for coherent quantum heat engines

Here, we re-derive the Lindblad master equation of coherent heat engine by
introducing counting fields, which will help us to evaluate current and power
statistics Esposito  _et al._ (2009). The total Hamiltonian for the system and
the baths (in the presence of driving) is given as

$H=H_{S}+H_{d}(t)+H_{B_{h}}+H_{B_{c}}+H_{SB_{h}B_{c}}^{C}.$ (58)

where $H_{d}(t)$ represents the external driving field acting on the three-
level system and the rest of the Hamiltonians defined in the previous section.
Here, we are considering a situation where the two baths continuously interact
with the system, and the interaction between the system and the baths is weak.
We choose the initial state as a product state, i.e.,
$\rho(0)=\rho_{S}(0)\otimes\rho_{B}$, where
$\rho_{B}=\rho_{\beta_{h}}\otimes\rho_{\beta_{c}}$ and the baths are prepared
in thermal states with respective Hamiltonians $H_{B_{h}}$, $H_{B_{c}}$ and
inverse temperatures $\beta_{h}$ and $\beta_{c}$, respectively. To measure the
observables are the Hamiltonians $H_{B_{h}}$ and $H_{B_{c}}$ and to get the
corresponding probability distributions of their measurement, we introduce
counting field $\chi_{j}$ $(j=h,c)$ to each bath. We introduce
$\chi\equiv{\\{\chi_{h},\chi_{c}\\}}$ to denote collectively both the counting
variables. The modified density matrix $\rho(\chi,t)$ of composite system is
given as

$\rho(\chi,t)=U(\chi,t)\rho(0)\bar{U}(-\chi,t),$ (59)

with

$\displaystyle
U(\chi,t)=e^{-i(\chi_{h}H_{B_{h}}+\chi_{c}H_{B_{c}})/2}U(t)e^{i(\chi_{h}H_{B_{h}}+\chi_{c}H_{B_{c}})/2}\
\ \mbox{and}\ \
\bar{U}(-\chi,t)=e^{i(\chi_{h}H_{B_{h}}+\chi_{c}H_{B_{c}})/2}U^{\dagger}(t)e^{-i(\chi_{h}H_{B_{h}}+\chi_{c}H_{B_{c}})/2}$

being the counting field-dressed evolution operator. Here $U(t)$ is the
unitary evolution operator generated by the total Hamiltonian $H$. The time
evolution of modified density matrix $\rho(\chi,t)$ is given by following
master equation

$\frac{d\rho(\chi,t)}{dt}=-i[H(\chi,t)\rho(\chi,t)-\rho(\chi,t)H(-\chi,t)],$
(60)

where,
$H(\chi,t)=e^{-i(\chi_{h}H_{B_{h}}+\chi_{c}H_{B_{c}})/2}He^{i(\chi_{h}H_{B_{h}}+\chi_{c}H_{B_{c}})/2}$.
In the interaction picture, one gets (the operators are labeled by tilde)

$\tilde{\rho}(\chi,t)=U_{0}\rho(\chi,t)U^{\dagger}_{0},$ (61)

where $U_{0}$ is the unitary operator generated by the Hamiltonian
$H_{0}(t)=H_{S}(t)+H_{B}$. Here we have denoted $H_{S}(t)=H_{S}+H_{d}(t)$ and
$H_{B}=H_{B_{h}}+H_{B_{c}}$. In the interaction picture, the dressed total
Hamiltonian is given by

$\tilde{H}_{I}(\chi,t)=U_{0}H_{SB}(\chi,t)U_{0}^{\dagger}=\sum_{\alpha,kk^{\prime}}A_{\alpha}(t)\otimes
B_{\alpha,kk^{\prime}}(\chi,t)\ \ \mbox{and}\ \
\tilde{H}_{I}(-\chi,t)=U_{0}H_{SB}(\chi,t)U_{0}^{\dagger}=\sum_{\alpha,kk^{\prime}}A_{\alpha}(t)\otimes
B_{\alpha,kk^{\prime}}(-\chi,t),$ (62)

where $B_{\alpha,kk^{\prime}}(\chi,t)=B^{h}_{\alpha,k}(\chi_{h},t)\otimes
B^{c}_{\alpha,k^{\prime}}(\chi_{c},t)$ and $B^{h(c)}$’s are bath operators
corresponding to hot (cold) bath. In the interaction picture, the evolution
equation can be written as

$\frac{d\tilde{\rho}(\chi,t)}{dt}=-i[\tilde{H}_{I}(\chi,t)\tilde{\rho}(\chi,t)-\tilde{\rho}(\chi,t)\tilde{H}_{I}(-\chi,t)].$
(63)

Next, considering the weak coupling assumption and performing the standard
Born-Markov approximation, we arrive at the following master equation

$\displaystyle\frac{d\tilde{\rho}_{S}(\chi,t)}{dt}=-\int_{0}^{\infty}d\tau{\rm
Tr}_{B}[\tilde{H}_{I}$
$\displaystyle(\chi,t)\tilde{H}_{I}(\chi,t-\tau)\tilde{\rho}_{S}(\chi,t)\rho_{B}-\tilde{H}_{I}(\chi,t)\tilde{\rho}_{S}(\chi,t)\rho_{B}\tilde{H}_{I}(-\chi,t-\tau)$
$\displaystyle-\tilde{H}_{I}(\chi,t-\tau)\tilde{\rho}_{S}(\chi,t)\rho_{B}\tilde{H}_{I}(-\chi,t)+\tilde{\rho}_{S}(\chi,t)\rho_{B}\tilde{H}_{I}(-\chi,t-\tau)\tilde{H}_{I}(-\chi,t)],$
(64)

where we have used ${\rm Tr}_{B}[\tilde{H}_{I}(\chi,t)\ \rho_{B}]=0$, and
$\rho_{B}=\rho_{\beta_{h}}\otimes\rho_{\beta_{c}}$. After simplification, the
above equation can be written as

$\displaystyle\frac{d\tilde{\rho}_{S}(\chi,t)}{dt}=-g_{0}^{2}\int_{0}^{\infty}d\tau\sum_{\alpha\beta
kk^{\prime}ss^{\prime}}$
$\displaystyle\left({A}_{\alpha}(t){A}_{\beta}(t-\tau)\tilde{\rho}_{S}(\chi,t)\Tr[{B}_{\alpha,kk^{\prime}}(\chi,t){B}_{\beta,ss^{\prime}}(\chi,t-\tau)\rho_{B}]\right.$
$\displaystyle-{A}_{\alpha}(t)\tilde{\rho}_{S}(\chi,t){A}_{\beta}(t-\tau)\Tr[{B}_{\alpha,kk^{\prime}}(\chi,t)\rho_{B}{B}_{\beta,ss^{\prime}}(-\chi,t-\tau)]$
$\displaystyle-{A}_{\alpha}(t-\tau)\tilde{\rho}_{S}(\chi,t){A}_{\beta}(t)\Tr[{B}_{\alpha,kk^{\prime}}(\chi,t-\tau)\rho_{B}{B}_{\beta,ss^{\prime}}(-\chi,t)]$
$\displaystyle\left.+\tilde{\rho}_{S}(\chi,t){A}_{\alpha}(t-\tau){A}_{\beta}(t)\Tr[\rho_{B}{B}_{\alpha,kk^{\prime}}(-\chi,t-\tau){B}_{\beta,ss^{\prime}}(-\chi,t)]\right).$

After further simplifying the bath correlation function, we obtain

$\displaystyle\frac{d\tilde{\rho}_{S}(\chi,t)}{dt}=$ $\displaystyle-
g_{0}^{2}\int_{0}^{\infty}d\tau\sum_{\alpha\beta}\sum_{kk^{\prime}ss^{\prime}}({A}_{\alpha}(t){A}_{\beta}(t-\tau)\tilde{\rho}_{S}(\chi,t)\tr[{B}_{\alpha,kk^{\prime}}(\tau){B}_{\beta,ss^{\prime}}(0)\rho_{B}]-{A}_{\alpha}(t)\tilde{\rho}_{S}(\chi,t){A}_{\beta}(t-\tau)\tr[{B}_{\beta,ss^{\prime}}(-2\chi,\tau){B}_{\alpha,kk^{\prime}}(0)\rho_{B}]$
$\displaystyle-{A}_{\alpha}(t-\tau)\tilde{\rho}_{S}(\chi,t){A}_{\beta}(t)\tr[{B}_{\beta,ss^{\prime}}(-2\chi,t){B}_{\alpha,kk^{\prime}}(0)\rho_{B}]+\tilde{\rho}_{S}(\chi,t){A}_{\alpha}(t-\tau){A}_{\beta}(t)\tr[{B}_{\alpha,kk^{\prime}}(-\tau){B}_{\beta,ss^{\prime}}(0)\rho_{B}]).$

Using the explicit form of system and bath operators
$A_{1}(t)=b_{hc}^{\dagger}(t)$, $A_{2}(t)=b_{hc}(t)$,
$B_{kk^{\prime},1}(t)=a_{k,h}(t)a^{\dagger}_{k^{\prime},c}(t)$,
$B_{kk^{\prime},2}(t)=a^{\dagger}_{k,h}(t)a_{k^{\prime},c}(t)$,
$b_{hc}(t)=b_{hc}e^{-i(\omega_{h}-\omega_{c})t}$,
$b_{hc}^{\dagger}(t)=b_{hc}^{\dagger}e^{i(\omega_{h}-\omega_{c})t}$,
$a_{p}(t)=a_{p}e^{-i\omega_{p}t}$ and
$a_{p}^{\dagger}(t)=a_{p}^{\dagger}e^{i\omega_{p}t}$, and solving the bath
correlation function and converting sums into integrals as considered in the
previous section B, we get the following dressed Lindblad master equation in
Schrodinger picture as

$\displaystyle\frac{d{\rho}(\chi,t)}{dt}=-i[H_{S}+H_{d}(t),\rho]+\gamma_{1}(e^{i(\omega_{h}\chi_{h}-\omega_{c}\chi_{c})}b_{hc}{\rho}(t)b_{hc}^{\dagger}-\frac{1}{2}\\{b_{hc}^{\dagger}b_{hc},{\rho}(t)\\})+\gamma_{2}(e^{-i(\omega_{h}\chi_{h}-\omega_{c}\chi_{c})}b_{hc}^{\dagger}{\rho}(t)b_{hc}-\frac{1}{2}\\{b_{hc}b_{hc}^{\dagger},{\rho}(t)\\}),$
(65)

where $\gamma_{1}=\gamma_{0}n_{c}(n_{h}+1)$,
$\gamma_{2}=\gamma_{0}n_{h}(n_{c}+1)$, $\gamma_{0}=2g^{2}\pi
D(\omega_{c})D(\omega_{h})$ is Weiskopf-Wigner decay constant, and
$n_{x}=1/(e^{\beta_{x}\omega_{x}}-1)$ is the average photon number in the bath
with inverse temperature $\beta_{x}$. In the rotating frame, the above master
equation reduces to

$\displaystyle\frac{d{\rho}(\chi,t)}{dt}=-i[H_{dR},\rho]+\gamma_{1}(e^{i(\omega_{h}\chi_{h}-\omega_{c}\chi_{c})}b_{hc}{\rho}(t)b_{hc}^{\dagger}-\frac{1}{2}\\{b_{hc}^{\dagger}b_{hc},{\rho}(t)\\})+\gamma_{2}(e^{-i(\omega_{h}\chi_{h}-\omega_{c}\chi_{c})}b_{hc}^{\dagger}{\rho}(t)b_{hc}-\frac{1}{2}\\{b_{hc}b_{hc}^{\dagger},{\rho}(t)\\}),$
(66)

and the corresponding full Liouvillian super-operator with counting fields is

$\mathcal{L}(\chi_{h},\chi_{c})=\left(\begin{array}[]{cccc}-\text{$\gamma$}_{1}&-i\alpha&i\alpha&\text{$\gamma$}_{2}e^{-i(\chi_{h}\text{$\omega$}_{h}-\chi_{c}\text{$\omega$}_{c})}\\\
-i\alpha&-\frac{\text{$\gamma$}_{1}}{2}-\frac{\text{$\gamma$}_{2}}{2}&0&i\alpha\\\
i\alpha&0&-\frac{\gamma_{1}}{2}-\frac{\text{$\gamma$}_{2}}{2}&-i\alpha\\\
\text{$\gamma$}_{1}e^{i(\chi_{h}\text{$\omega$}_{h}-\chi_{c}\text{$\omega$}_{c})}&i\alpha&-i\alpha&-\text{$\gamma$}_{2}\\\
\end{array}\right).$ (67)

For calculating power statistics, we set $\chi_{h}=\chi_{c}=\chi$. Following
the previous discussion in this section, we can determine the polynomial
factors with respective derivatives

$\displaystyle a_{1}$
$\displaystyle=2\alpha^{2}({\gamma}_{1}+{\gamma}_{2})+\frac{1}{4}({\gamma}_{1}+{\gamma}_{2})^{3},$
$\displaystyle a_{2}$
$\displaystyle=\frac{1}{4}\left(16\alpha^{2}+5({\gamma}_{1}+{\gamma}_{2})^{2}\right),$
$\displaystyle a^{\prime}_{0}$
$\displaystyle=\alpha^{2}({\gamma}_{1}-{\gamma_{2}})({\gamma}_{1}+{\gamma}_{2})({\omega}_{h}-{\omega}_{c}),$
$\displaystyle a^{\prime\prime}_{0}$
$\displaystyle=-\alpha^{2}({\gamma}_{1}+{\gamma}_{2})^{2}({\omega}_{h}-{\omega}_{c})^{2},$
$\displaystyle\mbox{and}\ \ \ a^{\prime}_{1}$
$\displaystyle=2\alpha^{2}({\gamma}_{1}-{\gamma}_{2})({\omega}_{h}-{\omega}_{c}).$

The expression for the average (mean) and variance of power are given by

$\displaystyle\langle
P_{C}\rangle=\frac{4\alpha^{2}(\text{$\gamma$}_{1}-\text{$\gamma$}_{2})}{8\alpha^{2}+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{2}}(\text{$\omega$}_{h}-\text{$\omega$}_{c}),\
\mbox{and}\ \Delta P_{C}=F_{p}(\langle
P_{C}\rangle-\frac{3}{2\alpha^{2}(\omega_{h}-\omega_{c})^{2}}\langle
P_{C}\rangle^{3})(\omega_{h}-\omega_{c}),$ (68)

where $F_{p}=\frac{2n_{h}n_{c}+n_{h}+n_{c}}{n_{h}-n_{c}}$. Similarly, we can
determine the average and variance of heat current corresponding to a bath
with inverse temperature $\beta_{x}$ by setting $\chi_{x}=\chi$ and
$\chi_{y}=0$ in the Liouvillian super-operator. The average heat currents from
the hot and cold baths are

$\displaystyle\langle\dot{J}^{h}_{C}\rangle=\frac{4\alpha^{2}(\text{$\gamma$}_{2}-\text{$\gamma$}_{1})}{8\alpha^{2}+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{2}}\text{$\omega$}_{h},\
\ \mbox{and}\ \
\langle\dot{J}^{c}_{C}\rangle=\frac{4\alpha^{2}(\text{$\gamma$}_{1}-\text{$\gamma$}_{2})}{8\alpha^{2}+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{2}}\text{$\omega$}_{c},$
(69)

respectively, and the corresponding variances in heat currents are

$\displaystyle\Delta\dot{J}^{h}_{C}$
$\displaystyle=\frac{4\alpha^{2}(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})\left(64\alpha^{4}-8\alpha^{2}\left(\text{$\gamma$}_{1}^{2}-10\text{$\gamma$}_{1}\text{$\gamma$}_{2}+\text{$\gamma$}_{2}^{2}\right)+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{4}\right)}{\left(8\alpha^{2}+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{2}\right)^{3}}\text{$\omega$}_{h}^{2},$
(70) $\displaystyle\mbox{and}\ \ \Delta\dot{J}^{c}_{C}$
$\displaystyle=\frac{4\alpha^{2}(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})\left(64\alpha^{4}-8\alpha^{2}\left(\text{$\gamma$}_{1}^{2}-10\text{$\gamma$}_{1}\text{$\gamma$}_{2}+\text{$\gamma$}_{2}^{2}\right)+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{4}\right)}{\left(8\alpha^{2}+(\text{$\gamma$}_{1}+\text{$\gamma$}_{2})^{2}\right)^{3}}\text{$\omega$}_{c}^{2}.$
(71)

With this, the heat-to-work conversion efficiency of CQHEs becomes

$\eta_{C}=-\frac{\langle
P_{C}\rangle}{\langle\dot{J}^{h}_{C}\rangle}=1-\frac{\omega_{c}}{\omega_{h}}.$
(72)

It is important to note that IQHEs and CQHEs have the same efficiency.
Further, the noise-to-signal ratio of the power of CQHEs is

$\mathcal{N}_{C}=\frac{\Delta P_{C}}{\langle
P_{C}\rangle^{2}}=F_{p}\big{(}\frac{1}{\langle
P_{C}\rangle}-\frac{3}{2\alpha^{2}(\omega_{h}-\omega_{c})^{2}}\langle
P_{C}\rangle\big{)}(\omega_{h}-\omega_{c})=\frac{F_{p}}{\alpha\mathcal{C}(\sigma_{{C}})}(1-\frac{3}{2}{\mathcal{C}(\sigma_{{C}})^{2}}),$
(73)

where $\langle
P_{C}\rangle=-\alpha(\omega_{h}-\omega_{c})\mathcal{C}(\sigma_{C})$, and
$\mathcal{C}(\sigma_{{C}})$ is $l$-1 norm of coherence of the steady state
$\sigma_{C}$. It is important to note that the noise-to-signal ratio of
currents, power, and photon number flux is the same for CQHEs.

###  D.2 Counting field statistics for Incoherent quantum heat engines

To determine the power statistics in incoherent heat engines, we again use the
Full Counting Statistics (FCS) technique, which includes counting fields in
the master equation. Let $\chi_{h}$ and $\chi_{c}$ be counting fields for the
hot and cold baths, respectively. The dressed Lindblad master equation (22) of
IQHEs in the rotating frame becomes

$\displaystyle\dot{\rho}_{R}=$
$\displaystyle-i[H_{dR},\rho_{R}]+\gamma_{h}(n_{h}+1)(e^{-i\omega_{h}\chi_{h}}b_{h}{\rho}_{R}b^{\dagger}_{h}-\frac{1}{2}\\{b^{\dagger}_{h}b_{h},{\rho}_{R}\\})+\gamma_{h}n_{h}(e^{i\omega_{h}\chi_{h}}b^{\dagger}_{h}{\rho}_{R}b_{h}-\frac{1}{2}\\{b_{h}b^{\dagger}_{h},{\rho}_{R}\\})$
(74)
$\displaystyle+\gamma_{c}(n_{c}+1)(e^{-i\omega_{c}\chi_{c}}b_{c}{\rho}_{R}b^{\dagger}_{c}-\frac{1}{2}\\{b^{\dagger}_{c}b_{c},{\rho}_{R}\\})+\gamma_{c}n_{c}(e^{i\omega_{c}\chi_{c}}b^{\dagger}_{c}{\rho}_{R}b_{c}-\frac{1}{2}\\{b_{c}b^{\dagger}_{c},{\rho}_{R}\\}).$

Accordinlgly, the full Liouvillian super-operator
$\mathcal{L}(\chi_{h},\chi_{c})$ with counting fields is

$\left(\begin{array}[]{ccccccccc}-\text{g}_{1}-\text{g}_{3}&0&0&0&\text{g}_{4}e^{i\chi_{c}\omega_{c}}&0&0&0&\text{g}_{2}e^{i\text{$\chi$}_{h}\text{$\omega$}_{h}}\\\
0&-\frac{\text{g}_{1}}{2}-\frac{\text{g}_{3}}{2}-\frac{\text{g}_{4}}{2}&-i\alpha&0&0&0&0&0&0\\\
0&-i\alpha&-\frac{\text{g}_{1}}{2}-\frac{\text{g}_{2}}{2}-\frac{\text{g}_{3}}{2}&0&0&0&0&0&0\\\
0&0&0&-\frac{\text{g}_{1}}{2}-\frac{\text{g}_{3}}{2}-\frac{\text{g}_{4}}{2}&0&0&i\alpha&0&0\\\
\text{g}_{3}e^{-i\chi_{c}\omega_{c}}&0&0&0&-\text{g}_{4}&-i\alpha&0&i\alpha&0\\\
0&0&0&0&-i\alpha&-\frac{\text{g}_{2}}{2}-\frac{\text{g}_{4}}{2}&0&0&i\alpha\\\
0&0&0&i\alpha&0&0&-\frac{\text{g}_{1}}{2}-\frac{\text{g}_{2}}{2}-\frac{\text{g}_{3}}{2}&0&0\\\
0&0&0&0&i\alpha&0&0&-\frac{\text{g}_{2}}{2}-\frac{\text{g}_{4}}{2}&-i\alpha\\\
\text{g}_{1}e^{-i\text{$\chi$}_{h}\text{$\omega$}_{h}}&0&0&0&0&i\alpha&0&-i\alpha&-\text{g}_{2}\\\
\end{array}\right),$

where $g_{1}=\gamma_{h}(n_{h}+1)$, $g_{2}=\gamma_{h}n_{h}$,
$g_{3}=\gamma_{c}(n_{c}+1)$ and $g_{4}=\gamma_{c}n_{c}$. We set
$\chi_{h}=\chi_{c}=\chi$ to calculate the power statistics. Following the
previous discussion in this section, we find the polynomial factors with
respective derivatives:

$\displaystyle a_{1}=$
$\displaystyle-\frac{1}{64}({\gamma}_{c}{n}_{c}+{\gamma}_{h}{n}_{h})\left(4\alpha^{2}+({\gamma}_{c}+{\gamma}_{h}+2{\gamma}_{c}{n}_{c}+{\gamma}_{h}{n}_{h})({\gamma}_{c}+{\gamma_{h}}+{\gamma}_{c}n_{c}+2{\gamma}_{h}n_{h})\right)^{2}\left(4\alpha^{2}({\gamma}_{c}(3n_{c}+2)+{\gamma}_{h}(3n_{h}+2))\right.$
$\displaystyle\left.+{\gamma}_{c}{\gamma}_{h}(3n_{c}n_{h}+n_{c}+n_{h})({\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})\right),$
$\displaystyle a_{2}=$
$\displaystyle-\frac{1}{64}(4\alpha^{2}+({\gamma}_{c}+{\gamma}_{h}+2{\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})({\gamma}_{c}+{\gamma}_{h}+{\gamma}_{c}n_{c}+2{\gamma}_{h}n_{h}))(({\gamma}_{c})^{5}n_{c}^{2}(n_{c}+1)(2n_{c}+1)^{2}+({\gamma}_{c})^{2}({\gamma}_{h})^{3}(5n_{c}^{2}+(n_{c}$
$\displaystyle(238n_{c}+157)+25)n_{h}^{3}+(6n_{c}(39n_{c}+19)+11)n_{h}^{2}+n_{c}(67n_{c}+18)n_{h})+({\gamma}_{c})^{3}({\gamma}_{h})^{2}(n_{c}^{3}(n_{h}(238n_{h}+157)+25)+n_{c}^{2}$
$\displaystyle(6n_{h}(39n_{h}+19)+11)+n_{c}n_{h}(67n_{h}+18)+5n_{h}^{2})+{\gamma}_{c}({\gamma}_{h})^{4}n_{h}(n_{c}(n_{h}(n_{h}(82n_{h}+113)+47)+6)+n_{h}(28n_{h}^{2}+30n_{h}$
$\displaystyle+7))+64\alpha^{4}({\gamma}_{c}+{\gamma}_{h}+2{\gamma}_{c}n_{c}+2n_{h}n_{h})+4\alpha^{2}((({\gamma}_{c})^{3}(3n_{c}+2)^{2}(6n_{c}+1)+(({\gamma}_{c})^{2}({\gamma}_{h}(n_{c}(2(91n_{c}+85)n_{h}+85n_{c}+72)$
$\displaystyle+36n_{h}+12)+({\gamma}_{c}(({\gamma}_{h})^{2}(2n_{c}(n_{h}(91n_{h}+85)+18)+n_{h}(85n_{h}+72)+12)+(({\gamma}_{h})^{3}(3n_{h}+2)^{2}(6n_{h}+1))+(({\gamma}_{c})^{4}({\gamma}_{h}n_{c}$
$\displaystyle(n_{c}(n_{c}(n_{c}(82n_{h}+28)+113n_{h}+30)+47n_{h}+7)+6n_{h})+(({\gamma}_{h})^{5}n_{h}^{2}(n_{h}+1)(2n_{h}+1)^{2}),$
$\displaystyle a^{\prime}_{0}=$
$\displaystyle\frac{1}{16}{\gamma}_{c}{\gamma}_{h}(n_{c}-n_{h})(\omega_{h}-\omega_{c})({\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})\left(4\alpha^{3}+\alpha({\gamma}_{c}+{\gamma}_{h}+2{\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})({\gamma}_{c}+{\gamma}_{h}+{\gamma}_{c}n_{c}+2{\gamma}_{h}n_{h})\right)^{2},$
$\displaystyle a^{\prime\prime}_{0}=$
$\displaystyle\frac{1}{16}\alpha^{2}{\gamma}_{c}{\gamma}_{h}(2n_{c}n_{h}+n_{c}+n_{h})({\omega}_{c}-{\omega}_{h})^{2}({\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})\left(4\alpha^{2}+({\gamma}_{c}+{\gamma}_{h}+2{\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})({\gamma}_{c}+{\gamma}_{h}+{\gamma}_{c}n_{c}+2{\gamma}_{h}n_{h})\right)^{2},$
$\displaystyle\mbox{and}\ \ \ a^{\prime}_{1}=$
$\displaystyle\frac{1}{8}\alpha^{2}{\gamma}_{c}{\gamma}_{h}(n_{c}-n_{h})({\omega}_{h}-{\omega}_{c})\left(4\alpha^{2}+({\gamma}_{c})^{2}(n_{c}(8n_{c}+7)+1)+{\gamma}_{c}{\gamma}_{h}(17n_{c}n_{h}+7n_{c}+7n_{h}+2)+({\gamma}_{h})^{2}(n_{h}(8n_{h}+7)+1)\right)$
$\displaystyle\ \ \ \ \ \ \
\left(4\alpha^{2}+({\gamma}_{c}+{\gamma}_{h}+2{\gamma}_{c}n_{c}+{\gamma}_{h}n_{h})({\gamma}_{c}+{\gamma}_{h}+{\gamma}_{c}n_{c}+2{\gamma}_{h}n_{h})\right).$

Utilizing these expressions, the average power and the variance in power of
IQHEs become

$\displaystyle\langle P_{I}\rangle$
$\displaystyle=-\frac{4\alpha^{2}\gamma_{h}\gamma_{c}(n_{h}-n_{c})}{4\alpha^{2}(\gamma_{c}(3n_{c}+2)+\gamma_{h}(3n_{h}+2))+\gamma_{c}\gamma_{h}(3n_{c}n_{h}+n_{c}+n_{h})(\gamma_{c}n_{c}+\gamma_{h}n_{h})}(\omega_{h}-\omega_{c}),$
(75) $\displaystyle\mbox{and}\ \ \ \Delta P_{I}$ $\displaystyle=(F_{p}\langle
P_{I}\rangle-\frac{k}{\alpha^{2}(\omega_{h}-\omega_{c})^{2}}\langle
P_{I}\rangle^{3})(\omega_{h}-\omega_{c}),$ (76)

where $F_{p}=\frac{2n_{h}n_{c}+n_{h}+n_{c}}{n_{h}-n_{c}}$ and
$k=\frac{4\alpha^{2}}{\gamma_{0}^{2}(n_{h}-n_{c})}+\frac{n_{h}n_{c}+n_{c}^{2}+n_{h}^{2}}{n_{h}-n_{c}}+2F_{p}$.
Now, the noise-to-signal ratio of the power of IQHEs is

$\mathcal{N}_{I}=\frac{\Delta P_{I}}{\langle
P_{I}\rangle^{2}}=\left(\frac{F_{p}}{\langle
P_{I}\rangle}-\frac{k}{\alpha^{2}(\omega_{h}-\omega_{c})^{2}}\langle
P_{I}\rangle\right)(\omega_{h}-\omega_{c})=\frac{F_{p}}{\alpha\mathcal{C}(\sigma_{{I}})}\left(1-\frac{k}{F_{p}}{\mathcal{C}(\sigma_{{I}})^{2}}\right),$
(77)

where $\langle
P_{I}\rangle=-\alpha(\omega_{h}-\omega_{c})\mathcal{C}(\sigma_{I})$, and
$\mathcal{C}(\sigma_{{I}})$ is $l$-1 norm of coherence of the steady state
$\sigma_{I}$. It is important to note that the noise-to-signal ratio of
currents, power, and photon number flux is the same for IQHEs.

##  Appendix E Classical thermodynamic uncertainty relation and power-
efficiency-constancy trade-off relation

Classical steady-state heat engines always exhibit trade-off relationships
between relative fluctuation in output power, the thermodynamic cost
(quantified by the rate of entropy production $\dot{S}$), and heat-to-work
conversion efficiency. There are two trade-off relations

$\displaystyle\mathcal{Q}$ $\displaystyle=\dot{S}\frac{\Delta P}{\langle
P\rangle^{2}}\geq 2,$ (78) $\displaystyle\mbox{and}\ \mathcal{D}$
$\displaystyle=({\eta_{Cor}}-{\eta})\frac{\rm\Delta P}{\langle
P\rangle}\frac{\beta_{c}\omega_{h}}{(\omega_{h}-\omega_{c})}\geq 2,$ (79)

where $\dot{S}$ the rate of entropy production,
$\eta=1-\frac{\omega_{c}}{\omega_{h}}$ is the engine efficiency for both
coherent and incoherent engines, and
$\eta_{Cor}=1-\frac{\beta_{h}}{\beta_{c}}$ is the Carnot efficiency. Note, Eq.
(78) is referred to as the classical thermodynamic uncertainty relation (cTUR)
Barato and Seifert (2015) and Eq. (79) is referred to as the power-efficiency-
constancy trade-off relation Pietzonka and Seifert (2018). The entropy
production rate $\dot{S}$ for coherent and incoherent engines can be written
as (for $X=C,I$)

$\dot{S}_{X}=-\beta_{h}\langle\dot{J}^{h}_{X}\rangle-\beta_{c}\langle\dot{J}_{X}^{c}\rangle=\ln{\left(\frac{n_{h}(n_{c}+1)}{n_{c}(n_{h}+1)}\right)}\langle\dot{N}_{X}\rangle>0,$
(80)

where
${\langle\dot{N}_{X}\rangle}={|\langle{P}_{X}\rangle}|/(\omega_{h}-\omega_{c})$
is the average photon number current, $\dot{J}^{X}_{h}$ and $\dot{J}^{X}_{c}$
are average heat currents corresponding hot and cold baths, respectively.
Moreover, we can write

$({\eta_{Cor}}-{\eta})\frac{\beta_{c}\omega_{h}}{(\omega_{h}-\omega_{c})}=\ln{\left(\frac{n_{h}(n_{c}+1)}{n_{c}(n_{h}+1)}\right)}.$
(81)

To obtain above expression we have used the relation
$n_{x}=1/(e^{\beta_{x}\omega_{x}}-1)$ for $x=h,c$. Using above relations, we
can show that

$\leavevmode\nobreak\
\mathcal{Q}_{X}=\mathcal{D}_{X}=\ln{\left(\frac{n_{h}(n_{c}+1)}{n_{c}(n_{h}+1)}\right)}F_{X}.$
(82)

Here $F_{X}=\frac{\Delta\dot{N}_{X}}{\langle\dot{N}_{X}\rangle}$ is known as
the Fano factor of photon number current ($\dot{N}$), where
${\langle\dot{N}_{X}\rangle}={|\langle{P}_{X}\rangle}|/(\omega_{h}-\omega_{c})$
and ${\Delta\dot{N}_{X}}=\Delta{P_{X}}/(\omega_{h}-\omega_{c})^{2}$ are
variance and average of photon number current for the steady state dynamics.
The Eq. (82) indicates that in the context of CQHEs and IQHEs, both the cTUR
and the power-efficiency-constancy trade-off relation coincide. By using the
expression of $\langle P_{X}\rangle$ and $\Delta P_{X}$, the Fano factors for
CQHEs and IQHEs can be respectively written in terms of population and
energetic coherence as,

$\displaystyle
F_{C}=F_{p}\left(1-\frac{3}{2}{(\mathcal{C}(\sigma_{{C}}))^{2}}\right),\ \ \
\mbox{and}\ \ \
F_{I}=F_{p}\left(1-\frac{k}{F_{p}}{(\mathcal{C}(\sigma_{{I}}))^{2}}\right).$
(83)

##  Appendix F Quantum Thermodynamic Uncertainty Relation

A quantum formulation of the thermodynamic uncertainty relation was recently
obtained for Markovian dynamics (described by the Lindblad master equation)
using the quantum Cramér-Rao bound. To read the steady-state version of qTUR,
one reads as follows Hasegawa (2020):

$\mathcal{N}=\frac{\Delta P}{\langle P\rangle^{2}}\geq
f=\frac{1}{\Upsilon+\Psi}.$ (84)

In the above bound (84), $\Upsilon$ denotes the quantum dynamical activity,
which is the average rate of transitions in the steady-state and reads

$\Upsilon=\sum_{k}\Tr(L_{k}^{\dagger}L_{k}\rho_{ss}),$ (85)

where $\rho_{ss}$ represent the steady state of the given system, $L_{k}$ and
$L_{k}^{\dagger}$ represent the jump operators and its ad-joint operators,
respectively. In the above bound (84), $\Psi$ denotes the coherent-dynamics
contribution and reads

$\Psi=-4(\langle\langle\mathbb{I}|\mathcal{L}_{L}\mathcal{L}^{+}\mathcal{L}_{R}|\rho_{ss}\rangle\rangle+\langle\langle\mathbb{I}|\mathcal{L}_{R}\mathcal{L}^{+}\mathcal{L}_{L}|\rho_{ss}\rangle\rangle),$
(86)

where $|\rho_{ss}\rangle\rangle$ denotes the vectorized steady-state density
matrix $\rho_{ss}$,
$|\mathbb{I}\rangle\rangle=\sum_{i}|i\rangle^{*}\otimes|i\rangle$ is the
vectorized identity. $\mathcal{L}^{+}$ denotes the Drazin inverse of
vectorized Liouvillian super operator
($\mathcal{L}=\mathcal{L}_{R}+\mathcal{L}_{L}$) and the expression of
$\mathcal{L}_{R}$ and $\mathcal{L}_{L}$ reads as follows

$\displaystyle\mathcal{L}_{R}$ $\displaystyle=-i\mathcal{I}\otimes
H+\frac{1}{2}\sum_{k}(L_{k}^{*}\otimes L_{k}-\mathcal{I}\otimes
L_{k}^{\dagger}L_{k}),$ and $\displaystyle\mathcal{L}_{L}$
$\displaystyle=iH^{T}\otimes\mathcal{I}+\frac{1}{2}\sum_{k}(L_{k}^{*}\otimes
L_{k}-(L_{k}^{\dagger}L_{k})^{T}\otimes\mathcal{I}),$

where $H$ is the Hamiltonian of the system and $\mathcal{I}$ is the identity
matrix. The vectorized Liouvillian super operator can be written as
$\mathcal{L}=\sum_{j\neq 0}\lambda_{j}|x_{j}\rangle\rangle\langle\langle
y_{j}|$, where $|x_{j}\rangle\rangle$ and $|y_{j}\rangle\rangle$ are right and
left eigenvectors of vectorized Liouvillian super operator, respectively and
$\lambda_{j}$ is eigen value of vectorized Liouvillian super operator. The
Drazin inverse of the Liouvillian super operator can be obtained by inverting
the eigen values $\mathcal{L}^{+}=\sum_{j\neq
0}\frac{1}{\lambda_{j}}|x_{j}\rangle\rangle\langle\langle y_{j}|$ Landi  _et
al._ (2024). The Drazin inverse also can be calculated using some alternative
methods, for more details see Ref. Landi  _et al._ (2024). Employing this
definition, we derived the Drazin inverse of vectorized Liouvillian
superoperators for CQHEs and IQHEs as

$\displaystyle\mathcal{L}^{+}_{C}=\left(\begin{array}[]{cccc}\frac{4\alpha^{2}(\gamma_{1}-3\gamma_{2})-\gamma_{1}(\gamma_{1}+\gamma_{2})^{2}}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}&\frac{2i\alpha}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}&-\frac{2i\alpha}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}&\frac{4\alpha^{2}(3\gamma_{1}-\gamma_{2})+\gamma_{2}(\gamma_{1}+\gamma_{2})^{2}}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}\\\
\frac{4i\alpha\left(4\alpha^{2}+(2\gamma_{1}-\gamma_{2})(\gamma_{1}+\gamma_{2})\right)}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}&-\frac{2\left(4\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)}{(\gamma_{1}+\gamma_{2})\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)}&-\frac{8\alpha^{2}}{(\gamma_{1}+\gamma_{2})\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)}&-\frac{4i\alpha\left(4\alpha^{2}-(\gamma_{1}-2\gamma_{2})(\gamma_{1}+\gamma_{2})\right)}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}\\\
-\frac{4i\alpha\left(4\alpha^{2}+(2\gamma_{1}-\gamma_{2})(\gamma_{1}+\gamma_{2})\right)}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}&-\frac{8\alpha^{2}}{(\gamma_{1}+\gamma_{2})\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)}&-\frac{2\left(4\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)}{(\gamma_{1}+\gamma_{2})\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)}&\frac{4i\alpha\left(4\alpha^{2}-(\gamma_{1}-2\gamma_{2})(\gamma_{1}+\gamma_{2})\right)}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}\\\
\frac{\gamma_{1}(\gamma_{1}+\gamma_{2})^{2}-4\alpha^{2}(\gamma_{1}-3\gamma_{2})}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}&-\frac{2i\alpha}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}&\frac{2i\alpha}{8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}}&\frac{4\alpha^{2}(\gamma_{2}-3\gamma_{1})-\gamma_{2}(\gamma_{1}+\gamma_{2})^{2}}{\left(8\alpha^{2}+(\gamma_{1}+\gamma_{2})^{2}\right)^{2}}\\\
\end{array}\right)$

and

$\displaystyle\mathcal{L}^{+}_{I}=\left(\begin{array}[]{ccccccccc}a_{11}&0&0&0&a_{15}&a_{16}&0&a_{18}&a_{19}\\\
0&a_{22}&a_{23}&0&0&0&0&0&0\\\ 0&a_{32}&a_{33}&0&0&0&0&0&0\\\
0&0&0&a_{44}&0&0&a_{47}&0&0\\\ a_{51}&0&0&0&a_{55}&a_{56}&0&a_{58}&a_{59}\\\
a_{61}&0&0&0&a_{65}&a_{66}&0&a_{68}&a_{69}\\\ 0&0&0&a_{74}&0&0&a_{77}&0&0\\\
a_{81}&0&0&0&a_{85}&a_{86}&0&a_{88}&a_{89}\\\
a_{91}&0&0&0&a_{95}&a_{96}&0&a_{98}&a_{99}\\\ \end{array}\right),$

respectively, where

$\displaystyle\gamma_{1}=\gamma_{0}n_{c}(n_{h}+1),$
$\displaystyle\gamma_{2}=\gamma_{0}n_{h}(n_{c}+1),$ $\displaystyle
a_{11}=-\frac{4\alpha^{2}\gamma_{0}^{2}(n_{c}+n_{h}+2)\left(n_{c}^{2}+6n_{c}n_{h}+n_{h}^{2}\right)+\gamma_{0}^{4}(n_{c}+n_{h})^{2}\left(n_{c}^{2}(n_{h}+1)+n_{c}n_{h}^{2}+n_{h}^{2}\right)+64\alpha^{4}(n_{c}+n_{h}+2)}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{61}=-a_{81}=\frac{2i\alpha(n_{c}-n_{h})\left(\gamma_{0}^{2}\left(3n_{c}^{2}(n_{h}+1)+n_{c}(3n_{h}(n_{h}+4)+4)+n_{h}(3n_{h}+4)\right)+12\alpha^{2}(n_{c}+n_{h}+2)\right)}{\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{23}=a_{32}=-a_{74}=-a_{47}=\frac{4i\alpha}{4\alpha^{2}+\gamma_{0}^{2}(2n_{c}+n_{h}+2)(n_{c}+2n_{h}+2)},$
$\displaystyle
a_{22}=a_{44}=-\frac{2\gamma_{0}(n_{c}+2n_{h}+2)}{4\alpha^{2}+\gamma_{0}^{2}(2n_{c}+n_{h}+2)(n_{c}+2n_{h}+2)},$
$\displaystyle
a_{33}=a_{77}=-\frac{2\gamma_{0}(2n_{c}+n_{h}+2)}{4\alpha^{2}+\gamma_{0}^{2}(2n_{c}+n_{h}+2)(n_{c}+2n_{h}+2)},$
$\displaystyle
a_{16}=-a_{18}=\frac{2i\alpha(n_{c}-n_{h})}{4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})},$
$\displaystyle
a_{56}=-a_{58}=\frac{2i\alpha(n_{c}+2n_{h}+2)}{4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})},$

$\displaystyle
a_{96}=a_{98}=-\frac{2i\alpha(2n_{c}+n_{h}+2)}{4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})},$
$\displaystyle
a_{86}=a_{68}=-\frac{4\alpha^{2}(3n_{c}+3n_{h}+4)}{\gamma_{0}(n_{c}+n_{h})\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)},$
$\displaystyle
a_{66}=a_{88}=\frac{-4\alpha^{2}(3n_{c}+3n_{h}+4)-2\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})}{\gamma_{0}(n_{c}+n_{h})\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)},$
$\displaystyle
a_{15}=\frac{4\alpha^{2}\gamma_{0}^{2}\left(-n_{c}^{3}+(5n_{c}+4)n_{h}^{2}+2(n_{c}-2)n_{c}n_{h}+2n_{h}^{3}\right)-\gamma_{0}^{4}n_{c}(n_{c}+n_{h})^{2}\left((n_{c}-1)n_{h}+n_{c}-2n_{h}^{2}\right)+32\alpha^{4}(n_{c}+n_{h})}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{19}=\frac{4\alpha^{2}\gamma_{0}^{2}\left(2n_{c}^{3}+n_{c}^{2}(5n_{h}+4)+2n_{c}(n_{h}-2)n_{h}-n_{h}^{3}\right)+32\alpha^{4}(n_{c}+n_{h})+\gamma_{0}^{4}n_{h}(n_{c}+n_{h})^{2}(n_{c}(2n_{c}+1)-(n_{c}+1)n_{h})}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{65}=-a_{85}=\frac{2i\alpha\left(4\alpha^{2}\left(3n_{c}^{2}+3n_{c}(3n_{h}+4)+6n_{h}(n_{h}+2)+8\right)+\gamma_{0}^{2}\left(3n_{c}^{3}(n_{h}+1)+3n_{c}^{2}(3n_{h}(n_{h}+2)+2)+n_{c}n_{h}\left(6n_{h}^{2}+3n_{h}+4\right)-2n_{h}^{2}\right)\right)}{\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{95}=\frac{4\alpha^{2}\gamma_{0}^{2}\left(n_{c}^{2}(2n_{c}+3)+(5n_{c}+7)n_{h}^{2}+(n_{c}+2)(5n_{c}+4)n_{h}+2n_{h}^{3}\right)-16\alpha^{4}(n_{c}+n_{h})+2\gamma_{0}^{4}n_{c}(n_{h}+1)(n_{c}+n_{h})^{2}(n_{c}+n_{h}+1)}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{59}=\frac{4\alpha^{2}\gamma_{0}^{2}\left(2n_{c}^{3}+n_{c}^{2}(5n_{h}+7)+n_{c}(n_{h}+2)(5n_{h}+4)+n_{h}^{2}(2n_{h}+3)\right)-16\alpha^{4}(n_{c}+n_{h})+2\gamma_{0}^{4}(n_{c}+1)n_{h}(n_{c}+n_{h})^{2}(n_{c}+n_{h}+1)}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{89}=-a_{69}=\frac{2i\alpha\left(4\alpha^{2}\left(6n_{c}^{2}+3n_{c}(3n_{h}+4)+3n_{h}(n_{h}+4)+8\right)+\gamma_{0}^{2}\left(6n_{c}^{3}n_{h}+n_{c}^{2}\left(9n_{h}^{2}+3n_{h}-2\right)+n_{c}n_{h}(3n_{h}(n_{h}+6)+4)+3n_{h}^{2}(n_{h}+2)\right)\right)}{\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{99}=-\frac{4\alpha^{2}\gamma_{0}^{2}\left((7n_{c}+3)n_{h}^{2}+10n_{c}(n_{c}+1)n_{h}+n_{c}(n_{c}(4n_{c}+11)+8)+n_{h}^{3}\right)+16\alpha^{4}(n_{c}+n_{h})+\gamma_{0}^{4}n_{h}(n_{c}+n_{h})^{2}(n_{c}(4n_{c}+n_{h}+5)+n_{h}+2)}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle
a_{55}=-\frac{4\alpha^{2}\gamma_{0}^{2}\left(n_{c}^{3}+n_{c}^{2}(7n_{h}+3)+10n_{c}n_{h}(n_{h}+1)+n_{h}(n_{h}(4n_{h}+11)+8)\right)+\gamma_{0}^{4}n_{c}(n_{c}+n_{h})^{2}\left((n_{c}+5)n_{h}+n_{c}+4n_{h}^{2}+2\right)+16\alpha^{4}(n_{c}+n_{h})}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$
$\displaystyle a_{51}=$
$\displaystyle\frac{4\alpha^{2}\gamma_{0}^{2}\left(n_{c}\left(n_{c}-n_{c}^{2}+4\right)+(5n_{c}+1)n_{h}^{2}+2n_{c}(n_{c}+3)n_{h}+2n_{h}^{3}-4n_{h}\right)-\gamma_{0}^{4}(n_{c}+1)(n_{c}+n_{h})^{2}\left((n_{c}-1)n_{h}+n_{c}-2n_{h}^{2}\right)+32\alpha^{4}(n_{c}+n_{h}+2)}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}},$

and

$\displaystyle a_{91}=$
$\displaystyle\frac{4\alpha^{2}\gamma_{0}^{2}\left(2n_{c}^{3}+n_{c}^{2}(5n_{h}+1)+2n_{c}(n_{h}(n_{h}+3)-2)+n_{h}\left(n_{h}-n_{h}^{2}+4\right)\right)+32\alpha^{4}(n_{c}+n_{h}+2)+\gamma_{0}^{4}(n_{h}+1)(n_{c}+n_{h})^{2}(n_{c}(2n_{c}+1)-(n_{c}+1)n_{h})}{\gamma_{0}\left(4\alpha^{2}(3n_{c}+3n_{h}+4)+\gamma_{0}^{2}(n_{c}+n_{h})(3n_{c}n_{h}+n_{c}+n_{h})\right)^{2}}.$

The superoperators $\mathcal{L}_{R}$ and $\mathcal{L}_{L}$ for CQHEs and IQHEs
can be computed using the corresponding jump operators
$\sqrt{\gamma_{0}n_{c}(n_{h}+1)}b_{h}$,
$\sqrt{\gamma_{0}n_{h}(n_{c}+1)}b_{hc}^{\dagger}$ and
$\sqrt{\gamma_{0}(n_{h}+1)}b_{h}$, $\sqrt{\gamma_{0}n_{h}}b_{h}^{\dagger}$,
$\sqrt{\gamma_{0}(n_{c}+1)}b_{c}$, $\sqrt{\gamma_{0}n_{c}}b_{c}^{\dagger}$
through a simple exercise. The expressions of the lower bounds ($f_{X}$) on
the noise-to-signal ratio of power for CQHEs and IQHEs in terms of driving and
bath parameters are as follows

$\displaystyle\frac{1}{f_{C}}$
$\displaystyle=\frac{2\left(2\alpha^{2}+\gamma_{0}^{2}{n_{h}}{n_{c}}({n_{c}}+1)({n_{h}}+1)\right)\left(32\alpha^{2}+\gamma_{0}^{2}(2{n_{c}}{n_{h}}+{n_{c}}+{n_{h}})^{2}\right)}{\gamma_{0}(n_{h}+n_{c}+2n_{h}n_{c})(8\alpha^{2}+\gamma_{0}^{2}(n_{h}+n_{c}+2n_{h}n_{c})^{2})},$
$\displaystyle\ \ \ \mbox{and}\ \ \ \frac{1}{f_{I}}$
$\displaystyle=\frac{2(n_{h}+n_{c}+2)(4\alpha^{2}+\gamma_{0}^{2}n_{h}n_{c})(16\alpha^{2}+\gamma_{0}^{2}(n_{h}+n_{c})^{2})}{\gamma_{0}(n_{h}+n_{c})(4\alpha^{2}(4+3(n_{h}+n_{c}))+\gamma_{0}^{2}(n_{h}+n_{c})(n_{h}+n_{c}+3n_{h}n_{c}))}.$

It is important to note that the noise-to-signal ratio of currents, power, and
photon number flux is the same for CQHEs as well as for IQHEs.

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