# Synchronization Lower Bounds the Efficiency of Near-Degenerate Thermal
Machines

Taufiq Murtadho  Center for Theoretical Physics of Complex Systems, Institute
for Basic Science (IBS), Daejeon 34126, Republic of Korea.  Basic Science
Program, Korea University of Science and Technology, Daejeon 34113, Republic
of Korea.  School of Physical and Mathematical Science, Nanyang Technological
University, Singapore 639798, Singapore  Juzar Thingna
<EMAIL_ADDRESS>Center for Theoretical Physics of Complex Systems,
Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea.  Basic
Science Program, Korea University of Science and Technology, Daejeon 34113,
Republic of Korea.  Department of Physics and Applied Physics, University of
Massachusetts, Lowell, MA 01854, USA.  Sai Vinjanampathy<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai
400076, India.  Centre of Excellence in Quantum Information, Computation,
Science and Technology, Indian Institute of Technology Bombay, Powai, Mumbai
400076, India.  Centre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, Singapore 117543, Singapore.

###### Abstract

We study the relationship between quantum synchronization and the
thermodynamic performance of a four-level near-degenerate extension of the
Scovil–Schulz-DuBois thermal maser. We show how the existence of interacting
coherences can potentially modify the relationship between synchronization and
the coherent power output of such a maser. In particular, the cooperation and
competition between interacting coherences, causes the coherent heat and
efficiency to be bounded by the synchronization measure in addition to the
well-studied power synchronization bound. Overall, our results highlight the
role of quantum synchronization in the working of a thermal machine.

##  I Introduction

Quantum coherence has been a vital resource [1] in quantum thermodynamics and
has been a topic of intense research in the recent past [2, 3, 4, 5]. The
presence of coherence typically boosts the performance metrics of thermal
machines such as engines [6], refrigerators [7], and batteries [8, 9, 10, 11].
Moreover, coherences, which can be thought of as phases, have also been
subject to resource counting studies from an information theoretic perspective
[12]. In the case of non-degenerate systems, that possess diagonal steady
states in the absence of driving, such coherences are usually generated using
a coherent drive. Such systems do not allow coherences to interact and thus a
thorough understanding of the performance of quantum thermal machines with
interacting coherences is lacking.

Classically, interacting phases are well studied in coupled Kuramoto models
where phase pulling can produce a well-known second-order phase transition
involving cooperative phase locking called synchronization [13]. Besides this,
differences in coupling of the Kuramoto model can lead to competition as well,
giving rise to a variety of behavior such as anti-synchronization and chimera,
a phenomenon explored even in the quantum regime [14]. Specifically, in
quantum systems, such coupled phase oscillator models arise due to a variety
of different reasons, one of them being degeneracies. Recently, for non-
degenerate quantum systems coupled to diagonal baths, it was highlighted that
synchronization measures fully account for all coherences in the system [15]
and can help explain the performance of quantum thermal machines.

In contrast to coherently driven quantum systems, incoherently driven near-
degenerate levels can generate bath-induced coherences. Since the energetic
cost of generating coherences is (nearly) zero for making transitions between
(near-)degenerate levels, such systems cause the thermodynamic analysis to be
decoupled from synchronization. Moreover, the (near-)degeneracies cause such
coherences to interact with each other and induce cooperation or competition
between the different phases.

In an accompanying manuscript, we show that synchronization of driven,
dissipative _exactly_ -degenerate open quantum systems exhibit a crossover
from cooperation to competition. In this manuscript, we highlight that this
synchronous behavior aids in the understanding of nanoscale thermal machines.
We show that while coherent driving of near-degenerate four-level thermal
masers has a tendency to entrain the quantum system to the external drive,
coupled coherences exert a simultaneous force that can be either competitive
or cooperative. This interplay between cooperation and competition leads to a
rich dynamical regime whose implications on the thermal maser are observed in
terms of its performance metrics (power, heat, and efficiency) being bounded
by the synchronization measure.

In Sec. II below, we begin with a discussion of the near-degenerate four-level
thermal maser. Then, we present an analysis of competition and cooperation in
four-level thermal masers taking into account near-degeneracy and bath-induced
coherence in Sec. III. Then, in Sec. IV and Sec. V we connect the competition
and cooperation to relevant thermodynamic quantities of the thermal machine,
specifically its power and heat current. In [16], the coherent power output of
a non-degenerate three-level maser was shown to be related to the measures of
synchronisation and it was noted in [17] that this relationship does not hold
for degenerate systems where there is no energetic cost of generating
coherences. Furthermore such degeneracies were related to synchronisation
blockade recently [18]. We investigate this relationship in detail and connect
synchronization to steady state heat current. The latter is then used to
derive lower-bound on the efficiency of near-degenerate thermal machines in
the mutual coupling dominant regime. Finally, we summarize our main results in
Sec. VI.

##  II Four-level Thermal Maser: Near degeneracy and noise-induced coherence

Figure 1: Schematic of the four-level Scovil–Schulz-DuBois thermal maser.
Here, $\Delta\equiv\omega_{3}-\omega_{2}$ is the near-degenerate energy gap
and $p$ is the noise-induced coherence strength that arises due to the hot
heat bath causing interference between the near-degenerate levels and the
ground state (red arrows). The relative phases $\varphi_{ij}$ between states
$|i\rangle$ and $|j\rangle$ are depicted on the circles on the right and the
arrows indicate the direction in which the non-degenerate phases
($\varphi_{31}$ and $\varphi_{21}$) move depending on the strength of mutual
coupling (orange arrows) and entrainment (green arrows). In the engine (top
circle) we observe both cooperation and competition, whereas in the
refrigerator regime (bottom circle) we find only cooperating behavior.

We begin with the analysis of a four-level thermal maser that operates under a
temperature difference to create a population inversion as depicted in Fig. 1.
The original Scovil–Schulz-DuBois thermal maser is a minimal setup that
comprises of three-levels and is one of the first models of a thermal heat
engine [19], which has been recently extended to four levels [20] and beyond
[21, 22] to include effects of bath-induced coherence. As a generalization of
this, we consider the four-level model consisting of a free Hamiltonian

$H_{0}=\omega_{1}\ket{1}\bra{1}+\sum_{j=2}^{3}\omega_{j}\ket{j}\bra{j},$ (1)

with $\omega_{3}>\omega_{2}>\omega_{1}>\omega_{0}=0$. The _near-degenerate_
levels ($\omega_{2},\omega_{3}$) are spread over a small interval
$\Delta\equiv\omega_{3}-\omega_{2}\ll(\omega_{1}-\omega_{0})$ and
$\Delta\ll(\omega_{2}-\omega_{1})$. The system is driven out-of-equilibrium by
a hot (temperature $T_{h}$) and a cold bath (temperature $T_{c}$) as well as
an external time-periodic drive of strength $\lambda$ and frequency
$\Omega\approx(\omega_{2}-\omega_{1})$. The role of the baths is to establish
population inversion between the near-degenerate manifold
$\\{\ket{2},\ket{3}\\}$ and the first-excited state $\ket{1}$. Specifically,
when population inversion is achieved, the external drive can easily trigger
stimulated emission, thereby producing power, i.e. the system acts as an
engine. On the other hand, if there is no population inversion, the system
absorbs power from the external drive, i.e. the system acts as a refrigerator.

When the baths are weakly coupled to the machine working fluid $H_{0}$, the
reduced dynamics of the system is governed by a quantum master equation for
the reduced density matrix $\rho$ [20] that reads,

$\frac{d\rho}{dt}=-i[H_{0}+V(t),\rho]+\mathcal{D}_{h}[\rho]+\mathcal{D}_{c}[\rho].$
(2)

The operator $V(t)$ is a collective drive [22],

$V(t)=\lambda e^{i\Omega t}\sum_{j=2}^{3}\ket{j}\bra{1}+\text{h.c.},$ (3)

that stimulates transitions between all the near-degenerate energy levels and
the first-excited state. The cold-bath dissipator $\mathcal{D}_{c}[\rho]$
takes the Gorini-Kassakowski-Sudarshan-Lindblad (GKSL) [23, 24] form

$\mathcal{D}_{c}[\rho]=\sum_{\mu=1}^{2}\Gamma_{c_{\mu}}(2c_{\mu}\rho
c_{\mu}^{\dagger}-\\{c_{\mu}^{\dagger}c_{\mu},\rho\\}),$ (4)

with jump operators $c_{1}=c_{2}^{\dagger}=\ket{0}\bra{1}$ and the decay rates
satisfying local detailed balance $\Gamma_{c_{1}}=\gamma_{c}(1+n_{c})$ and
$\Gamma_{c_{2}}=\gamma_{c}n_{c}$. Here, $\gamma_{c}$ is the coupling strength
squared between the system and the cold bath.

The hot-bath dissipator $\mathcal{D}_{h}[\rho]$ connects the first excited
state to the near-degenerate manifold. The near-degeneracy causes a breakdown
of the secular approximation [25] and results in a Bloch-Redfield form of the
dissipator [26, 27, 28, 29, 20],

$\mathcal{D}_{h}[\rho]=\sum_{\mu=1}^{2}\sum_{i,j=2}^{3}\Gamma_{\mu}^{ij}[h_{\mu}^{i},\rho\,h_{\mu}^{j\dagger}]+\Gamma_{\mu}^{ji}[h_{\mu}^{i}\,\rho,h_{\mu}^{j\dagger}].$
(5)

The operators $h_{1}^{i}=h_{2}^{i}=\ket{0}\bra{i}$ and pair-wise decay rates
$\Gamma_{1}^{ij}=P_{ij}\sqrt{\gamma_{h}^{i}\gamma_{h}^{j}}\,(1+n_{h}^{(j)})$
and $\Gamma_{2}^{ij}=P_{ij}\sqrt{\gamma_{h}^{i}\gamma_{h}^{j}}\,n_{h}^{(j)}$.
Here, $\gamma_{h}^{i}$ denotes the squared coupling strength of the bath that
induces transitions between the ground state $\ket{0}$ and the $i$th state of
the near-degenerate manifold $\ket{i}$ ($i=2,3$). The mean-bosonic hot-bath
distribution $n_{h}^{(i)}=(\exp[\omega_{i}/T_{x}]-1)^{-1}$ encodes information
about the hot bath temperature $T_{h}$ (similar definition for the cold-bath
with $\omega_{i}=\omega_{1}$). In this paper, we will throughout set
$\gamma_{h}^{(2)}=\gamma_{h}^{(3)}=\gamma_{h}$ and work with the Redfield form
that despite not being positive in all parameter ranges [30, 31] provides
accurate results when used appropriately in the weak system-bath coupling
regime [32, 33].

The coefficients $P_{ij}$ are elements of the symmetric _correlation matrix_
[34]

$P_{ij}=\begin{cases}1&\text{if }i=j\\\ p_{ij}&\text{for }i\neq j.\end{cases}$
(6)

where $|p_{ij}|\leq 1$. Above since we are focused on a four-level thermal
maser, the matrix $P$ is a $2\times 2$ matrix with its off-diagonal elements
$|p|\leq 1$. The elements $p$ originate from the dipole-alignment factor [28,
29] with
$p=\mathbf{d_{2}}\cdot\mathbf{d_{3}}/|\mathbf{d_{2}}||\mathbf{d}_{3}|$ such
that $\mathbf{d}_{i}=\braket{i}{\mathbf{d}}{i}$ and $\mathbf{d}$ being the
dipole operator. Intuitively, $|p|$ is the strength of quantum interference
between thermalization pathways $\ket{0}\leftrightarrow\ket{2}$ and
$\ket{0}\leftrightarrow\ket{3}$ while its sign encodes whether the
interference is constructive or destructive. The parameter $p$ determines the
strength of the noise-induced coherence with $p=0$ yielding no coherence due
to the hot bath and $|p|=1$ yielding maximum coherence. The above Redfield
hot-bath dissipator takes the completely positive Lindblad form 111By
‘Lindblad’ we refer to the general Lindblad form
$d\rho/dt=-i[H,\rho]+\sum_{i,j}K_{ij}(2A_{i}\rho
A_{j}^{\dagger}-\\{A_{j}^{\dagger}A_{i},\rho\\}$ where $A_{i}\equiv h_{i}$ are
the jump operators and $K_{ij}$ is the positive Kassakowski matrix. when
$\Gamma_{\mu}^{ij}=\Gamma_{\mu}^{ji}~{}\forall~{}i,j=2,3$, which can be
satisfied in the exact degenerate limit ($\Delta=0$ wherein
$n_{h}^{(i)}=n_{h}$). In this limit, the Kossakowski matrix becomes
proportional to the correlation matrix and thus positivity can be ensured if
the correlation matrix is positive definite, i.e., for $-1\leq p\leq 1$. When
the correlation matrix has a zero eigenvalue ($p=\pm 1$), the dissipator
possesses a dark state [36, 37] that can lead to multiple steady states [38,
39].

The above quantum master equation (2) can be transformed to a rotating frame
such that any operator $\tilde{O}=\exp[i\tilde{H}t]O\exp[-i\tilde{H}t]$ with

$\tilde{H}=\frac{\Omega}{2}\sum_{j=2}^{3}\ket{j}\bra{j}-\frac{\Omega}{2}\ket{1}\bra{1}.$
(7)

The above transformation eliminates the explicit time dependency from the
coherent evolution and leaves the dissipators invariant leading to a quantum
master equation in the rotating frame,

$\frac{d\tilde{\rho}}{dt}=-i[H_{0}-\tilde{H}+\tilde{V},\tilde{\rho}]+\mathcal{D}_{h}[\tilde{\rho}]+\mathcal{D}_{c}[\tilde{\rho}],$
(8)

with $\tilde{V}=\lambda\sum_{j=2}^{3}\ket{j}\bra{1}+\text{h.c.}$. Throughout
this work, we will focus our attention on the four-level thermal maser
investigating the effects of near degeneracy $\Delta$ and the dipole-alignment
factor $p$ (noise-induced coherence strength) on quantum synchronization and
the thermodynamic observables of the thermal maser. In particular, unlike Ref.
[22] we will not explore the effect of several degenerate levels (generalized
Scovil–Schulz-DuBois thermal maser) and restrict ourselves to the easily
tractable and physically intuitive four-level heat machine.

##  III Coexistence of entrainment and mutual coupling in a near-degenerate
thermal maser

Figure 2: Quasi-probability phase distribution $S(\varphi_{31},\varphi_{21})$
in the engine (a-b, $n_{h}^{(2)}/n_{c}=5$) and refrigerator (c-d,
$n_{h}^{(2)}/n_{c}=0.5$) regime. The noise-induced coherence strength $p=0.5$
in a ($k=4.5$) and c ($k=3.15$) and $p=-0.99$ in b ($k=0.03$) and d
($k=0.021$). Above, $\varphi_{ij}\equiv\phi_{i}-\phi_{j}$ is the phase
difference associated with states $\ket{i}$ and $\ket{j}$. In a and c, both
$\varphi_{21}$ and $\varphi_{31}$ are localized in the rotating frame,
indicating entrainment to the external drive. Meanwhile, in b and d, the
phases localize to a ring (on a torus), indicating that only the relative
phase $\varphi_{23}=\varphi_{21}-\varphi_{31}$ is fixed. In this regime,
synchronization is dominated by mutual coupling. We find that in the engine
(refrigerator) regime, the mutual coupling synchronization is out-of-phase
$\varphi_{23}\approx\pm\pi$ (in-phase $\varphi_{23}\approx 0$). Notice that
there is a mirror symmetry between the phase distribution in the engine and
refrigerator regime, i.e., the maximum (minimum) of
$S({\varphi_{21},\varphi_{31}})$ in the engine case is a minimum (maximum) in
the refrigerator case. The other model parameters are
$\omega_{2}=3\omega_{1}$, $\Omega=\omega_{2}-\omega_{1}+\Delta/2$,
$\Delta=0.05\omega_{1}$, $\gamma_{h}=\gamma_{c}=0.1\omega_{1}$,
$\lambda=0.05\omega_{1}$, and $n_{c}=0.1$.

The four-level thermal maser forms the minimal model in which entrainment and
mutual coupling coexist. In the accompanying work [22], we show how
competition and cooperation manifest in the synchronization measure for
degenerate multilevel thermal masers. Depending on the thermodynamic
functionality of the maser, the phases compete when the system behaves as an
engine and cooperate when it acts as a refrigerator.

The aim of this section is to show that the same phenomena persist even in
presence of noise-induced coherence ($p\neq 0$) and near-degeneracy
($\Delta\neq 0$) by analytically examining the minimal model. We derive the
formula for phase-space synchronization measure [40] $S_{max}$ for the
_exactly_ degenerate ($\Delta=0$) four-level maser in presence of noise-
induced coherence. This measure not only captures the strength of steady state
coherences, but also its phase-matching condition [22]. The general $SU(D)$
quantum synchronization measure applicable to $D$-level systems is presented
in Ref. [22]. Here, we apply the general result for our specific case of
$D=4$, to obtain,

$S_{max}=\frac{1}{16\pi^{2}}\times\begin{cases}|\tilde{\rho}_{12}^{ss}|+|\tilde{\rho}_{13}^{ss}|+|\tilde{\rho}_{23}^{ss}|&n_{c}\leq
n_{h}\\\
|\tilde{\rho}_{12}^{ss}+|\tilde{\rho}_{13}^{ss}|-|\tilde{\rho}_{23}^{ss}|&n_{h}>n_{c}\;\&\;k>2\\\
\left(1+\dfrac{k^{2}}{2}\right)|\tilde{\rho}_{23}^{ss}|&n_{h}>n_{c}\;\&\;k\leq
2,\\\ \end{cases}$ (9)

where $k=\gamma_{h}(1+n_{h})(1+p)/\lambda$ is the dissipation-to-driving
ratio. The superscript of $\tilde{\rho}$ denotes the steady state, whose
analytical formula can be derived for $\Delta=0$ and $p\neq 0$ (see Appendix
A). The analytical solution for $\tilde{\rho}^{ss}$ can then be used to derive
Eq. (9) by following the recipe in the Supplementary Material of [22]. In
fact, Eq. (9) has the same form as the result obtained in [22] with the
exception that the dissipation-to-driving ratio $k$ now depends on the noise-
induced coherence strength $p$.

Equation (9) displays cooperation and competition between entrainment and
mutual coupling in different thermodynamic regimes. The mutual-coupling
(entrainment) contribution is represented by (non-) degenerate coherences
$|\tilde{\rho}_{ij}^{ss}|(|\tilde{\rho}_{1j}^{ss}|)$ for $i,j=2,3$. In the
refrigerator regime ($n_{c}<n_{h}$), entrainment and mutual coupling cooperate
to increase the overall synchronization of the maser since all steady state
coherences contribute positively to $S_{max}$. Meanwhile, in the engine case,
we observe competition between coherences for $k<2$. The competition and
cooperation is due to different phase configurations preferred by the two
mechanisms, i.e., both prefer in-phase synchronization in the refrigerator
case while one prefers in-phase and the other prefers out-of-phase in the
engine case (see Fig.2). Moreover, in the engine case for $k<2$,
synchronization is dominated by mutual coupling contribution
$|\tilde{\rho}_{23}^{ss}|>|\tilde{\rho}_{12}^{ss}|+|\tilde{\rho}_{13}^{ss}|$.

The $p$-dependence of $k$ allows us to explore different synchronization
regimes by tuning the strength of noise-induced coherence. For example, in
absence of noise-induced coherence ($p=0$), the deep mutual coupling dominant
regime $k\ll 2$ can only be explored in the strong-driving regime $\lambda\gg
2\gamma_{h}(1+n_{h})$. However, if the driving is strong, one starts to deform
the limit cycle and deviate away from the typical synchronization paradigm
[13]. Yet, in presence of noise-induced coherence ($p\neq 0$), the deep mutual
coupling dominant regime of $k\ll 2$ can be easily explored in presence of
total destructive interference ($p\rightarrow-1$).

Figures 2a-b show the quasi-probability phase distribution
$S(\varphi_{21},\varphi_{31})$, where $\varphi_{ij}\equiv\phi_{i}-\phi_{j}$ is
the relative phase between states $\ket{i}$ and $\ket{j}$. They are computed
in the engine regime for $p=0.5$ and $p=-0.99$. Figure 2a shows the regime
where entrainment is stronger than mutual coupling so that despite their
competition, the phases localize in the rotating frame. On the other hand,
Fig. 2b shows the deep mutual-coupling dominant regime where entrainment is
effectively lost but the relative phase is still fixed. Note that in both
figures, the near-degenerate gap is set to be non-zero ($\Delta\neq 0$).

##  IV Thermodynamic Observables

In this section, we will derive the key thermodynamic observables,
specifically steady state power and heat current following the standard recipe
for weak-coupling thermodynamics [41]. We follow the standard energy
partitioning [42] that defines the internal energy as

$E=\mathrm{Tr}(\tilde{\rho}H_{0}).$ (10)

The internal energy here is defined as that of the bare Hamiltonian $H_{0}$
and differs from the standard definition that involves the full Hamiltonian
[41]. Importantly, such a definition remains the same in all pictures
(Schrödinger, Heisenberg, and Interaction) and thereby avoids the
inconsistencies when one moves from one picture to another. Using the quantum
master equation (8), the change in internal energy can be expressed as,

$\displaystyle\frac{dE}{dt}=-i\mathrm{Tr}([H_{0},\tilde{V}]\tilde{\rho})+\mathrm{Tr}(\mathcal{D}_{h}[\tilde{\rho}]H_{0})+\mathrm{Tr}(\mathcal{D}_{c}[\tilde{\rho}]H_{0}).$
(11)

The energy flux is separated into three terms. The first term in (11) is power
and the following terms are heat fluxes from the hot and cold bath
respectively [42, 43],

$\displaystyle P$ $\displaystyle=$
$\displaystyle-i\mathrm{Tr}([H_{0},\tilde{V}]\tilde{\rho})$ (12)
$\displaystyle=$ $\displaystyle
2\lambda\sum_{j=2}^{3}(\omega_{j}-\omega_{1})\text{Im}(\tilde{\rho}_{1j}),$
$\displaystyle\dot{Q}_{h}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}(\mathcal{D}_{h}[\tilde{\rho}]H_{0})$
$\displaystyle=$ $\displaystyle
2\gamma_{h}\Bigl{(}\sum_{j=2}^{3}\omega_{j}\left[n_{h}^{(j)}\tilde{\rho}_{00}-(1+n_{h}^{(j)})\tilde{\rho}_{jj}\right]\Bigr{.}$
$\displaystyle\Bigl{.}-[(1+n_{h}^{(2)})\omega_{3}+(1+n_{h}^{(3)})\omega_{2}]p\,\mathrm{Re}(\tilde{\rho}_{23})\Bigr{)},$
$\displaystyle\dot{Q}_{c}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}(\mathcal{D}_{c}[\tilde{\rho}]H_{0})$ (14)
$\displaystyle=$ $\displaystyle
2\omega_{1}\gamma_{c}\left[n_{c}\tilde{\rho}_{00}-(1+n_{c})\tilde{\rho}_{11}\right].$

In the _engine_ regime, heat flows from the hot bath to the cold bath and
power is produced in the steady state, i.e.
$\dot{Q}_{h}^{ss}>0,\;\dot{Q}_{c}^{ss}<0,P^{ss}<0$. Conversely, in the
_refrigerator_ regime, heat flows from the cold bath to the hot bath and power
is consumed, i.e. $\dot{Q}_{h}^{ss}<0,\;\dot{Q}_{c}^{ss}>0,\;P^{ss}>0$.

The heat current to the cold bath $\dot{Q}_{c}$ depends solely on populations
whereas the heat from the hot bath $\dot{Q}_{h}$ depends on populations and
coherences. Intuitively, since the hot bath connects the ground state and the
near-degenerate manifold, a finite dipole alignment factor $p$ leads to noise-
induced coherence causing the hot heat current to be dependent on both
populations and coherences. Thus, the incoherent and coherent contribution to
the hot bath’s heat current [44, 45] can be expressed as,

$\dot{Q}_{h}^{inc}=2\gamma_{h}\sum_{j=2}^{3}\omega_{j}\left[n_{h}^{(j)}\tilde{\rho}_{00}-(1+n_{h}^{(j)})\tilde{\rho}_{jj}\right]$
(15)
$\dot{Q}_{h}^{coh}=-2\gamma_{h}[(1+n_{h}^{(2)})\omega_{3}+(1+n_{h}^{(3)})\omega_{2}]p\,\mathrm{Re}(\tilde{\rho}_{23}).$
(16)

Note that the coherent heat current is proportional to $p$ which we set to be
zero in the accompanying work [22].

The coherent heat either suppresses or enhances the natural inclination of
heat flow. For example, in the exactly degenerate ($\Delta=0$) engine regime,
a net heat flow coming from the hot bath to the system creates a population
inverted state such that $\dot{Q}_{h}^{inc,ss}>0$. From Eq. (16), we see that
the sign of $\dot{Q}_{h}^{coh,ss}$ depends on the sign of $p$
[$\because\text{Re}(\tilde{\rho}^{ss}_{23})<0$, see Eq. (45)]. If noise-
induced coherence strength $p>0$ ($p<0$), then the heat flow is enhanced
(suppressed). Similarly, in the exactly degenerate ($\Delta=0$) refrigerator
regime, we observe enhancement or suppression of heat dumped into the hot bath
due to either constructive ($p>0$) or destructive ($p<0$) interference.

##  V Connection between thermodynamic observables and synchronization

In general the synchronization measure ($S_{max}$) and thermodynamic
observables ($P$, $\dot{Q}_{c}$, and $\dot{Q}_{h}$) are unrelated. However, in
the case of a three-level thermal maser, the steady state power of the maser
is bounded by the synchronization measure [16] (power-synchronization bound)
connecting two seemingly distinct quantities. In this section, we build the
general framework of connecting the mathematically abstract notion of
synchronization to physical thermodynamic observables for a four-level thermal
maser. Despite several efforts analyzing the thermodynamic properties of the
four-level thermal maser [20, 36, 21], there have so far been no connections
between the synchronizing ability of such a thermal maser and its
thermodynamic observables.

###  V.1 Power - synchronization bound

Figure 3: steady state power to the synchronization ratio $|P^{ss}|/(\kappa
S_{max})$ with $\kappa=32\pi^{2}\lambda\omega_{31}$ as a function of
$n_{h}^{(2)}/n_{c}$ (a) and noise-induced coherence parameter $p$ (b). In a,
the near-degenerate energy gap $\Delta=0.05\omega_{1}$ (red empty circles),
$\Delta=0.2\omega_{1}$ (black crosses), and the noise-induced coherence
strength $p=0.5$. In b, the red filled circles are computed in the engine
($n_{h}^{(2)}/n_{c}=5$) regime, and the black filled circles are computed in
the refrigerator ($n_{h}^{(2)}/n_{c}=0.5$) regime. The vertical dotted
($\Delta=0.05\omega_{1}$) and dash-dotted ($\Delta=0.2\omega_{1}$) lines in a
represent engine-to-refrigerator boundary defined by the change of sign in
power. They slightly deviate from the degenerate case ($n_{h}^{(2)}/n_{c}=1$).
Meanwhile, the dotted line in b marks the entrainment-dominant to the mutual-
coupling dominant boundary ($k=2$). Power-synchronization bound is satisfied
when $|P^{ss}|/(\kappa S_{max})\leq 1$ (below the dashed horizontal line) in
both panels. Other parameter values are the same as Fig. 2.

We begin by revisiting the relationship between quantum synchronization and
the power of a near-degenerate thermal four-level maser. In the case of a
three-level Scovil–Schulz-DuBois maser, the steady state power is bounded by
the synchronization measure [16], i.e.,

$|P^{ss}|\leq 2^{4}\pi\lambda(\omega_{2}-\omega_{1})S_{max},$ (17)

also known as the power-synchronization (P-S) bound. The bound above has a
different physical interpretation in the engine and refrigerator regimes: In
the engine regime, P-S bound implies that the engine’ power is limited by the
amount the working substance entrains with the external drive. Whereas, in the
refrigerator regime, since power is pumped into the system, the P-S bound
suggests that there exists a maximum energy cost to ensure that the working
substance is entrained to the external drive. In other words, in the engine
regime, it can be inferred from the P-S bound that synchronization enhances
the power output of the engine while in the refrigerator regime the P-S bound
suggests that the external power supplied to the machine is utilized to
synchronize the working substance.

We now investigate the P-S bound for a near-degenerate four-level thermal
maser. Using Eq. (12), the power of the maser can be expressed as,

$|P^{ss}|\leq
2\lambda\omega_{31}(|\tilde{\rho}_{12}^{ss}|+|\tilde{\rho}_{13}^{ss}|),$ (18)

where $\omega_{ij}\equiv\omega_{i}-\omega_{j}$. The steady state power
$|P^{ss}|$ depends only on those coherences that are influenced by the
external drive, whereas the synchronization measure $S_{max}$ is linearly
dependent on all coherences. Specifically, the coherence
$|\tilde{\rho}_{23}^{ss}|$, while it affects the synchronization measure
$S_{max}$ [see Eq. (9)], it does not appear in the expression for steady state
power [Eq. (18)]. Due to the additional dependence of
$\tilde{\rho}^{ss}_{23}$, we find that the P-S bound is violated in the engine
regime, while being respected in the refrigerator case, i.e.,

$\displaystyle|P^{ss}|$ $\displaystyle\leq
32\pi^{2}\lambda\omega_{31}S_{max}\quad\quad(n_{c}>n_{h}),$ (19)
$\displaystyle|P^{ss}|$ $\displaystyle\nleq
32\pi^{2}\lambda\omega_{31}S_{max}\quad\quad(n_{h}>n_{c}).$ (20)

The above expression is analytically obtained in the exact degeneracy limit
$\Delta=0$. The violation originates due to the presence of the near-
degenerate levels that induce competition between entrainment and mutual
coupling [22]. The origin of the violation can be clearly seen from the
expression for $S_{max}$ [Eq. (9)] that in the engine regime for $k>2$ reads,
$S_{max}\propto|\tilde{\rho}_{12}^{ss}|+|\tilde{\rho_{13}^{ss}}|-|\tilde{\rho}_{23}^{ss}|<|\tilde{\rho}_{12}^{ss}|+|\tilde{\rho_{23}^{ss}}|$.
Although Eqs. (19) and (20) are derived with the assumption $\Delta=0$ (exact
degeneracy), we show in Fig. a-b that the violation persists for small non-
zero values of $\Delta$ where we plot steady state power to synchronization
ratio $|P^{ss}|/(\kappa S_{max})$ with $\kappa=32\pi^{2}\lambda\omega_{31}$ as
a function of $n_{h}^{(2)}/n_{c}$ and noise-induced coherence strength $p$.
The violation occurs if the ratio exceeds unity.

In Fig. 3a, we observe that other than a sharp discontinuity near the engine-
to-refrigerator transition, $|P^{ss}|/(\kappa S_{max})$ does not strongly
depend on $n_{h}^{(2)}/n_{c}$. We also find that the P-S bound is always
satisfied in the refrigerator regime while being violated in the engine regime
for small values of $\Delta$. Interestingly, as the near-degeneracy is further
lifted by increasing $\Delta$, we find that the validity of the P-S bound is
restored in both the engine and refrigerator regimes.

We also plot $|P^{ss}|/(\kappa S_{max})$ as a function of noise-induced
coherence strength $p$ in both the engine and refrigerator regime (see Fig.
3b). We again observe that the bound is always satisfied in the refrigerator
case while it is violated for most values of $p$ in the engine case. As
$p\rightarrow-1$, the bound seems to be recovered again. However, this is not
guaranteed in general. As $p\rightarrow-1$ or equivalently $k\rightarrow 0$,
the expression for $S_{max}$ only contains degenerate coherences while that of
power only contains non-degenerate coherences [see Eq. (9)]. Thus, as
$p\rightarrow-1$, synchronization starts to decouple from power.

The violation of the power-synchronization bound implies that in degenerate
and near-degenerate four-level maser heat engines, more power can be generated
than the upper bound set by synchronization. In contrast, in the refrigerator
regime ($n_{c}>n_{h}$) the P-S bound is always satisfied but it is never
saturated. This implies that synchronization can be generated with an energy
cost less than the maximum cost imposed by the P-S bound.

###  V.2 Coherent-heat – Synchronization Bound

Figure 4: The coherent heat current to synchronization ratio
$\dot{Q}_{h}^{coh,ss}/(\alpha S_{max})$ with
$\alpha=(8\pi)^{2}\gamma_{h}\omega_{3}(1+n_{h}^{(2)})|p|$ as a function of
$n_{h}^{(2)}/n_{c}$ (a) and noise-induced coherence parameter $p$ (b). The
squares in a are computed for $p=0.5$ and the triangles are computed for
$p=-0.99$. The red filled circles in b are computed in the engine regime
($n_{h}^{(2)}/n_{c}=5$) and the black filled circles are in the refrigerator
regime ($n_{h}^{(2)}/n_{c}=0.5$). All the data points lie below the dashed
horizontal line which means Q-S bound is always satisfied. The dotted vertical
line in a represents the engine-to-refrigerator boundary and in b it
represents the transition from constructive (positive $p$) to destructive
(negative $p$) interference. The near-degenerate gap $\Delta=0.05\omega_{1}$.
Other parameter values are the same as Fig. 2.

Intuitively, when the working substance of a machine synchronizes it achieves
a low entropy state. It is therefore natural to expect that in order to
maintain this state the entropy of the bath should increase. Thus, it is
expected that synchronization will be coupled to heat. In this subsection, we
will solidify this intuition and rigorously show the connections between heat
and synchronization, leading to the heat-synchronization (Q-S) bound.

Specifically, we have seen in Sec. IV that in the presence of noise-induced
coherence, the heat contribution from the hot bath can be separated into
incoherent and coherent terms. We will now connect the coherent heat current
to synchronization by noting the inequality

$|\dot{Q}_{h}^{coh,ss}|\leq
4|p|\gamma_{h}(1+n_{h}^{(2)})\omega_{3}\text{Re}(\tilde{\rho}_{23}^{ss}).$
(21)

The above inequality is derived from Eq. (16) by making use of the fact that
$\omega_{3}\geq\omega_{2}$ and $n_{h}^{(2)}\geq n_{h}^{(3)}$. Note that in
contrast to power which is independent of degenerate coherences
$|\tilde{\rho}_{23}^{ss}|$, the coherent heat current is only a function of
degenerate coherences. Recall, also that synchronization measure $S_{max}$ is
a function of both degenerate and non-degenerate coherences. One may then
relate coherent heat current in the steady state with the synchronization
measure. In the limit of $\Delta=0$, we can use Eq. (9) to derive a coherent
heat-synchronization (Q-S) bound,

$|\dot{Q}_{h}^{coh,ss}|\leq(8\pi)^{2}|p|\gamma_{h}(1+n_{h}^{(2)})\omega_{3}S_{max}.$
(22)

The validity of this bound is trivial in the refrigerator regime where
$S_{max}$ is given by the $\ell_{1}$-norm $C_{\ell_{1}}$ and noting that
$\text{Re}(\tilde{\rho}_{23}^{ss})\leq|\tilde{\rho}_{23}^{ss}|\leq
C_{\ell_{1}}$. Even in the engine regime, when mutual coupling dominates
($k<2$), the bound is easily satisfied since
$\text{Re}(\tilde{\rho}_{23}^{ss})\leq|\tilde{\rho}_{23}^{ss}|\leq(1+k^{2}/2)|\tilde{\rho}_{23}^{ss}|$.
Yet, perhaps more surprisingly, this bound is also valid in the engine
entrainment dominant case ($k>2$). In this regime, since
$S_{max}\propto(2k-1)|\tilde{\rho}_{23}^{ss}|\geq 3|\tilde{\rho}_{23}^{ss}|$
and
$\dot{Q}^{coh,ss}_{h}\propto\mathrm{Re}(\tilde{\rho}_{23}^{ss})\leq|\tilde{\rho}_{23}^{ss}|$,
the Q-S bound is always satisfied. In other words, unlike the P-S bound which
can be violated in presence of near-degeneracy, the Q-S bound is always
satisfied for $\Delta=0$.

Although we derived the bound using the exact degeneracy assumption
($\Delta=0$), we show numerically in Fig. 4 that the bound is still valid for
small values of $\Delta\neq 0$. In Fig. 4a, we plot the coherent heat to
synchronization ratio $|\dot{Q}_{h}^{coh,ss}|/(\alpha S_{max})$ with
$\alpha=(8\pi)^{2}\gamma_{h}\omega_{3}|p|(1+n_{h}^{(2)})$ for different values
of $n_{h}^{(2)}/n_{c}$ and noise-induced coherence strength $p$. We observe
that $|\dot{Q}_{h}^{coh,ss}|/(\alpha S_{max})$ does not depend strongly on
$n_{h}^{(2)}/n_{c}$ except for a discontinuity near engine-to-refrigerator
boundary. Furthermore, we find that the bound is always satisfied and is not
strongly dependent on $\Delta$. However, the bound is tight only deep in the
mutual coupling regime, e.g. when $p\rightarrow-1$.

###  V.3 Lower Bound on Efficiency and Coefficient of Performance (COP)

Efficiency is one of the most well-known performance metrics for thermal
engines. It is defined by $\eta=-P^{ss}/\dot{Q}_{h}^{ss}$. Previously, we have
demonstrated that both steady state power $P^{ss}$ and the coherent part of
heat current $\dot{Q}_{h}^{coh,ss}$ are connected to synchronization measure
$S_{max}$. It is then natural to ask whether efficiency is also connected to
synchronization. The efficiency can be computed analytically in the degenerate
limit ($\Delta=0$) and it is given by the standard expression
$\eta=1-(\omega_{1}/\omega_{2})$ (see Appendix B), which is also obtained for
the three-level thermal maser. Thus, the efficiency in the exact degenerate
limit depends only on the system’s energy scale and it is independent of the
synchronization measure $S_{max}$ or any other parameters. However, this is no
longer guaranteed in the near-degenerate case ($\Delta\neq 0$), where the
steady state is not analytically solvable. In such cases, the efficiency is
generally a function of all the parameters of the system, baths, and drive.

It is well-known that the efficiency of a heat engine is upper-bounded by the
Carnot efficiency. Here, we show that synchronization sets a lower bound to
the efficiency of a near-degenerate thermal maser. We will use the violation
of P-S bound and the validity of Q-S bound which have been demonstrated for
the near-degenerate $\Delta\neq 0$ case in subsections V.1-V.2. Below, we
state the efficiency-synchronization (E-S) bound as,

$\eta\geq\frac{\kappa S_{max}}{\dot{Q}_{h}^{inc,ss}+\alpha S_{max}}.$ (23)

The above inequality is derived under the assumption that P-S bound is
violated $|P^{ss}|>\kappa S_{max}$ and the Q-S bound is respected
$\dot{Q}_{h}^{inc,ss}+\dot{Q}_{h}^{coh,ss}\leq\dot{Q}_{h}^{inc,ss}+|\dot{Q}_{h}^{coh,ss}|\leq\dot{Q}_{h}^{inc,ss}+\alpha
S_{max}$ with $\kappa=32\pi^{2}\lambda\omega_{31}$ and
$\alpha=(8\pi)^{2}|p|\gamma_{h}(1+n_{h}^{(2)})\omega_{3}$. Note that the
incoherent heat current $\dot{Q}_{h}^{inc,ss}$ is only a function of
populations [see Eq. (15)] and so it is unrelated to synchronization.

We check the validity of the E-S bound Eq. (23) for various parameter values
in Fig. 5a-b. Similar to the P-S and Q-S bound, we find that the ratio
$\eta_{S}/\eta$ with

$\eta_{S}=\frac{\kappa S_{max}}{\dot{Q}_{h}^{inc,ss}+\alpha S_{max}},$ (24)

does not strongly depend on the baths mean occupation number ratio
$n_{h}^{(2)}/n_{c}$ (Fig. 5a). Instead, it strongly depends on the noise-
induced coherence parameter $p$ (Fig. 5b). In the limit when there is no
interference effect from the hot bath, i.e., $p\rightarrow 0$, we find that
the bound is saturated.

Similarly, the coefficient of performance (COP) in the _exact_ degenerate
($\Delta=0$) refrigerator regime can be analytically computed to be
$\chi=\omega_{1}/(\omega_{2}-\omega_{1})$ (see Appendix B). In the near-
degenerate ($\Delta\neq 0$) case, the COP is lower-bounded by the inverse of
synchronization measure $S_{max}$,

$\chi=\frac{\dot{Q}_{c}^{ss}}{P^{ss}}\geq\frac{\dot{Q}_{c}^{ss}}{\kappa
S_{max}}.$ (25)

The lower bound is valid because the consumed power is always upper-bounded by
$S_{max}$ in the refrigerator regime [see Eq. (19)]. We also note that
$\dot{Q}_{c}^{ss}$ is only a function of populations [see Eq. (14)] and hence
is unrelated to the synchronization.

We check the validity of the bound Eq. (25) for various values of parameters
in Fig. 5c-d. Similar to the E-S bound, we find that the ratio
$\chi_{S}/\chi$, with

$\chi_{S}=\frac{\dot{Q}_{c}^{ss}}{\kappa S_{max}},$ (26)

does not strongly depend on the baths mean occupation number ratio
$n_{h}^{(2)}/n_{c}$ (Fig. 5c), but depends strongly on the noise-induced
coherence parameter $p$ (Fig. 5d). However, different from the E-S bound, the
bound (25) is saturated in the limit $p\rightarrow 1$, i.e., when the hot bath
causes constructive interference.

Figure 5: Synchronization lower bounds engine’s efficiency $\eta$ (a-b) and
refrigerator’s COP $\chi$ (c-d) in near-degenerate four-level thermal maser.
The red (black) data points are computed in the engine (refrigerator) regime.
The circles (crosses) are computed with the value of near-degenerate energy
gap $\Delta=0.05\omega_{1}$ ($\Delta=0.1\omega_{1}$). Panels a-b show
efficiency ratio $\eta_{S}/\eta$ [see Eq. (24)] as a function of
$n_{h}^{(2)}/n_{c}$ (a) and noise-induced coherence parameter $p$ (b).
Similarly, panels c-d show the COP ratio $\chi_{S}/\chi$ [see Eq. (26)] as a
function of $n_{h}^{(2)}/n_{c}$ (c) and $p$ (d). In panels a and c the noise-
induced coherence parameter is kept constant at $p=0.1$. In panel b (d), the
baths mean occupation number ratio is fixed $n_{h}^{(2)}/n_{c}=2$
($n_{h}^{(2)}/n_{c}=0.5$). The bounds (23)-(25) are satisfied if all the data
points lie below the dashed horizontal line. Other parameters are the same as
in Fig. 2

.

##  VI Summary and Discussion

There is intense debate regarding the optimal design of quantum thermal
machines. Almost all design paradigms assume the systems are well modeled by
Markovian master equations and furthermore pursue coherence from an
information-theoretic resource cost point of view or a non-equilibrium cost
point of view. While such analyses are definitely sufficient for non-
degenerate systems, secular approximation begins to fail to describe near-
degenerate systems interacting with a bath. Furthermore, interacting
coherences can be well understood under the paradigm of coupled oscillator
models, an analysis that has tremendously benefited classical dynamical
systems.

In this manuscript, we argue fundamentally that such a coupled oscillator
model is indispensable to understanding the subtle dynamics of interacting
coherences, which arise due to a variety of situations like those described in
this manuscript. We developed the synchronization dynamics for a four-level
atom coupled to two baths, generalizing the canonical thermal maser analysis
by Scovil and Schulz-DuBois. We show that coherences interact with each other
in two distinct qualitative ways. The first of these is a cooperative phase-
locking phenomenon understood as entrainment dynamics of coherences being
driven by an external field. Such entrainment dynamics and their contribution
to the performance of a thermal maser were discussed earlier [16]. In this
manuscript, we add to this entrainment dynamics and highlight the competition
between coherences that can arise due to the bath coupling in our model. This
coexistence of cooperation and competition gives rise to a richer tapestry of
dynamics, uncoupling the power output from the thermal maser from the
synchronization measure. The bound on power from the synchronization measure
of the standard maser is modified to include coherent contributions to heat.

In presence of noise-induced coherence, in addition to the power-
synchronization bound, we also find that coherent heat is bounded by
synchronization measure showing that synchronization not only influences the
useful work [46, 47] but can also bound the wasteful heat. The result of this
work can be summarized in the following table namely,

| Engine | Refrigerator
---|---|---
P-S bound | x | ✓
Q-S bound | ✓ | ✓
E-S bound | ✓ | ✓

where the check mark indicates that the bound is satisfied whereas the cross
means the bound is violated. The lower bound of COP is included in the
refrigerator column of E-S bound for compactness. Unlike the standard upper
bounds like the Carnot efficiency, our analysis shows that efficiency can be
_lower_ bounded in presence of quantum synchronization. In other words, a
quantum synchronous working substance can boost the efficiency of the machine.
Even though our analysis was limited to a four-level thermal maser, it would
be an interesting future avenue to investigate other coupled oscillator models
for interacting coherences to inform the future designs of quantum thermal
machines.

###### Acknowledgements.

This research was supported by the Institute for Basic Science in South Korea
(IBS-R024-Y2). S.V. acknowledges support from a Government of India DST-QUEST
grant number DST/ICPS/QuST/Theme-4/2019. The authors would like to thank V.
Singh for the useful discussions.

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##  Appendix A Steady state of Four-Level Thermal Maser with Interference

The quantum master equation (8) can be expanded into a series of linear first-
order differential equations for each density matrix element. We divide the
elements into three groups: populations ($\rho_{ii}$ with $i=0,\cdots,3$),
non-degenerate coherences ($\rho_{12}$ and $\rho_{13}$), and degenerate
coherences ($\rho_{23}$). In the following derivation, we have ignored the
coherences $\rho_{0i}$ with $i=1,2,3$ since their dynamics are decoupled from
other density matrix elements and they decay to zero in the steady state.

We restrict ourselves to the exact degenerate case scenario with $\Delta=0$
and $n_{h}^{(i)}=n_{h}$ and obtain the equations for the populations, namely,

$\displaystyle\frac{d\tilde{\rho}_{11}}{dt}$ $\displaystyle=$ $\displaystyle
i\lambda\sum_{j=2}^{3}(\tilde{\rho}_{1j}-\tilde{\rho}_{j1})-2\xi_{c}\tilde{\rho}_{11}+2\gamma_{c}n_{c}\tilde{\rho}_{00},$
(27) $\displaystyle\frac{d\tilde{\rho}_{22}}{dt}$ $\displaystyle=$
$\displaystyle-i\lambda(\tilde{\rho}_{12}-\tilde{\rho}_{21})-\xi_{h}p(\tilde{\rho}_{23}+\tilde{\rho}_{32})-2\xi_{h}\tilde{\rho}_{22}$
(28) $\displaystyle+2\gamma_{h}n_{h}\tilde{\rho}_{00},$
$\displaystyle\frac{d\tilde{\rho}_{33}}{dt}$ $\displaystyle=$
$\displaystyle-i\lambda(\tilde{\rho}_{13}-\tilde{\rho}_{31})-\xi_{h}p(\tilde{\rho}_{23}+\tilde{\rho}_{32})-2\xi_{h}\tilde{\rho}_{33}$
(29) $\displaystyle+2\gamma_{h}n_{h}\tilde{\rho}_{00}.$

Above we have introduced $\xi_{x}=\gamma_{x}(1+n_{x})$ with $x=c,h$ to
simplify our notation. The equations for the non-degenerate and degenerate
coherences read,

$\displaystyle\frac{d\tilde{\rho}_{12}}{dt}$ $\displaystyle=$ $\displaystyle
i(\omega_{2}-\omega_{1}-\Omega)\tilde{\rho}_{12}-(\xi_{c}+\xi_{h})\tilde{\rho}_{12}-\xi_{h}p\tilde{\rho}_{13}+i\lambda\left(\tilde{\rho}_{11}-\tilde{\rho}_{22}-\tilde{\rho}_{32}\right),$
(30) $\displaystyle\frac{d\tilde{\rho}_{13}}{dt}$ $\displaystyle=$
$\displaystyle
i(\omega_{2}-\omega_{1}+\Delta-\Omega)\tilde{\rho}_{13}-(\xi_{c}+\xi_{h})\tilde{\rho}_{13}-\xi_{h}p\tilde{\rho}_{12}+i\lambda\left(\tilde{\rho}_{11}-\tilde{\rho}_{33}-\tilde{\rho}_{23}\right),$
(31) $\displaystyle\frac{d\tilde{\rho}_{23}}{dt}$ $\displaystyle=$
$\displaystyle-i\Delta\tilde{\rho}_{23}-2\xi_{h}\tilde{\rho}_{23}-i\lambda(\tilde{\rho}_{13}-\tilde{\rho}_{21})+2\gamma_{h}n_{h}p\rho_{00}-\xi_{h}p(\tilde{\rho}_{22}+\tilde{\rho}_{33}).$
(32)

To find an analytical steady state ($d\tilde{\rho}_{ij}/dt=0\;\forall i,j$)
solution, we further restrict ourselves to the resonant drive
$\Omega=\omega_{2}-\omega_{1}$ and degenerate ($\Delta=0$) limit. Using Eqs.
(30) and (31) and splitting them into their real and imaginary parts we
obtain,

$\displaystyle\text{Re}(\tilde{\rho}^{ss}_{12})+\text{Re}(\tilde{\rho}^{ss}_{13})$
$\displaystyle=$ $\displaystyle 0,$ (33)
$\displaystyle\left[\xi_{c}+\xi_{h}(1-p)\right]\text{Re}(\tilde{\rho}^{ss}_{12})+i\lambda\;\text{Im}(\tilde{\rho}^{ss}_{23})$
$\displaystyle=$ $\displaystyle 0.$ (34)

Furthermore, using Eq. (33) and computing the imaginary part of Eq. (32) we
obtain,

$2\xi_{h}\;\text{Im}(\tilde{\rho}^{ss}_{23})+i\lambda\;\text{Re}(\tilde{\rho}^{ss}_{12})=0.$
(35)

Note that both Eqs. (34) and (35) are linear, involving the same density
matrix elements. Therefore, combining these two equations yields,

$\text{Re}(\tilde{\rho}^{ss}_{12})=\text{Re}(\tilde{\rho}^{ss}_{13})=\text{Im}(\tilde{\rho}^{ss}_{23})=0,$
(36)

i.e., the steady state (non-) degenerate coherences are real (imaginary).
Further simplification can be sought by subtracting the imaginary part of Eq.
(30) from Eq. (31) to yield,

$\left[\xi_{c}+\xi_{h}(1-p)\right]\left[\text{Im}(\tilde{\rho}^{ss}_{12})-\text{Im}(\tilde{\rho}^{ss}_{13})\right]+i\lambda(\tilde{\rho}^{ss}_{22}-\tilde{\rho}^{ss}_{33})=0.$
(37)

The equation above connects the difference between non-degenerate coherences
and the population difference between the degenerate states. We can find
another linearly independent equation by computing the population differences
between degenerate states using Eqs. (28) and (29), i.e.,

$i\lambda\left[\text{Im}(\tilde{\rho}^{ss}_{12})-\text{Im}(\tilde{\rho}^{ss}_{13})\right]+\xi_{h}(\tilde{\rho}^{ss}_{22}-\tilde{\rho}^{ss}_{33})=0.$
(38)

Combining the above two equations gives,

$\text{Im}(\tilde{\rho}^{ss}_{12})=\text{Im}(\tilde{\rho}^{ss}_{13})\qquad\rm{and}\qquad\tilde{\rho}^{ss}_{22}=\tilde{\rho}^{ss}_{33}.$
(39)

Thus, the population and the coherences associated with the degenerate states
are equal. This makes intuitive sense because we are working in the exact
degenerate limit. Using the constraints derived thus far together with a
trace-preserving condition
$\tilde{\rho}^{ss}_{00}=1-\tilde{\rho}^{ss}_{11}-\tilde{\rho}^{ss}_{22}-\tilde{\rho}^{ss}_{33}$,
the problem is reduced to four independent variables:
$\tilde{\rho}^{ss}_{11},\;\tilde{\rho}^{ss}_{22},\;\tilde{\rho}^{ss}_{12},$
and $\tilde{\rho}^{ss}_{23}$. The corresponding four equations to solve for
these variables can be derived from Eqs. (27) – (32) and they can be expressed
as,

$\displaystyle
2i\lambda\tilde{\rho}^{ss}_{12}-\xi_{c}\tilde{\rho}^{ss}_{11}+\gamma_{c}n_{c}(1-\tilde{\rho}^{ss}_{11}-2\tilde{\rho}^{ss}_{22})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left[\xi_{c}+\xi_{h}(1+p)\right]\tilde{\rho}^{ss}_{12}-i\lambda\left(\tilde{\rho}^{ss}_{11}-\tilde{\rho}^{ss}_{22}-\tilde{\rho}^{ss}_{23}\right)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
i\lambda\;\tilde{\rho}^{ss}_{12}+\xi_{h}p\tilde{\rho}^{ss}_{23}+\xi_{h}\tilde{\rho}^{ss}_{22}-\gamma_{h}n_{h}(1-\tilde{\rho}^{ss}_{11}-2\tilde{\rho}^{ss}_{22})$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
i\lambda\tilde{\rho}^{ss}_{12}+\xi_{h}\tilde{\rho}^{ss}_{23}+\xi_{h}p\tilde{\rho}^{ss}_{22}-\gamma_{h}pn_{h}(1-\tilde{\rho}^{ss}_{11}-2\tilde{\rho}^{ss}_{22})$
$\displaystyle=$ $\displaystyle 0.$

Solving the above linear system of equations for the four-independent steady
state density matrix elements we first obtain the ratio between non-degenerate
and degenerate coherence,

$\frac{\tilde{\rho}^{ss}_{12}}{\tilde{\rho}^{ss}_{23}}=i\frac{\gamma_{h}(1+n_{h})(1+p)}{\lambda}.$
(41)

The norm of the above quantity is the dissipation-to-driving ratio $k$
[defined below Eq. (9)]. It is a crucial quantity to determine the
synchronization regime of the system (entrainment-dominant or mutual-coupling
dominant) and in observing competition between these mechanisms in the engine
regime [see Eq. (9)]. Solving for the populations gives us,

$\displaystyle\tilde{\rho}^{ss}_{11}$ $\displaystyle=$
$\displaystyle\frac{(1+n_{h})\Big{[}2\lambda^{2}\big{(}n_{c}\gamma_{c}+\gamma_{h}(1+p)n_{h}\big{)}+\xi_{h}(1+p)\gamma_{c}n_{c}\big{(}\xi_{c}+\xi_{h}(1+p)\big{)}\Big{]}}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)},$
$\displaystyle\tilde{\rho}^{ss}_{22}$ $\displaystyle=$
$\displaystyle\tilde{\rho}^{ss}_{33}=\frac{\lambda^{2}[n_{h}+n_{c}+2n_{h}n_{c}+2\xi_{h}n_{h}(1+p)]+\xi_{c}\xi_{h}(1+p)n_{h}[\xi_{c}+\xi_{h}(1+p)]}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)},$
(42)

with the denominator

$\displaystyle F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)$
$\displaystyle=$ $\displaystyle
2\lambda^{2}\left[\gamma_{c}(1+3n_{c}+2n_{h}+4n_{h}n_{c})+\xi_{h}(1+p)(1+4n_{h})\right]$
(43)
$\displaystyle+\gamma_{c}\xi_{h}(1+p)(1+3n_{h}+2n_{c}+4n_{h}n_{c})\left[\xi_{c}+\xi_{h}(1+p)\right],$

On the other hand, using the population results, the steady state coherences
read,

$\tilde{\rho}^{ss}_{12}=\tilde{\rho}^{ss}_{13}=i\frac{\lambda\gamma_{c}\xi_{h}(1+p)(n_{c}-n_{h})}{F(n_{h},n_{c},\gamma_{h},\gamma_{c}\lambda,p)},$
(44)
$\tilde{\rho}^{ss}_{23}=\frac{\lambda^{2}\gamma_{c}(n_{c}-n_{h})}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)}.$
(45)

These steady state coherences are the important quantities in deriving the
synchronization measure $S_{max}$ [see (9)]. The derivation of $S_{max}$ from
the steady state coherences has been explained in detail in our accompanying
work [22].

##  Appendix B Thermodynamic Observables in Steady State

In Appendix A, we derived the expressions for the steady state of a degenerate
four-level thermal maser with interference ($p\neq 0$). We can use those
formulas to express thermodynamic observables such as power and heat current
in the steady-state [see (12)-(14)],

$P^{ss}=-\frac{2\lambda^{2}\gamma_{c}\xi_{h}(\omega_{2}-\omega_{1})(1+p)(n_{h}-n_{c})}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)},$
(46)
$\dot{Q}_{c}^{ss}=-\frac{2\lambda^{2}\gamma_{c}\xi_{h}\omega_{1}(1+p)(n_{h}-n_{c})}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)},$
(47)
$\dot{Q}_{h}^{inc,ss}=\frac{2\lambda^{2}\omega_{2}\gamma_{c}\xi_{h}(n_{h}-n_{c})}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)},$
(48)
$\dot{Q}_{h}^{coh,ss}=\frac{2p\lambda^{2}\omega_{2}\gamma_{c}\xi_{h}(n_{h}-n_{c})}{F(n_{h},n_{c},\gamma_{h},\gamma_{c},\lambda,p)},$
(49)

where the total heat current from the hot bath is the total of coherent and
incoherent contribution
$\dot{Q}_{h}^{ss}=\dot{Q}_{h}^{inc,ss}+\dot{Q}_{h}^{coh,ss}$. It is
interesting to note that $\dot{Q}_{h}^{coh,ss}/\dot{Q}_{h}^{inc,ss}=p$. This
relation gives a thermodynamic meaning to noise-induced coherence parameter
$p$ as the ratio between coherent and incoherent heat current in the steady
state. It also implies the suppression or enhancement of heat flow due to
coherence is fully determined by the sign of $p$. When there is destructive
interference ($p<0$), the direction of coherent heat flow is opposite to the
incoherent one (suppression) whereas for constructive interference ($p>0$)
they are in the same direction (enhancement).

Recall that the direction of power and heat flow determines whether the system
operates as an engine or refrigerator. In the engine regime, heat flows from
the hot bath to the cold bath, and power is produced
($\dot{Q}_{h}^{ss}>0,\;\dot{Q}_{c}^{ss}<0,\;P^{ss}<0$). Conversely, in the
refrigerator regime, heat flows from the cold bath to the hot bath, and power
is consumed ($\dot{Q}_{h}^{ss}<0,\;\dot{Q}_{c}^{ss}>0,\;P^{ss}>0$). Equations
(46) - (49) implies that in the exact degenerate case ($\Delta=0$), the
engine-to-refrigerator transition is fully determined by the baths bosonic
mean-occupation number. Specifically, $n_{h}>n_{c}$ ($n_{h}<n_{c}$) implies
the maser is functioning as an engine (refrigerator).

In the engine regime ($n_{h}>n_{c}$), the efficiency $\eta$ is calculated as

$\eta=-\frac{P^{ss}}{\dot{Q}_{h}^{ss}}=1-\frac{\omega_{1}}{\omega_{2}}\leq\eta_{C},$
(50)

where $\eta_{C}=1-T_{c}/T_{h}$ is the Carnot’s efficiency ($T_{c,h}$ are the
temperatures of the cold and hot bath respectively). The Carnot bound can be
obtained by writing $\omega_{1}=T_{c}\ln[(1+n_{c})/n_{c}]$ and
$\omega_{2}=T_{h}\ln[(1+n_{h})/n_{h}]$.

Meanwhile, in the refrigerator regime ($n_{h}<n_{c}$), we calculate the COP,

$\chi=\frac{\dot{Q}_{c}^{ss}}{P^{ss}}=\frac{\omega_{1}}{\omega_{2}-\omega_{1}}.$
(51)

We emphasize that the simple forms of efficiency and COP derived in Eqs.
(50)-(51) are only valid in the degenerate limit ($\Delta=0$). In the main
text, we lift the degeneracy and show that the efficiency and the COP are
lower bounded by synchronization measure $S_{max}$.