# Two-body quantum absorption refrigerators with optomechanical-like
interactions

M. Tahir Naseem  Department of Physics, Koç University, Sarıyer, İstanbul,
34450, Turkey  Avijit Misra  Department of Chemical and Biological Physics,
Weizmann Institute of Science, Rehovot 7610001, Israel  Özgür E.
Müstecaplıoğlu<EMAIL_ADDRESS>Department of Physics, Koç University,
Sarıyer, İstanbul, 34450, Turkey

###### Abstract

Quantum absorption refrigerator (QAR) autonomously extracts heat from a cold
bath and dumps into a hot bath by exploiting the input heat from a higher
temperature reservoir. QARs typically require three-body interactions. We
propose and examine a two-body QAR model based upon optomechanical-like
coupling in the working medium composed of either two two-level systems or two
harmonic oscillators or one two-level atom and a harmonic oscillator. In the
ideal case without internal dissipation, within the experimentally realizable
parameters, our model can attain the coefficient of performance that is
arbitrarily close to the Carnot bound. We study the efficiency at maximum
power, a bound for practical purposes, and show that by using suitable
reservoir engineering and exploiting the nonlinear optomechanical-like
coupling, one can achieve efficiency at maximum power close to the Carnot
bound, though the power gradually approaches zero as the efficiency approaches
the Carnot bound. Moreover, we discuss the impact of non-classical
correlations and the size of Hilbert space on the cooling power. Finally, we
consider a more realistic version of our model in which we consider heat leaks
that makes QAR non-ideal and prevent it to achieve the Carnot efficiency.

††preprint: APS/123-QED

##  I Introduction

Thermal machines with quantum working substances, such as quantum heat engines
Kosloff and Levy (2014); Roßnagel  _et al._ (2016); Hardal  _et al._ (2017)
and refrigerators Karimi and Pekola (2016); Manikandan  _et al._ (2019), have
been attracted much attention recently. A particularly promising class of
quantum thermal machines are those that do not require any external work. For
compact and energetically cheap implementations of quantum technologies,
autonomous quantum cooling schemes are highly desirable. For that aim, so-
called quantum absorption refrigerators (QARs) Mitchison (2019) have been
proposed Palao  _et al._ (2001); Linden  _et al._ (2010); Levy and Kosloff
(2012); Levy  _et al._ (2012); Correa  _et al._ (2013); Venturelli  _et al._
(2013); Correa (2014); Leggio  _et al._ (2015); Silva  _et al._ (2015, 2016);
Hofer  _et al._ (2016); Doyeux  _et al._ (2016); Roulet  _et al._ (2017);
Erdman  _et al._ (2018); Härtle  _et al._ (2018); Segal (2018); Kilgour and
Segal (2018); Seah  _et al._ (2018); Mitchison  _et al._ (2016); Chen and Li
(2012); He  _et al._ (2017); Mukhopadhyay  _et al._ (2018); Das  _et al._
(2019); Latune  _et al._ (2019); Hewgill  _et al._ (2020) and recently a
three-ion QAR has been experimentally demonstrated Maslennikov  _et al._
(2019). While typical QARs rely upon three-body interaction Levy and Kosloff
(2012); Chen and Li (2012), smaller QARs, such as two-atoms Linden  _et al._
(2010); Levy  _et al._ (2012) or two-resonators Levy  _et al._ (2012), have
also been proposed.

The ideal QAR based on two-body interaction can saturate the Carnot bound, at
the cost of vanishing cooling power Mitchison (2019). A more practical bound
for QAR then can be efficiency at maximum cooling power $\epsilon_{*}$ Correa
_et al._ (2014). Surprisingly, such a bound has been established in Correa
_et al._ (2014) for the QARs that can be broken up in three-level masers. Two-
body QAR Levy  _et al._ (2012) has been shown to obey this bound, which is
much smaller than the Carnot efficiency. Here we consider two-body QAR with
directly interacting subsystems of the working medium. We show that by using
reservoir engineering, our model can surpass this bound. In the strong
coupling regime, the efficiency at maximum power even can saturate the Carnot
efficiency (the power gradually goes to zero at the Carnot point). In addition
to discrete and continuous variable (CV) models, we investigate a hybrid
system in which a two-level atom is coupled with a single resonator. The
coupling then becomes identical to the single-mode case of electron-phonon
Fröhlich (1952); Holstein (1959) or spin-boson interaction Weiss (2012). In
our model, we observe that larger Hilbert space sizes enhance the cooling
power Brunner  _et al._ (2014), and quantum correlations play no role in the
effecient performance of the refrigerator Correa  _et al._ (2013). Finally, we
consider a more realistic case, in which our model suffers from heat leaks,
consequently becomes non-ideal Palao  _et al._ (2001).

To give analytical and physical explanations of our results, we use the atomic
analog of the optomechanical model. Such two-atom QARs can be of interest per
se and can be realized by quantum optical methods Kargı  _et al._ (2019).
Using the atomic model, we explain the cooling mechanism by the energy cycles
that consist of thermally induced transitions between the dressed states of
the working medium. We identify similar transition structures within CV models
to justify the emergence of the same cooling mechanism in them. In suitable
parameters regime, according to the second law of thermodynamics, the energy
cycles that remove the heat from the cold bath become more favorable than the
reciprocal process. We use a thermodynamically consistent master equation that
has been recently derived for optomechanical systems Naseem  _et al._ (2018).
We verify that proposed devices operate within the Carnot bound but can get
arbitrarily close to Carnot efficiency, where unfortunately but necessarily
the cooling power vanishes.

The rest of the paper is organized as follows. We describe the three models of
QARs that we examine in Sec. II. They are all based upon the standard
nonlinear optomechanical model, which is the pure CV case. The other two cases
are for its hybrid and atomic analogs. The mechanism of the cooling is
explained for the three working mediums in Sec. III, where main analytical
results are based upon exactly solvable atomic analog. Detailed analysis of
thermally induced transitions among the dressed states is given and compared
for justification of the common physics behind QAR for the three models. We
discuss non-ideal model of our QAR in Sec. IV and conclude in Sec. V.

##  II Quantum absorption refrigerator model

Figure 1: (Color online) Schematic illustration of a quantum absorption
refrigerator in which the working medium is composed of two coupled qubits or
two coupled harmonic oscillators or a single qubit (A) and a harmonic
oscillator (B). The frequency of subsystem A (B) is $\omega_{h}$
($\omega_{c}$), and the coupling strength between them is $g$. The subsystem A
is coupled with two thermal baths, hot bath and work reservoir namely, at
temperatures $T_{h}$ and $T_{w}$, and subsystem B is coupled with the cold
bath at temperature $T_{c}$. We also have $T_{w}>T_{h}>T_{c}$.

We propose a model of QAR in which the working medium is composed of two
coupled systems (A) and (B), as shown in Fig. 1. We consider three different
cases viz. (i) A and B both are two-level systems (TLS) (ii) A is a TLS and B
is a harmonic oscillator and (iii) Both A and B are harmonic oscillators. The
subsystem A is coupled with a hot bath at temperature $T_{h}$, besides, it is
also coupled with a thermal bath that plays the role of work reservoir with a
temperature of $T_{w}$. The subsystem B is coupled with a cold bath at
temperature $T_{c}$. We assume that all three thermal baths are independent
with temperatures $T_{w}>T_{h}>T_{c}$.

The Hamiltonian of the working medium (A + B) of our QAR is given by
Aspelmeyer  _et al._ (2014)

$\hat{H}_{S}=\omega_{h}\hat{a}^{\dagger}\hat{a}+\omega_{c}\hat{b}^{\dagger}\hat{b}+g\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger}),$
(1)

where $g$ is the coupling strength, $\omega_{h}$ and $\omega_{c}$ are the
frequencies of the A and B subsystems, respectively. The bosonic annihilation
(creation) operator for the subsystem A is represented by $\hat{a}$
($\hat{a}^{\dagger}$), and those of subsystem B with $\hat{b}$
($\hat{b}^{\dagger}$). For the case of coupled two two-level system (TLS),
$\hat{a},\hat{a}^{\dagger}$ and $\hat{b},\hat{b}^{\dagger}$ are replaced Pauli
operators $\hat{\sigma}^{\pm}_{h,c}$, and the last term becomes atomic analog
of optomechanical coupling, $g\hat{\sigma}^{z}_{h}\hat{\sigma}^{x}_{c}$, which
can be realized by quantum optical methods Kargı  _et al._ (2019). For case of
two-level-oscillator system (TLOS), we replace $\hat{a},\hat{a}^{\dagger}$ by
the Pauli operators $\hat{\sigma}^{\pm}_{h}$, yielding hybrid optomechanical-
like model $g\hat{\sigma}_{h}^{z}(\hat{b}+\hat{b}^{\dagger})$, which is
identical with the single-mode case of electron-phonon Fröhlich (1952);
Holstein (1959) or spin-boson coupling Weiss (2012).

The Hamiltonian of the independent thermal baths is given by

$\hat{H}_{B\alpha}=\sum_{\mu,\alpha}\omega_{\mu}\hat{a}^{\dagger}_{\mu,\alpha}\hat{a}_{\mu,\alpha},$
(2)

where sum is taken over infinite number of bath modes, indexed by $\mu$, and
$\alpha=w,h,c,$ representing the work-like reservoir, hot and cold thermal
baths, respectively. The interaction of system (A+B) with the baths are
defined as

$\hat{H}_{SB}=\sum_{\mu}g_{k,\mu}(\hat{a}+\hat{a}^{\dagger})\otimes(\hat{a}_{k,\mu}+\hat{a}^{\dagger}_{k,\mu})\\\
+\sum_{\mu}g_{c,\mu}(\hat{b}+\hat{b}^{\dagger})\otimes(\hat{a}_{c,\mu}+\hat{a}^{\dagger}_{c,\mu}),$
(3)

where $k=w,h$. In case of TLS the term $(\hat{a}+\hat{a}^{\dagger})$ is
replaced by $\sigma^{x}$.

In order to derive the master equation, first we diagnolize the Hamiltonian
given in Eq. (1) using the transformation

$\hat{U}=e^{-\beta\hat{a}^{\dagger}\hat{a}(\hat{b}^{\dagger}-\hat{b})},$ (4)

where $\beta=g/\omega_{c}$ for both hybrid and pure optomechanical systems
(OMS). It represents $\beta=\theta/2$ for TLS, where the angle is defined by
$\sin\theta:=2g/\tilde{\omega_{c}}$,
$\cos\theta:=\omega_{c}/\tilde{\omega_{c}}$ and
$\tilde{\omega_{c}}=\sqrt{\omega^{2}_{c}+4g^{2}}$. The master equation in the
interaction picture is given by

$\displaystyle\dot{\tilde{\rho}}$
$\displaystyle=\hat{\mathcal{L}}_{w}+\hat{\mathcal{L}}_{h}+\hat{\mathcal{L}}_{c},$
(5)

where $\hat{\mathcal{L}}_{\alpha}$ are Liouville superoperators that describe
the energy exchange of the working medium with the baths, which are of the
form Kargı  _et al._ (2019); Naseem  _et al._ (2018); Gelbwaser-Klimovsky  _et
al._ (2013)

$\displaystyle\hat{\mathcal{L}}_{k}$ $\displaystyle=$ $\displaystyle
G_{k}(\omega_{h})c^{2}\hat{\mathcal{D}}[\tilde{a}]+G_{k}(-\omega_{h})c^{2}\hat{\mathcal{D}}[\tilde{a}^{\dagger}]$
$\displaystyle+$ $\displaystyle
G_{k}(\omega_{w})s^{2}\hat{\mathcal{D}}[\tilde{a}\tilde{b}^{\dagger}]+G_{k}(-\omega_{w})s^{2}\hat{\mathcal{D}}[\tilde{a}^{\dagger}\tilde{b}]$
$\displaystyle+$ $\displaystyle
G_{k}(\omega_{+})s^{2}\hat{\mathcal{D}}[\tilde{a}\tilde{b}]+G_{k}(-\omega_{+})s^{2}\hat{\mathcal{D}}[\tilde{a}^{\dagger}\tilde{b}^{\dagger}],$
$\displaystyle\hat{\mathcal{L}}_{c}$ $\displaystyle=$ $\displaystyle
G_{c}(\tilde{\omega}_{c})c^{2}\hat{\mathcal{D}}[\tilde{b}]+G_{c}(-\tilde{\omega}_{c})c^{2}\hat{\mathcal{D}}[\tilde{b}^{\dagger}],$
(7)

where $\omega_{+}=\omega_{h}+\tilde{\omega}_{c}$, and
$\omega_{w}=\omega_{h}-\tilde{\omega}_{c}$. For TLS, we write
$c^{2}=\cos^{2}\theta$, and $s^{2}=\sin^{2}\theta$; while for both hybrid and
pure optomechanical systems, we like to emphasize, the notations represent
$\tilde{\omega}_{c}=\omega_{c}$, $c^{2}=1$, and
$s^{2}=(g/\tilde{\omega}_{c})^{2}$. We ignore the pure decoherence (dephasing)
dissipators, because they do not change the diagonal elements of the density
matrix in the energy eigenbasis and therefore, do not contribute to steady-
state heat flow. We present the explicit expressions for the transformed
operators $\tilde{a}$ and $\tilde{b}$ with eigenvalues and eigenvectors for
the dressed Hamiltonians in the Appendix A. The bath spectral response
functions are given by

$\displaystyle
G_{\alpha}(\omega)=\begin{cases}\gamma_{\alpha}(\omega)[1+\bar{n}_{\alpha}(\omega)]&\omega>0,\\\
\gamma_{\alpha}(\left|\omega\right|)\bar{n}_{\alpha}(\left|\omega\right|)&\omega<0,\end{cases}$
(8)

and $\bar{n}_{\alpha}(\omega):=1/(\exp{\omega/T_{\alpha}}-1)$ is the average
excitation number of the baths, and the energy damping rate Leggett  _et al._
(1987)

$\displaystyle\gamma_{\alpha}(\omega)$ $\displaystyle=$ $\displaystyle
2\pi\sum_{\mu}g^{2}_{\mu,\alpha}\delta(\omega-\omega_{\mu,\alpha})=2\pi
J_{\alpha}(\omega),and$ $\displaystyle J_{\alpha}(\omega)$ $\displaystyle=$
$\displaystyle\kappa_{\alpha}\frac{\omega^{s}}{\omega_{\text{ct}}^{1-s}}e^{-\omega/\omega_{\text{ct}}},$
(9)

where, $\omega_{\text{ct}}$ is the cutoff frequency. In this work, we consider
the Ohmic ($s=1$) spectral density (OSD) of the baths, unless otherwise
mentioned. The Lindblad dissipators in the Eqs. (II) and (7)
$\hat{\mathcal{D}}[\hat{o}]$ are defined as

$\hat{\mathcal{D}}[\hat{o}]=\frac{1}{2}(2\hat{o}\hat{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}\hat{\rho}+\hat{\rho}\hat{o}^{\dagger}\hat{o}).$
(10)

We assume OSD for the cold bath, and spectrally filtered response functions
for the work-like and hot reservoirs, which is taken to be Kofman  _et al._
(1994); Gelbwaser-Klimovsky and Kurizki (2014)

$G^{k}_{f}=\frac{\kappa_{f}}{\pi}\frac{(\pi
G^{k}(\omega))^{2}}{[\omega-\big{(}\omega^{k}_{f}+\Delta^{k}(\omega)\big{)}^{2}]+(\pi
G^{k}(\omega))^{2}},$ (11)

where $G^{k}(\omega)$ is unfiltered coupling spectrum, and

$\Delta^{k}(\omega)=P\bigg{[}\int^{\infty}_{o}d\omega^{{}^{\prime}}\frac{G^{k}(\omega^{{}^{\prime}})}{\omega-\omega^{{}^{\prime}}}\bigg{]},$
(12)

where $P$ is the principal value. For the OMS, filtering of the bath spectrum
can be achieved using photonic bandgap materials and phononic cavities Naseem
_et al._ (2020). The motivation for using spectrally filtered response
functions of hot and work baths are to avoid the unwanted overlap of these
baths. This can induce heat-leaks, consequently, the efficiency of the QAR
reduces, and the details of which are given in Sec. IV.

##  III Ideal QAR

In this section, we discuss the parameters regime in which our device work as
a fridge and underlying mechanism of cooling.

###  III.1 The cooling window

We are interested in the steady-state analysis for our QAR, and the steady-
state heat currents from the bath $\alpha$ into the system is given by
Nieuwenhuizen and Allahverdyan (2002)

$\mathcal{J}_{\alpha}=\text{Tr}\\{(\mathcal{L}_{\alpha}\tilde{\rho})\tilde{H}_{\text{S}}\\}.$
(13)

The heat flow into the system is taken as positive, and $\tilde{H}_{S}$ is the
diagonlized Hamiltonian of the system in terms of transformed operators. At
steady-state, the first and second law of thermodynamics is respectively
written as

$\displaystyle\sum_{\alpha}\mathcal{J}_{\alpha}$ $\displaystyle=$
$\displaystyle 0,$ (14)
$\displaystyle\sigma=-\sum_{\alpha}\frac{\mathcal{J}_{\alpha}}{T_{\alpha}}$
$\displaystyle\geq$ $\displaystyle 0,$ (15)

where $\sigma$ is the total entropy production of the system and reservoirs at
the steady state.

The figure of merit for the performance of QAR is obtained by comparing the
ratio of heat extracted from the cold bath to heat injected into the system
from the work-like reservoir. Accordingly, the coefficient of performance
(COP) for the absorption refrigerator is defined as
$\epsilon=\mathcal{J}_{c}/\mathcal{J}_{w}$. The COP is colloquially referred
as ‘efficiency’ of the QAR, although, unlike heat engine it can be greater
than one. For refrigeration, we must have $\mathcal{J}_{w}$,
$\mathcal{J}_{c}>0$ and $\mathcal{J}_{h}<0$, then by using the Eq. (14) to
remove the $\mathcal{J}_{h}$ from the Eq. (15), we get

$\epsilon=\frac{\mathcal{J}_{c}}{\mathcal{J}_{w}}\leq\frac{(T_{w}-T_{h})T_{c}}{(T_{h}-T_{c})T_{w}}\equiv\epsilon_{c},$
(16)

where $\epsilon_{c}$ is upper bound on the COP for the heat-driven QAR
operating between three independent thermal baths, which is nothing but the
Carnot refrigerator efficiency.

Figure 2: (Color online) Panel (a) shows the dressed energy levels of the
coupled TLS and the arrows describe the possible transitions induced by the
thermal baths whose spectral densities are presented in (c). The orange
dashed, red dot-dashed and blue solid arrows illustrate the transitions
induced by the work-like reservoir, hot and cold baths, respectively. (b) The
qutrit breakup of our coupled qubits QAR is presented, it shows that our QAR
can be considered as two three-level refrigerators functioning in parallel.
The two qutrits share the work transition $2\leftrightarrow 3$, and note that
the resonance condition $\omega_{h}=\tilde{\omega}_{c}+\omega_{w}$ holds. (c)
Shows the spectrally separated work (orange dashed) and hot (red dot-dashed)
baths spectra given by Eq. (11). Figure 3: (Color online) The possible Raman
cycles for the heat transfer. Panels (a) and (b) show the only possible Raman
cycles (4324) and (3213), respectively, the panels (c) and (d) show the
inverses of these cycles. The orange dashed, red dot-dashed and blue solid
arrows represent the transitions induced by the work-like reservoir, hot and
cold baths, respectively. For $T_{w}>T_{h}>T_{c}$, within the cooling window,
Raman cycles (4324) and (3213) are completed and their inverses are prohibited
by the second law of thermodynamics, and only these two cycles are responsible
for cooling.

For the case of coupled TLS, transitions among the eigenstates of the
Hamiltonian has shown in Fig. 2(a), and details of which have given in
Appendix A. Breakdown of TLS model in two three-level (qutrit) subsystems is
presented in Fig. 2(b). Our model can be considered as two qutrit
refrigerators functioning simultaneously that share a common work-like
transition $|3\rangle\leftrightarrow|2\rangle$. Here, the resonance condition
$\omega_{h}=\tilde{\omega}_{c}+\omega_{w}$ holds, and no transition is induced
by more than one bath. Consequently, we have

$\left|\frac{\mathcal{J}_{\alpha}}{\mathcal{J}_{\beta}}\right|=\frac{\omega_{\alpha}}{\omega_{\beta}}.$
(17)

Similar result has also been shown for the case of ideal single-qutrit
refrigerator in Ref. Geusic  _et al._ (1967), and we show that it also holds
for our coupled qubit design presented in the Fig. 2. In the high temperature
limit $T_{w}\to\infty$, the steady-state heat currents Levy  _et al._ (2012)
is given by

$\mathcal{J}_{\alpha}=\omega_{\alpha}K,$ (18)

(See Appendix B for details.) The Eq. (18) also validates Eq. (17) for the
case of coupled TLS. However, in general, Eq. (17) does not remain valid for
the case of TLOS and OMS, because the resonance condition is not satisfied.
This can be seen from the dressed Hamiltonians of TLOS and OMS, presented in
Appendix. A. The presence of the $g^{2}$ dependent terms in Eqs. (30) and (32)
are responsible to move the system away from resonance condition. However, in
the weak-coupling regime $g\ll\omega_{c}$, we have confirmed numerically, Eq.
(17) is valid for both TLOS and OMS, as expected intuitively. The COP is then
evaluated and by using these equations and found to be

$\epsilon=\frac{\mathcal{J}_{c}}{\mathcal{J}_{w}}=\frac{\tilde{\omega}_{c}}{\omega_{w}}.$
(19)

Using Eqs. (15) and (17) we can write

$\tilde{\omega}_{c}\leq\tilde{\omega}_{c,max}\equiv\frac{(T_{w}-T_{h})T_{c}}{(T_{h}-T_{c})T_{w}}\omega_{w},$
(20)

which gives the “cooling window”. Maximum COP is obtained when
$\tilde{\omega}_{c}\to\tilde{\omega}_{c,max}$, for which the entropy
production $\sigma\to 0$.

###  III.2 Mechanism of cooling

The spectral response functions of the work and hot baths are illustrated in
the Fig. 2(c), and we consider OSD for the cold bath. With the selection of
these spectra, transitions induced by all three baths are not overlapped. This
makes our QAR model ideal so we can highlight the basic mechanism of cooling
in our device. Additionally, for the case of spectrally separated thermal
baths, Eq. (19) holds and at resonance condition we can have efficiency
arbitrarily close to the Carnot bound. Under the selection of different
spectrally filtered bath spectra, we present our discussion in Sec. IV and
Appendix C.

Figure 4: (Color online) (a), (b) The cooling power $\mathcal{J}_{c}$ and (c),
(d) normalized coefficient of performance (COP) $\epsilon/\epsilon_{c}$ as a
function of the subsytem B frequency $\omega_{c}$. In all panels, solid,
dashed, and dash-dotted lines are for TLS, TLOS and OMS, respectively.
Parameters: (a), (c) $g=0.001$, (b), (d) $g=0.005$ and rest of the parameters
are same for all panels. $\omega_{h}=1$,
$\kappa_{w}=\kappa_{h}=\kappa_{c}=0.005$, $T_{w}=0.75$, $T_{h}=0.50$ and
$T_{c}=0.125$. All the parameters are scaled with subsystem A frequency
$\omega_{h}/2\pi=10$ GHz.

For the selection of the bath spectra in Fig. 2(c), only two global Raman
cycles (4324) [here (4324) means the sequence of transitions
$|4\rangle\to|3\rangle\to|2\rangle\to|4\rangle$, etc.] and (3213) and their
inverses are possible as illustrated in Fig. 3. If the cycles in Figs. 3(a)
and 3(b) dominate, we have refrigeration effect, and we refer to these cycles
as “cooling cycles”. However, if the inverse (‘heating’) cycles in Figs. 3(c)
and 3(d) are more favorable, then we get the heating effect. To decide which
cycles are more likely, one needs to calculate the entropy production using
the second law of thermodynamics. For example, using Eq. (15), the entropy
production for the cooling cycle in Fig. 3(a) yields

$\sigma=-\frac{\tilde{\omega}_{c}}{T_{c}}-\frac{\omega_{w}}{T_{w}}+\frac{\omega_{h}}{T_{h}},$
(21)

so that the entropy production of this cycle $(4324)$ becomes $\sigma>0$ for
system parameters within the cooling window presented in Eq. (20). On the
contrary, the entropy production for the heating cycles in Figs. 3(c) and in
3(d) are negative for the same set of parameters. Consequently, cooling cycles
dominate and we get refrigeration effect for the considered system parameters.
Similar cooling and heating cycles can be identified for the TLOS and OMS by
the breakdown of the system into many qutrit subsystems working collectively.
For example, we show two cooling cycles in Fig. 11 in Appendix A. For the
choice of parameters within the cooling window, such cooling cycles dominate
over their inverse heating cycles.

Figure 5: (Color online) (a) The cooling power $\mathcal{J}_{c}$, (b) total
correlations $\mathcal{I}(\rho_{AB})$, (c) quantum correlations
$\mathcal{D}(\rho_{AB})$, and (d) classical correlations
$\mathcal{I}(\sigma_{AB})$ as a function of the $\omega_{c}$ and the coupling
strength $g$, for the coupled TLS. Parameters: $\omega_{h}=1$,
$\kappa_{w}=\kappa_{h}=\kappa_{c}=0.005$, $T_{w}=3$, $T_{h}=2$ and $T_{c}=1$.

In Figs. 4(a) and 4(c), the cooling power and scaled efficiency are plotted
against normalized $\omega_{c}$, respectively, for the TLS, TLOS, and OMS in
the weak coupling regime. Note that the efficiency curves are not far apart,
because in the parameter regime $g\ll\omega_{c}$, for all three cases the
efficiency becomes $\epsilon\approx\omega_{c}/\omega_{w}$. According to Eq.
(19), the COP can be varied arbitrarily close to the Carnot limit via
controlling the frequency of subsystem B within the cooling window. In the
limit $\epsilon\to\epsilon_{c}$, the entropy production $\sigma\to 0$ and the
cooling power of QAR becomes zero, as expected for a reversible the Carnot
refrigerator-type operation. The cooling power and scaled efficiency are
plotted against normalized $\omega_{c}$ for the TLS,TLOS, and OMS in Figs.
4(b) and 4(d) for larger value of coupling strength $g$. The cooling power
$\mathcal{J}_{c}$ vanishes both at $\tilde{\omega}_{c}=0$ and
$\tilde{\omega}_{c}=\tilde{\omega}_{c,max}$. For given system parameters,
there exists a optimum value of $\tilde{\omega}_{c}$ for which the cooling
power is maximum. This can be explained from the energy transitions shown in
Fig. 3. For optimal cooling effect following two conditions must hold, (i) the
work bath must be hot enough to induce $|3\rangle\to|2\rangle$ transition;
$T_{w}\gg\omega_{h}-\tilde{\omega}_{c}$, and (ii) the cold bath must have
$T_{c}>\tilde{\omega}_{c}$. For given system parameters, at small values of
$\tilde{\omega_{c}}$, condition (i) may not be satisfied, and for higher
values of $\tilde{\omega_{c}}$, (ii) may not be true. Hence, the optimal
cooling can be achieved when both of these conditions are satisfied.

###  III.3 Hilbert space size, quantum correlations and cooling power

We compare the cooling power $\mathcal{J}_{c}$ of three considered working
mediums in Figs. 4(a) and 4(b). The cooling power of the working mediums are
found to be in the following order:
$\mathcal{J}^{\text{OMS}}_{c}>\mathcal{J}^{\text{TLOS}}_{c}\gg\mathcal{J}^{\text{TLS}}_{c}$.
This is because the size of Hilbert space for the coupled TLS is smallest
among all three working mediums, and the Hilbert space dimensions should be
considered as a thermodynamic resource Correa (2014); Silva  _et al._ (2016).

Figure 6: (Color online) (a), (b) The cooling power $\mathcal{J}_{c}$ and (c),
(d) mutual information $\mathcal{I}({\rho_{AB}})$ between subsystems A and B
as a function of coupling strength $g$. In all panels, solid, dashed, and
dash-dotted lines are for TLS, TLOS and OMS, respectively. Results in panels
(a) and (c) are for weak coupling regime $\omega_{c}=0.1$, and in panels (b),
(d) we have $\omega_{c}=0.025$ for the TLS, and shaded regions show the weak
coupling regime. Parameters: $\omega_{h}=1$,
$\kappa_{w}=\kappa_{h}=\kappa_{c}=0.005$, $T_{w}=3$, $T_{h}=2$ and $T_{c}=1$.

Now we explore the role of quantum correlations on the performance of our QAR.
We find no entanglement between the subsystems A and B in the considered
parameters regime. To explain this, we express the density matrix of the
coupled TLS in the individual eigenstates of the qubits (Appendix A), given by

$\rho_{AB}=\begin{pmatrix}\rho_{11}&\rho_{12}&0&0\\\
\rho_{12}&\rho_{22}&0&0\\\ 0&0&\rho_{33}&\rho_{34}\\\
0&0&\rho_{34}&\rho_{44}\end{pmatrix}.$ (22)

For the considered parameters, the least eigenvalue of the partial-transpose
of the density matrix $\rho_{AB}$ remains positive;
$\rho_{11}+\rho_{22}-((\rho_{11}-\rho_{22})^{2}+4\rho_{12}^{2})^{1/2}>0$.
Consequently, $\rho_{AB}$ is not entangled following the positivity-of-the-
partial-transpose separability criterion Peres (1996). To quantify the quantum
correlations between the coupled TLS, that are beyond entanglement-
separability paradigm, we consider the quantum discord, which is defined as
Ollivier and Zurek (2001); Henderson and Vedral (2001)

$D(\rho_{AB}):=\mathcal{I}(\rho_{AB})-\mathcal{I}(\sigma_{AB}).$ (23)

Where, $\mathcal{I}(\rho_{AB})$ is the quantum mutual information which
quantifies the total correlations in the system and given by

$\mathcal{I}(\rho_{AB})=S_{\rho_{A}}+S_{\rho_{B}}-S_{\rho_{AB}}.$ (24)

$S_{\rho_{i}}=-\text{Tr}[\rho_{i}\text{log}\rho_{i}]$ is the von Neumann
entropy of the reduced density matrix of the subsystem $i$. In Eq. (23),
$\mathcal{I}(\sigma_{AB})$ quantifies the classical correlations in the
system, where $\sigma_{AB}$ is the resultant state of minimally disturbing
projective measurement on qubit B of $\rho_{AB}$ Ollivier and Zurek (2001);
Henderson and Vedral (2001). To look for the possible role of correlations on
the maximization of the cooling power, we plot the cooling power
$\mathcal{J}_{c}$, quantum correlations $D(\rho_{AB})$, classical correlations
$\mathcal{I}(\sigma_{AB})$, and total correlations $\mathcal{I}(\rho_{AB})$ in
Fig. 5. The quantum correlations appear to be much weaker than the classical
correlations. The region of the maximum cooling power does not coincide with
the quantum or classical correlations. Similar results are obtained for COP at
maximum cooling power $\epsilon_{*}$. We find zero discord, $D(\rho_{BA})=0$,
when minimally disturbing projective measurement is performed on qubit A.

Figure 7: (Color online) Parametric plot of cooling power $\mathcal{J}_{c}$
vs. normalized coefficient of performance (COP) $\epsilon/\epsilon_{c}$ for
different values of the coupling strength $g$. Solid, dashed, horizontal
dashed, dash-dotted and dotted lines are for
$g={0.005,0.01,0.05,0.09,and~{}0.11}$, respectively. The rest of the
parameters are the same as in Fig. 4. The shaded area shows the efficiency at
maximum power $\epsilon_{*}$ bound for one dimensional bosonic reservoirs
given in Eq. (25).

The computation of the quantum discord is numerically intractable, as the
computing time grows exponentially with the size of the Hilbert space Huang
(2014). Except for Gaussian states Giorda and Paris (2010), there is no known
method of quantum discord calculation for continuous variable states. Due to
the nonlinear nature of optomechanical coupling, our OMS may not have a
Gaussian steady-state in general. Accordingly, while we report both the
quantum discord and quantum mutual information for coupled TLS in Fig. 5, we
are limited to present only the quantum mutual information for OMS in Fig. 6.
In Figs. 6(a) and 6(c), we plot the cooling power $\mathcal{J}_{c}$ and
quantum mutual information $\mathcal{I}(\rho_{AB})$ as a function of coupling
strength $g$ in the weak coupling limit. For the coupled TLS, we plot the
cooling power $\mathcal{J}_{c}$ and mutual information
$\mathcal{I}(\rho_{AB})$ in Figs. 6(b) and 6(d), respectively. Here both the
weak and strong coupling regimes are considered. Our numerical observations
based upon the quantum discord for coupled TLS (Fig. 5) and the quantum mutual
information for all three working mediums (Fig. 6), implies that the maximum
cooling power $\mathcal{J}^{*}_{c}$ is not associated with the maximum of
total correlations in the system (c.f. with the Ref. Correa  _et al._ (2013)
which also supports our results).

###  III.4 Efficiency at maximum power

Figure 8: (Color online) Parametric plot of cooling power $\mathcal{J}_{c}$
vs. normalized coefficient of performance (COP) $\epsilon/\epsilon_{c}$ for
different bath spectral densities. Solid, dashed, and dash-dotted lines are
for $s={1,2,3}$, respectively, for details see Eq. (II). The dotted lines show
COP at maximum cooling power. Parameters: $T_{w}=3$, $T_{h}=2$, $T_{c}=1$,
$\omega_{h}=1$, $\kappa_{w}=\kappa_{h}=\kappa_{c}=0.005$, and $g=0.01$.

In addition to the Carnot bound, a more practical bound for quantum thermal
machines is efficiency at maximum power $\epsilon_{*}$. At the limit of the
Carnot bound, the cooling power vanishes due to the reversibility of the ideal
QAR as shown in Figs. 4(c) and 4(d). In Ref. Correa  _et al._ (2014), it has
been argued that the efficiency at maximum cooling power of any ideal QAR
composed of elementary qutrit fridges is tightly upper bounded by

$\epsilon_{*}\leq\frac{d_{c}}{d_{c}+1}\epsilon_{c},$ (25)

where $d_{c}$ is the spatial dimensionality of the cold bath. We show that one
can surpass this bound in a strong coupling regime, contrary to previously
proposed models Linden  _et al._ (2010); Levy  _et al._ (2012); Correa  _et
al._ (2013). However, in the case of TLOS and OMS, Eq. (5) is only valid in
weak coupling regime
$g^{2}\langle\tilde{b}^{\dagger}\tilde{b}\rangle\ll\omega^{2}_{c}$. Therefore,
in the rest of the paper, we only consider coupled TLS in the strong coupling
regime with one dimensional OSD of the baths.

Figure 9: (Color online) Histogram of efficiency at maximum cooling power
$\epsilon_{*}/\epsilon$ for the coupled two-level system. The shaded area
shows the efficiency at maximum power $\epsilon_{*}/\epsilon$ bound for one-
dimensional bosonic reservoirs given in Eq. (25). The solid line on the right
of the plot shows the Carnot bound. We have considered $5\times 10^{6}$
randomly selected system parameters $\\{T_{\alpha},\kappa_{\alpha},g\\}$ sets
within the cooling window. We fix the value of subsystem A frequency;
$\omega_{h}=1$. For every choice of one of these system parameters set, we
vary $\omega_{c}$ to find the optimum value that maximized the cooling power
$\mathcal{J}_{c}$. It can be seen that we can beat the bound of Eq. (25) and
even can reach arbitrarily close to the Carnot limit. Inset shows the values
of coupling strength $g$ and subsystem B frequency $\omega^{*}_{c}$ obtained
at $\epsilon_{*}$. Solid and dashed lines are associated with above and below
the bound given in Eq. (25). It is clear that the efficiency at maximum power
can be reached close to the Carnot limit operating at $g\gg\omega^{*}_{c}$ and
$\omega^{*}_{c}\to 0$, which makes cooling power $\mathcal{J}^{*}_{c}\to 0$.

In Fig. 7, we plot the normalized COP versus cooling power for different
values of the coupling strength $g$. The shaded region shows the efficiency at
maximum power bound $\epsilon_{*}$ presented in Eq. (25). The increment in the
coupling strength $g$ results in the increment of $\epsilon_{*}$, and one can
surpass the bound given in Eq. (25) for higher values of $g$. The efficiency
at maximum cooling power can be made arbitrarily close to the Carnot bound
$\epsilon_{*}\to\epsilon_{c}$, though the power gradually approaches to zero
as the efficiency approaches the Carnot bound. The bound $\epsilon_{*}$
depends on the dimensionality of the cold bath, with the increase in the
dimensions of the cold bath, this bound goes higher. For $d$-dimensional bath
the power spectrum is $G(\tilde{\omega_{c}})\propto\tilde{\omega}^{d}_{c}$,
and in our case, the presence of $c=\omega_{c}/\tilde{\omega}_{c}$ coefficient
in Eq. (5), effectively changes the dimensionality of the cold bath.
Consequently, the increase in $\epsilon_{*}$ as compared to previously
proposed models is due to the non-linear coupling between the subsystem A and
B. In Fig. 8, parametric plot of cooling power $\mathcal{J}_{c}$ as a function
of normalized coefficient of performance $\epsilon/\epsilon_{c}$ is shown for
bath spectrums different than Ohmic ($s\neq 1$), for details see Eq. (II). The
qualitative behavior of cooling power and coefficient of performance remains
the same, however, efficiency at maximum power $\epsilon_{*}$ becomes
quantitatively different.

Here we like to point out that, for $g=N\omega_{c}$, where $N$ is a real
number, the coefficient $c$ (Eq. (5)) no longer depends on $\omega_{c}$. Then
it is not possible to beat the bound in Eq. (25) even for arbitrarily large
coupling strengths $g$, as elaborated in Appendix D. In Fig. 9, we fix the
system parameters such that $T_{w}>\omega_{w}$ and $T_{h}\gg T_{c}$. For every
value of $g$, we find the optimum value of $\omega_{c}$ at which we obtain
$\epsilon_{*}$. The $\omega^{*}_{c}$ scales quadratically with the coupling
strength $g$, and to operate at $\epsilon_{*}\to\epsilon_{c}$ we should have
$g\gg\omega_{c}$ and $\omega_{c}\to 0$. The histogram of scaled efficiency at
maximum cooling power $\epsilon_{*}/\epsilon$ of coupled TLS for randomly
selected system parameters $\\{T_{\alpha},\kappa_{\alpha},g\\}$ has been
presented in Fig. 9. All system parameters are considered within the cooling
window such that the Eq. (5) remains valid. It is clear that $\epsilon_{*}$
can be made arbitrarily close to the Carnot limit. To achieve the efficiency
at maximum power close to the Carnot bound, the sufficient condition on system
parameters is $\omega_{w}\ll T_{w}$, $T_{h}\gg T_{c}$, and $g\gg\omega_{c}$.

##  IV Heat leaks: non-ideal QAR

Figure 10: (Color online) (a), (c) Parametric plot of normalized cooling power
$\mathcal{J}_{c}/\mathcal{J}_{0}$ vs. normalized coefficient of performance
(COP) $\epsilon/\epsilon_{c}$ for different values of the coupling strength
$g$. In panel (b), $\omega_{w}=\omega_{h}-\tilde{\omega_{c}}$ transition is
induced by both work and hot baths. The weight of the arrows shows the
transition rates. In panel (d), $\omega_{h}$ transitions are induced by both
hot and work baths. In (c) and (d), solid, dashed, dash-dotted arrows
represent the transitions induced by cold, work and hot baths, respectively.
In panels (a) and (c), solid, dashed, dash-dotted and dotted lines are for
$g={0.04,0.06,0.08,and~{}0.10}$, respectively. Rest of the parameters are same
as in Fig. 4.

Thermal devices are normally characterized by their efficiency vs. power
curves. The ideal QAR has open characteristic curves as shown in Fig. 7. It
indicates that in the reversible limit, the COP saturates to the Carnot bound
at the cost of the vanishing of cooling power. However, more realistic models
of QAR suffer through heat leaks and internal dissipation that deteriorates
the COP of the fridge Correa  _et al._ (2015). For the case of these non-ideal
QARs, the characteristic curves for power vs. efficiency are closed.

Here we consider a more realistic scenario compared to previous sections. We
model the irreversible process in our device by allowing one or more energy
transitions to be induced by two or more thermal baths. This can happen if the
thermal baths coupled to subsystem A are not spectrally well separated,
instead they overlap on one or more transitions. For example, if the work
transition ($\omega_{w}$) is induced by both work and hot baths, it can be a
source of irreversibility Palao  _et al._ (2001). In our model, this is shown
in Fig. 10(b) and corresponding efficiency vs. power plot has been shown in
Fig. 10(a) for different values of coupling strength $g$. As expected, we have
closed efficiency vs. power curves, and with the increase in the coupling
strength the irreversibility increases. The coupling of a hot bath with work
transition is parasitic, which introduces the heat leaks in the device, hence
the irreversibility. Due to this parasitic coupling, in addition to cycles
shown in Fig. 3, three additional cycles and their inverses are possible;
$4\xrightarrow{T_{c}}3\xrightarrow{T_{h}}2\xrightarrow{T_{h}}4$,
$3\xrightarrow{T_{h}}2\xrightarrow{T_{c}}1\xrightarrow{T_{h}}3$,
$2\xrightarrow{T_{w}}3\xrightarrow{T_{h}}2$, and this leads to thermal short-
circuit. In Fig. 10(d), $\omega_{h}$ transition is coupled with both work and
a hot bath, and corresponding irreversible efficiency vs. cooling power curves
are shown in Fig. 10(c). Qualitatively similar results are obtained for TLOS
and OMS in the weak coupling regime.

##  V Conclusions

We have proposed a model of quantum absorption refrigerator based upon two-
body interaction instead of widely used three-body interaction. The model is
analyzed for a working medium that consists of two coupled qubits or two
harmonic oscillators or one qubit and one harmonic oscillator. A global master
equation is employed to study the dynamics of QAR in both the weak and strong
coupling regime of the working medium. The refrigeration is explained by Raman
transitions between the dressed energy levels of the interacting working
medium. By reservoir engineering and selection of suitable system parameters,
the net effect of these cycles results in the transfer of heat from cold to a
hot bath. For all three working mediums, the coefficient of performance
arbitrarily close to the Carnot bound has been achieved at which the cooling
power vanishes. In the strong coupling regime, the efficiency at maximum power
has saturated the Carnot bound (the power gradually goes to zero at the Carnot
point), which is not possible with previously proposed models of QARs. We
investigated a more realistic model of our QAR that suffers from heat leaks.
We use the experimentally realizable system parameters in our numerical
simulations.

Our proposal of QAR presents a versatile experimental platform for its
realization. For example, it can be implemented in circuit QED architectures;
for coupled qubits Kargı  _et al._ (2019), for atom-cavity model Gelbwaser-
Klimovsky and Kurizki (2014) and for the optomechanical system as working
medium Eichler and Petta (2018). Unlike previously proposed optomechanical QAR
models Mitchison  _et al._ (2016); Mari and Eisert (2012) which require three-
body interaction, our model require only two-body interaction. From the
conceptual point of view, we provide a simple interpretation of the mechanism
of cooling in non-linear optomechanical systems. Compared to the previous two-
body QARs proposals Correa  _et al._ (2014); Silva  _et al._ (2015); Levy  _et
al._ (2012); Mitchison (2019), in our model, using reservoir engineering, both
the efficiency and efficiency at maximum power can be made very close to the
Carnot limit.

Acknowledgements: M.T.N acknowledges O. Pusuluk for the fruitful discussions,
and C. L. Latune for the useful comments on the the manuscript.

##  Appendix A  Eigenoperators and dressed Hamiltonians

Figure 11: (Color online) Examples of possible cooling cycles in two-level-
oscillator and optomechanical systems. $|n_{h},m_{c}\rangle$ are the dressed
states, where $n_{h}=0,1$ represent the ground and excited state of the two-
level system for TLOS, and $n_{h}=0,1,2,...$ give the photon number for OMS.
Furthermore, $m_{c}=0,1,2,...$ for both TLOS and OMS. The orange dashed, red
dot-dashed and blue solid arrows represent the transitions induced by the
work-like reservoir, hot and cold baths, respectively. For
$T_{w}>T_{h}>T_{c}$, within the cooling window, these cycles (cooling) are
completed and their inverses (heating) are prohibited by the second law of
thermodynamics.

Here we present the explicit form of eigenoperators and diagonalized
Hamiltonians for all three cases.

(i) Coupled two two-level system: by applying the unitary transformation given
in Eq. (4) on the Hamiltonian of Eq. (1) we get

$\tilde{H}_{qq}=\frac{\omega_{h}}{2}\tilde{\sigma}_{h}^{z}+\frac{\tilde{\omega_{c}}}{2}\tilde{\sigma}_{c}^{z},$
(26)

where the transformed Pauli operators read

$\displaystyle{\tilde{\sigma}}_{h}^{z}$ $\displaystyle=$ $\displaystyle
U\hat{\sigma}_{h}^{z}U^{\dagger}=\hat{\sigma}_{h}^{z},$ (27)
$\displaystyle{\tilde{\sigma}}_{c}^{z}$ $\displaystyle=$ $\displaystyle
U\hat{\sigma}_{c}^{z}U^{\dagger}=\cos\theta\hat{\sigma}_{c}^{z}-\sin\theta\hat{\sigma}_{h}^{z}\hat{\sigma}_{c}^{x}.$
(28)

The dressed Hamiltonian has eigenstates $|i\rangle$, $i=1,2,3,4$ and described
in terms of individuals eigenstates of the qubits as

$\displaystyle\ket{1}=\cos\frac{\theta}{2}\ket{++}-\sin\frac{\theta}{2}\ket{+-}$
$\displaystyle\ket{2}=\sin\frac{\theta}{2}\ket{++}+\cos\frac{\theta}{2}\ket{+-}$
$\displaystyle\ket{3}=\cos\frac{\theta}{2}\ket{-+}+\sin\frac{\theta}{2}\ket{--}$
$\displaystyle\ket{4}=\cos\frac{\theta}{2}\ket{--}-\sin\frac{\theta}{2}\ket{-+},$
(29)

the corresponding eigenvalues are
$\omega_{1}=(\omega_{h}+\tilde{\omega}_{c})/2$,
$\omega_{2}=(\omega_{h}-\tilde{\omega}_{c})/2$,
$\omega_{3}=(-\omega_{h}+\tilde{\omega}_{c})/2$, and
$\omega_{4}=(-\omega_{h}-\tilde{\omega}_{c})/2$, respectively.

Figure 12: (Color online) (a) Shows the spectrally separated work (orange
dashed) and hot (red dot-dashed) baths spectra given by Eq. (11). Panel (b)
shows the dressed energy levels of the coupled TLS and the double-ended arrows
describe the possible transitions induced by the thermal baths. The orange
dashed, red dot-dashed and blue solid arrows illustrate the transitions
induced by the work-like reservoir, hot and cold baths, respectively. There is
only one possible cooling cycle (43214), that consists of four transitions and
it is shown in (b) by dotted single-ended arrows. (c), (d) cooling power
$\mathcal{J}_{c}$ is plotted as a function of the subsystem B frequency
$\omega_{c}$. In panel (c), solid, dashed and dash-dotted lines are for TLS,
TLOS and OMS, respectively. In (d), solid and dashed lines are for the choice
of spectrally separated work and hot baths spectra given in panel (a) and Fig.
2(c), respectively, for TLS.

(ii) Coupled two-level oscillator system: the diagonalized Hamiltonian for
this system excluding thermal baths is given by

$\tilde{H}_{qo}=\frac{\omega_{h}}{2}\tilde{\sigma^{z}_{h}}+\omega_{c}\tilde{b}^{\dagger}\tilde{b}-\frac{g^{2}}{\omega_{c}},$
(30)

the transformed operators in terms of the bare ones read as

$\displaystyle\tilde{\sigma}^{-}_{h}$ $\displaystyle=$
$\displaystyle\hat{U}\hat{\sigma}_{h}^{-}\hat{U}^{\dagger}=\hat{\sigma}^{-}_{h}e^{-\beta(\hat{b}^{\dagger}-\hat{b})}$
(31) $\displaystyle\tilde{b}$ $\displaystyle=$
$\displaystyle\hat{U}\hat{b}\hat{U}^{\dagger}=\hat{b}-\beta\hat{\sigma}^{+}_{h}\hat{\sigma}^{-}_{h},$

where the unitary transformation is described in Eq. (4) with
$\beta=g/\omega_{c}$. The eigenenergies of this system are
$E_{n_{h},m_{c}}=n_{h}\omega_{h}/2+m_{c}\omega_{{c}}-g^{2}/\omega_{c}$, where
$n_{a}=0,1$ corresponds to ground and excited state of the two level system
(TLS), respectively. Furthermore, $m_{c}$ is the cavity photon number and the
cavity Fock state is displaced by a factor of $\beta$.

(iii) optomechanical system: The diagonalized Hamiltonian of an isolated
optomechanical system consisting of a cavity and a mechanical mode is given by

$\tilde{H}_{om}=\omega_{h}\tilde{a}^{\dagger}\tilde{a}+\omega_{c}\tilde{b}^{\dagger}\tilde{b}-\frac{g^{2}}{\omega_{c}}(\tilde{a}^{\dagger}\tilde{a})^{2},$
(32)

where the dressed operators in terms of bare operators given by

$\displaystyle\tilde{a}$ $\displaystyle=$
$\displaystyle\hat{U}\hat{a}\hat{U}^{\dagger}=\hat{a}e^{-\beta(\hat{b}^{\dagger}-\hat{b})},$
(33) $\displaystyle\tilde{b}$ $\displaystyle=$
$\displaystyle\hat{U}\hat{b}\hat{U}^{\dagger}=\hat{b}-\beta\hat{a}^{\dagger}\hat{a}.$

The eigenenergies for the dressed optomechanial Hamiltonian are
$E_{n_{h},m_{c}}=n_{h}\omega_{h}+m_{c}\omega_{{c}}-ng^{2}/\omega_{c}$, where
$n_{h}$ is the number of photons in the cavity and $m_{c}$ is phonon number.

##  Appendix B  Heat Currents for coupled qubits

The equations of motions to calculate the steady-state heat currents in case
of coupled TLS for the choice of bath spectra shown in Fig. 2(c) are
determined from , Eq. (5), and given by

$\displaystyle\frac{d}{dt}\langle\tilde{\sigma}^{+}_{h}\tilde{\sigma}^{-}_{h}\rangle$
$\displaystyle=$
$\displaystyle-c^{2}\Gamma_{h}(1+e^{-\frac{\omega_{h}}{T_{h}}})\langle\tilde{\sigma}^{+}_{h}\tilde{\sigma}^{-}_{h}\rangle+c^{2}\Gamma_{h}e^{-\frac{\omega_{h}}{T_{h}}}$
(34) $\displaystyle+$ $\displaystyle
s^{2}\Gamma_{w}(\langle\tilde{\sigma}^{+}_{c}\tilde{\sigma}^{-}_{c}\rangle-\langle\tilde{\sigma}^{+}_{h}\tilde{\sigma}^{-}_{h}\rangle),$
$\displaystyle\frac{d}{dt}\langle\tilde{\sigma}^{+}_{c}\tilde{\sigma}^{-}_{c}\rangle$
$\displaystyle=$
$\displaystyle-c^{2}\Gamma_{c}(1+e^{-\frac{\Omega}{T_{c}}})\langle\tilde{\sigma}^{+}_{c}\tilde{\sigma}^{-}_{c}\rangle+c^{2}\Gamma_{c}e^{-\frac{\Omega}{T_{c}}}$
(35) $\displaystyle+$ $\displaystyle
s^{2}\Gamma_{w}(\langle\tilde{\sigma}^{+}_{h}\tilde{\sigma}^{-}_{h}\rangle-\langle\tilde{\sigma}^{+}_{c}\tilde{\sigma}^{-}_{c}\rangle),$

where we have considered $T_{w}\to\infty$, consequently, higher order
correlations such as
$\langle\tilde{\sigma}^{+}_{h}\tilde{\sigma}^{-}_{h}\tilde{\sigma}^{+}_{c}\tilde{\sigma}^{-}_{c}\rangle$
vanish. In addition, $c=$ cos$\theta$, $s=$ sin$\theta$, and

$\displaystyle\Gamma_{\alpha}=\omega_{\alpha}\kappa_{\alpha}\bar{n}(\omega_{\alpha}),$
(36)

The steady-state heat currents in the limint $T_{w}\to\infty$ are described as

$\mathcal{J}_{\alpha}=\omega_{\alpha}K,$ (37)

and

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{c^{2}s^{2}\Gamma_{w}\Gamma_{h}\Gamma_{c}(e^{\frac{\tilde{\omega}_{c}}{T_{c}}}-e^{\frac{\tilde{\omega}_{c}}{T_{h}}})}{c^{2}\Gamma_{w}\Gamma_{c}(1+e^{\frac{\tilde{\omega}_{c}}{T_{c}}})(1+e^{\frac{\omega_{h}}{T_{h}}})+s^{2}\Gamma_{w}[e^{\frac{\omega_{h}}{T_{h}}}\Gamma_{c}+e^{\frac{\tilde{\omega}_{c}}{T_{c}}}(\Gamma_{w}+e^{\frac{\omega_{h}}{T_{h}}}(\Gamma_{w}+\Gamma_{c}))]}.$
(38)

##  Appendix C  Single cooling cycle QAR

Here we investigate the working of our QAR for the selection of different
spectra of hot and work reservoirs than presented in the main text. We
consider spectrally filtered spectra shown in Fig. 12(a), and the cold bath
has OSD. For the case of TLS, under the selection of these bath spectra, only
one energy cycle (43214) and its inverse are possible. The (43214) cycle
consists of four transitions and if parameters are selected within the cooling
window, then it is responsible for cooling.

Figure 13: (Color online) Schematic illustration of the QAR model, here work
and cold baths are swapped compared to the model presented in Fig. 1, and
subsystem A (B) has a frequency $\omega_{c}$ ($\omega_{w}$). Panel (b) shows
the dressed energy levels of the coupled TLS and the double-ended arrows
describe the possible transitions induced by the thermal baths. The solid,
dashed and dot-dashed arrows illustrate the transitions induced by the work-
like reservoir, cold and hot baths, respectively. (c) shows the spectrally
separated cold (dashed) and hot (dot-dashed) baths spectra given by Eq. (11)

For the same baths spectra, similar four transitions energy cycles can also be
identified for TLOS and OMS which lead to cooling. To show it explicitly, we
plot the cooling power $\mathcal{J}_{c}$ as a function of the subsystem B
frequency $\omega_{c}$ for TLS, TLOS, and OMS in Fig. 12(c). The higher
dimensional OMS outperforms the lower dimensional TLOS and TLS models. For
comparison between the performance of three (Fig. 2(a)) and four transitions
(Fig. 12(b)) energy cycles models of TLS, we plot the cooling power
$\mathcal{J}_{c}$ as a function of the subsystem B frequency $\omega_{c}$,
shown in Fig. 12(d). The model based on three transitions energy cycles
presented in the main text outperforms the four transitions energy cycles
model. Because the model presented Fig. 2(a) can be considered as two three-
level QARs working together, which can outperform a single four-level QAR. A
four-level QAR, similar to the system presented in Fig. 12(b) has been
proposed in the Ref. Silva  _et al._ (2015), without giving a specific
physical implementation. Our study shows that this can be implemented via
optomechanical-like coupling between the coupled two two-level systems or two
harmonic oscillators or a two-level system and a harmonic oscillator.

##  Appendix D  Efficiency at maximum power and cold bath spectrum

Figure 14: (Color online) Parametric plot of cooling power $\mathcal{J}_{c}$
vs. normalized coefficient of performance (COP) $\epsilon/\epsilon_{c}$ for
different values of the coupling strength $g$. Solid, dashed, horizontal
dashed, dash-dotted and dotted lines are for
$g={0.025,0.05,0.075,0.1,and~{}0.125}$, respectively. Rest of the parameters
are $T_{w}=10$, $T_{h}=6$, $T_{c}=5$, $\omega_{w}=0.1$. The shaded area shows
the efficiency at maximum power $\epsilon_{*}$ bound for one-dimensional
bosonic reservoirs given in Eq. (25).

In the main text, we argue that the increase in the efficiency at maximum
power $\epsilon_{*}$ in our model is because the dimensionality of the cold
bath spectrum effectively changes, which bounds the $\epsilon_{*}$, as given
in Eq. (25). To emphasize this, we consider a case shown in Fig. 13, in which
the cold bath dimensionality does not change effectively. Note that, we have
swapped the cold and work reservoir compared to the model in Fig. 1, in
addition, now subsystem A (B) frequency is $\omega_{c}$ ($\omega_{w}$). To
avoid confusion, we re-write the master equation given in Eq. (5) for this
case,

$\displaystyle\hat{\mathcal{L}}_{k}$ $\displaystyle=$ $\displaystyle
G_{k}(\omega_{c})c^{2}\hat{\mathcal{D}}[\tilde{a}]+G_{k}(-\omega_{c})c^{2}\hat{\mathcal{D}}[\tilde{a}^{\dagger}]$
$\displaystyle+$ $\displaystyle
G_{k}(\omega_{-})s^{2}\hat{\mathcal{D}}[\tilde{a}\tilde{b}^{\dagger}]+G_{k}(-\omega_{-})s^{2}\hat{\mathcal{D}}[\tilde{a}^{\dagger}\tilde{b}]$
$\displaystyle+$ $\displaystyle
G_{k}(\omega_{h})s^{2}\hat{\mathcal{D}}[\tilde{a}\tilde{b}]+G_{k}(-\omega_{h})s^{2}\hat{\mathcal{D}}[\tilde{a}^{\dagger}\tilde{b}^{\dagger}],$
$\displaystyle\hat{\mathcal{L}}_{w}$ $\displaystyle=$ $\displaystyle
G_{w}(\tilde{\omega}_{w})c^{2}\hat{\mathcal{D}}[\tilde{b}]+G_{w}(-\tilde{\omega}_{w})c^{2}\hat{\mathcal{D}}[\tilde{b}^{\dagger}],$
(40)

where, $s=2g/\omega_{w}$, $c=\omega_{w}/\tilde{\omega}_{w}$,
$\omega_{-}=\omega_{c}-\omega_{h}$,
$\tilde{\omega}_{w}=\sqrt{\omega_{w}+4g^{2}}$, and $k=h,c$ for hot and cold
baths, respectively. The possible transitions induced by the three baths are
shown in Fig. 13(b), under the selection of the hot and cold baths spectrums
presented in Fig. 13(c). Note that the dimensonality of cold bath spectrum
$G_{c}(\omega_{c})$ does not change by the presence of $s$ and $c$ terms in
the Eq. (D). In this case, the bound given in Eq. (25) should hold, and this
is verified in Fig. 14, where we plot cooling power $\mathcal{J}_{c}$ as a
function of the normalized coefficient of performance $\epsilon/\epsilon_{c}$
for different values of coupling strengths. Unlike the case presented in Fig.
7, here the coupling strength $g$ does not change $\omega_{c}$, consequently
does not affect the $\epsilon_{*}$.

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