# Optomechanical cooling with coherent and squeezed light: the thermodynamic
cost of opening the heat valve

Juliette Monsel  Department of Microtechnology and Nanoscience (MC2), Chalmers
University of Technology, S-412 96 Göteborg, Sweden  Nastaran Dashti
Department of Microtechnology and Nanoscience (MC2), Chalmers University of
Technology, S-412 96 Göteborg, Sweden  Sushanth Kini Manjeshwar  Department of
Microtechnology and Nanoscience (MC2), Chalmers University of Technology,
S-412 96 Göteborg, Sweden  Jakob Eriksson  Department of Microtechnology and
Nanoscience (MC2), Chalmers University of Technology, S-412 96 Göteborg,
Sweden  University of Gothenburg, S-412 96 Göteborg, Sweden  Henric Ernbrink
Department of Microtechnology and Nanoscience (MC2), Chalmers University of
Technology, S-412 96 Göteborg, Sweden  Ebba Olsson  Department of
Microtechnology and Nanoscience (MC2), Chalmers University of Technology,
S-412 96 Göteborg, Sweden  Emelie Torneus  Department of Microtechnology and
Nanoscience (MC2), Chalmers University of Technology, S-412 96 Göteborg,
Sweden  Witlef Wieczorek  Department of Microtechnology and Nanoscience (MC2),
Chalmers University of Technology, S-412 96 Göteborg, Sweden  Janine
Splettstoesser  Department of Microtechnology and Nanoscience (MC2), Chalmers
University of Technology, S-412 96 Göteborg, Sweden

(August 27, 2024)

###### Abstract

Ground-state cooling of mechanical motion by coupling to a driven optical
cavity has been demonstrated in various optomechanical systems. In our work,
we provide a so far missing thermodynamic performance analysis of
optomechanical sideband cooling in terms of a heat valve. As performance
quantifiers we examine not only the lowest reachable effective temperature
(phonon number), but also the evacuated-heat flow as an equivalent to the
cooling power of a standard refrigerator, as well as appropriate thermodynamic
efficiencies, which all can be experimentally inferred from measurements of
the cavity output light field. Importantly, in addition to the standard
optomechanical setup fed by coherent light, we investigate two recent
alternative setups for achieving ground-state cooling: replacing the coherent
laser drive by squeezed light or using a cavity with a frequency-dependent
(Fano) mirror. We study the dynamics of these setups within and beyond the
weak-coupling limit and give concrete examples based on parameters of existing
experimental systems. By applying our thermodynamic framework, we gain
detailed insights into these three different optomechanical cooling setups,
allowing a comprehensive understanding of the thermodynamic mechanisms at
play.

##  I Introduction

Nano- and micromechanical resonators constitute an excellent platform for
exploring thermodynamics on the nanoscale [1, 2, 3]. Importantly, stochastic
and quantum fluctuations can play a major role determining nanomechanical
motion [4, 5], enabling tests of stochastic [1] and quantum thermodynamics
concepts [6] in experiments. In this respect, optomechanical systems [7] are a
pertinent platform as they allow for precise control over the classical and
quantum dynamics of nanomechanical motion by using electromagnetic fields.
This capability motivated proposals of optomechanics-based work measurement
schemes [8, 9, 10] or heat engines [11, 12, 13, 14, 15, 16, 17, 18] and
already lead to the experimental quantification of non-equilibrium
thermodynamic processes such as irreversible entropy production [19, 20].

An ubiquitous thermodynamic process is cooling. In the context of
optomechanics, cooling is employed to reduce the entropy of eigenmodes of the
mechanical resonator, which is a necessary step for exerting quantum control
over mechanical motion and, thus, a crucial requirement for the implementation
of any optomechanics-based quantum technology. Furthermore, any optomechanics-
based heat engine exploits cooling in its operation cycle. Cooling to the
ground state of mechanical motion has been theoretically analyzed (see, e.g.,
Refs. [21, 22, 23, 24]) and experimentally realized by direct thermal cooling
[5], optomechanical sideband cooling [25, 26, 27] or feedback cooling [28,
29]. In the context of thermodynamics, optomechanical cooling has been
examined in the light of entropy production [19, 20] or as a quantum
absorption refrigerator [30, 31, 32] fed by thermal light. However, a detailed
analysis of the desired output, such as the reduction in phonon number,
related to the thermodynamic cost of optomechanical cooling with non-thermal
light sources in the steady-state has so far been missing.

Figure 1:  (a) Thermodynamic framework: a mechanical resonator (in black) is
in contact with a hot phonon bath (in red). To cool it down, it is coupled to
a photon bath (in blue) whose temperature is much smaller than the relevant
photon energies. The coupling can be regarded as opening a valve (in green),
allowing heat to flow to the cold bath. The arrows indicate the chosen
direction for positive heat flows, but they do not reflect the actual
direction of the heat flows in the setup. (b) Standard setup for
optomechanical cooling: a cavity with a moving-end mirror is driven by a
laser. The mechanical resonator is in contact with a thermal phonon reservoir
at temperature $T_{\text{mec}}$ and the cavity with a photon reservoir at
temperature $T_{\text{cav}}$. The light blue dashed box represents the
effective phonon bath that can be identified in the weak-coupling regime. See
Sec. II.1.1 for definitions of the notations.

In our work, we present this missing analysis of the thermodynamic performance
of different optomechanical sideband cooling schemes, which is of crucial
relevance to optimize the performance of future devices. We therefore provide
a comprehensive theoretical framework for sideband cooling beyond previously
performed approximations. We find that this analysis not only yields
information about actual performance benchmarks, but also provides insights
into the thermodynamic mechanisms at play and the required resources for
realizing optomechanical cooling in different experimental setups. First, we
notice that optomechanical cooling does not correspond to the commonly
employed refrigerator framework, in which the work provided to a system makes
heat flow out of a cold bath111See also Appendix E for an overview of the
differences between our framework and absorption refrigerators.. The
thermodynamic system to be cooled consists of a mechanical resonator (from
which heat should be taken away) that is in contact with a hot phonon bath,
which leads to Brownian motion of the resonator. On the other hand, there is a
photon bath, which is determined by the noise properties of the light field
and which appears as cold, since its temperature is much smaller than the
energy scale of the photons. This photon bath couples to the mechanical
degrees of freedom via a driven optomechanical cavity [see Fig. 1]. We
identify that the light field acts as a tunable valve that controls the flow
of heat from the mechanical resonator to the cold bath, i.e., the noise of the
light field. Therefore, unlike as in usual refrigerators, heat flows from hot
to cold, but, importantly, there is a cost associated with maintaining the
valve open (via the driving of the cavity). The heat-valve scenario that we
suggest here thereby gives a very clear picture of this setting with two heat
baths and an additional work input. It is particularly helpful for the
understanding of the mechanisms at play when comparing sideband cooling in
different optomechanical setups.

We furthermore find it insightful to look at optomechanical cooling as a
special example of reservoir engineering [33]: a low temperature photon bath
is prepared with the help of a laser-irradiated cavity, such that it is
transformed into an effective cold phonon bath [21]. Engineered reservoirs
have shown to be especially appealing to thermodynamics as they allow to
explore non-thermal resources [34, 35, 36, 37, 38] and study the impact of
genuine quantum features, such as squeezing [39, 40, 41, 42], entanglement
[43] or quantum coherence [44, 45, 46] on energy and entropy exchanges.
Recently, it was demonstrated that a classically engineered reservoir can be
used to beat the classical Carnot bound of heat machines [4]. Here, the
situation is different: we start from a photon bath, in which thermal
fluctuations are negligible at optical frequencies, in order to engineer a
thermal bath of a completely different type, namely an effective phonon bath.
In this sense the valve that is established is an important ingredient of the
engineered bath. In most cases, the cost to create the engineered bath is not
taken into account in the efficiency.

In our work, we provide and calculate appropriate cooling efficiencies of
optomechanical sideband cooling, which account for the cost of the heat valve
(i.e. the bath engineering) and which yield an additional benchmark in
optomechanics, besides the resonator’s phonon occupation. We then apply our
comprehensive thermodynamic analysis of optomechanical cooling to three
different setups, which can be used to cool mechanical motion to the ground
state. These are: (i) a standard optomechanical setup driven by a coherent
laser input [Fig. 1], (ii) a standard optomechanical setup driven by squeezed-
light [Fig. 3(a)], and (iii) an optomechanical cavity that employs a
frequency-dependent (Fano) mirror as one of the cavity boundaries and that is
driven by a coherent laser [Fig. 3(b)]. Importantly, we illustrate our
analysis with concrete experimental parameters, which allow us to benchmark
different optomechanical platforms against each other. Setup (i) [21, 22, 23,
24] is widely employed for ground-state cooling provided the optomechanical
system is in the resolved-sideband regime [25, 26]. Setup (ii) [47, 48, 49] is
motivated by shaping the quadrature noise of the cold photon bath. This setup
can lead to a reduced backaction of the light field onto the mechanical
resonator and, thus, to ground-state cooling in a regime, where a coherent
drive fails, as we here illustrate. Finally, setup (iii) [50] results in a
Fano-like cavity line shape. This setup aims at suppressing the Stokes process
and, thus, the light-induced heating of the mechanical motion originating from
the optomechanical interaction itself. Importantly, our theoretical analysis,
which employs a Langevin as well as a master equation approach, is valid both
for weak and strong optomechanical coupling222We are referring here to the
effective optomechanical coupling $g=g_{0}\absolutevalue{\alpha}$, enhanced by
the light field in the cavity, of amplitude $\absolutevalue{\alpha}$, see Eq.
(4). On the contrary, the single-photon coupling $g_{0}$ is assumed to be very
small compared to the other frequencies involved. It thereby extends the
validity of previous work [47].

This paper is structured as follows. In Sec. II we describe the theoretical
models we use for each setup, outline the thermodynamic picture that we
develop here and present the specific experimental platforms we use as
examples. In Sec. III we derive the dynamics of the three setups and apply
these results to a thermodynamic analysis of optomechanical cooling
illustrated with realistic optomechanics experiments. We conclude in Sec. IV.
The appendix contains detailed information on the employed theoretical methods
and approximations and thereby allows to straightforwardly follow all
calculations performed in this work.

##  II Setups and models

###  II.1 Sideband cooling in cavity optomechanical setups

In the following, we introduce the models for the three different setups that
we consider for sideband cooling and start with the standard setup of driving
an optomechanical cavity with coherent light.

####  II.1.1 Standard setup

A typical optomechanical setup consists of a Fabry-Pérot cavity of frequency
$\omega_{\text{cav}}$ with a moving-end mirror of mechanical resonance
frequency $\Omega_{\text{mec}}$ [see Fig. 1(b)]. The cavity is driven by a
laser of frequency $\omega_{\text{las}}$. It is described by the Hamiltonian
[7], in the frame rotating at $\omega_{\text{las}}$,

$\displaystyle\hat{H}=\,$
$\displaystyle\hbar\Omega_{\text{mec}}\hat{b}^{\dagger}\hat{b}+\hbar\Delta_{0}\hat{a}^{\dagger}\hat{a}+\hbar(\epsilon\hat{a}^{\dagger}+\epsilon^{*}\hat{a})$
$\displaystyle-\hbar
g_{0}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger}),$ (1)

where $\Delta_{0}=\omega_{\text{cav}}-\omega_{\text{las}}$ is the bare
detuning between the cavity and the laser. We have denoted $\hat{a}$ the
photon-annihilation operator of the cavity and $\hat{b}$ the phonon-
annihilation operator. The single-photon optomechanical coupling strength is
given by $g_{0}$ and $\epsilon$ is the drive amplitude.

The cavity is coupled to a photon and a phonon bath, where $\kappa$ denotes
the optical loss rate and $\gamma$ the mechanical damping rate. The evolution
of this system can be described with a set of Langevin equations [23]

$\displaystyle\dot{\hat{q}}$ $\displaystyle=\Omega_{\text{mec}}\hat{p},$ (2a)
$\displaystyle\dot{\hat{p}}$
$\displaystyle=-\Omega_{\text{mec}}\hat{q}-\gamma\hat{p}+g_{0}\sqrt{2}\hat{a}^{\dagger}\hat{a}+\sqrt{\gamma}\hat{\xi},$
(2b) $\displaystyle\dot{\hat{a}}$
$\displaystyle=-(\kappa+i\Delta_{0})\hat{a}+ig_{0}\sqrt{2}\hat{a}\hat{q}-i\epsilon+\sqrt{2\kappa}\hat{a}_{\text{in}},$
(2c)

where we have defined the mechanical quadratures
$\hat{q}=\left(\hat{b}+\hat{b}^{\dagger}\right)/\sqrt{2}$ and
$\hat{p}=\left(\hat{b}-\hat{b}^{\dagger}\right)/i\sqrt{2}$. The thermal noise
of the mechanics is given by the operator $\hat{\xi}$, while vacuum noise of
the light field is associated with the operator $\hat{a}_{\text{in}}$.

We assume in the following that the mechanical quality factor
$Q_{\text{mec}}=\Omega_{\text{mec}}/\gamma$ is large and that the temperature
$T_{\text{mec}}$ of the mechanical environment is high, that is
$\hbar\Omega_{\text{mec}}\ll k_{\text{B}}T_{\text{mec}}$, see Sec. II.3 for
concrete implementations in this regime. We can then make the white-noise
approximation for the correlation function for the thermal noise of the
mechanics

$\langle\hat{\xi}(t)\hat{\xi}(t^{\prime})\rangle=(2\bar{n}_{\text{mec}}+1)\delta(t-t^{\prime}),$
(3)

where
$\bar{n}_{\text{mec}}=(\text{e}^{\hbar\Omega_{\text{mec}}/k_{\text{B}}T_{\text{mec}}}-1)^{-1}$
is the average number of phonons in the mechanical environment. This
approximation is equivalent to the Born-Markov approximation made in the
derivation of the master equation [24] (see Appendix B.5 for a comparison of
the different approaches and level of approximations frequently used in
optomechanics). In the examples studied in this article (presented in Sec.
II.3), this approximation yields satisfactory results, but we explain how to
relax it in Appendix B.2.3.

In order to quantify the ratio between the optomechanical coupling and the
coupling to the optical and mechanical baths, we use the quantum
cooperativity,

$C=\frac{2g^{2}}{\kappa\gamma\bar{n}_{\text{mec}}}.$ (4)

Here, $g=g_{0}|\alpha|$ is the effective coupling with $|\alpha|$ being the
amplitude of the coherent field in the cavity, which is proportional to the
driving amplitude of the laser $\epsilon$, see Eq. (14a).

Figure 2:  Principle of resolved-sideband cooling in the weak-coupling limit
in a Raman-scattering picture. The Stokes and anti-Stokes processes are
represented by the magenta and cyan arrows, respectively. (a) Cavity density
of states. $\Delta$ is the effective detuning (defined in Sec. II.2) between
the laser frequency (orange line) and the cavity resonance frequency (dashed
black line), which is shifted with respect to $\Delta_{0}$ by the
optomechanical interaction. (b) Partial energy diagram of the optomechanical
system around the state $\ket{p,n}$ with $p$ photons and $n$ phonons.

The laser driving the cavity is modeled as a classical drive, included in the
Hamiltonian (II.1.1). Its drive amplitude $\epsilon$ is related to the input
laser power ${P}_{\text{las}}$ by

${P}_{\text{las}}=\hbar\omega_{\text{las}}\frac{\absolutevalue{\epsilon}^{2}}{2\kappa}.$
(5)

The cavity and the resonator are typically at the same temperature, i.e.
$T_{\text{mec}}=T_{\text{cav}}$, however the cavity frequency is typically
orders of magnitude larger than the mechanical frequency and
$\hbar\omega_{\text{cav}}\gg k_{\text{B}}T_{\text{cav}}$ (see Sec. II.3).
Therefore the average number of photons in the optical environment is
negligible, $\bar{n}_{\text{cav}}\simeq 0$, and when driving with coherent
light, there is only one non-zero correlation function of the cavity input
noise, in contrast to the squeezed case, see Eqs. (8). This correlation
function is

$\langle\hat{a}_{\text{in}}(t)\hat{a}_{\text{in}}^{\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime}).$
(6)

To sketch the principle of sideband cooling, we consider temporarily the
resolved-sideband regime, that is $\kappa\ll\Omega_{\text{mec}}$, and a weak
optomechanical coupling: $g\ll\kappa,\Omega_{\text{mec}}$. In this regime, as
illustrated by Fig. 2, three processes can occur: A photon from the laser can
(1) be absorbed by the cavity without changing the state of the resonator (in
orange), (2) create a photon in the cavity at lower frequency
$\omega_{\text{las}}-\Omega_{\text{mec}}$ and a phonon (Stokes process, in
magenta), or (3) in combination with a phonon, create a photon at higher
frequency $\omega_{\text{las}}+\Omega_{\text{mec}}$ (anti-Stokes process, in
cyan). The Stokes process heats up the resonator while the anti-Stokes process
cools it down. By choosing a cavity-laser detuning close to the mechanical
frequency, the rates of the first two processes can be reduced, allowing
effective cooling [51]. Note that the results presented in this article focus
on, but are not restricted to, the resolved-sideband regime and they are valid
beyond the weak-coupling limit.

Figure 3:  (a) Squeezed-light setup: We add to the laser driving the cavity a
squeezed vacuum field. Since the control of the environment of the cavity is
not perfect, there is some residual vacuum noise entering the cavity. The
squeezed noise is generated with a degenerate parametric amplifier (DPA)
driven at $2\omega_{\text{las}}$ thanks to a second harmonic generator (SHG).
See Sec. II.1.2 for the definitions of the notations. (b) Fano-mirror setup:
The left-hand side mirror of the cavity is replaced by a Fano mirror. Unlike
in the canonical setup [Fig. 1(b)], we differentiate the noise coming through
the left and right mirrors. See Sec. II.1.3 for the definitions of the
notations. (c) Cavity density of states (solid gray line) which is
asymmetrical in the Fano-mirror setup. The dashed gray line represents the
density of states of a standard cavity of equivalent linewidth.

####  II.1.2 Squeezed-light setup

To improve the cooling scheme presented above, one can use squeezed light
instead of coherent light to go beyond the quantum backaction limit [47, 48].
In the model considered here, as depicted in Fig. 3(a), this means replacing
the vacuum noise input by squeezed noise, generated with a degenerate
parametric amplifier (DPA) (see Ref. [52], chapter 10). This is an example of
a cascaded open quantum system [53], where the output $\hat{c}_{\text{out,s}}$
of the DPA is used as an input for the cavity, namely
$\hat{c}_{\text{out,s}}(t)=\hat{a}_{\text{in,s}}(t)$ (we neglect the delay due
to propagation, which is irrelevant for our purposes). We consider that the
DPA mode, with annihilation operator $\hat{c}$, is resonant with the cavity
pump, i.e., at frequency $\omega_{\text{las}}$. The DPA pump is tuned with the
parametric oscillator, i.e., at frequency $2\omega_{\text{las}}$.

The Langevin equation for the DPA reads [47]

$\dot{\hat{c}}=-\varkappa\hat{c}+\chi\hat{c}^{\dagger}+\sqrt{2\varkappa}\hat{c}_{\text{in}},$
(7)

where
$\hat{c}_{\text{in}}=\frac{1}{\sqrt{\varkappa}}(\sqrt{\varkappa_{\text{s}}}\hat{c}_{\text{in,s}}+\sqrt{\varkappa_{\text{v}}}\hat{c}_{\text{in,v}})$
is the total input field,
$\varkappa=\varkappa_{\text{s}}+\varkappa_{\text{v}}$ the corresponding total
loss rate and $\chi=\absolutevalue{\chi}\text{e}^{-i2\theta}$ is the squeezing
parameter with $\theta$ the squeezing angle [see Fig. 3(a)]. The input vacuum
noise $\hat{c}_{\text{in,v}}$ with associated coupling $\varkappa_{\text{v}}$
corresponds to the uncontrolled losses of the DPA. The desired output from the
DPA is then finally obtained from
$\hat{c}_{\text{out,s}}=\sqrt{2\varkappa_{\text{s}}}\hat{c}-\hat{c}_{\text{in,s}}$.
The Langevin equations for the optomechanical system are still Eqs. (2), but
now the total optical input noise reads
$\hat{a}_{\text{in}}=\frac{1}{\sqrt{\kappa}}(\sqrt{\kappa_{\text{s}}}\hat{c}_{\text{out,s}}+\sqrt{\kappa_{\text{v}}}\hat{a}_{\text{in,v}})$,
where $\kappa_{\text{s}}$ is the coupling to squeezed vacuum noise and
$\hat{a}_{\text{in,v}}$ corresponds to the uncontrolled losses of the cavity
with associated coupling $\kappa_{\text{v}}$ [47]. The total loss rate of the
cavity is therefore $\kappa=\kappa_{\text{s}}+\kappa_{\text{v}}$. In the
white-noise approximation, the correlation functions of the input noise read
[52] (see Appendix C.1)

$\displaystyle\langle\hat{a}^{\dagger}_{\text{in}}(t)\hat{a}_{\text{in}}(t^{\prime})\rangle$
$\displaystyle=\pi_{\text{s}}N_{\text{s}}\delta(t-t^{\prime}),$ (8a)
$\displaystyle\langle\hat{a}_{\text{in}}(t)\hat{a}_{\text{in}}(t^{\prime})\rangle$
$\displaystyle=\pi_{\text{s}}M_{\text{s}}\delta(t-t^{\prime}).$ (8b)

The purity of the squeezing is
$\pi_{\text{s}}=\varkappa_{\text{s}}\kappa_{\text{s}}/\varkappa\kappa$ with
$\pi_{\text{s}}=1$ for pure squeezing and $\pi_{\text{s}}=0$ for vacuum noise
only. Furthermore, we have

$\displaystyle N_{\text{s}}$
$\displaystyle=\frac{4r_{\text{s}}^{2}}{(1-r_{\text{s}}^{2})^{2}},$ (9a)
$\displaystyle M_{\text{s}}$
$\displaystyle=\frac{2r_{\text{s}}(1+r_{\text{s}}^{2})}{(1-r_{\text{s}}^{2})^{2}}\text{e}^{-i2\theta}.$
(9b)

with the squeezing ratio $r_{\text{s}}=\absolutevalue{\chi}/\varkappa$ and
this ratio is such that $0<r_{\text{s}}<1$ so that the white-noise
approximation holds.

####  II.1.3 Fano-mirror setup

An alternative strategy to improve the optomechanical cooling is to make the
cavity density of states asymmetric in order to further suppress the
detrimental Stokes process [see Fig. 3(c)]. This can be achieved by replacing
the left-hand side mirror of the cavity by a Fano mirror (e.g., a photonic
crystal membrane) [50]. We then treat separately the cavity input noise
$\hat{a}_{\text{in,L}}$ and $\hat{a}_{\text{in,R}}$ coming through the left-
hand side and right-hand side mirrors, denoting $\kappa_{\text{L}}$ and
$\kappa_{\text{R}}$ the corresponding loss rates. Both are vacuum noise and
their correlation functions obey Eq. (6). The setup is depicted in Fig. 3(b)
and modeled by adding the internal mode of the Fano mirror to the
optomechanical description. We consider an even Fano-mirror mode333This is the
one studied in Ref. [50], but it could also be an odd mode, as treated in the
Supplementary material of the same article., so the setup’s Hamiltonian reads
[50]

$\hat{H}_{\text{Fano}}=\hat{H}+\hbar\omega_{d}\hat{d}^{\dagger}\hat{d}+\hbar\lambda(\hat{a}^{\dagger}\hat{d}+\hat{a}\hat{d}^{\dagger}),$
(10)

where $\hat{H}$ is the Hamiltonian of the standard setup, given by Eq.
(II.1.1). We have denoted $\hat{d}$ the annihilation operator for the
considered mode of the Fano mirror, $\omega_{d}$ its frequency and $\lambda$
the coupling strength between this mode and the cavity mode.

The Langevin equations for such a system read [50] (in the frame rotating at
$\omega_{\text{las}}$)

$\displaystyle\dot{\hat{q}}=\,$ $\displaystyle\Omega_{\text{mec}}\hat{p},$
(11a) $\displaystyle\dot{\hat{p}}=\,$
$\displaystyle-\Omega_{\text{mec}}\hat{q}-\gamma\hat{p}+g_{0}\sqrt{2}\hat{a}^{\dagger}\hat{a}+\sqrt{\gamma}\hat{\xi},$
(11b) $\displaystyle\dot{\hat{a}}=\,$
$\displaystyle-(\kappa+i\Delta_{0})\hat{a}+ig_{0}\sqrt{2}\hat{a}\hat{q}-i\epsilon-\mathcal{G}\hat{d}$
$\displaystyle+\sqrt{2\kappa_{\text{R}}}\hat{a}_{\text{in,R}}+\sqrt{2\kappa_{\text{L}}}\hat{a}_{\text{in,L}},$
(11c) $\displaystyle\dot{\hat{d}}=\,$
$\displaystyle-(\gamma_{d}+i\Delta_{d})\hat{d}-\mathcal{G}\hat{a}+\sqrt{2\gamma_{d}}\hat{a}_{\text{in,L}},$
(11d)

with $\gamma_{d}$ the coupling strength between the Fano-mirror mode and the
left-hand side input, $\Delta_{d}=\omega_{d}-\omega_{\text{las}}$ the detuning
with the laser drive and $\kappa=\kappa_{\text{L}}+\kappa_{\text{R}}$ the
total loss rate of the cavity. We have also defined
$\mathcal{G}=i\lambda+\sqrt{\kappa_{\text{L}}\gamma_{d}}$. In the model
considered here, the coupling between the Fano-mirror mode and the cavity mode
is given by $\lambda=\sqrt{\kappa_{0}\gamma_{d}}$ [50], where $\kappa_{0}$ is
the loss rate that the left-hand side mirror would have if it were frequency
independent. For simplicity we take $\kappa_{0}=\kappa_{\text{R}}$.

For the Fano-mirror setup, the total loss rate of the cavity is very large (a
lot larger than for the standard setup). Such a large loss rate would place a
standard setup in the non-sideband resolved limit. However, this does not
reflect the actual linewidth of the cavity. Instead, the presence of the Fano
mirror leads to a different effective loss rate, corresponding to the
effective linewidth of the cavity, $\kappa_{\text{eff}}$ (see Sec. II.3.3). We
therefore use this effective linewidth in the cooperativity

$C\rightarrow\frac{2g^{2}}{\kappa_{\text{eff}}\gamma\bar{n}_{\text{mec}}}\ ,$
(12)

for this setup, in what follows.

###  II.2 Thermodynamic picture

The thermodynamic system of interest is the mechanical resonator. The phonon
number of this mechanical resonator fluctuates, due to its coupling to a
mechanical environment, which constitutes a hot phonon bath at temperature
$T_{\text{mec}}$. The goal of optomechanical (sideband) cooling is to reduce
the amount of fluctuations in the resonator, thereby “cooling it down”. In
order to do so, we have at our disposal a cold photon bath, in the sense that
it contains negligibly few thermal excitations.444Laser phase noise may be
treated as a non-zero temperature bath, see [54].. However, heat can flow from
the fluctuating mechanical resonator to this zero-temperature environment,
only if a coupling mechanism between phonons and photons is established. This
coupling is provided by the laser that is driving the cavity, thereby
constituting a heat valve [see Fig. 1 for a sketch of this process]. Hence,
the established heat flow from hot to cold comes with a cost, arising from
coupling the system to the cold bath, i.e., from keeping the valve opened.
Note that this cooling process is very different from a typical thermodynamic
refrigerator, where one has the goal to make heat flow from a cold bath to a
hot bath by providing work to the system.

In order to characterize this cooling process, we analyze three thermodynamic
quantities, which are complementary to each other and to a certain extent
equivalent to those typically quantifying the performance of a thermodynamic
cooling device. This is first of all the coldest temperature that can be
reached, here represented by the lowest steady-state phonon number of the
mechanical resonator. This quantity is experimentally accessible via
calibrated optomechanical thermometry or sideband asymmetry determined from
the cavity output light field. This is the quantity that is commonly
considered when studying optomechanical cooling. Note, however, that the
mechanical resonator is a microscopic system and does not constitute a bath
itself. By its coupling to the hot mechanical phonon bath and the cold photon
bath it is brought into a nonequilibrium state and the lowest reachable
steady-state phonon number can hence only be related to an effective
temperature.

Importantly, in our work we complement this study by analyzing two further
thermodynamic quantities. Typically, in a refrigerator the cooling power is
analyzed as a performance quantifier, namely the heat flow out of the bath to
be cooled. As mentioned above, the mechanical resonator that is intended to be
cooled is, however, not a bath, and the total heat flow out of or into the
resonator is always zero in the steady state. We therefore introduce instead
the evacuated-heat flow as an equivalent to the cooling power, namely the heat
current, $J_{\text{c}}$, carried by phonons at frequency $\Omega_{\text{mec}}$
flowing out of the mechanical resonator and into the effective cold bath. We
define $J_{\text{c}}=\hbar\Omega_{\text{mec}}I_{\text{c}}^{\text{phonon}}$
with the associated phonon flux, $I_{\text{c}}^{\text{phonon}}$. The latter
quantity is experimentally accessible from the cavity output light field by
measuring the amplitude and phase quadrature of the light field, see Appendix
B.4.

Finally, we analyze the efficiency of this cooling process. As described
above, maintaining the valve opened in a given position has a cost. The full
cost is given by the laser power ${P}_{\text{las}}$ required to create the
coherent field in the cavity mode. This motivates us to define a cooling
efficiency

$\eta_{\text{full}}=\frac{J_{\text{c}}}{{P}_{\text{las}}}\ ,$ (13)

comparing the evacuated-heat flow (the desired output) with the full laser
power (the exploited resource). We define an alternative cooling efficiency
later in Eq. (36), which, at the cost of neglecting parts of the resource,
gives a better picture of the conversion process from phonons to photons. This
alternative efficiency [Eq. (36)] can actually be more relevant for device
realizations, when the laser light, which was not used in the conversion
process, is partly re-used for further operations and, hence, is not lost.
Note that standard refrigerators or heat pumps are often characterized by
“coefficients of performance”; here we refrain from using this terminology and
rather use the more generic notion “(cooling) efficiency”, since the desired
output of the present process is not the heat flow into or out of one of the
baths, but rather the relevant part of the flux out of the microscopic system
attached to them.

In the following, we want to consolidate the above described thermodynamic
picture and therefore start with an analysis of the standard setup. As a first
step, we linearize the Hamiltonian (II.1.1) around the system’s semiclassical
steady state (see Appendix B.1 for details)555This approximation is justified
in most optomechanical setups as the optomechanical coupling $g_{0}$ is
typically very small: $g_{0}\ll\gamma(2\bar{n}_{\text{mec}}+1),\kappa$ [7,
55], see Tab. 2., in which the cavity mode is in the coherent state of
amplitude $\alpha$ and the resonator’s rest position is $\bar{q}$ [23], with

$\displaystyle\alpha$ $\displaystyle=\dfrac{-i\epsilon}{\kappa+i\Delta},$
(14a) $\displaystyle\bar{q}$
$\displaystyle=\sqrt{2}\dfrac{g_{0}}{\Omega_{\text{mec}}}\absolutevalue{\alpha}^{2}.$
(14b)

We define the effective detuning $\Delta=\Delta_{0}-g_{0}\sqrt{2}\bar{q}$ and
choose the phase in $\epsilon$ so that $\alpha=|\alpha|$ is real and positive.
We split the operators into their average and fluctuation terms:
$\hat{a}=\alpha+\delta\hat{a}$, $\hat{q}=\bar{q}+\delta\hat{q}$,
$\hat{p}=\delta\hat{p}$. The linearized Hamiltonian reads

$\displaystyle\hat{H}_{\text{lin}}=\,$
$\displaystyle\hbar{\Delta}\delta\hat{a}^{\dagger}\delta\hat{a}+\hbar\Omega_{\text{mec}}\delta\hat{b}^{\dagger}\delta\hat{b}$
$\displaystyle-\hbar
g(\delta\hat{a}+\delta\hat{a}^{\dagger})(\delta\hat{b}+\delta\hat{b}^{\dagger}),$
(15)

where we have denoted $\delta\hat{b}=(\delta\hat{q}+i\delta\hat{p})\sqrt{2}$.
Indeed, we see from the expression of the effective coupling
$g=g_{0}\absolutevalue{\alpha}$ that the light field in the cavity plays a key
role in the cooling process, acting like a valve that allows phonons to flow
towards the cavity.

For the sake of obtaining a simple and more usual thermodynamics picture,
we—in this section—furthermore resort to the weak-coupling regime
($g\ll\Omega_{\text{mec}},\kappa$). Tracing out the cavity degrees of freedom,
we then find the master equation [24] describing the dynamics of the
mechanical resonator,

$\displaystyle\derivative{\hat{\rho}_{\text{mec}}}{t}=\,$
$\displaystyle-i[\Omega_{\text{mec}}^{\text{eff}}\delta\hat{b}^{\dagger}\delta\hat{b},\hat{\rho}_{\text{mec}}]+(\gamma\bar{n}_{\text{mec}}+A_{+})D[\delta\hat{b}^{\dagger}]\hat{\rho}_{\text{mec}}$
$\displaystyle+(\gamma(\bar{n}_{\text{mec}}+1)+A_{-})D[\delta\hat{b}]\hat{\rho}_{\text{mec}}\
.$ (16)

Here, $\hat{\rho}_{\text{mec}}$ is the density operator of the resonator,
$\Omega_{\text{mec}}^{\text{eff}}$ the effective mechanical frequency (see
Appendix B.2.1) and
$D[\hat{O}]\hat{\rho}=\hat{O}\hat{\rho}\hat{O}^{\dagger}-\frac{1}{2}\\{\hat{O}^{\dagger}\hat{O},\hat{\rho}\\}$.
We have defined the rates

$A_{\pm}=\frac{2g^{2}\kappa}{\kappa^{2}+(\Omega_{\text{mec}}\pm{\Delta})^{2}},$
(17)

which correspond to the Stokes and anti-Stokes processes, respectively. This
is the situation depicted in Fig. 2. Equivalently, Eq. (II.2) can be rewritten
in the form

$\displaystyle\\!\\!\derivative{\hat{\rho}_{\text{mec}}}{t}=\,$
$\displaystyle-i[\Omega_{\text{mec}}^{\text{eff}}\delta\hat{b}^{\dagger}\delta\hat{b},\hat{\rho}_{\text{mec}}]$
$\displaystyle+\Gamma_{\text{opt}}\left(\bar{n}_{\text{opt}}D[\delta\hat{b}^{\dagger}]+(\bar{n}_{\text{opt}}+1)D[\delta\hat{b}]\right)\hat{\rho}_{\text{mec}}\quad\quad\,\,\,$
$\displaystyle+\gamma\left(\bar{n}_{\text{mec}}D[\delta\hat{b}^{\dagger}]+(\bar{n}_{\text{mec}}+1)D[\delta\hat{b}]\right)\hat{\rho}_{\text{mec}}.$
(18)

This equation shows that the driven cavity acts on the resonator like an
effective thermal phonon bath of associated damping rate
$\Gamma_{\text{opt}}\equiv A_{-}-A_{+}$ and containing a finite number of
phonons $\bar{n}_{\text{opt}}=A_{+}/\Gamma_{\text{opt}}$. Hence, the cavity
and its environment play the role of an effective cold bath (dashed box in
Fig. 1(b)), of finite effective temperature $T_{\text{opt}}$ such that
$\bar{n}_{\text{opt}}=(\text{e}^{\hbar\Omega_{\text{mec}}/k_{\text{B}}T_{\text{opt}}}-1)^{-1}$.
As a result, the average phonon number associated to the fluctuations of the
resonator in the steady state
$\bar{n}_{\text{fin}}=\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle_{\text{ss}}$
can be interpreted as the steady state of a system coupled to two thermal
baths [7],

$\bar{n}_{\text{fin}}=\frac{\gamma\bar{n}_{\text{mec}}+\Gamma_{\text{opt}}\bar{n}_{\text{opt}}}{\gamma+\Gamma_{\text{opt}}}.$
(19)

We can see from Eq. (17) that $\Gamma_{\text{opt}}$ is proportional to
$g^{2}=g_{0}^{2}|\alpha|^{2}$. This confirms that the light field in the
cavity is like a heat valve allowing more or less heat to flow between the
resonator and the cold phonon bath. Note that the effective cold bath is an
engineered bath and therefore it comes with a cost. This further motivates the
above described thermodynamics picture and the resulting definition for the
cooling efficiency as given in Eq. (13).

In the weak-coupling regime, the evacuated-heat flow is explicitly given by

$\displaystyle J_{\text{c}}$
$\displaystyle=\hbar\Omega_{\text{mec}}\Gamma_{\text{opt}}(\bar{n}_{\text{fin}}-\bar{n}_{\text{opt}})$
$\displaystyle=\hbar\Omega_{\text{mec}}\frac{\gamma\Gamma_{\text{opt}}}{\gamma+\Gamma_{\text{opt}}}(\bar{n}_{\text{mec}}-\bar{n}_{\text{opt}}).$
(20)

Using Eq. (14), the full efficiency, Eq. (13), becomes

$\eta_{\text{full}}=\frac{\Omega_{\text{mec}}2\kappa}{\omega_{\text{las}}\alpha^{2}(\kappa^{2}+\Delta^{2})}\frac{\gamma\Gamma_{\text{opt}}}{\gamma+\Gamma_{\text{opt}}}(\bar{n}_{\text{mec}}-\bar{n}_{\text{opt}})\
.$ (21)

This weak-coupling thermodynamic picture also gives us further insights into
the improvements provided by alternative setups. Squeezing the input noise, as
described in Sec. II.1.2, allows to reduce the position quadrature noise of
the intra-cavity field, which couples to the mechanics. As the cavity transfer
function converts part of the input momentum noise into position noise and
conversely, it is necessary to rotate the input squeezed state by the adequate
angle $\theta$ (given in Sec. III.2.2) so as to compensate for this effect.
This reduction of the intra-cavity position noise can be interpreted as
lowering the temperature of the effective cold phonon bath, i.e.,
$\bar{n}_{\text{opt}}$ is smaller while $\Gamma_{\text{opt}}$ remains the same
as in the standard setup. On the other hand, with the Fano-mirror setup from
Sec. II.1.3, the imbalance between rates of the Stokes and anti-Stokes
processes is increased, which results in a change in $\Gamma_{\text{opt}}$ and
a decrease in $\bar{n}_{\text{opt}}$. Therefore, the effective cold bath gets
colder and its coupling to the resonator is modified.

###  II.3 Examples for specific realizations

We now turn to inspect specific optomechanical realizations in order to
illustrate our thermodynamic analysis of optomechanical cooling with concrete
examples and also to benchmark these realizations against each other. With the
evacuated-heat flow and the cooling efficiency, our analysis yields
complementary insights beyond the achievable phonon occupation as a benchmark
only. To this end, we select four representative optomechanical systems, whose
parameters are summarized in Tab. 2 in the appendix. From here on, we refer to
them as system (1) to system (4).

System (1) employs mechanical resonators in the MHz regime and is placed in
the sideband-resolved regime. Such systems can be realized, for example, with
free-space Fabry-Pérot-based optomechanics [56, 57]. Importantly, ground-state
cooling has already been demonstrated with such a system in the membrane-in-
the-middle configuration [58]. An advantage of the Fabry-Pérot system is its
versatility as it can be directly modified to realize a Fano-mirror setup [50,
59]. System (2) is an integrated optomechanics device made of an
optomechanical crystal with mechanical motion in the GHz regime that is placed
in the sideband-resolved regime. This system is a prominent implementation for
realizing optomechanics-based quantum networks with demonstrations of ground-
state cooling [25] and non-classical state generation [60, 61]. Our
thermodynamic analysis of this system yields complementary insights beyond its
use in quantum networks. System (3) is an optically levitated nanoparticle
coupled to an optical cavity. Recently, ground-state cooling of the center-of-
mass motion of such a levitated particle has been demonstrated [27]. Levitated
optomechanics constitutes a pertinent platform for exploring thermodynamic
heat engines or stochastic and quantum thermodynamics phenomena. Multiple
experiments are already performed along these lines [62]. System (4) is based
on the membrane-in-the-middle configuration [56], but in the non-sideband-
resolved regime. Recently, feedback-based cooling to the ground state [63] has
been demonstrated in such a system. We here show that one can actually achieve
ground-state cooling when employing a squeezed-light drive instead of a
coherent one, even in the absence of feedback, as evidenced in Refs. [47, 48,
49].

We analyze the considered four systems in the context of three different
setups: (i) the standard setup, (ii) the squeezed-light setup and (iii) the
Fano-mirror setup. The corresponding choice of parameters is shortly discussed
in the following.

####  II.3.1 Standard setup

For the standard setup, we assume driving of the optomechanical system with
coherent laser light and use the parameters of the four systems as given in
Tab. 2. In this way, we basically follow the experimental implementations of
Refs.[64, 25, 27, 65, 63].

####  II.3.2 Squeezed-light setup

In principle, the four systems can also be driven with a squeezed-light
source. However, this has so far only been done and used for demonstrating
ground-state cooling in microwave optomechanics [48]. Squeezed states of light
are customarily generated using non-linear optics processes. For example,
squeezing levels of 12.7 dB [66] and 13 dB [67] at wavelengths of 1064 nm and
1550 nm, respectively, have been generated. These wavelengths are also
typically used in optomechanics experiments, see Tab. 2. Squeezing in these
experiments [66, 67] is customarily observed at MHz sideband frequencies,
which matches optomechanics implementations with mechanical motion in the MHz
regime. Squeezing at GHz sideband frequencies, which is required when
employing mechanical motion at GHz frequencies, has been generated at 5 dB
levels [68, 69, 70]. In our analysis, we assume a squeezed-light source with
realistic squeezing levels666The amount of squeezing in dB is calculated as
$-10\log_{10}(2\langle\Delta P^{\theta}_{\text{in,s}}\rangle)$ with the
variance of shot noise being $\frac{1}{2}$ and $\langle\Delta
P^{\theta}_{\text{in,s}}\rangle$ the variance of the squeezed quadrature. of
0.87 dB, 0.59 dB, 2.7 dB and 15.4 dB for the four systems, respectively.

####  II.3.3 Fano-mirror setup

Free-space optomechanical devices can be directly adapted to realize a Fano-
mirror setup, see, e.g., Refs. [50, 59]. To be able to compare the Fano-mirror
setup with the standard setup, we use the fact that the effective linewidth of
the cavity with the Fano mirror is
$\kappa_{\text{eff}}=\gamma_{d}/\zeta_{0}^{2}$ [50], where $\zeta_{0}$ is the
polarizability of the left mirror at frequency $\omega_{d}$. Since we have
chosen $\kappa_{\text{R}}=\kappa_{0}$, $\zeta_{0}$ is also the (frequency-
independent) polarizability of the right mirror. Denoting $\Gamma$ the free
spectral range of the cavity that is given in Tab. 2, we have
$\kappa_{\text{L}}=2\Gamma$ and $\kappa_{0}=\Gamma/2\zeta_{0}^{2}$. Moreover,
choosing to use the optimal value
$\gamma_{d}^{*}=4\Omega_{\text{mec}}\zeta_{0}$ [50], we can determine
$\zeta_{0}$ for each system such that $\kappa_{\text{eff}}$ is equal to the
system’s cavity loss rate.

Optomechanical cavities with Fano mirrors can be realized by using suspended
photonic crystal slabs, see for example Refs. [71, 72, 73, 74, 75]. The
parameters of the photonic crystal slab [76] allow engineering of a desired
reflectivity at frequency $\omega_{d}$ and, thus, a corresponding
$\kappa_{\text{eff}}$. Fine tuning of the optical resonance of the photonic
crystal mirror can be achieved in post-fabrication, for instance by using
thermal tuning [77], strain tuning [78] or by depositing [79] or etching [80]
atomic layers of the suspended photonic crystal. Therefore it is reasonable to
assume that $\gamma_{d}$ close to the optimal value can be achieved in an
experimental setup.

##  III Thermodynamic performance

In Sec. II we have described the cooling process in the three setups and have
outlined how to set up a consistent thermodynamic picture. In this section, we
analyze the thermodynamic performance of four example realizations and compare
their performance for the operation in the three different setups. The
parameters used for the four example realizations are given in Tab. 2. The
analysis of the thermodynamic performance includes the lowest reachable phonon
number, the evacuated-heat flow as well as the cooling efficiency. In order to
obtain all these quantities, we first need to analyze the dynamics of the
systems in the three setups.

###  III.1 Dynamics

The first step to the study of the systems’ dynamics in each of the three
setups is to linearize the Langevin equations, as we have previously done in
Sec. II.1.1 for the standard setup. Here we present a short overview for all
three setups. We then derive the evolution of the second-order moments, and in
particular of the photon and phonon numbers.

####  III.1.1 Standard setup

Applying the linearization introduced in Sec. II.2 to the Langevin equations
for the standard setup, Eq. (2), we obtain

$\displaystyle\delta\dot{\hat{q}}$
$\displaystyle=\Omega_{\text{mec}}\delta\hat{p},$ (22a)
$\displaystyle\delta\dot{\hat{p}}$
$\displaystyle=-\Omega_{\text{mec}}\delta\hat{q}-\gamma\delta\hat{p}+g\sqrt{2}(\delta\hat{a}^{\dagger}+\delta\hat{a})+\sqrt{\gamma}\hat{\xi},$
(22b) $\displaystyle\delta\dot{\hat{a}}$
$\displaystyle=-(\kappa+i\Delta)\delta\hat{a}+ig\sqrt{2}\delta\hat{q}+\sqrt{2\kappa}\hat{a}_{\text{in}}.$
(22c)

From these equations and the noise correlation functions, given by Eqs. (3)
and (6), we derive the evolution of the photon and phonon numbers associated
to the fluctuations777Note that the environment contribution in Eq. (23b) has
an extra term in $\langle\delta\hat{b}^{2}+\delta\hat{b}^{\dagger 2}\rangle$
compared to the one in Ref. [24] because Ref. [24] used a different noise
model. Both models give very similar results in the high mechanical quality
factor regime considered here. See Appendix B.5 for a discussion of the
differences between the two approaches.

$\displaystyle\\!\derivative{\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle}{t}=\,$
$\displaystyle
ig\left(\langle\delta\hat{a}^{\dagger}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle-\langle\delta\hat{a}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle\right)$
$\displaystyle-2\kappa\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle,$
(23a)
$\displaystyle\\!\derivative{\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle}{t}=\,$
$\displaystyle
ig\left(\langle(\delta\hat{a}+\delta\hat{a}^{\dagger})\delta\hat{b}^{\dagger}\rangle-\langle(\delta\hat{a}^{\dagger}+\delta\hat{a})\delta\hat{b}\rangle\right)$
$\displaystyle\\!\\!\\!+\gamma\left(\bar{n}_{\text{mec}}-\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle+\frac{\langle\delta\hat{b}^{2}+\delta\hat{b}^{\dagger
2}\rangle}{2}\right).\\!$ (23b)

Details about the derivation and the evolution equations of the other second-
order moments can be found in Appendix B.2.2. For each of the above equations,
two contributions stand out: the optomechanical one, proportional to $g$, and
the one from the optical and mechanical environment, proportional to $\kappa$
and $\gamma$, respectively. When one neglects the non-resonant processes
$\delta\hat{a}\delta\hat{b}$ and
$\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}$ (rotating wave approximation,
valid for small coupling $g$, see Appendix B.5), the optomechanical
contribution corresponds to the scattering picture from Fig. 2(b), with
heating from the Stokes process and cooling from the anti-Stokes process.

####  III.1.2 Squeezed-light setup

In the case of the squeezed-light setup, described in Sec. II.1.2, the DPA and
cavity are only coupled through the noise. Therefore, the average fields in
the cavity and resonator are still given by Eqs. (14) and the linearized
Langevin equations are the same as Eqs. (22). In contrast, the cavity input
noise correlation functions are given by Eqs. (8). As a consequence, the
photon number evolution becomes

$\displaystyle\derivative{\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle}{t}=\,$
$\displaystyle
ig\left(\langle\delta\hat{a}^{\dagger}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle-\langle\delta\hat{a}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle\right)$
$\displaystyle+2\kappa(\pi_{\text{s}}N_{\text{s}}-\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle),$
(24)

while the evolution equation of the phonon number is unchanged (Eq. (23b)).
Further changes to the evolution of the second-order moments are detailed in
Appendix C.2.1.

####  III.1.3 Fano-mirror setup

Like in Sec. II.2, we linearize the Langevin equations (11) by splitting
operators into their average terms and fluctuation terms. But in this case, we
have one extra operator, $\hat{d}=\delta+\delta\hat{d}$, which modifies the
semiclassical steady state into (see Appendix D.1 for details)

$\displaystyle\alpha$
$\displaystyle=\frac{-i\epsilon}{\kappa+i\Delta-\mathcal{G}^{2}/(\gamma_{d}+i\Delta_{d})}$
(25a) $\displaystyle\bar{q}$
$\displaystyle=\sqrt{2}\dfrac{g_{0}}{\Omega_{\text{mec}}}\absolutevalue{\alpha}^{2},$
(25b) $\displaystyle\delta$
$\displaystyle=\frac{-\mathcal{G}}{\gamma_{d}+i\Delta_{d}}\alpha.$ (25c)

The linearized Langevin equations read

$\displaystyle\delta\dot{\hat{q}}=\,$
$\displaystyle\Omega_{\text{mec}}\delta\hat{p},$ (26a)
$\displaystyle\delta\dot{\hat{p}}=\,$
$\displaystyle-\Omega_{\text{mec}}\delta\hat{q}-\gamma\delta\hat{p}+g\sqrt{2}(\delta\hat{a}^{\dagger}+\delta\hat{a})+\sqrt{\gamma}\hat{\xi},$
(26b) $\displaystyle\delta\dot{\hat{a}}=\,$
$\displaystyle-(\kappa+i\Delta)\delta\hat{a}+ig\sqrt{2}\delta\hat{q}-\mathcal{G}\delta\hat{d}$
$\displaystyle+\sqrt{2\kappa_{\text{L}}}\hat{a}_{\text{in,L}}+\sqrt{2\kappa_{0}}\hat{a}_{\text{in,R}},$
(26c) $\displaystyle\delta\dot{\hat{d}}=\,$
$\displaystyle-(\gamma_{d}+i\Delta_{d})\delta\hat{d}-\mathcal{G}\delta\hat{a}+\sqrt{2\gamma_{d}}\hat{a}_{\text{in,L}}.$
(26d)

The free parameter $\Delta_{d}$ of the Fano-mirror mode was found to be
constrained by the cavity mode and the optical loss rates by
$\Delta_{d}=\Delta-2\sqrt{\kappa_{0}\kappa_{\text{L}}}$ [50].

From these equations and the noise correlation functions, given by Eqs. (3)
and (6), we derive the evolution of the photon number (see Appendix D.2.2):

$\displaystyle\derivative{\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle}{t}=\,$
$\displaystyle
ig\left(\langle\delta\hat{a}^{\dagger}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle-\langle\delta\hat{a}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle\right)$
$\displaystyle-2\Re{\mathcal{G}\langle\delta\hat{d}\delta\hat{a}^{\dagger}\rangle}-2\kappa\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle,$
(27)

The evolution equation of the phonon number is unchanged (Eq. (23b)).

###  III.2 Steady-state phonon number

The first performance indicator we consider is the steady-state phonon number
$\bar{n}_{\text{fin}}=\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle_{\text{ss}}$,
which is the standard quantity that is used to characterize sideband cooling
and experimentally accessible via calibrated noise thermometry or sideband
asymmetry. In a thermodynamic picture, this quantity would correspond to the
lowest temperature that can be reached by a refrigerator. Indeed, the average
phonon number in the fluctuations can be related to $T_{\text{eff}}$, the
effective temperature of the resonator, by

$\bar{n}_{\text{fin}}=\left(\text{e}^{\hbar\Omega_{\text{mec}}/k_{\text{B}}T_{\text{eff}}}-1\right)^{-1}.$
(28)

Since the position of the resonator has been shifted (see Sec. II.2), we
actually have the mean stationary phonon number
$\langle\hat{b}^{\dagger}\hat{b}\rangle_{\text{ss}}=(g/\Omega_{\text{mec}})^{2}|\alpha|^{2}+\bar{n}_{\text{fin}}$.

In the following, we describe how to obtain the steady-state phonon number,
$\bar{n}_{\text{fin}}$, in the three setups and discuss some of its general
properties. We then discuss differences between the different setups and
example realizations in the end of this section.

####  III.2.1 Standard setup

For the standard setup, $\bar{n}_{\text{fin}}$ can be found analytically, by
solving the equations for the second-order moments for the steady state.
However, the obtained expression is rather complex and we therefore present it
in Appendix B.2.2, see also Refs. [23, 24]. Instead, in order to illustrate
our results, we here present plots, see Fig. 4, using the realistic
experimental parameters from Tab. 2. The only free parameter in the standard
setup is the detuning $\Delta$. In the resolved-sideband regime,
$\kappa\ll\Omega_{\text{mec}}$, the lowest phonon number is obtained for
$\Delta\simeq\Omega_{\text{mec}}$, as illustrated in Fig. 2. Note however,
that the optimal detuning is exactly given by the mechanical frequency only in
the weak-coupling approximation. All the plots for the standard setup are done
at $\Delta=\Omega_{\text{mec}}$, except for the experimental system (4) (see
Tab. 2), where we use $\Delta=\kappa$. System (4) is not in the resolved-
sideband regime and the cavity linewidth is larger than the mechanical
frequency. As a consequence, if $\Delta=\Omega_{\text{mec}}$, the Stokes and
anti-Stokes rates are almost equal, not leading to effective cooling. The best
cooling is obtained when the difference between the two rates is the largest,
that is on the slope of the cavity density of states, for a detuning close to
$\kappa$.

####  III.2.2 Squeezed-light setup

As in the standard case, we obtain an analytical expression of the steady-
state phonon number (see Appendix C.2.1) for the squeezed-light setup. Again,
the lowest phonon number is obtained for $\Delta\simeq\Omega_{\text{mec}}$.
However, we have three additional parameters: the squeezing purity
$\pi_{\text{s}}$, the squeezing ratio $r_{\text{s}}$ and the squeezing angle
$\theta$ (see Sec. II.1.2). As expected, the best results with respect to the
lowest reachable phonon number are obtained for pure squeezing, i.e.
$\pi_{\text{s}}=1$. Besides, there exists an optimal value of the squeezing
angle $\theta^{*}$ that minimizes the phonon number. Interestingly, this angle
does not depend on the other squeezing parameters (see Eq. (83) in Appendix
C.2.2 for the full expression) and, in the limit of weak coupling and
neglecting the mechanical damping rate, it is given by [47]

$\tan(2\theta^{*})=\frac{2{\Delta}{\kappa}}{{\Delta}^{2}-{\Omega_{\text{mec}}}^{2}-{\kappa}^{2}}.$
(29)

Therefore it is possible to determine this optimal angle from the
optomechanical setup’s parameters. There exists also an optimal value
$r^{*}_{\text{s}}$ of the squeezing ratio which depends on the squeezing angle
(see Appendix C.2.2 for details). All the plots for this setup are done using
the parameters from Tab. 2, with $\Delta=\Omega_{\text{mec}}$ (except for
system (4) for which $\Delta=\kappa$), $\pi_{\text{s}}=1$ and the values of
$\theta$ and $r_{\text{s}}$, which are optimal for reducing the steady-state
phonon number.

Going to the limit of weak coupling and neglecting the mechanical damping
rate, we derive the phonon number in the effective cold bath (see Sec. II.2)

$\displaystyle\bar{n}_{\text{opt}}^{\text{squ}}$ $\displaystyle=$
$\displaystyle\bar{n}_{\text{opt}}\left(1+\frac{2\pi_{\text{s}}N_{\text{s}}(\Delta^{2}+\Omega_{\text{mec}}^{2}+\kappa^{2})}{\kappa^{2}+(\Delta-\Omega_{\text{mec}})^{2}}\right.$
$\displaystyle\left.-2\pi_{\text{s}}\absolutevalue{M_{\text{s}}}\sqrt{\frac{\kappa^{2}+(\Delta+\Omega_{\text{mec}})^{2}}{\kappa^{2}+(\Delta-\Omega_{\text{mec}})^{2}}}\cos(2\theta-2\theta^{*})\right),$

where $\bar{n}_{\text{opt}}$ is the phonon number in the effective cold bath
in the absence of squeezing, in agreement with the results from [47]. This
explicitly shows that, with the appropriate squeezing parameters, we can get a
colder effective cold phonon bath than with the standard setup, thus allowing
to reach a lower $\bar{n}_{\text{fin}}$. In this limit, the optimal squeezing
ratio $r_{\text{s}}^{*}$ is such that

$\frac{\absolutevalue{M_{\text{s}}(r_{\text{s}}^{*})}}{N_{\text{s}}(r_{\text{s}}^{*})}=\sqrt{\frac{\kappa^{2}+(\Delta+\Omega_{\text{mec}})^{2}}{\kappa^{2}+(\Delta-\Omega_{\text{mec}})^{2}}}$
(31)

and then
$\bar{n}_{\text{opt}}^{\text{squ}}=\bar{n}_{\text{opt}}(1-\pi_{\text{s}})$
[47]. Therefore, in this ideal case and with perfect squeezing, the lowest
attainable temperature of the effective cold phonon bath is zero.

####  III.2.3 Fano-mirror setup

In this setup, the cavity density of states is centered around $\omega_{d}$
[50] [see Fig. 3(c)], therefore the lowest phonon number can be obtained for
$\Delta_{d}\simeq\Omega_{\text{mec}}$ [50]. As explained in Sec. II.3.3, Fano
mirrors with the desired loss rate $\gamma_{d}$ can be fabricated by
engineering the photonic crystal lattice parameters. It is therefore realistic
to use the loss rate $\gamma_{d}^{*}$, for which the phonon number gets
minimized, see Tab. 2 for the numerical values, and
$\Delta_{d}=\Omega_{\text{mec}}$ in the plots. Once again, to obtain the best
cooling, we use a different detuning for system (4):
$\Delta_{d}=2.5\Omega_{\text{mec}}$ (see Appendix D.2.5). We solved the
equations for the second-order moments in order to obtain
$\bar{n}_{\text{fin}}$ numerically (see Appendix D.2.2).

####  III.2.4 Discussion

Figure 4:  Steady-state phonon number as a function of cooperativity for the
three setups (indicated by different lines in each plot). The dotted and
dashed black lines respectively indicate the average phonon number in the hot
mechanical bath and in the effective cold bath (weak-coupling limit) for the
standard setup. The vertical solid lines indicate the cooperativities at which
the standard and squeezed-light setups (in blue) and Fano-mirror setup (in
green) become unstable (see Appendices B.1 and D.1).The number in each sub-
figure corresponds to the system number as given in Tab. 2.

We have plotted $\bar{n}_{\text{fin}}$ as a function of cooperativity for each
setup and for each system from Tab. 2 in Fig. 4. For all four example systems
in all three cooling setups, we notice that at low cooperativity, the thermal
noise prevails over the optomechanical coupling and
$\bar{n}_{\text{fin}}\simeq\bar{n}_{\text{mec}}$. Then, when the cooperativity
increases, the optomechanical coupling allows to reach lower phonon numbers,
in most cases down to the ground state of the mechanical resonator (namely
$\bar{n}_{\text{fin}}<1$). The minimum is reached, when the optomechanical
coupling becomes so large that $\bar{n}_{\text{fin}}$ is governed by the
occupation of the effective cold phonon bath, namely
$\bar{n}_{\text{fin}}\approx\bar{n}_{\text{opt}}$ (see Eq. (19) obtained in
the weak-coupling limit). Finally, when further increasing the cooperativity,
the coupling becomes so strong that the optomechanical back-action increases
the fluctuations of the resonator and the phonon number increases as well888In
the strong coupling limit, lower phonon numbers can be obtained in an
instantaneous instead of the steady state, see Ref. [81].. This is the case
when non-resonant processes of the type $\delta\hat{a}\delta\hat{b}$ and
$\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}$ start to play a role. This
behavior due to back-action is not captured by the weak-coupling approximation
(Eq. (19), see also Appendix B.5). Eventually, the phonon number diverges999
The system is no longer stable, see Appendices B.1 and B.2.2.. Interestingly,
Fig. 5 shows that the divergence of
$\bar{n}_{\text{fin}}=(\langle\delta\hat{q}^{2}\rangle+\langle\delta\hat{p}^{2}\rangle-1)/2$
fully comes from the position fluctuations while the momentum fluctuations
keep decreasing with cooperativity.

There are, however, also differences between the different systems and between
the different setups. Comparing the different setups, we find that both the
driving with squeezed light as well as the use of a Fano mirror allow to reach
lower phonon numbers than the standard setup, as expected. Interestingly, the
steady-state phonon number is however only little decreased with respect to
the standard setup. For systems (1) to (3), the difference is hardly visible
on the log-scale used in Fig. 4; while the absolute difference in the phonon
number is indeed small, the relative difference with respect to the standard
setup is quite significant, as visible in Tab. 1.

System | (1) | (2) | (3) | (4)
---|---|---|---|---
$x^{\text{squeezed}}$ | 14.4% | 22.5% | 54.8% | 99.4%
$x^{\text{Fano}}$ | 5.0% | 22.7% | 56.5% | 66.4%
Table 1:  Relative difference between the minimum phonon number
$\bar{n}_{\text{fin}}$ reached with the standard and the squeezed-light/Fano-
mirror setup,
$x^{\text{squeezed/Fano}}=\left(\bar{n}_{\text{fin}}^{\text{standard}}-\bar{n}_{\text{fin}}^{\text{squeezed/Fano}}\right)/\bar{n}_{\text{fin}}^{\text{standard}}$.
Figure 5:  Fluctuations of the mechanical position (solid lines) and momentum
(dashed lines) quadratures as a function of cooperativity for the three
setups: standard setup in blue (dark gray), squeezed-light setup in red and
orange (medium gray) and Fano-mirror setup in green (light gray). The number
in each sub-figure corresponds to the system number as given in Tab. 2. For
systems (1) to (3) the curves for the standard and squeezed-light setups are
almost identical.

Furthermore, the dips in the curves for the Fano-mirror setup are always
shifted to larger cooperativity values, which means that a larger
optomechanical coupling is required to cool the resonator. This can be reached
by increasing the power of the laser driving. The shift can be understood by
looking at the weak-coupling picture. For the systems we consider, the
coupling $\Gamma_{\text{opt}}$ to the effective cold phonon bath, at a given
$C$, is weaker in the Fano-mirror setup than in the standard setup (see
Appendix D.2.4).

For all systems in the resolved-sideband regime (the first three in Tab. 2),
all setups enable ground-state cooling. While the overall behavior seems very
similar for systems (1) to (3), the initial phonon number has very different
orders of magnitude in the different systems, meaning that also the overall
reduction in order to reach the ground state is not the same. In contrast
system (4), which is not in the resolved-sideband regime, does not allow for
ground state cooling when using a coherent drive101010This would require
feedback cooling (see for instance [23]), as was done in the experiment using
this setup [63].. However, driving with squeezed light reduces
$\bar{n}_{\text{fin}}$ below one. Numerical values for the lowest reachable
phonon numbers in all setups and systems are given in Tab. 2.

###  III.3 Evacuated-heat flow

Traditionally when studying refrigerators, one looks at the cooling power,
namely the heat that is extracted from the object to be cooled. Here, heat is
extracted from the mechanical resonator by phonons flowing into the
engineered, cold environment, thereby compensating the heat flow from the hot
mechanical bath. The evacuated-heat flow, constituting an equivalent to the
cooling power here, hence corresponds to the phonon flow into this cold
environment times the frequency of the mechanical resonator, namely the energy
that each of these phonons carries away,
$J_{\text{c}}=\hbar\Omega_{\text{mec}}I_{\text{c}}^{\text{phonon}}$. This
quantity can be inferred from measurements of the cavity output light field ,
see Appendix B.4 for details.

The flux of phonons from the resonator to the cavity,
$I_{\text{c}}^{\text{phonon}}$, can be identified in the evolution of the
phonon number (Eq. (23b)) as

$I_{\text{c}}^{\text{phonon}}=ig\langle(\delta\hat{a}+\delta\hat{a}^{\dagger})(\delta\hat{b}-\delta\hat{b}^{\dagger})\rangle.$
(32)

Note that this expression is very similar to the contribution $\mu_{b}$ of the
entropy production rate analyzed in Ref. [19] with
$I_{\text{c}}^{\text{phonon}}=-\mu_{b}(\bar{n}_{\text{mec}}+\frac{1}{2})$ in
the steady state. The flux $-I_{\text{c}}^{\text{phonon}}$ is then equal to
the flow of phonons going into the mechanical environment, which is an
irreversible process that contributes to the entropy production rate and,
thus, is reflected in $\mu_{b}$.

Figure 6:  Evacuated-heat flow as a function of cooperativity for the three
setups (indicated by different lines in each plot). The insets show the
evacuated-heat flow at low cooperativity with a linear scale. The number in
each sub-figure corresponds to the system number in the parameter Tab. 2.

We obtain the steady-state evacuated-heat flow in the three setups and for the
four different systems and show $J_{\text{c}}$ in Fig. 6 as a function of the
cooperativity, up to those values, where $\bar{n}_{\text{fin}}$ diverges. For
small cooperativity values, the evacuated-heat flow is linear (visible in the
insets) but vanishingly small, namely when the optomechanical coupling is
small. The result for the evacuated-heat flow from the weak-coupling
approximation (Eq. (II.2)),
$J_{\text{c}}\approx\hbar\Omega_{\text{mec}}\Gamma_{\text{opt}}\bar{n}_{\text{mec}}$,
clearly shows that the phonon flow for small $C$ is constrained by the small
coupling to the effective cold phonon bath. It then rapidly increases at
cooperativity values at which $\Gamma_{\text{opt}}\approx\gamma$. This is the
same parameter range for which the steady-state phonon number
$\bar{n}_{\text{fin}}$ starts to differ from the value imposed by the hot
environment, $\bar{n}_{\text{mec}}$. The evacuated-heat flow then saturates at
a constant value,
$J_{\text{c}}\approx\hbar\Omega_{\text{mec}}\gamma\bar{n}_{\text{mec}}$,
namely when the coupling to the effective optical bath is so large that the
flow of phonons into the cold bath is only constrained by the inflow of
phonons from the hot mechanical bath. This is the maximum amount of (phononic)
heat that can flow through the system, as long as the effective optical phonon
bath is colder than the mechanical bath. Note that this saturation happens at
a much smaller cooperativity than the one at which the minimum
$\bar{n}_{\text{fin}}$ is reached. Indeed, the bottleneck of the heat flow
through the system is reached when $\Gamma_{\text{opt}}$ exceeds $\gamma$;
however, equilibration of the resonator at some $\bar{n}_{\text{fin}}$ is
governed by the effective cold bath only when the exchange rate
$\Gamma_{\text{opt}}\bar{n}_{\text{opt}}$ exceeds
$\gamma\bar{n}_{\text{mec}}$. Importantly, the evacuated-heat flow is not
affected by the backaction effect leading to an increase of
$\bar{n}_{\text{fin}}$ at large cooperativity. This means that the opposite
heat flow from the non-resonant terms $\delta\hat{a}\delta\hat{b}$ and
$\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}$ exactly counterbalance each
other, see Appendix B.3 and the plot therein. The strong hybridization between
cavity photons and phonons in the limit of large cooperativity can hence be
understood as leading to a direct steady-state heat flow between mechanical
bath and light field, with less cooling effect on the local mechanical sub-
system. Note that this crucial difference between the behaviors of the lowest
reachable phonon number and the evacuated heat flow clearly show the
complementarity of these two performance quantifiers.

The described qualitative behavior is globally the same in all setups and for
all studied systems. The threshold, which is needed to significantly cool down
the resonator, is however higher for the Fano mirror, since, as discussed
previously, higher cooperativities are needed to establish a
$\Gamma_{\text{opt}}$ that is comparable to the one of the standard and the
squeezed-light setup. The standard setup and the setup fed by squeezed light
only differ by small numerical values. This means that the evacuated-heat flow
is not influenced by the noise properties of the input light. Indeed, in the
weak-coupling regime that governs the physics of the evacuated-heat flow,
squeezing only enters via $\bar{n}_{\text{opt}}$, see Eq. (2), which is
negligibly small both in the standard and the squeezing setup.

###  III.4 Efficiencies

The squeezed-light and Fano-mirror setups allow to reach lower phonon numbers
than the standard setup, without altering the saturation value of the
evacuated-heat flow. In this section, we aim to understand if these cooling
schemes are more or less efficient, namely whether they require extra
resources to evacuate the same amount of heat and reach resonator states close
to the ground state. We therefore look at the cooling efficiency
$\eta_{\text{full}}$ defined as the evacuated-heat flow divided by the laser
power (Eq. (13)) as well as a complementary cooling efficiency
$\eta_{\text{conv}}$, characterizing the efficiency of the phonon-photon
conversion process.

####  III.4.1 Full cooling efficiency

In order to obtain the full cooling efficiency for all setups and systems, we
insert the expression for the laser power, Eq. (5), and for the evacuated-heat
flow, Eq. (32), into the full efficiency,
$\eta_{\text{full}}=J_{\text{c}}/{P}_{\text{las}}$. The drive amplitude
$\epsilon$ of the laser, entering the expression for the laser power is
determined from the steady-state amplitude of the field in the cavity,
$|\alpha|=g/g_{0}$, using Eq. (14a) in the standard and squeezed-light setups,
and Eq. (25a) for the Fano-mirror setup. The resulting full cooling efficiency
is plotted in Fig. 7 as a function of the cooperativity.

For all setups and systems, the cooling efficiency is constant until some
cooperativity threshold, namely the one at which the evacuated-heat flow
displays a crossover to a plateau. The reason for this is that both the
evacuated-heat flow as well as the laser power are linear in $C$ for small
cooperativities. The small magnitude of the full cooling efficiency derives
from the low phonon energy compared to the energy of the laser photons and the
weak optomechanical single-photon coupling $g_{0}$. At cooperativities at
which the evacuated-heat flow sets in, the full cooling efficiency decreases
rapidly. We chose to cut the curves at the cooperativities, in which the
phonon number $\bar{n}_{\text{fin}}$ diverges.

Figure 7:  Full cooling efficiency $\eta_{\text{full}}$ as a function of
cooperativity for the three setups (indicated by different lines in each
plot). The number in each sub-figure corresponds to the system number in
parameter Tab. 2. Note that the full cooling efficiency of the squeezed-light
setup differs from the plotted one by a factor $R$ as given in Eq. (33).

The full cooling efficiencies for the three setups have similar features, but
differ by their magnitude. The largest efficiency is the one of the standard
setup, since it requires the least laser-power input. For the squeezed-light
setup, only the reflected fraction $R$ of the input laser power is actually
used for driving the cavity, while the remaining transmitted part feeds the
DPA, generating the squeezing, see Fig. 3(a). Therefore, the actual
efficiency, $\eta_{\text{full}}^{\prime}$, is smaller than the plotted one,
$\eta_{\text{full}}$, by a factor $R$,

$\eta_{\text{full}}^{\prime}=\frac{J_{\text{c}}}{{P}_{\text{las}}/R}=R\eta_{\text{full}}$
(33)

The factor $R$ by which the full cooling efficiency of the squeezed-light
setup is reduced with respect to the one of the standard setup hence depends
on the cost of the squeezing generation. We refrain from giving a specific
value for $R$, because quantifying this cost requires to fix the details of
the experimental setup used to generate the squeezed vacuum. With Eq. (33), we
provide an instruction on how to include the cost of squeezing in a concrete
experimental realization.

The full cooling efficiency of the Fano-mirror setup is orders of magnitude
smaller than the one of the standard (and squeezed-light) setup. Indeed, as
visible in Eqs. (25), for the Fano-mirror setup, the laser field not only
creates the steady-state field in the cavity but also the one in the mirror
mode. So more laser power is required to reach the same value of $\alpha$, and
therefore the same cooperativity. Thus, $\eta_{\text{full}}$ illustrates how
much extra resources are required to cool down the resonator using a Fano
mirror.

Overall, reaching lower phonon numbers, either with squeezed light (as one has
to account for the factor $R<1$ taking into account the cost to create
squeezed light) or a Fano mirror, is hence more costly and reduces the
efficiency of the cooling process.

Comparing the efficiencies of the example realizations of different systems,
we note that the full cooling efficiency of the non-resolved sideband system
(4) is similar to systems (1) and (3), even though the phonon number
$\bar{n}_{\text{fin}}$ of system (4) is larger than the one for the standard
and Fano-mirror setups. This motivates us to define in the following section
another cooling efficiency that is more sensitive to whether the system is in
the resolved-sideband regime.

Figure 8:  Full cooling efficiency $\eta_{\text{full}}$ as a function of the
effective phonon number for the three setups (indicated by different lines in
each plot). The number in each sub-figure corresponds to the system number in
parameter Tab. 2. Note that the full cooling efficiency of the squeezed-light
setup differs from the plotted one by a factor $R$ as given in Eq. (33).

As is commonly the case in thermodynamics, there is a trade-off between
optimizing the efficiency and the desired output. While in a standard
refrigerator, this trade-off is typically analyzed between the efficiency and
the cooling power, we here identify the lowest reachable phonon number as the
relevant quantity which should be optimized, possibly together with the
efficiency of the heat evacuation process. We illustrate this trade-off in
Fig. 8. Indeed, $\eta_{\text{full}}$ starts to decrease at a cooperativity
orders of magnitude lower than the one at which $\bar{n}_{\text{fin}}$ reaches
its minimum value.

####  III.4.2 Conversion efficiency

The full cooling efficiency, $\eta_{\text{full}}$, discussed above compares
the evacuated-heat flow to the full laser power input needed to set up the
cooling mechanism. While this yields a rather complete benefit/cost ratio, it
gives less insight about the cooling efficiency of the optomechanical
conversion process from phononic to photonic heat. Note however, that the
light leaking out of the cavity could in principle partly be reused; thus,
this part would further be available as a resource. In this subsection, we
therefore introduce an alternative cooling efficiency that we call conversion
efficiency.

When looking at the microscopic process, it is of interest to consider the
photons involved in the optomechanical interaction as the resource. We
therefore start by identifying the flux of photons interacting with the
mechanics

$I^{\text{photon}}=ig\left(\langle\delta\hat{a}^{\dagger}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle-\langle\delta\hat{a}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle\right).$
(34)

We have obtained this quantity by analyzing the evolution of the photon
number, Eq. (23a). Note that this expression is very similar to the
contribution $\mu_{a}$ of the entropy production rate analyzed in Ref. [19]
with $I^{\text{photon}}=\mu_{a}/2$ in the steady state. Again, this is not
surprising as $I^{\text{photon}}$ describes the flux of photons that
ultimately leak out of the cavity to the optical environment, which is an
irreversible process that contributes to the entropy production rate. Further,
also $I^{\text{photon}}$ can be inferred from measurements of the output
cavity light field, for details see Appendix B.4.

With this, the conversion efficiency can be defined as the evacuated-heat flow
(phonons flowing out of the resonator into the cold bath) with respect to the
flow of photons that actually interact with the mechanics,

$\displaystyle\eta_{\text{conv}}$
$\displaystyle=\frac{I_{\text{c}}^{\text{phonon}}}{I^{\text{photon}}}\ .$ (35)

Using Eqs. (32) and (34), we obtain

$\displaystyle\eta_{\text{conv}}$
$\displaystyle=\frac{\langle\delta\hat{a}^{\dagger}\delta\hat{b}-\delta\hat{a}\delta\hat{b}^{\dagger}\rangle-\langle\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}-\delta\hat{a}\delta\hat{b}\rangle}{\langle\delta\hat{a}^{\dagger}\delta\hat{b}-\delta\hat{a}\delta\hat{b}^{\dagger}\rangle+\langle\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}-\delta\hat{a}\delta\hat{b}\rangle}\
.$ (36)

In sideband cooling schemes, the detuning of the laser drive is close to
resonance with the beam-splitter terms, $\delta\hat{a}^{\dagger}\delta\hat{b}$
and $\delta\hat{a}\delta\hat{b}^{\dagger}$, while the two-mode-squeezing
terms, $\delta\hat{a}\delta\hat{b}$ and
$\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}$, are non-resonant. The latter
are detrimental to the cooling, i.e.
$\langle\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}-\delta\hat{a}\delta\hat{b}\rangle>0$
(a plot of the beam-splitter and two-mode-squeezing contributions to the heat
flow is included in Appendix B.3), leading to $\eta_{\text{conv}}<1$. However
when the beam-splitter terms are perfectly resonant, that is for
$\Delta=\Omega_{\text{mec}}$ in the resolved-sideband regime the two-mode-
squeezing contributions become negligible in the weak-coupling limit and the
conversion efficiency goes to 1.

Figure 9:  Conversion efficiency $\eta_{\text{conv}}$ as well as
$\eta_{\text{conv}}^{\prime}$ as a function of cooperativity for the three
setups (indicated by different lines in each plot). The insets show the same
plots but in log-log scale. The number in each sub-figure corresponds to the
system number in parameter Tab. 2. The insets show sections of the main plots
as log-log plots.

Note that in the Fano-mirror setup, there is an extra term in the evolution of
the photon number (Eq. (27)) due to the added mirror mode. This extra term
does neither interact with the mechanics nor is it leaking out of the cavity.
If interested in this extra cost of the interaction between the cavity and the
Fano-mirror mode as well, one can take this term into account as one of the
resources in the efficiency, which then gives

$\\!\\!\\!\eta_{\text{conv}}^{\prime}=\frac{g\imaginary(\langle\delta\hat{b}^{\dagger}(\delta\hat{a}^{\dagger}+\delta\hat{a})\rangle)}{g\imaginary(\langle\delta\hat{a}(\delta\hat{b}^{\dagger}+\delta\hat{b})\rangle)-\real(\mathcal{G}\langle\delta\hat{d}\delta\hat{a}^{\dagger}\rangle)}.\\!$
(37)

Figure 9 shows the conversion efficiency, $\eta_{\text{conv}}$, as a function
of cooperativity for all setups and systems. One first of all observes that
the conversion efficiency is by orders of magnitude larger than the full
cooling efficiency and can even reach values up to 1. This means that the
largest reduction of resources provided by the laser comes from the re-usable
laser power, which is accounted for in $\eta_{\text{conv}}$ but not in
$\eta_{\text{full}}$. At the same time, it shows that it is the conversion
efficiency, $\eta_{\text{conv}}$, which provides the deepest insights into the
fundamental physics of our cooling process.

Interestingly, while the full efficiency $\eta_{\text{full}}$ rapidly
decreases at those cooperativity values at which the evacuated-heat flow
starts to be considerably large, the conversion efficiency,
$\eta_{\text{conv}}$ stays close to maximal up to those cooperativity values,
at which the phonon number, $\bar{n}_{\text{fin}}$ approaches its minimum.

The conversion efficiency is overall the largest for the Fano-mirror setup,
while it is very similar for the squeezed-light and standard setups. This
similarity between standard and squeezed-light setup reflects that the effect
of squeezing is mainly a reduction of the temperature of the cold effective
phonon bath. Instead, the presence of the Fano mirror improves the conversion
process. However, the modified conversion efficiency of the Fano-mirror setup,
$\eta_{\text{conv}}^{\prime}$, which takes into account the cost of the
interaction between the cavity and Fano-mirror modes (light green dotted
line), is reduced by up to several orders of magnitude. That can be seen best
in the insets of the plot panels of Fig. 9.

Figure 10:  Conversion efficiency $\eta_{\text{conv}}$ as a function of the
effective phonon number for the three setups (indicated by different lines in
each plot). The number in each sub-figure corresponds to the system number in
parameter Tab. 2.

This is different only for system (4), which is not in the sideband-resolved
regime. For system (4), as expected, the conversion efficiency is smaller than
for the other systems. Surprisingly, the Fano-mirror setup is significantly
more efficient than the other two setups, even when looking at
$\eta_{\text{conv}}^{\prime}$, which means that the asymmetry in the density
of states of the cavity created by the Fano mirror can provide the same
evacuated-heat flow out of the non-sideband resolved system, at the cost of
much less resources than what is required for the standard setup using
coherent or squeezed light.

The relation between the conversion efficiency and the smallest reachable
steady-state phonon number, $\bar{n}_{\text{fin}}$, is plotted in Fig. 10.
While also here a trade-off can be observed, importantly, the conversion
efficiency is shown to be large down to phonon numbers that are very close to
the lowest reachable values.

##  IV Conclusion

In this work, we have provided a detailed thermodynamic analysis of
optomechanical sideband cooling for different setups of cavity optomechanics—a
standard setup employing a coherent laser drive or a squeezed laser drive, as
well as a setup with a Fano mirror as one of the cavity mirrors. We have
demonstrated how optomechanical cooling is realized by powering a heat valve
that connects the mechanical resonator, heated by its coupling to a mechanical
bath, to an engineered cold phonon bath provided by the optical driving. We
have analyzed the lowest steady-state phonon number that can be reached for
the mechanical system, together with the evacuated-heat flow and the
efficiencies of the cooling process. The latter are additional, complementary
performance quantifiers that we have identified in our heat-valve setting and
that turn out to be particularly insightful in the analysis put forward here.

This comprehensive theoretical analysis, which goes beyond the validity regime
of previous studies of sideband cooling [47], employs realistic numbers from a
broad range of possible experimental implementations [56, 57, 64, 25, 27, 65,
63]. Our analysis has two main benefits. First, it provides an improved
understanding of the underlying processes and the required mechanisms and
resources for different optomechanical cooling setups. Second, the
understanding of the thermodynamics underlying sideband cooling, as discussed
in this work, will enable further development of thermodynamic machines
exploiting the heat flow between hot phonon bath and cold bath, engineered by
the laser-light induced phonon-photon coupling.

## Acknowledgments

We thank Alexia Auffèves, Andrew Jordan and Nikolai Kiesel for helpful
discussions. We acknowledge funding from the Knut and Alice Wallenberg
foundation through two Wallenberg Academy fellowships (J.S. and W.W.), from
the Vetenskapsrådet, Swedish VR, under project numbers 2018-05061 (J.S. and
J.M.) and 2019-04946 (W.W.) and the QuantERA project C’MON-QSENS! This work
was performed in the context of the Excellence Initiative Nano at Chalmers.

##  Appendix A Specific parameters used in the figures

All the parameters used in the figures are gathered in Tab. 2. The columns
correspond to each of the studied systems. The uppermost section of the table
contains the actual experimental parameters, the second one gives some
insightful parameter ratios, the third one corresponds to the assumed
parameters we used for the squeezed-light and Fano-mirror setups and the last
one indicates the minimum achievable phonon number $\bar{n}_{\text{fin}}$ in
the mechanical resonator for each of the setups.

Quantity of interest | Symbol |  System (1) |  System (2) |  System (3) |  System (4)
---|---|---|---|---|---
|  |  |  [64, 25] |  [27, 65] |  [63]
|  |  sideband-resolved, MHz |  sideband-resolved, GHz |  sideband-resolved, levitated |  non-sideband-resolved
Mechanical frequency | $\Omega_{\text{mec}}/2\pi$ |  $1\cdot 10^{6}$ Hz |  $3.7\cdot 10^{9}$ Hz |  $3.05\cdot 10^{5}$ Hz |  $1.14\cdot 10^{6}$ Hz
Mechanical damping rate | $\gamma/2\pi$ |  0.1 Hz |  $3.5\cdot 10^{4}$ Hz |  $1.6\cdot 10^{-4}$ Hz |  $1.1\cdot 10^{-3}\,$ Hz
Mechanical environment temperature | $T_{\text{mec}}$ |  4 K |  $20$ K |  $300$ K |  $10$ K
Average phonon number at $T_{\text{mec}}$ | $\bar{n}_{\text{mec}}$ |  $8.3\cdot 10^{4}$ |  $7.1\cdot 10^{2}$ |  $1.3\cdot 10^{8}$ |  $1.1\cdot 10^{6}$
Laser wavelength | $\lambda_{\text{las}}$ |  1550 nm |  $1537$ nm |  $1064$ nm |  $795$ nm
Cavity loss rate | $\kappa/2\pi$ |  $2\cdot 10^{5}$ Hz |  $5\cdot 10^{8}$ Hz |  $1.93\cdot 10^{5}$ Hz |  $1.59\cdot 10^{7}$ Hz
Cavity length | $L_{\text{cav}}$ |  $3.75\cdot 10^{-3}$ m |  $3\cdot 10^{-6}\,$m |  $1.07\cdot 10^{-2}$ m |  $1.6\cdot 10^{-3}$ m
Cavity free spectral range | $\Gamma/2\pi$ |  $4.0\cdot 10^{10}$ Hz |  $1.44\cdot 10^{13}$ Hz |  $1.40\cdot 10^{10}$ Hz |  $9.38\cdot 10^{10}$ Hz
Single-photon coupling strength | $g_{0}/2\pi$ |  10 Hz |  $9.1\cdot 10^{5}$ Hz |  $0.3$ Hz |  $129$ Hz
Sideband resolution | $\Omega_{\text{mec}}/\kappa$ |  $5$ |  $7.4$ |  $1.58$ |  $0.07$
Granularity parameter | $g_{0}/\kappa$ |  $5.0\cdot 10^{-5}$ |  $1.8\cdot 10^{-3}$ |  $1.5\cdot 10^{-6}$ |  $8.1\cdot 10^{-6}$
Mechanical quality factor | $Q_{\text{mec}}=\Omega_{\text{mec}}/\gamma$ |  $1.0\cdot 10^{7}$ |  $1\cdot 10^{5}$ |  $1.9\cdot 10^{9}$ |  $1.03\cdot 10^{9}$
Single-photon quantum cooperativity | $C_{0}=2g_{0}^{2}/\kappa\gamma\bar{n}_{\text{mec}}$ |  $6.0\cdot 10^{-8}$ |  $6.6\cdot 10^{-5}$ |  $2.2\cdot 10^{-11}$ |  $8.6\cdot 10^{-7}$
Number of coherent oscillations | $Q_{\text{mec}}/\bar{n}_{\text{mec}}$ |  $120.0$ |  $140.8$ |  $14.8$ |  $936.7$
Effective detuning | $\Delta=\Delta_{0}-g_{0}\sqrt{2}\bar{q}$ |  $\Omega_{\text{mec}}$ |  $\Omega_{\text{mec}}$ |  $\Omega_{\text{mec}}$ |  $\kappa$
Squeezing angle | $\theta$ |  $0.835$ rad |  $0.819$ rad |  $0.939$ rad |  $1.11$ rad
Squeezing ratio | $r_{\text{s}}$ |  $0.050$ |  $0.034$ |  $0.154$ |  $0.711$
Squeezing level111111This is the amount of squeezing in dB below shot noise, with the variance of shot noise being $1/2$ and $\langle\Delta P^{\theta}_{\text{in,s}}\rangle$ the variance of the squeezed quadrature. | $\\!\\!\\!-10\log_{10}(2\langle\Delta P^{\theta}_{\text{in,s}}\rangle)$ |  0.87 dB |  0.59 dB |  2.7 dB |  15.4 dB
Fano mode loss rate | $\gamma_{d}/2\pi$ |  $8.0\cdot 10^{7}$ Hz |  $4.38\cdot 10^{11}$ Hz |  $7.71\cdot 10^{6}$ Hz |  $1.31\cdot 10^{6}$ Hz
Fano setup, left-side loss rate | $\kappa_{\text{L}}/2\pi$ |  $8.0\cdot 10^{10}$ Hz |  $2.88\cdot 10^{13}$ Hz |  $2.80\cdot 10^{10}$ Hz |  $1.88\cdot 10^{11}$ Hz
Fano setup, right-side loss rate | $\kappa_{\text{R}}/2\pi$ |  $5.0\cdot 10^{7}$ Hz |  $8.22\cdot 10^{9}$ Hz |  $1.75\cdot 10^{8}$ Hz |  $5.70\cdot 10^{11}$ Hz
Minimum phonon number, standard setup | $\min(\bar{n}_{\text{fin}}^{\text{standard}})$ |  0.077 |  0.021 |  0.21 |  6.7
Minimum phonon number, squeezed-light setup | $\min(\bar{n}_{\text{fin}}^{\text{suqeezing}})$ |  0.066 |  0.016 |  0.096 |  0.039
Minimum phonon number, Fano-mirror setup | $\min(\bar{n}_{\text{fin}}^{\text{Fano}})$ |  0.073 |  0.016 |  0.091 |  2.4
Table 2:  Parameters of four different optomechanical systems that we consider
for our thermodynamic analysis of optomechanical cooling. The last section in
the table indicates the minimum achievable steady-state phonon number
$\bar{n}_{\text{fin}}$ for each of the setups and the section above
corresponds to the assumed parameters we used in the figures.

##  Appendix B Standard setup

###  B.1 Steady state and linearization

The semi-classical steady state of the optomechanical system is given by Eqs.
(14), which are actually a non-linear system of equations because the
effective detuning $\Delta$ depends on the steady-state mechanical position
$\bar{q}$ [23]. As a consequence, the average photon number in the cavity,
$|\alpha|^{2}$, is a solution of the cubic equation

$|\alpha|^{2}\left(\kappa^{2}+\left(\Delta_{0}-\frac{2g_{0}^{2}}{\Omega_{\text{mec}}}|\alpha|^{2}\right)^{2}\right)=\absolutevalue{\epsilon}^{2}.$
(38)

Therefore there are two different regimes depending on the optomechanical
parameters and the laser power and detuning: one where the above equation has
a single real root and one where it has three. The systems (see Tab. 2) with
the respective detunings considered in this article are in the first regime,
therefore there is a unique steady state $(\alpha,\bar{q})$. On the other
hand, in the second regime there are multiple possible steady states, see for
instance [55], Chap. 2, for more details.

We study the evolution of the optomechanical system around this steady state
by splitting the Heisenberg operators in the Langevin equations (2) into a
steady-state value and a fluctuation operator: $\hat{a}=\alpha+\delta\hat{a}$,
$\hat{q}=\bar{q}+\delta\hat{q}$ and $\hat{p}=\bar{p}+\delta\hat{p}$, with
$\bar{p}=0$. Then, we use Eqs. (14) and neglect the second order terms (in
$\delta\hat{a}\delta\hat{q}$ and $\delta\hat{a}^{\dagger}\delta\hat{a}$) to
obtain the linearized Langevin equations (22). In these equations, the non-
dissipative part of the evolution is governed by the quadratic Hamiltonian
$\hat{H}_{\text{lin}}$, given in Eq. (II.2).

Figure 11:  Stability condition $S_{\text{RH}}$ as a function of the
cooperativities considered in this article for each of the systems from Tab.
2. $S_{\text{RH}}$ has been normalized by its value at zero coupling,
$S_{\text{RH},0}=\Omega_{\text{mec}}(\kappa^{2}+\Delta^{2})$.

The stability of the optomechanical system can be determined by applying the
Routh-Hurwitz criterion, from which emerges the stability condition
$S_{\text{RH}}>0$ [24, 23], with

$S_{\text{RH}}=\Omega_{\text{mec}}(\kappa^{2}+\Delta^{2})-4g^{2}\Delta.$ (39)

As can be seen in Fig. 11, the four systems studied in this article fulfill
this stability condition for all the considered values of the coupling $g$.

###  B.2 Solving the dynamics

####  B.2.1 Solution of the Langevin equations

The system of equations (22) can be solved in the frequency domain [23]. We
have used the following convention for the Fourier transform:
$\hat{f}[\omega]=\int_{-\infty}^{+\infty}\differential t\text{e}^{i\omega
t}\hat{f}(t)$. In particular, the expression of the mechanical position is
given by

$\displaystyle\chi_{\text{mec}}^{\text{eff}}[\omega]^{-1}\delta\hat{q}[\omega]=$
$\displaystyle\frac{\sqrt{2\kappa}2g\left((\kappa-i\omega)\hat{X}_{\text{in}}[\omega]+\Delta\hat{P}_{\text{in}}[\omega]\right)}{(\kappa-i\omega)^{2}+\Delta^{2}}$
$\displaystyle+\sqrt{\gamma}\hat{\xi}[\omega],$ (40)

with the input noise quadratures
$\hat{X}_{\text{in}}=(\hat{a}_{\text{in}}+\hat{a}_{\text{in}}^{\dagger})/\sqrt{2}$
and
$\hat{P}_{\text{in}}=i(\hat{a}_{\text{in}}^{\dagger}-\hat{a}_{\text{in}})/\sqrt{2}$.
The mechanical susceptibility is given by

$\chi_{\text{mec}}^{\text{eff}}[\omega]=\Omega_{\text{mec}}\left(\Omega_{\text{mec}}^{2}-\omega^{2}-i\omega\gamma-\frac{4g^{2}\Delta\Omega_{\text{mec}}}{(\kappa-i\omega)^{2}+\Delta^{2}}\right)^{-1}\\!\\!\\!.$
(41)

We can put it in the usual form for a harmonic oscillator [7], that is
$\chi_{\text{mec}}^{\text{eff}}[\omega]=\Omega_{\text{mec}}\left(\Omega_{\text{mec}}^{\text{eff}}[\omega]^{2}-\omega^{2}-i\omega\gamma^{\text{eff}}[\omega]\right)^{-1}$,
identifying the effective mechanical frequency
$\Omega_{\text{mec}}^{\text{eff}}[\omega]=\Omega_{\text{mec}}+\delta\Omega_{\text{mec}}[\omega]$
and damping rate
$\gamma^{\text{eff}}[\omega]=\gamma+\Gamma_{\text{opt}}[\omega]$. We have
denoted

$\displaystyle\delta\Omega_{\text{mec}}[\omega]$
$\displaystyle=-\frac{2g^{2}\Delta(\kappa^{2}-\omega^{2}+\Delta^{2})}{(\kappa^{2}+(\omega-\Delta)^{2})(\kappa^{2}+(\omega+\Delta)^{2})},$
(42) $\displaystyle\Gamma_{\text{opt}}[\omega]$
$\displaystyle=\frac{8g^{2}\Delta\Omega_{\text{mec}}\kappa}{(\kappa^{2}+(\omega-\Delta)^{2})(\kappa^{2}+(\omega+\Delta)^{2})}.$
(43)

$\delta\Omega_{\text{mec}}[\omega]$ is the frequency shift due to the optical
spring effect [7]. In the weak-coupling regime, we can neglect the
contributions to the evolution of the non-resonant frequencies, therefore the
effective frequency from the master equation (II.2) is equal to
$\Omega_{\text{mec}}^{\text{eff}}[\Omega_{\text{mec}}]$ and the cold bath
damping rate is given by $\Gamma_{\text{opt}}[\Omega_{\text{mec}}]$.

####  B.2.2 Lyapunov equation

From the Langevin equations (22) and using the correlation functions of the
noise, Eqs. (3) and (6), we obtain the evolution of the second-order moments
of the quadratures:

$\displaystyle\derivative{\langle\delta\hat{X}_{a}^{2}\rangle}{t}=\,$
$\displaystyle
2{\Delta}\langle\delta\hat{X}_{a}\delta\hat{P}_{a}\rangle-2\kappa\langle\delta\hat{X}_{a}^{2}\rangle+\kappa,$
(44a)
$\displaystyle\derivative{\langle\delta\hat{X}_{a}\delta\hat{P}_{a}\rangle}{t}=\,$
$\displaystyle{\Delta}(\langle\delta\hat{P}_{a}^{2}\rangle-\langle\delta\hat{X}_{a}^{2}\rangle)+2g\langle\delta\hat{X}_{a}\delta\hat{q}\rangle-2\kappa\langle\delta\hat{X}_{a}\delta\hat{P}_{a}\rangle,$
(44b)
$\displaystyle\derivative{\langle\delta\hat{X}_{a}\delta\hat{q}\rangle}{t}=\,$
$\displaystyle\Omega_{\text{mec}}\langle\delta\hat{X}_{a}\delta\hat{p}\rangle+{\Delta}\langle\delta\hat{P}_{a}\delta\hat{q}\rangle-\kappa\langle\delta\hat{X}_{a}\delta\hat{q}\rangle,$
(44c)
$\displaystyle\derivative{\langle\delta\hat{X}_{a}\delta\hat{p}\rangle}{t}=\,$
$\displaystyle-\Omega_{\text{mec}}\langle\delta\hat{X}_{a}\delta\hat{q}\rangle+{\Delta}\langle\delta\hat{P}_{a}\delta\hat{p}\rangle+2g\langle\delta\hat{X}_{a}^{2}\rangle-(\gamma+\kappa)\langle\delta\hat{X}_{a}\delta\hat{p}\rangle,$
(44d) $\displaystyle\derivative{\langle\delta\hat{P}_{a}^{2}\rangle}{t}=\,$
$\displaystyle-2{\Delta}\langle\delta\hat{X}_{a}\delta\hat{P}_{a}\rangle+4g\langle\delta\hat{P}_{a}\delta\hat{q}\rangle-2\kappa\langle\delta\hat{P}_{a}^{2}\rangle+\kappa,$
(44e)
$\displaystyle\derivative{\langle\delta\hat{P}_{a}\delta\hat{q}\rangle}{t}=\,$
$\displaystyle-{\Delta}\langle\delta\hat{X}_{a}\delta\hat{q}\rangle+\Omega_{\text{mec}}\langle\delta\hat{P}_{a}\delta\hat{p}\rangle-\kappa\langle\delta\hat{P}_{a}\delta\hat{q}\rangle+2g\langle\delta\hat{q}^{2}\rangle,$
(44f)
$\displaystyle\derivative{\langle\delta\hat{P}_{a}\delta\hat{p}\rangle}{t}=\,$
$\displaystyle-{\Delta}\langle\delta\hat{X}_{a}\delta\hat{p}\rangle-\Omega_{\text{mec}}\langle\delta\hat{P}_{a}\delta\hat{q}\rangle-(\gamma+\kappa)\langle\delta\hat{P}_{a}\delta\hat{p}\rangle+2g(\langle\delta\hat{X}_{a}\delta\hat{P}_{a}\rangle+\langle\delta\hat{q}\delta\hat{p}\rangle),$
(44g) $\displaystyle\derivative{\langle\delta\hat{q}^{2}\rangle}{t}=\,$
$\displaystyle 2\Omega_{\text{mec}}\langle\delta\hat{q}\delta\hat{p}\rangle,$
(44h)
$\displaystyle\derivative{\langle\delta\hat{q}\delta\hat{p}\rangle}{t}=\,$
$\displaystyle\Omega_{\text{mec}}(\langle\delta\hat{p}^{2}\rangle-\langle\delta\hat{q}^{2}\rangle)+2g\langle\delta\hat{X}_{a}\delta\hat{q}\rangle-\gamma\langle\delta\hat{q}\delta\hat{p}\rangle,$
(44i) $\displaystyle\derivative{\langle\delta\hat{p}^{2}\rangle}{t}=\,$
$\displaystyle-2\Omega_{\text{mec}}\langle\delta\hat{q}\delta\hat{p}\rangle+4g\langle\delta\hat{X}_{a}\delta\hat{p}\rangle-2\gamma\langle\delta\hat{p}^{2}\rangle+{\gamma}(2\bar{n}_{\text{mec}}+1),$
(44j)

where we have defined the optical quadratures
$\delta\hat{X}_{a}=(\delta\hat{a}+\delta\hat{a}^{\dagger})/\sqrt{2}$ and
$\delta\hat{P}_{a}=(\delta\hat{a}-\delta\hat{a}^{\dagger})/i\sqrt{2}$. From
these equations, we get the evolution of the photon and phonon numbers,
$\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle=\frac{1}{2}(\langle\delta\hat{X}_{a}^{2}\rangle+\langle\delta\hat{P}_{a}^{2}\rangle-1)$
and
$\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle=\frac{1}{2}(\langle\delta\hat{q}^{2}\rangle+\langle\delta\hat{p}^{2}\rangle-1)$:

$\displaystyle\derivative{\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle}{t}$
$\displaystyle=2g\langle\delta\hat{P}_{a}\delta\hat{q}\rangle-2\kappa\langle\delta\hat{a}^{\dagger}\delta\hat{a}\rangle,$
(45a)
$\displaystyle\derivative{\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle}{t}$
$\displaystyle=2g\langle\delta\hat{X}_{a}\delta\hat{p}\rangle+\gamma\left(\bar{n}_{\text{mec}}+\frac{1}{2}-\langle\delta\hat{p}^{2}\rangle\right)$
$\displaystyle=-I^{\text{phonon}}_{\text{c}}-I^{\text{phonon}}_{\text{h}}.$
(45b)

We have identified the phonon flows
$I^{\text{phonon}}_{\text{c}}=-2g\langle\delta\hat{X}_{a}\delta\hat{p}\rangle$
and
$I^{\text{phonon}}_{\text{h}}=-\gamma\left(\bar{n}_{\text{mec}}+\frac{1}{2}-\langle\delta\hat{p}^{2}\rangle\right)$
directed towards the cold and hot bath, respectively. Similarly, the flow of
photons coming from the cavity and interacting with the mechanics reads
$I^{\text{photon}}=2g\langle\delta\hat{P}_{a}\delta\hat{q}\rangle$. Rewriting
Eqs. (45) fully in terms of the annihilation and creation operators, we obtain
Eqs. (23).

The linearized Hamiltonian $\hat{H}_{\text{lin}}$ (Eq. (II.2)) is Gaussian.
Therefore, the system of equations (44) can be rewritten in the form of a
Lyapunov equation for $V$, the covariance matrix of the quadratures,

$\displaystyle\derivative{V}{t}=AV+VA^{T}+B.$ (46)

The elements of covariance matrix $V$ are defined as

$\displaystyle V_{ij}=\dfrac{1}{2}\langle\\{Y_{i},Y_{j}\\}\rangle-\langle
Y_{i}\rangle\langle Y_{j}\rangle,$ (47)

where $\vec{Y}$ is the vector of operators
$\left(\delta\hat{X}_{a},\delta\hat{P}_{a},\delta\hat{q},\delta\hat{p}\right)$.
We have defined the matrices

$\displaystyle A$ $\displaystyle=\begin{bmatrix}-\kappa&\Delta&0&0\\\
-\Delta&-\kappa&2g&0\\\ 0&0&0&\Omega_{\text{mec}}\\\
2g&0&-\Omega_{\text{mec}}&-\gamma\end{bmatrix},$ (48a) $\displaystyle B$
$\displaystyle=\begin{bmatrix}\kappa&0&0&0\\\ 0&\kappa&0&0\\\ 0&0&0&0\\\
0&0&0&\gamma\left(2\bar{n}+1\right)\end{bmatrix}.$ (48b)

We then determine the steady-state $\bar{V}$ by solving
$0=A\bar{V}+\bar{V}A^{T}+B$ and obtain the steady-state phonon number

$\bar{n}_{\text{fin}}=\frac{1}{2}(\bar{V}_{33}+\bar{V}_{44}-1).$ (49)

Furthermore, the steady-state efficiencies (Eqs. (36) and (13)) can be
expressed as

$\displaystyle\eta_{\text{full}}$
$\displaystyle=-\frac{4\kappa\Omega_{\text{mec}}g\bar{V}_{14}}{\omega_{\text{las}}\absolutevalue{\epsilon}^{2}},$
(50) $\displaystyle\eta_{\text{conv}}$
$\displaystyle=-\frac{\bar{V}_{14}}{\bar{V}_{23}}.$ (51)

We therefore obtain the following analytical expressions

$\displaystyle\bar{n}_{\text{fin}}$
$\displaystyle=\frac{g^{2}c_{1}+\gamma\bar{n}_{\text{mec}}c_{2}}{c_{3}S_{\text{RH}}},$
(52) $\displaystyle\eta_{\text{full}}$
$\displaystyle=\frac{(4\Omega_{\text{mec}}g_{0}^{2}{\gamma}{\kappa}^{2})(c_{4}+\bar{n}_{\text{mec}}c_{5})}{{\left({\Delta}^{2}+{\kappa}^{2}\right)}\omega_{\text{las}}c_{3}},$
(53) $\displaystyle\eta_{\text{conv}}$ $\displaystyle=\frac{\gamma
S_{\text{RH}}(c_{4}+\bar{n}_{\text{mec}}c_{5})}{\Omega_{\text{mec}}(c_{6}+\gamma\bar{n}_{\text{mec}}c_{7})},$
(54)

with $S_{\text{RH}}$ defined by Eq. (39) and

$\displaystyle c_{1}=\,$
$\displaystyle\Big{[}{\Omega_{\text{mec}}}({\Delta}^{2}+{\kappa}^{2})^{2}+{\left({\Delta}^{2}-4\,{\Delta}{\Omega_{\text{mec}}}+2\,{\Omega_{\text{mec}}}^{2}+{\kappa}^{2}\right)}S_{\text{RH}}\Big{]}\kappa^{2}$
$\displaystyle+\gamma\Big{[}{\Delta}^{2}({\Delta}^{3}+{\Omega_{\text{mec}}}^{3})-2\,{\Delta}{\Omega_{\text{mec}}}^{2}g^{2}+(2{\Delta}^{3}+2{\Delta}^{2}{\Omega_{\text{mec}}}+{\Omega_{\text{mec}}}^{3}){\kappa}^{2}+{\left({\Delta}+2\,{\Omega_{\text{mec}}}\right)}{\kappa}^{4}$
$\displaystyle\hskip
22.76228pt+{\left({\Delta}^{2}-3\,{\Delta}{\Omega_{\text{mec}}}+{\kappa}^{2}\right)}S_{\text{RH}}\Big{]}\kappa$
$\displaystyle+\gamma^{2}\Big{[}2{\Delta}^{2}\Omega_{\text{mec}}g^{2}+{\left(2{\Delta}+\Omega_{\text{mec}}\right)}{\kappa}^{4}+{\Delta}^{2}{\left(2{\Delta}+\Omega_{\text{mec}}\right)}{\kappa}^{2}\Big{]}+\gamma^{3}(\Delta^{2}+\kappa^{2})\kappa\Delta,$
(55a) $\displaystyle c_{2}=\,$
$\displaystyle\Big{[}\Omega_{\text{mec}}{\kappa}^{6}+{\left(3{\Delta}^{2}\Omega_{\text{mec}}+2\Omega_{\text{mec}}^{3}-2{\Delta}g^{2}\right)}{\kappa}^{4}+{\left((3{\Delta}^{4}+\Omega_{\text{mec}}^{4})\Omega_{\text{mec}}-2(2{\Delta}^{2}+3\Omega_{\text{mec}}^{2})\Delta
g^{2}\right)}{\kappa}^{2}$ $\displaystyle\hskip
5.69046pt+\Delta{\left(({\Delta}^{2}-\Omega_{\text{mec}}^{2})^{2}\Delta\Omega_{\text{mec}}-2({\Delta}^{4}-5{\Delta}^{2}\Omega_{\text{mec}}^{2}+2\Omega_{\text{mec}}^{4})g^{2}-8{\Delta}\Omega_{\text{mec}}g^{4}\right)}\Big{]}\kappa$
$\displaystyle+\gamma\Big{[}2\Omega_{\text{mec}}{\kappa}^{6}+2{\left(2{\Delta}^{2}\Omega_{\text{mec}}+\Omega_{\text{mec}}^{3}-2{\Delta}g^{2}\right)}{\kappa}^{4}+2\Delta{\left(\Delta\Omega_{\text{mec}}({\Delta}^{2}+\Omega_{\text{mec}}^{2})-(2{\Delta}^{2}+3\Omega_{\text{mec}}^{2})g^{2}\right)}{\kappa}^{2}$
$\displaystyle\hskip
21.33955pt+2g^{2}{\Delta}^{2}\Omega_{\text{mec}}\left({\Delta}\Omega_{\text{mec}}-2g^{2}\right)\Big{]}+\gamma^{2}\Big{[}\Omega_{\text{mec}}{\kappa}^{4}+2{\left({\Delta}\Omega_{\text{mec}}-g^{2}\right)}\Delta{\kappa}^{2}+{\left({\Delta}\Omega_{\text{mec}}-2g^{2}\right){\Delta}^{3}}\Big{]}\kappa,$
(55b) $\displaystyle c_{3}=\,$ $\displaystyle
8{\Delta}\Omega_{\text{mec}}g^{2}{\kappa}^{2}+\gamma\left[{\kappa}^{4}+2{\left({\Delta}^{2}+\Omega_{\text{mec}}^{2}\right)}{\kappa}^{2}+{({\Delta}^{2}-\Omega_{\text{mec}}^{2})^{2}+8{\Delta}\Omega_{\text{mec}}g^{2}}\right]\kappa$
$\displaystyle+2{\gamma}^{2}{\left[{\Delta}\Omega_{\text{mec}}g^{2}+({\Delta}^{2}+\Omega_{\text{mec}}^{2}){\kappa}^{2}+{\kappa}^{4}\right]}+{\gamma}^{3}{\left({\Delta}^{2}+{\kappa}^{2}\right)}\kappa$
(55c) $\displaystyle c_{4}=\,$
$\displaystyle-\left({\kappa}^{2}+{\left({\Delta}-\Omega_{\text{mec}}\right)}^{2}\right){\kappa}+{\left({\Delta}\Omega_{\text{mec}}+{\kappa}^{2}-{\Delta}^{2}\right)}{\gamma}$
(55d) $\displaystyle c_{5}=\,$ $\displaystyle
2{\Delta}\Omega_{\text{mec}}{\left({\gamma}+2{\kappa}\right)}$ (55e)
$\displaystyle c_{6}=\,$ $\displaystyle
4{\left({\Delta}^{2}+{\kappa}^{2}\right)}g^{2}{\kappa}^{2}+\gamma\Big{[}({\kappa}^{2}+\Delta^{2})^{2}+2g^{2}(4\kappa^{2}+\Omega_{\text{mec}}^{2})+(\Omega_{\text{mec}}-2\Delta)S_{\text{RH}}\Big{]}\kappa$
$\displaystyle+\gamma^{2}\Big{[}2{\kappa}^{4}+{\left({\Delta}^{2}+({\Delta}-\Omega_{\text{mec}})^{2}+4g^{2}\right)}{\kappa}^{2}+{\Delta}^{2}\Omega_{\text{mec}}(\Omega_{\text{mec}}-{\Delta})+2{\Delta}{\left(2{\Delta}-\Omega_{\text{mec}}\right)}g^{2}\Big{]}+\gamma^{3}{\left({\Delta}^{2}+{\kappa}^{2}\right)}{\kappa}$
(55f) $\displaystyle c_{7}=\,$
$\displaystyle\Big{[}({\kappa}^{2}+\Delta^{2})^{2}+\Omega_{\text{mec}}S_{\text{RH}}\Big{]}2\kappa+\gamma\Big{[}(\Omega_{\text{mec}}^{2}+4\kappa^{2})(\kappa^{2}+\Delta^{2})+\Omega_{\text{mec}}S_{\text{RH}}\Big{]}+\gamma^{2}{\left({\Delta}^{2}+{\kappa}^{2}\right)}2{\kappa}$
(55g)

When $g$ reaches the critical value

$g_{\textbf{crit}}=\sqrt{\frac{\Omega_{\text{mec}}}{4\Delta}(\kappa^{2}+\Delta^{2})},$
(56)

the matrix $A$ is not invertible and the Lyapunov equation (46) does not have
a steady state. In particular, $\bar{n}_{\text{fin}}$ diverges, as visible in
Fig. 4. This critical value of the coupling $g$ corresponds to the point at
which the stability condition $S_{\text{RH}}>0$ (see Appendix B.1) is no
longer fulfilled.

####  B.2.3 Beyond the white-noise approximation

According to [82], the white-noise approximation for the mechanical noise (Eq.
(3)) gives inconsistent results, even at high temperature because the
commutation relation $[\hat{q},\hat{p}]=i$ is not preserved and it yields a
spurious term in the phase fluctuation spectrum. Depending on the quantities
of interest this does not necessarily impact the results, like in our case
(see Appendix B.5). Nevertheless, to restore the correct commutation relation,
the thermal noise operator should fulfill the following relations

$\displaystyle\\!\\!\\!\langle\hat{\xi}(t)\hat{\xi}(t^{\prime})\rangle$
$\displaystyle=\int\\!\\!\differential\omega\frac{\omega\text{e}^{-i\omega(t-t^{\prime})}}{2\pi}\frac{\coth(\frac{\hbar\omega}{2k_{\text{B}}T_{\text{mec}}})+1}{\Omega_{\text{mec}}},\\!\\!$
(57) $\displaystyle\\!\\!\\![\hat{\xi}(t),\hat{\xi}(t^{\prime})]$
$\displaystyle=2i\frac{1}{\Omega_{\text{mec}}}\derivative{t}\delta(t-t^{\prime}).$
(58)

To compute the steady-state phonon number without making the white-noise
approximation, we use the same method as in [23]. The effective phonon number
is given by
$\frac{1}{2}(\langle\delta\hat{q}^{2}\rangle+\langle\delta\hat{p}^{2}\rangle-1)$,
therefore we can compute it from the position and momentum fluctuations:

$\displaystyle\langle\delta\hat{q}^{2}\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{\differential\omega}{2\pi}S_{q}[\omega],$
(59) $\displaystyle\langle\delta\hat{p}^{2}\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{\differential\omega}{2\pi}\frac{\omega^{2}}{\Omega_{\text{mec}}^{2}}S_{q}[\omega].$
(60)

The position fluctuation spectrum,

$\displaystyle S_{q}[\omega]$
$\displaystyle=\int_{-\infty}^{+\infty}\differential t\text{e}^{i\omega
t}\langle\delta\hat{q}(t)\delta\hat{q}(0)\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{\differential\omega^{\prime}}{2\pi}\langle\delta\hat{q}[\omega]\delta\hat{q}[\omega^{\prime}]\rangle,$
(61)

is obtained from the solution of the Langevin equations (B.2.1). In this case,
the noise correlation functions are

$\langle\hat{\xi}[\omega]\hat{\xi}[\omega^{\prime}]\rangle=2\pi\frac{\omega}{\Omega_{\text{mec}}}\left[1+\coth(\frac{\hbar\omega}{2k_{\text{B}}T_{\text{mec}}})\right]\delta(\omega+\omega^{\prime}),$
(62)

and

$\displaystyle\langle\hat{X}_{\text{in}}[\omega]\hat{X}_{\text{in}}[\omega^{\prime}]\rangle$
$\displaystyle=\langle\hat{P}_{\text{in}}[\omega]\hat{P}_{\text{in}}[\omega^{\prime}]\rangle=\pi\delta(\omega+\omega^{\prime})\\!\\!$
(63a)
$\displaystyle\langle\hat{X}_{\text{in}}[\omega]\hat{P}_{\text{in}}[\omega^{\prime}]\rangle$
$\displaystyle=-\langle\hat{P}_{\text{in}}[\omega]\hat{X}_{\text{in}}[\omega^{\prime}]\rangle=i\pi\delta(\omega+\omega^{\prime})\\!\\!\\!$
(63b)

So, we get

$S_{q}[\omega]=\absolutevalue{\chi_{\text{mec}}^{\text{eff}}[\omega]}^{2}(S_{\text{th}}[\omega]+S_{\text{rp}}[\omega])$
(64)

where

$\displaystyle S_{\text{th}}[\omega]$
$\displaystyle=\gamma\frac{\omega}{\Omega_{\text{mec}}}\coth(\frac{\hbar\omega}{2k_{\text{B}}T_{\text{mec}}}),$
(65) $\displaystyle S_{\text{rp}}[\omega]$
$\displaystyle=\frac{4g^{2}\kappa}{\kappa^{2}+(\omega-\Delta)^{2}},$ (66)

are the thermal and radiation pressure noise spectra. To avoid lengthy
computation we did not calculate the analytical expression for the phonon
number, but integrated it numerically instead. We confirmed that it gives
almost identical results as the white-noise approximation for the parameters
considered in this article [see Fig. 13].

For the other quantities, namely $J_{\text{c}}$, $\eta_{\text{full}}$ and
$\eta_{\text{conv}}$, we would also need to compute
$\langle\delta\hat{X}_{a}\delta\hat{p}\rangle$ and
$\langle\delta\hat{P}_{a}\delta\hat{q}\rangle$. This can be done by inserting
the solution of the Langevin equations (22) in the frequency domain into

$\displaystyle\langle\delta\hat{X}_{a}\delta\hat{p}\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{\differential\omega}{2\pi}\int_{-\infty}^{+\infty}\frac{\differential\omega^{\prime}}{2\pi}\langle\delta\hat{X}_{a}[\omega]\delta\hat{p}[\omega^{\prime}]\rangle,$
(67) $\displaystyle\langle\delta\hat{P}_{a}\delta\hat{q}\rangle$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{\differential\omega}{2\pi}\int_{-\infty}^{+\infty}\frac{\differential\omega^{\prime}}{2\pi}\langle\delta\hat{P}_{a}[\omega]\delta\hat{q}[\omega^{\prime}]\rangle.$
(68)

Note that this approach could be used to treat a case where the input noise in
the cavity is not vacuum noise by replacing Eqs. (63) by the appropriate
correlation functions.

###  B.3 Contributions to the evacuated-heat flow

The evacuated-heat flow $J_{\text{c}}$, Eq. (32), can be split into two
contributions:
$J_{\text{c}}=J_{\text{c}}^{\text{BS}}+J_{\text{c}}^{\text{TMS}}$, where
$J_{\text{c}}^{\text{BS}}=ig\langle\delta\hat{a}^{\dagger}\delta\hat{b}-\delta\hat{a}\delta\hat{b}^{\dagger}\rangle$
comes from the beam-splitter interaction term $-\hbar
g(\delta\hat{a}^{\dagger}\delta\hat{b}+\delta\hat{a}\delta\hat{b}^{\dagger})$
in the Hamiltonian $\hat{H}_{\text{lin}}$ (Eq. (II.2)) and
$J_{\text{c}}^{\text{TMS}}=ig\langle\delta\hat{a}\delta\hat{b}-\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}\rangle$
from the two-mode-squeezing term $-\hbar
g(\delta\hat{a}\delta\hat{b}+\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger})$.
In the rotating wave approximation, valid in the weak coupling regime (see
next section B.5) for $\Delta$ close to $\Omega_{\text{mec}}$, only the beam-
splitter contribution is taken into account. Figure 12 indeed shows that in
the resolved-sideband regime (systems (1) to (3)), $J_{\text{c}}^{\text{TMS}}$
is negligible at low $C$, then both contributions diverge while $J_{\text{c}}$
remains constant. In the non-resolved-sideband regime (system (4)),
$J_{\text{c}}$ has the same behavior but the two contributions are always of
the same order of magnitude.

Figure 12:  Beam splitter (dashed dark blue line) and two-mode squeezing
(dotted orange line) contributions to the evacuated-heat flow $J_{\text{c}}$
(solid cyan line) as a function of cooperativity for each of the systems from
Tab. 2. We have actually plotted $-J_{\text{c}}^{\text{TMS}}$ so as to see all
the curves on the same plot in log-log scale.

###  B.4 Measuring the photon and phonon flows

In an experiment, the power spectra of the quadratures $\hat{X}_{\text{out}}$
and $\hat{P}_{\text{out}}$ of the light field leaking out of the cavity can be
measured using a double homodyne detection scheme. They are related to the
fluctuation spectra of the quadratures of the intra-cavity field by

$\displaystyle S_{{X}_{\text{out}}}[\omega]$
$\displaystyle=S_{{X}_{\text{in}}}[\omega]+4\kappa|\alpha|^{2}\delta(\omega)+2\kappa
S_{{X}_{a}}[\omega],$ (69) $\displaystyle S_{{P}_{\text{out}}}[\omega]$
$\displaystyle=S_{{P}_{\text{in}}}[\omega]+2\kappa S_{{P}_{a}}[\omega],$ (70)

where we have used the input-output relation
$\hat{a}_{\text{out}}=\hat{a}_{\text{in}}-\sqrt{2\kappa}\hat{a}$. The power
spectra are defined like in Eq. (61).

Besides, using the equations for the second-order moment (44) in the steady-
state, the phonon and photon flows, $I_{\text{c}}^{\text{phonon}}$ from Eq.
(32) and $I^{\text{photon}}$ from Eq. (34), can be expressed as functions of
$\langle\delta\hat{X}_{a}^{2}\rangle$ and
$\langle\delta\hat{P}_{a}^{2}\rangle$:

$\displaystyle\\!\\!I_{\text{c}}^{\text{phonon}}$
$\displaystyle=\frac{2\kappa}{\Delta\Omega}\left[\kappa^{2}\langle\delta\hat{X}_{a}^{2}\rangle-\Delta^{2}\langle\delta\hat{P}_{a}^{2}\rangle+\frac{\Delta^{2}-\kappa^{2}}{2}\right],\\!$
(71) $\displaystyle\\!\\!I^{\text{photon}}$
$\displaystyle=\kappa\left[\langle\delta\hat{X}_{a}^{2}\rangle+\langle\delta\hat{P}_{a}^{2}\rangle-1\right].$
(72)

For a given operator $\hat{A}$,
$\langle\hat{A}^{2}\rangle=\int_{-\infty}^{+\infty}\frac{\differential{\omega}}{2\pi}S_{A}[\omega]$,
so we obtain

$\displaystyle I_{\text{c}}^{\text{phonon}}=\,$
$\displaystyle\frac{2\kappa}{\Delta\Omega}\int_{-\infty}^{+\infty}\frac{\differential{\omega}}{2\pi}\left(\kappa^{2}S_{{X}_{a}}[\omega]-\Delta^{2}S_{{P}_{a}}[\omega]\right)$
$\displaystyle+\frac{\kappa(\Delta^{2}-\kappa^{2})}{\Delta\Omega},$ (73)
$\displaystyle I^{\text{photon}}=\,$
$\displaystyle\kappa\int_{-\infty}^{+\infty}\frac{\differential{\omega}}{2\pi}\left(S_{{X}_{a}}[\omega]+S_{{P}_{a}}[\omega]\right)-\kappa.$
(74)

The cavity loss rate $\kappa$ and the effective detuning $\Delta$ can be
measured using optomechanically induced transparency, therefore
$I_{\text{c}}^{\text{phonon}}$ and $I^{\text{photon}}$ are accessible
experimentally. Finally, since the laser input power ${P}_{\text{las}}$ can
also be determined, we have access to the evacuated-heat flow $J_{\text{c}}$
and both efficiencies $\eta_{\text{full}}$ and $\eta_{\text{conv}}$.

###  B.5 Comparison of the different approximations

Different approaches involving various approximations are used in the
literature when studying optomechanical systems. Here we provide a quick
overview of their differences and how this impacts the steady-state phonon
number in the case of the setups considered in this article.

The simplest case is to consider the weak-coupling regime (see Ref. [21, 7]
and Sec. II.2) where $g\ll\kappa,\Omega_{\text{mec}}$. This allows to neglect
the mechanical backaction in the fluctuation spectrum of the radiation
pressure force and the steady-state phonon number in the resonator is given by
Eq. (19).

When going beyond the weak coupling limit, two commonly used approaches are
master equations (e.g. [24]) or Langevin equations (e.g. [23, 55]). Modeling
of the Brownian motion of a mechanical resonator is a complex problem, as
highlighted in [82, 83] for instance, and these two approaches usually do it
differently which leads to different values for $\bar{n}_{\text{fin}}$.
Nevertheless, these differences vanish in the limit of high temperature
$T_{\text{mec}}$ and high quality factor $Q_{\text{mec}}$, which is relevant
for the setups studied in this article (see Tab. 2). A detailed derivation of
both the master equation and the Langevin equations valid beyond the high
temperature regime for the mechanical bath is provided in Ref. [83].
Typically, in the master equation approach, the Brownian motion is modeled by
the dissipative term
$\gamma(1+\bar{n}_{\text{mec}})D[\hat{b}]\hat{\rho}+\gamma\bar{n}_{\text{mec}}D[\hat{b}^{\dagger}]\hat{\rho}$
in the evolution of the density operator $\hat{\rho}$ , where
$D[\hat{O}]\hat{\rho}=\hat{O}\hat{\rho}\hat{O}^{\dagger}-\frac{1}{2}\\{\hat{O}^{\dagger}\hat{O},\hat{\rho}\\}$.
This term corresponds to symmetrical noise affecting identically the $\hat{q}$
and $\hat{p}$ quadratures. Conversely, in the Langevin equations, the thermal
noise typically only acts on $\hat{p}$ (like in Eqs. (2)). However,
symmetrical noise can be obtained by making a rotating wave approximation
(RWA) during the derivation of the Langevin equations (see Chap. 1 in Ref.
[55]).

Figure 13:  Comparison of the resonator’s steady-state phonon number
$\bar{n}_{\text{fin}}$ obtain with different methods for each of the systems
from Tab. 2.

Figure 13 shows the resonator’s steady-state phonon number
$\bar{n}_{\text{fin}}$ computed with different methods: the weak-coupling
limit using Eq. (19) (dashed green line), the master equation approach from
Ref. [24] (dotted red line), the same master equation approach but making an
extra RWA (yellow dash-dotted line) and a Langevin approach, going beyond the
white-noise approximation for the mechanical noise like in Appendix B.2.3 and
Ref. [23] (solid blue line). We can see that the master equation and Langevin
approaches give almost identical results for the experimental systems
considered in this article. Note that the approach followed in the main text
to obtain, among others, the plots of $\bar{n}_{\text{fin}}$ in Fig. 4 uses an
intermediate level of approximation: Langevin equations with noise on the
resonator’s momentum only (more accurate than the symmetrical noise in the
master equation) but using the white-noise approximation for the correlations
of the mechanical noise.

The green and yellow curves show us at which cooperativity the weak-coupling
approximation and RWA respectively stop working. The RWA in question here
consists in neglecting the non-resonant terms $\delta\hat{a}\delta\hat{b}$ and
$\delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger}$, and therefore leads to the
Hamiltonian

$\hat{H}_{\text{RWA}}=\hbar{\Delta}\delta\hat{a}^{\dagger}\delta\hat{a}+\hbar\Omega_{\text{mec}}\delta\hat{b}^{\dagger}\delta\hat{b}-\hbar
g(\delta\hat{a}^{\dagger}\delta\hat{b}+\delta\hat{a}\delta\hat{b}^{\dagger}),$
(75)

which corresponds to an intuitive scattering picture since the total number of
particles is conserved.

##  Appendix C Squeezed-light setup

###  C.1 Squeezed noise

The correlation functions for the squeezed noise generated by the setup
described in Sec. II.1.2 read [47, 52]

$\displaystyle\langle\hat{a}^{\dagger}_{\text{in}}(t)\hat{a}_{\text{in}}(t^{\prime})\rangle$
$\displaystyle=\pi_{\text{s}}n_{\text{s}}(t-t^{\prime}),$ (76a)
$\displaystyle\langle\hat{a}_{\text{in}}(t)\hat{a}_{\text{in}}(t^{\prime})\rangle$
$\displaystyle=\pi_{\text{s}}m_{\text{s}}(t-t^{\prime}).$ (76b)

We have defined

$\displaystyle n_{\text{s}}(\tau)$
$\displaystyle=\frac{r_{+}^{2}-r_{-}^{2}}{4}\left(\frac{\text{e}^{-r_{-}\absolutevalue{\tau}}}{2r_{-}}-\frac{\text{e}^{-r_{+}\absolutevalue{\tau}}}{2r_{+}}\right),$
(77a) $\displaystyle m_{\text{s}}(\tau)$
$\displaystyle=\frac{r_{+}^{2}-r_{-}^{2}}{4}\left(\frac{\text{e}^{-r_{-}\absolutevalue{\tau}}}{2r_{-}}+\frac{\text{e}^{-r_{+}\absolutevalue{\tau}}}{2r_{+}}\right)\text{e}^{-i2\theta},$
(77b)

where we have denoted $r_{\pm}=\varkappa\pm\absolutevalue{\chi}$ the decay
rates of the fluctuations of the squeezed and anti-squeezed quadratures. We
have assumed that we have $\varkappa>\absolutevalue{\chi}$. Then, we make the
white-noise approximation ([52], Chap. 10). Namely, we take the limit
$r_{+},r_{-}\rightarrow\infty$ in Eqs. (77) to obtain the expressions of
$N_{\text{s}}$ and $M_{\text{s}}$ (Eqs. (9)) used in the noise autocorrelation
functions (Eqs. (8)). Rewriting $r_{\pm}=\varkappa(1\pm r_{\text{s}})$, where
$r_{\text{s}}$ is the squeezing ratio defined in Sec. II.1.2, we see that the
white-noise approximation can be made for any value of $r_{\text{s}}$ in
$[0,1[$ provided that $\varkappa$ is sufficiently large.

###  C.2 Solving the dynamics

The only differences between the squeezed-light and the standard setup are the
correlation functions of the optical noise, given by Eqs. (8). As a
consequence, the results from Appendix B.2.1 still hold while there are some
small modifications to the evolution of the second-order moments. In
particular, the effective mechanical susceptibility is unchanged (Eq. (41)).
This shows us that in the weak-coupling limit, the resonator’s coupling
strength to the effective cold bath, $\Gamma_{\text{opt}}$, is the same while
$\bar{n}_{\text{opt}}$ can be made smaller than in the standard setup.

####  C.2.1 Lyapunov equation

Like in Section B.2.2, we derive the Lyapunov equation
$\derivative{V}{t}=AV+VA^{T}+B^{\text{squ}}$ for the covariance matrix $V$ of
the quadratures. $A$ is unchanged (Eq. (48a)) while $B^{\text{squ}}=B+\delta
B$, where $B$ is the matrix for the usual vacuum noise, given by Eq. (48b) and
$\delta B$ the additional term coming from the squeezing:

$\\!\\!\\!\frac{\delta
B}{2\kappa\pi_{\text{s}}}=\begin{bmatrix}\\!N_{\text{s}}+\absolutevalue{M_{\text{s}}}\cos(2\theta)\\!\\!&-\absolutevalue{M_{\text{s}}}\sin(2\theta)&\\!0&\\!0\\\
-\absolutevalue{M_{\text{s}}}\sin(2\theta)&\\!\\!N_{\text{s}}-\absolutevalue{M_{\text{s}}}\cos(2\theta)&\\!0&\\!0\\\
0&0&\\!0&\\!0\\\ 0&0&\\!0&\\!0\end{bmatrix}\\!\\!.\\!\\!\\!\vspace{0.01cm}$
(78)

As previously, we determine the steady state from the Lyapunov equation and
compute $\bar{n}_{\text{fin}}$ from Eq. (49) and the efficiencies
$\eta_{\text{conv}}$ and $\eta_{\text{full}}$ from Eqs. (51) and Eq. (50). We
obtain the following analytical expressions

$\displaystyle\\!\\!\bar{n}_{\text{fin}}$
$\displaystyle=\frac{g^{2}(c_{1}+\delta
c_{1})+\gamma\bar{n}_{\text{mec}}c_{2}}{c_{3}S_{\text{RH}}},$ (79)
$\displaystyle\\!\\!\eta_{\text{full}}$
$\displaystyle=\frac{(4\Omega_{\text{mec}}g_{0}^{2}{\gamma}{\kappa}^{2})(c_{4}+\delta
c_{4}+\bar{n}_{\text{mec}}c_{5})}{{\left({\Delta}^{2}+{\kappa}^{2}\right)}\omega_{\text{las}}c_{3}},$
(80) $\displaystyle\\!\\!\eta_{\text{conv}}$ $\displaystyle=\frac{\gamma
S_{\text{RH}}(c_{4}+\delta
c_{4}+\bar{n}_{\text{mec}}c_{5})}{\Omega_{\text{mec}}(c_{6}+\delta
c_{6}+\gamma\bar{n}_{\text{mec}}c_{7})},\\!\\!$ (81)

where $c_{1},\,c_{2},\,c_{3},\,c_{4},\,c_{5},\,c_{6}$ and $c_{7}$ are given by
Eqs. (55) and $S_{\text{RH}}$ is defined by Eq. (39). We have defined the
corrections coming from the squeezing

$\displaystyle\delta c_{1}=\,$ $\displaystyle
2\kappa\pi_{\text{s}}N_{\text{s}}\Big{[}\left(\Omega_{\text{mec}}({\kappa}^{2}+\Delta^{2})^{2}+(\Delta^{2}+2\Omega_{\text{mec}}^{2})S_{\text{RH}}\right)\kappa$
$\displaystyle\hskip
34.14322pt+\gamma\left(\Omega_{\text{mec}}\kappa^{2}(2\kappa^{2}+2\Delta^{2}+\Omega_{\text{mec}}^{2})+(\Delta^{2}+\kappa^{2})S_{\text{RH}}+\Delta\Omega_{\text{mec}}^{2}(\Delta\Omega_{\text{mec}}-2g^{2})\right)+\gamma^{2}{\left({\Delta}^{2}+{\kappa}^{2}\right)}\Omega_{\text{mec}}{\kappa}\Big{]}$
$\displaystyle-2\kappa\pi_{\text{s}}\absolutevalue{M_{\text{s}}}\left(c^{*}\cos(2\theta)+s^{*}\sin(2\theta)\right),$
(82a) $\displaystyle\delta c_{4}=\,$ $\displaystyle
2\pi_{\text{s}}\absolutevalue{M_{\text{s}}}\Big{[}{\left({\Delta}^{2}-\Omega_{\text{mec}}^{2}-{\kappa}^{2}-{\gamma}{\kappa}\right)}{\cos(2\theta)}+\Delta{\left({\gamma}+2{\kappa}\right)}{\sin(2\theta)}\Big{]}\kappa-2\pi_{\text{s}}N_{\text{s}}\Big{[}({\Delta}^{2}+\Omega_{\text{mec}}^{2}+{\kappa}^{2})\kappa+\gamma{\left({\Delta}^{2}+{\kappa}^{2}\right)}\Big{]}$
$\displaystyle=\,$ $\displaystyle\delta c_{6},$ (82b)

with

$\displaystyle s^{*}=$
$\displaystyle\Big{(}2{\left[\Omega_{\text{mec}}({\Delta}^{2}+{\kappa}^{2})+S_{\text{RH}}\right]}{\kappa}+\gamma\left[5\Omega_{\text{mec}}(4{\kappa}^{2}+\Omega_{\text{mec}}^{2})+S_{\text{RH}}\right]+2\gamma^{2}\Omega_{\text{mec}}{\kappa}\Big{)}{\Delta}{\kappa},$
(83a) $\displaystyle c^{*}=\,$
$\displaystyle\Big{[}\Omega_{\text{mec}}({\Delta}^{4}-{\kappa}^{4})+{\left({\Delta}^{2}-2\,{\Omega_{\text{mec}}}^{2}-{\kappa}^{2}\right)}S_{\text{RH}}\Big{]}\kappa+\gamma\Big{[}-3\Omega_{\text{mec}}{\kappa}^{4}+{\left({\Delta}^{2}\Omega_{\text{mec}}-\Omega_{\text{mec}}^{3}+4{\Delta}g^{2}\right)}{\kappa}^{2}+2{\Delta}\Omega_{\text{mec}}^{2}g^{2}\Big{]}$
$\displaystyle+\gamma^{2}{\left({\Delta}+{\kappa}\right)}{\left({\Delta}-{\kappa}\right)}\Omega_{\text{mec}}{\kappa}.$
(83b)

We can see that the stability of the optomechanical system is still given by
the condition $S_{\text{RH}}>0$.

####  C.2.2 Parameter optimization

The optimal squeezing angle $\theta^{*}$ is obtained by finding the minimum of
$\bar{n}_{\text{fin}}(\theta)$. We find
$\cos(2\theta^{*})=c^{*}/\sqrt{c^{*2}+s^{*2}}$ and
$\sin(2\theta^{*})=s^{*}/\sqrt{c^{*2}+s^{*2}}$, where $s^{*}$ and $c^{*}$ are
defined in Eqs. (83). Therefore the optimal squeezing angle depends only on
the optomechanical parameters and not on the other squeezing parameters.

We also minimize $\bar{n}_{\text{fin}}(r_{\text{s}})$ to find the optimal
squeezing ratio $r_{\text{s}}^{*}$. We find that $r_{\text{s}}^{*}$ is a root
of the fourth degree equation

$a^{*}r_{\text{s}}^{4}-4b^{*}r_{\text{s}}^{3}+6a^{*}r_{\text{s}}^{2}-4b^{*}r_{\text{s}}+a^{*}=0,$
(84)

with

$\displaystyle a^{*}=$ $\displaystyle\,c^{*}\cos(2\theta)+s^{*}\sin(2\theta)$
(85a) $\displaystyle b^{*}=$
$\displaystyle\Big{[}\Omega_{\text{mec}}({\kappa}^{2}+\Delta^{2})^{2}+S_{\text{RH}}(\Delta^{2}+\kappa^{2}+2\Omega_{\text{mec}}^{2})\Big{]}\kappa$
$\displaystyle+\gamma\Big{[}\Omega_{\text{mec}}(\kappa^{2}+\Delta^{2})(2\kappa^{2}+\Delta^{2})+(\Omega_{\text{mec}}^{2}+\kappa^{2})S_{\text{RH}}$
$\displaystyle\hskip
21.33955pt-2\Delta^{3}g^{2}\Big{]}+\gamma^{2}{\left({\Delta}^{2}+{\kappa}^{2}\right)}\Omega_{\text{mec}}{\kappa}$
(85b)

The only root that fulfills $0\leq r_{\text{s}}<1$ is

$\displaystyle r_{\text{s}}^{*}=\,$
$\displaystyle\frac{b^{*}}{a^{*}}+\sqrt{\left(\frac{b^{*}}{a^{*}}\right)^{2}-1}$
$\displaystyle-\sqrt{2}\sqrt{\left(\frac{b^{*}}{a^{*}}\right)^{2}+\frac{b^{*}}{a^{*}}\sqrt{\left(\frac{b^{*}}{a^{*}}\right)^{2}-1}-1},$
(86)

which depends on both the optomechanical parameters and the squeezing angle.

####  C.2.3 Beyond the white-noise approximation

Like for the standard setup in Appendix B.2.3, we can go beyond the white-
noise approximation, but this time for both noises. The correlation functions
for the optical noise, in the frequency domain, read

$\displaystyle\langle(\hat{a}^{\dagger}_{\text{in}})[\omega]\hat{a}_{\text{in}}[\omega^{\prime}]\rangle$
$\displaystyle=\pi_{\text{s}}n_{\text{s}}[\omega]2\pi\delta(\omega+\omega^{\prime}),$
(87a)
$\displaystyle\langle\hat{a}_{\text{in}}[\omega]\hat{a}_{\text{in}}[\omega^{\prime}]\rangle$
$\displaystyle=\pi_{\text{s}}m_{\text{s}}[\omega]2\pi\delta(\omega+\omega^{\prime}),$
(87b)

with

$\displaystyle n_{\text{s}}[\omega]$
$\displaystyle=\frac{r_{+}^{2}-r_{-}^{2}}{4}\left(\frac{1}{r_{-}^{2}+\omega^{2}}-\frac{1}{r_{+}^{2}+\omega^{2}}\right),$
(88a) $\displaystyle m_{\text{s}}[\omega]$
$\displaystyle=\frac{r_{+}^{2}-r_{-}^{2}}{4}\left(\frac{1}{r_{-}^{2}+\omega^{2}}+\frac{1}{r_{+}^{2}+\omega^{2}}\right)\text{e}^{-i2\theta}.$
(88b)

As previously, the steady-state phonon number $\bar{n}_{\text{fin}}$ can be
computed using the position fluctuation spectrum

$S_{q}[\omega]=\absolutevalue{\chi_{\text{mec}}^{\text{eff}}[\omega]}^{2}(S_{\text{th}}[\omega]+S^{\text{squ}}_{\text{rp}}[\omega]).$
(89)

$S_{\text{th}}[\omega]$ is given by Eq. (65) and the radiation pressure noise
spectrum reads

$\displaystyle S^{\text{squ}}_{\text{rp}}[\omega]=\,$ $\displaystyle
S_{\text{rp}}[\omega]\left(1+2\pi_{\text{s}}\frac{n_{\text{s}}[\omega](\kappa^{2}+\Delta^{2}+\omega^{2})-\absolutevalue{m_{\text{s}}[\omega]}\left[(\Delta^{2}-\kappa^{2}-\omega^{2})\cos(2\theta)+2\Delta\kappa\sin(2\theta)\right]}{\kappa^{2}+(\omega+\Delta)^{2}}\right)\\!\\!\\!\\!$
(90)

where $S_{\text{rp}}[\omega]$ is the radiation pressure noise spectrum for
unsqueezed vacuum noise (Eq. (66)).

##  Appendix D Fano-mirror setup

###  D.1 Steady state and linearization

We determine the semi-classical steady state of Eqs. (11) and derive the
Langevin equations (26) like for the standard setup in Appendix B.1. The
difference is the presence of the Fano-mirror mode, whose annihilation
operator is split into $\hat{d}=\delta+\delta\hat{d}$. The steady state
$(\alpha,\bar{q},\delta)$ is given by the system of non-linear equations (25).

In this case, the stability condition is $S_{\text{RH}}^{\text{Fano}}>0$, with

$\displaystyle S_{\text{RH}}^{\text{Fano}}=\,$
$\displaystyle{\Delta}^{2}\Omega_{\text{mec}}{\gamma_{d}}^{2}+4\Omega_{\text{mec}}{\gamma_{d}}^{2}({\kappa_{0}}^{2}-{\Delta}\sqrt{{\kappa_{0}\kappa_{\text{L}}}}+{\kappa_{0}}{\kappa_{\text{L}}})$
$\displaystyle+2\Omega_{\text{mec}}{\gamma_{d}}(\kappa_{\text{L}}-{\kappa_{0}}){\Delta_{d}}({\Delta}+2\sqrt{\kappa_{0}{\kappa_{\text{L}}}})$
$\displaystyle+\Omega_{\text{mec}}{\Delta_{d}}^{2}\left({\Delta}^{2}+({\kappa_{0}}+{\kappa_{\text{L}}})^{2}\right)$
$\displaystyle-4\big{(}{\gamma_{d}}^{2}(\Delta-2\sqrt{{\kappa_{0}}{\kappa_{\text{L}}}})+{\gamma_{d}}({\kappa_{\text{L}}}-{\kappa_{0}}){\Delta_{d}}$
$\displaystyle\hskip 21.33955pt+{\Delta}{\Delta_{d}}^{2}\big{)}g^{2}.$ (91)

$S_{\text{RH}}^{\text{Fano}}>0$ corresponds, up to a factor
$\Omega_{\text{mec}}$, to the determinant of the $A$ matrix (Eq. (98a)) of the
Lyapunov equation (see Section D.2.2). As can be seen in Fig. 14, the Fano-
mirror setup for the four systems studied in this article fulfill this
stability condition for all the considered values of the coupling $g$.

Figure 14:  Stability condition $S_{\text{RH}}^{\text{Fano}}$ of the Fano-
mirror setup as a function of the cooperativities considered in this article
for each of the systems from Tab. 2. $S_{\text{RH}}^{\text{Fano}}$ has been
normalized by its value at zero coupling $S_{\text{RH},0}^{\text{Fano}}$.

###  D.2 Solving the dynamics

####  D.2.1 Solution of the Langevin equations

Like in Appendix B.2.1, the Langevin equations (26) can be solved in the
frequency domain and the expression of the mechanical position is given by

$\displaystyle\chi_{\text{mec}}^{\text{eff}}[\omega]^{-1}\delta\hat{q}[\omega]=\;$
$\displaystyle
g\sum_{\mu=\text{L},\text{R}}\Big{\\{}(t_{\mu}[\omega]+t^{*}_{\mu}[-\omega])\hat{X}_{\text{in},\mu}[\omega]$
$\displaystyle\hskip
35.56593pt+i(t_{\mu}[\omega]-t^{*}_{\mu}[-\omega])\hat{P}_{\text{in},\mu}[\omega]\Big{\\}}$
$\displaystyle+\sqrt{\gamma}\hat{\xi}[\omega],$ (92)

with the input noise quadratures
$\hat{X}_{\text{in},\mu}=(\hat{a}_{\text{in},\mu}+\hat{a}^{\dagger}_{\text{in},\mu})/\sqrt{2}$
and
$\hat{P}_{\text{in},\mu}=i(\hat{a}^{\dagger}_{\text{in},\mu}-\hat{a}_{\text{in},\mu})/\sqrt{2}$.
We have defined $t_{\mu}[\omega]$ as

$\displaystyle t_{\text{L}}[\omega]$
$\displaystyle=\frac{\sqrt{2\kappa_{\text{L}}}-\sqrt{2\gamma_{d}}\mathcal{G}\varepsilon_{d}[\omega]^{-1}}{\varepsilon_{a}[\omega]-\mathcal{G}^{2}\varepsilon_{d}[\omega]^{-1}},$
(93) $\displaystyle t_{\text{R}}[\omega]$