# Optical coherent feedback control of a mechanical oscillator

Maryse Ernzer  These authors contributed equally to this work.  Manel Bosch
Aguilera  These authors contributed equally to this work.  Matteo Brunelli
Gian-Luca Schmid  Thomas M. Karg  Present address: IBM Research Europe,
Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland.  Christoph Bruder
Patrick P. Potts<EMAIL_ADDRESS>Philipp Treutlein
<EMAIL_ADDRESS>Department of Physics and Swiss Nanoscience
Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

###### Abstract

Feedback is a powerful and ubiquitous technique both in classical and quantum
system control. Its standard implementation relies on measuring the state of a
system, processing the classical signal, and feeding it back to the system. In
quantum physics, however, measurements not only read out the state of the
system but also modify it irreversibly. Coherent feedback is a different kind
of feedback that coherently processes and feeds back quantum signals without
actually measuring the system. Here, we report on the experimental realization
and the theoretical analysis of an optical coherent feedback platform to
control the motional state of a nanomechanical membrane in an optical cavity.
The coherent feedback loop consists of a light field interacting twice with
the same mechanical mode through different cavity modes, without performing
any measurement. Tuning the optical phase and delay of the feedback loop
allows us to control the motional state of the mechanical oscillator, its
resonance frequency and also its damping rate, which we use to cool the
membrane close to the quantum ground state. Our theoretical analysis provides
the optimal cooling conditions, showing that this new technique enables
ground-state cooling. Experimentally, we show that we can cool the membrane to
a state with $\bar{n}_{m}=4.89\pm 0.14$ phonons
($480\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$) in a
$20\text{\,}\mathrm{K}$ environment. This lies below the theoretical limit of
cavity dynamical backaction cooling in the unresolved sideband regime and is
achieved with only 1$\%$ of the optical power required for cavity cooling. Our
feedback scheme is very versatile, offering new opportunities for quantum
control in a variety of optomechanical systems.

Figure 1: Sketch of the experimental setup and optical coherent feedback loop.
(a): An incoming light beam $\hat{a}^{\mathrm{in}}_{1}$ is injected into an
optomechanical cavity, where it drives the cavity field $\hat{c}_{1}$ that
interacts with a mechanical oscillator with quadratures
$\hat{X}_{m},\hat{P}_{m}$. The back-reflected beam
$\hat{a}^{\mathrm{out}}_{1}$ is combined with an auxiliary local oscillator
mode $\hat{a}_{\mathrm{aux}}$ to control the phase of the feedback loop
$\varphi$. The combined field is delayed by $\tau$ with the help of an optical
fiber, before being sent back as input $\hat{a}^{\mathrm{in}}_{2}$ for a
second interaction with the mechanical oscillator in an orthogonal
polarization cavity mode $\hat{c}_{2}$. The outgoing light after the second
interaction leaves the loop. A small fraction of $\hat{a}^{\mathrm{out}}_{1}$
is picked up for detection and phase locking of the loop. The color coding of
the light beams is used for visual guidance. Dashed lines are used for visual
distinction between the incoming and back-reflected beams. (b): Phase-space
visualisation of the feedback loop. The sketch represents the amplitude
$(X_{L})$ and phase $(P_{L})$ quadratures of the light outside the cavity in
an arbitrary frame. On resonance, the coherent cavity output field after the
first interaction $\alpha_{1}^{\mathrm{out}}$ is phase-modulated (dashed line)
with the membrane displacement signal $X_{m}(t)$. This is converted into an
amplitude modulation of $\alpha_{2}^{\mathrm{in}}$ by mixing the coherent
field with an auxiliary local oscillator $\alpha_{\mathrm{aux}}$ with the
appropriate relative phase $\phi$, to achieve the desired value of $\varphi$
between $\alpha_{1}$ and $\alpha_{2}$. After a delay $\Omega_{m}\tau=\pi/2$,
the amplitude modulation becomes proportional to the momentum of the membrane
$P_{m}$ at time $t$ (dashed line in $\alpha_{2}^{\mathrm{in}}$) and exerts a
force on the mechanical oscillator.

##  I Introduction

Quantum feedback is a powerful technique for cooling and controlling quantum
systems [1]. The conventional strategy relies on quantum-limited measurements
followed by classical processing and feedback actuation onto the system.
However, quantum mechanics also allows for _coherent_ feedback of _quantum_
signals [2, 3], without destroying coherence in the process. This kind of
feedback may exploit the information stored in non-commuting observables while
circumventing the decoherence and back-action noise associated with a
measurement [2, 4, 1]. Coherent feedback has thus the potential to improve
quantum control and provide new capabilities across a broad range of physical
systems [5, 6]. Coherent feedback strategies have so far been adopted to
assist in a variety of different tasks [1], e.g. for noise cancellation [7,
8], pure-state preparation [6], optical squeezing [9, 10], stabilization and
enhancement of entanglement [11, 12], sympathetic cooling [13, 14, 15], swaps
of arbitrary states [16], qubit state control [17], and generating large
optical nonlinearities at the single-photon level [18, 19].

Optomechanical systems are very well suited for coherent feedback control, as
they offer a clean and tailored interface between highly coherent mechanical
and electromagnetic field modes [20]. Indeed, various coherent feedback
protocols have been theoretically proposed to enhance the cooling of
optomechanical systems [21, 22, 23], to reduce the added noise in the low
phonon-number regime of optomechanical precision measurements [24], to enable
or enhance entanglement generation, verification, as well as state transfer
[25, 26, 27, 22]. Coherent feedback can thus facilitate and extend the
capabilities of quantum transducers between optics and mechanics [28].

Despite this wide range of possibilities, there have been surprisingly few
experiments investigating coherent feedback in optomechanics [15, 29]. An
optical coherent feedback loop acting directly onto a mechanical oscillator
has not yet been realized. Moreover, while measurement-based feedback has been
studied in some depth from a theory point of view [30, 31, 32, 33, 34],
essential questions regarding the performance and limitations of coherent
feedback in actual optomechanics experiments remain open.

In this work, we present both a theoretical description and experimental
realization of a simple, all-optical coherent feedback platform to control a
single vibrational mode of a mechanical oscillator. We use a double-pass
scheme where an optical signal interacts twice with the same mechanical mode
through two different cavity modes of orthogonal polarization. The entire
control of the phase and delay of the feedback signal is implemented purely
via the optical field, without introducing measurements and subsequent
electronic processing. Our approach is thus able to generate a variety of
different interactions, ranging from Hamiltonian couplings to dissipative and
non-reciprocal dynamics [35, 36, 37].

As a first application of the extended control offered by the coherent
feedback loop, we investigate the cooling of the mechanical mode close to its
quantum ground state, which is a prerequisite for many applications in quantum
science and technology [20, 38, 39, 28]. Theoretically, we show that coherent
feedback enables ground-state cooling even in the unresolved sideband regime,
where cavity dynamical backaction cooling with a single or two independent
optomechanical interactions cannot reach the ground state [20].
Experimentally, we demonstrate the advantage of the coherent feedback loop by
cooling below the theoretical limit of cavity dynamical backaction cooling in
our system. This is particularly interesting for optomechanical systems with
cavities of large bandwidth, which induce only a small delay and are
frequently encountered in optomechanical displacement sensing, quantum
interfaces and hybrid setups [40].

The remainder of this paper is structured as follows: We first provide an
overview of the working principle of our coherent feedback platform for
controlling a mechanical oscillator in an optical cavity. Next, we develop a
theoretical model of the feedback scheme, followed by our experimental results
on motional state control and its application to cooling. Finally, we compare
our theoretical results with those of measurement-based feedback for the
specific task of cooling.

##  II Overview of the coherent feedback scheme

We start by illustrating the working principle of our coherent feedback
scheme, sketched in Fig. 1. The goal is to control the motional state of a
mechanical oscillator by designing an optical feedback loop that preserves the
quantum coherent properties of the light field, which acts as the controller.
To this end, the mechanical oscillator is radiation-pressure coupled to _two_
cavity modes in a cascaded double-pass interaction. The first interaction
takes place between the mechanical oscillator and the cavity mode
$\hat{c}_{1}$, which is driven by a strong local oscillator, realizing the
standard cavity optomechanical interaction [20]. Due to the optomechanical
coupling, information about the mechanical position $\hat{X}_{m}$ is imprinted
onto the phase quadrature of $\hat{c}_{1}$. This mode is then cascaded into
the second cavity mode $\hat{c}_{2}$ via an all-optical feedback loop.
Specifically, the output light of the first mode, with mean amplitude
$\alpha_{1}^{\mathrm{out}}$, is mixed with a second local oscillator and fed
back as the input of the second cavity mode, with amplitude
$\alpha_{2}^{\mathrm{in}}$, as shown in Fig. 1. The resulting optical feedback
loop is characterized by two parameters, the relative phase $\varphi$ and the
in-loop delay time $\tau$. The phase $\varphi$ is controlled by the second
local oscillator $\alpha_{\mathrm{aux}}$, which implements a displacement in
the optical phase space of the modes traveling within the loop, see Fig. 1
(b).

Both feedback parameters are crucial for controlling the mechanical
oscillator. The phase $\varphi$ is adjusted so that the phase quadrature of
the outgoing mode, which contains information on the mechanical position, is
turned into the amplitude quadrature of the incoming mode, such that it exerts
a feedback force by radiation pressure on the mechanical oscillator. As
sketched in Fig. 1 (b), this occurs for $\varphi=\pi/2$. Adjusting the delay
$\tau$ allows to either feed back the instantaneous position [when
$\hat{X}_{m}(t-\tau)\simeq\hat{X}_{m}(t)$], momentum [when
$\hat{X}_{m}(t-\tau)\simeq\hat{P}_{m}(t)$, as represented in Fig. 1 (b)], or a
superposition thereof. While feeding back the position enables control of the
mechanical oscillator frequency, feeding back the momentum allows control of
its damping, which can be exploited for ground-state cooling.

Previous theoretical proposals for coherent feedback cooling of mechanical
oscillators [21, 27, 22, 23] rely on coherently enhancing the interaction of
the cavity light with the mechanics, mostly by modifying the effective cavity
linewidth [27, 22], and on loops that impart only a delay (plus unavoidable
coupling losses). In contrast, our scheme applies the coherent feedback
directly to the mechanical oscillator, such that the feedback can be generated
with a single cavity driven in two independent modes. Moreover, it allows
tuning of the loop phase $\varphi$, which strongly influences the effect of
the feedback.

Our scheme requires no additional optical devices such as cavities and only
minor modifications of the optical path, resulting in a modular scheme that is
optimally suited for incorporation into various types of optomechanical
systems.

Furthermore, our double-pass scheme does not require non-classical input light
states [41, 42, 39], additional interactions with other physical systems [13,
43, 15], nor the overall very high detector efficiency of measurement-based
feedback schemes [30, 31, 32, 33, 34], which is now replaced by the
requirement of small optical losses in the loop. The relaxation of the
requirements on measurement efficiency renders our scheme valuable for systems
working in wavelength ranges where efficient photodetectors are not available,
e.g. in integrated circuit platforms [44].

##  III Theoretical model

In this section, we provide a theoretical model for the coherent feedback
scheme described above.

###  III.1 Langevin equations

The mechanical oscillator that is to be controlled is described by the
linearized Langevin equations

$\displaystyle\partial_{t}\hat{X}_{m}(t)=\Omega_{m}\hat{P}_{m}(t),$ (1)
$\displaystyle\partial_{t}\hat{P}_{m}(t)=-\Omega_{m}\hat{X}_{m}(t)-\gamma_{m}\hat{P}_{m}(t)-2\sum_{j=1}^{2}g_{j}\hat{x}_{j}(t)-\sqrt{2}\,\hat{\xi}_{\rm
th}(t),$

where $\hat{X}_{m}(t)$ and $\hat{P}_{m}(t)$ denote the dimensionless position
and momentum operators of the mechanical oscillator, $\Omega_{m}$ its
frequency, and $\gamma_{m}$ its energy damping rate. The mechanical oscillator
is driven by thermal noise $\hat{\xi}_{\rm th}$, which has zero average and is
fully described by its spectral density [45]

$S_{\rm
th}(\omega)=\gamma_{m}\frac{|\omega|}{\Omega_{m}}[n_{B}(|\omega|)+\Theta(\omega)],$
(2)

where $n_{B}(\omega)=[\exp(\hbar\omega/k_{B}T)-1]^{-1}$ denotes the Bose-
Einstein distribution and $\Theta(\omega)$ the Heaviside step function.

The oscillator furthermore couples to the amplitude quadrature of two optical
modes, $\hat{x}_{j}=(\hat{c}_{j}^{\dagger}+\hat{c}_{j})/\sqrt{2}$ with
strength $g_{j}$, where $j=1,2$. As discussed in more detail below, the
coupling strengths depend on the average displacements of the optical modes
and the operators $\hat{c}_{j}$ describe fluctuations around these
displacements [20].

The first cavity mode is described by the Langevin equation (in a frame
rotating at the laser frequency $\omega_{L}$)

$\partial_{t}\hat{c}_{1}(t)=\left(i\Delta-\frac{\kappa}{2}\right)\hat{c}_{1}(t)-i\sqrt{2}g_{1}\hat{X}_{m}(t)-\sqrt{\kappa}\hat{a}_{1}^{\rm
in}(t),$ (3)

where $\Delta=\omega_{L}-\omega_{c}$ denotes the detuning from the cavity mode
frequency $\omega_{c}$ and $\kappa$ the cavity linewidth. This mode is driven
by a local oscillator with frequency $\omega_{L}$ and average displacement
$\alpha_{1}^{\rm in}$. Fluctuations around this displacement are described by
the operator $\hat{a}_{1}^{\rm in}$. Due to the optomechanical coupling, the
outgoing field leaving this cavity mode

$\hat{a}_{1}^{\rm out}(t)=\hat{a}_{1}^{\rm
in}(t)+\sqrt{\kappa}\hat{c}_{1}(t),$ (4)

contains information on the position of the mechanical oscillator. From Eq.
(3), it follows that on resonance ($\Delta=0$), this information is only
contained in the phase quadrature
$\hat{p}_{1}=i(\hat{c}_{1}^{\dagger}-\hat{c}_{1})/\sqrt{2}$.

To implement the coherent feedback, the output of the first cavity mode is fed
back into the input of the second cavity mode. Before it is coupled into the
cavity, it undergoes a displacement by combining it with an auxiliary local
oscillator with the same frequency $\omega_{L}$ and average displacement
$\alpha_{\rm aux}$, and it is delayed by the time $\tau$. The second cavity
mode is then driven by the input mode [46]

$\hat{a}_{2}^{\rm in}(t)=\sqrt{\eta}\,e^{i\varphi}\hat{a}_{1}^{\rm
out}(t-\tau)+\sqrt{1-\eta}\,\hat{a}_{\rm aux}(t),$ (5)

where the phase $\varphi={\rm arg}(\alpha_{1}/\alpha_{2})$ denotes the phase
difference between the average displacements of the cavity modes. As shown in
App. B.1, $\eta$ takes into account any losses in the loop. We note that we
chose to define the mode $\hat{a}_{\rm aux}$ such that no time-shift appears
in its argument. The Langevin equation for the second cavity mode then reads

$\displaystyle\partial_{t}\hat{c}_{2}(t)=\left(i\Delta-\frac{\kappa}{2}\right)\hat{c}_{2}(t)-i\sqrt{2}g_{2}\hat{X}_{m}(t)-\sqrt{\eta}\kappa\,e^{i\varphi}\hat{c}_{1}(t-\tau)$
(6) $\displaystyle\hskip
28.45274pt-\sqrt{\eta\kappa}\,e^{i\varphi}\hat{a}_{1}^{\rm
in}(t-\tau)-\sqrt{(1-\eta)\kappa}\,\hat{a}_{\rm aux}(t).$

The average displacements $\alpha_{j}$ of the cavity modes are given in Eq.
(43). In terms of these, the optomechanical coupling strengths can be written
as $g_{j}=g_{0}|\alpha_{j}|$, where $g_{0}$ is the bare coupling strength. The
amplitudes of the local oscillators are related to their input powers as
$P_{1}=\hbar\omega_{L}|\alpha_{1}^{\rm in}|^{2}$ and $P_{\rm
aux}=\hbar\omega_{L}|\alpha_{\rm aux}|^{2}$ respectively.

To better understand the effect of the coherent feedback loop, it is
illustrative to eliminate the cavity modes from the Langevin equations. For a
high-quality oscillator $\gamma_{m}\ll\Omega_{m}$ and delay times that obey
$\tau\gamma_{m}\ll 1$, we find

$\displaystyle\partial_{t}^{2}\hat{X}_{m}(t)=$
$\displaystyle-(\Omega_{m}+\delta\Omega_{m})^{2}\hat{X}_{m}(t)-(\gamma_{m}+\Gamma_{m})\,\partial_{t}\hat{X}_{m}(t)$
(7) $\displaystyle+\Omega_{m}\sqrt{2}\,\hat{\xi}_{\rm
th}(t)+\Omega_{m}\sqrt{2}\,\hat{\xi}_{\rm fb}(t),$

where $\Gamma_{m}$ and $\delta\Omega_{m}$ denote the effective damping and the
frequency shift that are controlled by the coherent feedback loop. To derive
Eq. (7), we assumed that these are small compared to the frequency of the
resonator, i.e., $\Gamma_{m},\delta\Omega_{m}\ll\Omega_{m}$. In the unresolved
sideband regime and on cavity resonance, we find

$\displaystyle\Gamma_{m}=16\sqrt{\eta}\frac{g_{1}g_{2}}{\kappa}\sin(\varphi)\sin(\Omega_{m}\tau),$
(8)
$\displaystyle\delta\Omega_{m}=-8\sqrt{\eta}\frac{g_{1}g_{2}}{\kappa}\sin(\varphi)\cos(\Omega_{m}\tau).$

Expressions for the general scenario are given in App. B.

From these expressions, we may understand the physical significance of the
parameters that determine the coherent feedback loop. To maximize the effect
that the optical field exerts on the mechanics, we should choose
$\varphi=\pi/2$. The reason for this is sketched in Fig. 1 (b) and was already
discussed qualitatively in Sec. II: For $\Delta=0$, the first optical mode
contains the information on $\hat{X}_{m}$ in the phase quadrature,
$\hat{p}_{1}=i(\hat{c}_{1}^{\dagger}-\hat{c}_{1})/\sqrt{2}$, the quadrature
that does not exert a radiation pressure force on the mechanical oscillator.
Through the feedback loop, the $\hat{c}_{1}$ mode will be fed into the second,
$\hat{c}_{2}$ mode. In order for the feedback to be effective, the phase
quadrature of the $\hat{c}_{1}$ mode has to be fed into the amplitude
quadrature of the $\hat{c}_{2}$ mode, $\hat{x}_{2}$, the quadrature that does
couple to the mechanical oscillator. For this to occur, the field has to be
displaced by the auxiliary local oscillator, such that $\varphi=\pi/2$.

Furthermore, Eqs. (8) show that by tuning the delay, the feedback can either
result in a frequency shift or damping. Since for $\tau\gamma_{m}\ll 1$ we
have approximately

$\hat{X}_{m}(t-\tau)\simeq\cos(\Omega_{m}\tau)\hat{X}_{m}(t)-\sin(\Omega_{m}\tau)\hat{P}_{m}(t),$
(9)

we can see that the delay determines which quadrature, $\hat{X}_{m}$ or
$\hat{P}_{m}$, is being fed back to the oscillator. In the limit of no delay,
we are feeding back a force proportional to the position, resulting in a
strong frequency shift. Maximal damping can be achieved by feeding back a
force proportional to the momentum, which occurs at $\Omega_{m}\tau=\pi/2$. We
note that the induced damping can become negative as
$\gamma_{m}+\Gamma_{m}<0$, in which scenario the system becomes unstable and
our linearized description fails.

It is instructive to carry out a similar analysis in frequency space. This is
done in App. B.2.

Figure 2: Sideband picture of optomechanical cooling. (a)
$S_{\mathrm{fb}}(\omega)$ in the standard optomechanical cavity cooling for
$\kappa/\Omega_{m}\ll 1$ and $\Delta=-\Omega_{m}$. The Stokes processes are
suppressed because they lie outside the cavity resonance. (b)
$S_{\mathrm{fb}}(\omega)$ in the coherent feedback scenario, for
$\kappa/\Omega_{m}\gg 1$ and $\Delta=0$, see Eq. (57). The Stokes processes in
the two optical passes interfere destructively, such that the pump beam
predominantly extracts phonons and cools the mechanical oscillator. The dotted
line corresponds to the ideal case in the absence of losses with $\eta=1$.

###  III.2 Sideband picture

Just as for cavity dynamical backaction cooling [20], we may develop a
sideband picture and write the damping induced by the coherent feedback as

$\Gamma_{m}=A^{-}-A^{+},$ (10)

where $A^{\pm}=S_{\rm fb}(\mp\Omega_{m})$ denote the rates for the Stokes
$(A^{+})$ and anti-Stokes $(A^{-})$ processes which are determined by the
spectral density of the feedback noise

$S_{\rm
fb}(\omega)=\frac{8}{\kappa}\left[\frac{g_{1}^{2}+g_{2}^{2}}{2}-\sqrt{\eta}g_{1}g_{2}\cos(\varphi+\omega\tau)\right],$
(11)

where the last equality holds for the unresolved sideband regime on cavity
resonance [see Eq. (57) for the full expression]. Equation (57) is plotted in
Fig. 2 (b).

The phonon number of the mechanical oscillator may then be written as

$\bar{n}_{m}=\frac{\gamma_{m}n_{\rm th}+A^{+}}{\gamma_{m}+\Gamma_{m}},$ (12)

where $n_{\mathrm{th}}=n_{B}(\Omega_{m})$ denotes the thermal occupation. From
the last equation we find that in order to reach a phonon number close to
zero, two conditions need to be met. First, the quantum regime, where thermal
excitations can be neglected, needs to be reached. This requires a large
quantum cooperativity

$\mathcal{C}_{\rm qu}=\frac{4g_{1}g_{2}}{\kappa\gamma_{m}n_{\mathrm{th}}}\gg
1.$ (13)

Note that the quantum cooperativity usually only includes a single
optomechanical coupling. Here, the product of the two couplings $g_{1}$ and
$g_{2}$ is relevant.

The second condition to reach the ground state is a suppression of the Stokes
processes (i.e., $A^{+}\ll A^{-}$), such that phonons are predominantly
absorbed by Anti-Stokes scattering into the pump beam. For cavity dynamical
backaction cooling in the sideband resolved regime, this is achieved by
detuning the drive by an amount $\Delta=-\Omega_{m}$, see Fig. 2 (a). In
contrast, in our coherent feedback scheme, which operates in the unresolved
sideband regime, this suppression results from the interference between
scattering processes in the first and second optical passes, which results in
a frequency-dependent modulation of the noise spectral density $S_{\rm
fb}(\omega)$ as illustrated in Fig. 2 (b) and thus a suppression of quantum
backaction heating.

In the unresolved sideband limit and on resonance, we find the minimum phonon
number

$\bar{n}_{m}=\frac{A^{+}}{\Gamma_{m}}=\frac{A^{+}}{A^{-}-A^{+}}\geq\frac{1-\sqrt{\eta}}{2\sqrt{\eta}},$
(14)

where the lower limit is reached for a large quantum cooperativity,
$\Omega_{m}\tau=\varphi$, and $g_{1}=g_{2}$. Interestingly, this lower bound
has the same form as the limit of measurement-based feedback cooling [47, 31],
with the efficiency of the feedback loop $\eta$ replacing the measurement
efficiency. As discussed in more detail in Sec. VI, we can draw an analogy
between coherent feedback cooling and measurement-based cooling: the coherent
feedback cooling can, in the ideal limit, be understood as a measurement-free
implementation where the readout signal is coherently transformed into a
feedback actuation. In App. D we compare the performance of coherent feedback
cooling with standard cavity backaction cooling for different cavity regimes.

##  IV Coherent feedback control of a nanomechanical membrane

Our experimental setup consists of a mechanical oscillator inside a cavity in
a cryogenic environment provided by a low-noise liquid-Helium flow cryostat.
The mechanical oscillator is the (2,2) square drum mode of a silicon nitride
membrane [48] with a vibrational frequency
$\Omega_{m}=2\pi\times$1.9\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$. The
membrane is surrounded by a silicon phononic bandgap structure which shields
this mode, leading to intrinsic quality factors that range from
$Q=\Omega_{m}/\gamma_{m}=$1.9\text{\times}{10}^{6}\text{\,}$$ at room
temperature to $Q=$3.2\text{\times}{10}^{6}\text{\,}$$ at
$20\text{\,}\mathrm{K}$. The membrane is placed inside a single-sided optical
cavity of free spectral range $150\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$,
finesse $\mathcal{F}=1200$ and linewidth
$\kappa=2\pi\times$55\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$, such that the
optomechanical system operates in the unresolved sideband regime
$\kappa\gg\Omega_{m}$. The bare optomechanical coupling strength is
$g_{0}=2\pi\times$160\text{\,}\mathrm{H}\mathrm{z}$$, calibrated via a phase
modulation tone [49].

The overall efficiency of the feedback loop is determined by a combination of
different losses that accumulate along the optical path. Following the optical
path illustrated in Fig. 1 (a), for the first beam we have to consider the
finite cavity incoupling efficiency $\eta_{1}$ = 0.91. For the second pass, it
includes the unavoidable loss at the beamsplitter that combines the auxiliary
local oscillator $\hat{a}_{\mathrm{aux}}$ and the back reflection of the first
beam $\hat{a}^{\mathrm{out}}_{1}$, which has a splitting ratio
$\eta_{\mathrm{aux}}$ = 0.87 in our experiment. Note that this loss can be
made arbitrarily small in principle by using a strongly unbalanced beam
splitter and higher incoming optical power for $\hat{a}_{\mathrm{aux}}$.
Additionally, there is a cumulative loss due to the propagation in the optical
fiber and other optical elements, leading to a transmission efficiency
$\eta_{T}\simeq 0.3$ together with the cavity incoupling efficiency of the
second beam in orthogonal polarization $\eta_{2}$ = 0.9. As discussed in App.
B.1, these losses can be fully taken into account by the overall efficiency of
the feedback loop $\eta=0.22$ and by an appropriate rescaling of the average
displacements. Since the powers are measured in front of the cavity, $P_{1}$
is measured directly but the measured auxiliary power is given by
$\tilde{P}_{\mathrm{aux}}=(1-\eta_{\mathrm{aux}})\eta_{T}{P}_{\mathrm{aux}}$.
In the following experiments, we use the delay and the phase of the coherent
feedback loop as the tuning knobs that allow us to control the mechanical
state of the membrane, as described by Eq. (8). This shows up in the recorded
mechanical power spectral densities as a change of both the mechanical
linewidth and the oscillation frequency, which we extract from Lorentzian fits
to our data.

###  IV.1 Control via the loop delay

In a first set of experiments, we study the effect of delay alone without an
auxiliary local oscillator in the feedback loop. We generate different delays
between the two interactions with the mechanical oscillator by sending the
light after the first pass through optical fibers of different lengths.
Interesting situations arise once the delay is significant, i.e. of order
$\Omega_{m}\tau\sim$ $\pi$. We investigate the effect generated by different
delays starting from a fiber length of $2\text{\,}\mathrm{m}$ going up to
$80\text{\,}\mathrm{m}$, which corresponds to $\Omega_{m}\tau=0.07\pi$ up to
$\Omega_{m}\tau=1.55\pi$.

At zero detuning, the motion of the membrane is imprinted purely as a phase
modulation onto the output light such that in the absence of the auxiliary
local oscillator this results in $\varphi=\pi$ (due to the back reflection
from the cavity) and we expect no effect from the coherent feedback loop [cf.
Eqs. (8)]. With a finite detuning however, a phase shift $\varphi\neq\pi$ is
introduced even without any auxiliary local oscillator [cf. Eqs. (43)].
Therefore, in that case, the motion is imprinted onto both the amplitude and
phase quadratures and the effect of different delays due to feedback becomes
apparent. Additionally, the standard cavity dynamical backaction effects that
are not captured by Eqs. (8) modify the frequency shift and damping rate, see
App. B for details.

Figure 3 shows experimental data where we study the effect of different
feedback delays while scanning the detuning for an input power
$P_{1}=$60\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$$. The coherent
feedback onto the mechanical oscillator results in both in a shift of the
mechanical frequency [Fig. 3(a)] and in a broadening or narrowing of the
mechanical linewidth [Fig. 3(b)], leading to damping or driving, respectively.
This is consistent with a picture in which the membrane motion couples via the
light to a delayed version of itself, leading to feedback forces
$F_{\mathrm{fb}}\propto\pm P_{m}(t)$ for certain delays as shown in Eq. (7)
and in Fig. 3(c).

Indeed, we observe that for a delay close to $\Omega_{m}\tau\sim\pi/2$ (i.e. a
quarter of the oscillation period) the coupling is mostly proportional to
$+P_{m}$ and we observe driving (narrowing of the linewidth) even with a red
detuned beam. Half a period later, for $\Omega_{m}\tau\sim 3\pi/2$, the
feedback force is mostly proportional to $-P_{m}$ and the motional damping is
amplified by more than a factor 3 as compared to a single interaction, leading
to additional cooling of the mechanical oscillator. Finally, we see that for
the smaller delays $\Omega_{m}\tau\sim 0$, the effect of the second
interaction on the broadening is small $\Gamma_{m}\simeq 0$, since the
feedback force, in this case, is mostly $\propto X_{m}$.

The agreement with the theoretical predictions in Eqs. (55) and (56) (solid
lines in Fig. 3) is excellent. The theory lines for the feedback interaction
contain no free parameters. The detuning axis is calibrated from the recorded
linewidths in the single-pass interaction under the effect of the standard
cavity dynamical backaction [20] and can be extracted with an uncertainty of
$\pm 5\%$.

Figure 3: Mechanical frequency shift (a) and damping rate (b) as a function of
the cavity detuning for different feedback delays, with
$\hat{a}_{\mathrm{aux}}=0$. The data points correspond to the results of
Lorentzian fits to the mechanical power spectral density. The solid lines
correspond to the theoretical predictions in Eqs. (55) and (56) evaluated at
$\omega=\Omega_{m}$ with no free parameters. The detuning axis is calibrated
from the measured linewidth in the single-pass interaction (dashed red line).
(c) The mechanical linewidth at a detuning of $\Delta/\kappa$ = -0.57 for the
different fibres and respective delays.

###  IV.2 Control via the loop phase

Control over the feedback phase is a handy knob in a feedback platform,
allowing to modify the effect of the feedback on the system under control.
Here, we investigate how the loop phase modifies the membrane motion at a
fixed delay and cavity detuning. As previously discussed, this phase allows us
to control the amount of motional information that is transferred onto the
amplitude quadrature of the second interaction beam, thereby maximizing or
minimizing the feedback force on the membrane, as well as the overall sign of
the interaction. Experimentally, we vary the loop phase $\varphi$ by adjusting
the phase of the auxiliary local oscillator
$\phi=\arg(\alpha_{\mathrm{aux}}/\alpha^{\mathrm{out}}_{1})$, which is
selected and stabilized by locking at a specific angle of the interferometric
signal between a small leak of $\alpha^{\mathrm{out}}_{1}$ and
$\alpha_{\mathrm{aux}}$ [see Fig. 1].

The measured frequency shifts and linewidths are shown in Fig. 4. In this
measurement, the delay is held constant at $\Omega_{m}\tau\sim 0.07\pi$ and
the detuning at $\Delta/\kappa=-0.2$, the input powers were set to
$\tilde{P}_{1}=$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$$ and
$\tilde{P}_{\mathrm{aux}}=$3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$$.
This detuning is experimentally chosen such that the amount of standard cavity
dynamical backaction cooling is strong. This allows us to show the coherent
feedback effect by scanning the full range $2\pi$ of the loop phase without
running into instabilities when approaching negative effective linewidths. In
practice, when we reach this unstable regime [blue dashed lines in Fig. 4
(b)], the system is driven and the measured linewidth is close to zero.

Scanning the phase, we observe that both the resonance frequency and the
linewidth can be modified to either higher or lower values compared to the
optical spring and broadening that occur even without the coherent feedback.
We exploit this aspect in the next section to optimally feedback cool the
mechanical oscillator. Here again, we find an excellent agreement between the
experimental data points and the theoretical prediction in Eqs. (55) and (56)
with no free parameters.

Figure 4: Mechanical frequency shift (a) and damping rate (b) as a function of
the phase of the auxiliary local oscillator. The red dotted lines correspond
to the broadening expected in the absence of feedback, but with an equivalent
power in a single beam. The solid lines correspond to Eqs. (55) and (56)
evaluated at $\omega=\Omega_{m}$ with no free parameters. The solid line in
(b) is set to zero for negative values (dashed line), where the mechanical
oscillator is driven by the feedback.

##  V Coherent feedback cooling below the dynamical backaction limit

Figure 5: Coherent feedback cooling below the dynamical backaction limit. (a):
Phonon occupation plotted as a function of the phase of the auxiliary local
oscillator at different cryostat temperatures for input powers
${P}_{1}=$0.4\text{\,}\mathrm{m}\mathrm{W}$$ and
$\tilde{P}_{\mathrm{aux}}=$1.2\text{\,}\mathrm{m}\mathrm{W}$$ and detuning
$\Delta/\kappa=-0.35$. The red shaded area indicates the limit of cavity
dynamical backaction cooling. The shaded areas around the theory lines
correspond to a $\pm 5\%$ uncertainty in the inferred detuning. The error bars
take into account both the numerical uncertainty from the fit of the raw data
and the propagation of uncertainties from the calibrated quantities and are
small compared to the point size. (b): Shot-noise normalized mechanical power
spectral densities corresponding to the data points at
$T=$20\text{\,}\mathrm{K}$$ in (a), the frequency origin is centered so that
the $\delta\Omega=0$ corresponds to the single-pass mechanical resonance
frequency. (c): Coherent feedback cooling compared to standard cavity
dynamical backaction cooling. Blue line and data points: Phonon occupation at
a constant phase $\phi=130^{\circ}$, scanning the total input power resulting
from the double-pass interaction
$P={P}_{1}(1+\eta_{T}\eta_{\mathrm{aux}})+\tilde{P}_{\mathrm{aux}}+2({\eta_{\mathrm{aux}}\eta_{2}{P}_{1}\tilde{P}_{\mathrm{aux}}})^{1/2}\cos{(\phi)}$
while keeping the ratio $\tilde{P}_{\mathrm{aux}}/{P}_{1}=3$ fixed, at a
detuning $\Delta/\kappa=-0.35$. Red line: cooling by standard cavity dynamical
backaction given the same total input power $P$ at the optimal detuning
$\Delta/\kappa=-0.5$ for dynamical backaction.

For cavity optomechanical systems within the so-called resolved sideband
regime, it has been established theoretically and demonstrated in multiple
platforms that a red-detuned drive allows cooling the mechanical oscillator
close to its motional ground state [20]. Outside this regime, cavity dynamical
backaction cooling to the ground state is no longer attainable and the most
widely used cooling technique is measurement-based feedback [30, 50], where
the optical signal is measured and converted into a classical electronic
signal that drives the feedback actuator.

Here, we exploit the control provided by the coherent feedback loop
demonstrated in the previous section to cool the membrane close to the ground
state in the unresolved sideband regime. The available tuning knobs are the
loop phase $\varphi$ and delay $\tau$ as well as the detuning $\Delta$ and the
powers of the first and auxiliary local oscillators $P_{1}$ and
$P_{\mathrm{aux}}$. In standard cavity cooling, the minimal number of phonons
achievable in the unresolved sideband regime is bounded by
$\kappa/(4\Omega_{m})$ (cf. App. B.4), which in our case corresponds to about
7 phonons. In order to reach this dynamical backaction limit with our
mechanical quality factor, we would need a laser power on the order of
$100\text{\,}\mathrm{m}\mathrm{W}$ [see Fig. 5 (c)]. The coherent feedback
loop dramatically relaxes this power constraint and we are able to cool the
motion of the membrane below the dynamical backaction limit, approaching the
ground state.

In Fig. 5 (a) we show experiments where we reach our lowest membrane phonon
occupation by scanning the experimental feedback loop phase. We present
measurements at different cryostat temperatures, where the delay is set to
$\Omega_{m}\tau\sim\pi/4$ and the detuning is kept fixed at
$\Delta/\kappa=-0.35$. The powers for the first and auxiliary local
oscillators are set to $400\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$ and
$1.2\text{\,}\mathrm{m}\mathrm{W}$, respectively. With these experimental
parameters, the feedback loop drives the mechanical oscillator towards a state
with phonon occupation of $\bar{n}_{m}=4.89\pm 0.14$ phonons for a cryostat
temperature of $20\text{\,}\mathrm{K}$, reaching a phonon number below the
theoretical limit of cavity dynamical backaction cooling for our membrane-
cavity assembly. The coherent feedback cooling rate is
$\Gamma_{m}>10\,\Gamma_{\mathrm{dyn}}$, where $\Gamma_{\mathrm{dyn}}$ is the
cooling rate of dynamical backaction cooling at the same power.

In these experiments, the membrane phonon occupation is determined from the
area of the mechanical power spectral density, recorded via phase-sensitive
homodyne detection. By determining the reduction in area with respect to a
single interaction [see Fig. 5 (b) and App. A] we extract $\bar{n}_{m}$.

We note that higher powers, smaller detunings and slightly smaller phases
should further reduce the final number of phonons, but these parameter regimes
were not accessible to us due to technical instabilities related to the cavity
lock. Similarly, the optimal delay $\Omega_{m}\tau\sim\pi/2$ could not be
implemented, most likely due to the increased phase noise in longer fibers. We
also note that, due to the cavity detuning we use, incident phase noise will
be rotated into the light amplitude quadrature, which can in turn heat the
mechanical oscillator mode. Furthermore, the membrane sustains a multitude of
higher-order mechanical modes which experience coherent feedback with
different phase shifts depending on their resonance frequencies. For those
modes with sufficiently strong optomechanical coupling [51], the feedback
effect might turn into amplification instead of cooling and drive the cavity
to an unstable regime. Although this could be a potential issue, we did not
observe any effect in our measurements.

##  VI Comparison with measurement-based feedback cooling

It is illustrative to compare our coherent feedback scheme to well-known
measurement-based feedback (mf) schemes for the specific task of cooling
mechanical motion [30, 50, 31, 33, 32]. In measurement-based feedback, the
all-optical loop is replaced by an optoelectronic loop. An estimate of the
mechanical displacement is obtained from a measurement of the phase quadrature
of the output light from cavity mode $\hat{c}_{1}$, and then a mechanical
force proportional to the derivative of the estimated displacement is applied.
In the case where the feedback force is optomechanically actuated, the
electronic signal modulates the amplitude of a laser driving the second cavity
mode $\hat{c}_{2}$, which results in a time-dependent coupling $g_{2}$.
Irrespective of the particular implementation of the feedback force, the
effect of the feedback can be effectively modeled by replacing the coupling to
the second cavity in the equation of motion for $\hat{P}_{m}$ with a feedback
force, entering Eq. (1) as a convolution term
$F_{\mathrm{mf}}(t)=(h_{\mathrm{mf}}*p_{1}^{\mathrm{est}})(t)$, where
$h_{\mathrm{mf}}$ denotes the feedback transfer function and
$p_{1}^{\mathrm{est}}=p^{\mathrm{out,\,\eta_{\mathrm{det}}}}_{\mathrm{1}}/\sqrt{\eta_{\mathrm{det}}\kappa}$.
Notice that the measurement is limited by the finite quantum efficiency of the
detector $\eta_{\mathrm{det}}$, i.e.,
$p^{\mathrm{out},\,\eta_{\mathrm{det}}}_{1}=\sqrt{\eta_{\mathrm{det}}}p^{\mathrm{out}}_{1}-\sqrt{1-\eta_{\mathrm{det}}}p_{0}$,
where $p_{0}$ is an uncorrelated vacuum field. For concreteness, here we focus
on the case of the so-called cold damping scheme [50], while a more general
treatment is provided in App. C. Cold damping is characterized by the
following spectral filter function

$h_{\mathrm{mf}}(\omega)=-i\frac{g_{\mathrm{mf}}\,\omega}{1-i{\omega}/{\omega_{\mathrm{mf}}}}\,,$
(15)

which is expressed in terms of the bandwidth $\omega_{\mathrm{mf}}$ and by the
dimensionless quantity $g_{\mathrm{mf}}$, which quantifies the feedback gain.
The bandwidth $\omega_{\mathrm{mf}}$ describes the finite time response of the
feedback, while any explicit delay in the feedback loop is neglected [52].
From Eq. (15) we can already understand the regime of interest for feedback by
observing that
$\arg(h_{\mathrm{mf}})=-\arctan({\omega_{\mathrm{mf}}}/{\omega})$. For
$\omega_{\mathrm{mf}}\gg\omega\approx\Omega_{m}$, the argument tends to
$-\pi/2$, so that the feedback force becomes proportional to momentum, thus
providing damping of the mechanical motion. Therefore, the relevant regime for
feedback cooling is that of large feedback bandwidth.

Similar to coherent feedback cooling, the effect of measurement-based feedback
cooling is fully taken into account by introducing a modified mechanical
frequency and a modified damping rate, which are given by

$\displaystyle\Gamma_{\mathrm{mf}}$
$\displaystyle=\frac{2\Omega_{m}g_{1}g_{\mathrm{mf}}\omega_{\mathrm{mf}}[({\kappa}/{2})\,\omega_{\mathrm{mf}}-\Omega_{m}^{2}]}{\left[\left({\kappa}/{2}\right)^{2}+\Omega_{m}^{2}\right](\Omega_{m}^{2}+\omega_{\mathrm{mf}}^{2})}\,,$
(16) $\displaystyle\delta\Omega_{\mathrm{mf}}$
$\displaystyle=\frac{\Omega_{m}^{2}g_{1}g_{\mathrm{mf}}\omega_{\mathrm{mf}}\,({\kappa}/{2}+\omega_{\mathrm{mf}})}{\left[\left({\kappa}/{2}\right)^{2}+\Omega_{m}^{2}\right](\Omega_{m}^{2}+\omega_{\mathrm{mf}}^{2})}\,.$
(17)

In the relevant limit $\omega_{\mathrm{mf}},\,\kappa\gg\Omega_{m}$, the
residual phonon occupation, as obtained from the corresponding noise power
spectral density, is given by

$\displaystyle\bar{n}_{m,\mathrm{mf}}$
$\displaystyle=\frac{g_{1}}{g_{\mathrm{mf}}\Omega_{m}}+\frac{g_{\mathrm{mf}}\Omega_{m}}{16g_{1}\eta_{\mathrm{det}}}-\frac{1}{2}\geq\frac{1-\sqrt{\eta_{\mathrm{det}}}}{2\sqrt{\eta_{\mathrm{det}}}}\,,$
(18)

where the inequality is saturated for a value of the feedback gain
$g_{\mathrm{mf}}=4\sqrt{\eta_{\mathrm{det}}}g_{1}/\Omega_{m}$. Remarkably, the
above expression has the same form as the residual occupation of coherent
feedback cooling, given by Eq. (14).

To better appreciate this correspondence, we can evaluate Eqs. (8) and (14)
for $\Omega_{m}\tau=\varphi=\pi/2$, i.e. the parameters that result in optimal
cooling. We find

$\displaystyle\Gamma_{m}=16\sqrt{\eta}\frac{g_{1}g_{2}}{\kappa},\hskip
14.22636pt\delta\Omega_{m}=0,$ (19)
$\displaystyle\bar{n}_{m}=\frac{g_{1}}{4\sqrt{\eta}g_{2}}+\frac{g_{2}}{4\sqrt{\eta}g_{1}}-\frac{1}{2}\,.$

In the limit $\omega_{\mathrm{mf}},\,\kappa\gg\Omega_{m}$, we can recover
these expressions from Eqs. (16), (17), and (18) upon setting $g_{\rm
mf}=4\sqrt{\eta}g_{2}\Omega_{m}$ and $\eta_{\rm det}=\eta$. We, therefore,
conclude that in this limit the two schemes are equivalent. In App. C we show
that this equivalence can be extended to arbitrary delays $\tau$ and
linewidths $\kappa$.

We note that beyond cooling, in applications where both light quadratures
might play a role, coherent and measurement-based feedback control are not
equivalent anymore [1].

##  VII Conclusions & Outlook

We implemented an all-optical coherent feedback platform to control the motion
of a mechanical oscillator and demonstrated full control via the parameters of
the feedback loop, namely the phase and the delay. We showed theoretically
that this scheme can be used for ground-state cooling in the unresolved
sideband regime without the need for measurements. We demonstrated
experimentally that even with a moderate mechanical $Q-$factor, we can beat
the theoretical lowest-phonon number limit of cavity dynamical backaction
cooling in the unresolved sideband regime, with only 1$\%$ of the optical
power required for the latter. In contrast to previous proposals, where
feedback is performed on the optical cavity mode [27, 22], we perform feedback
directly on a mechanical oscillator mode , using orthogonal polarizations for
first and second passes. But our scheme does not rely on the availability of
same-frequency orthogonal cavity modes: When experimentally possible, one
could use another longitudinal mode of the cavity. Alternatively, the scheme
could also be implemented via a loop that is opened or closed by an optical
switch, with a switching rate $1/(2\tau)$, in such a way that first and second
passes are never at the same time in the cavity. As such, the double-pass
scheme can be adapted to a variety of different physical systems and is not
restricted to optomechanics. In the present configuration, ground state
cooling would be achievable by improving the thermal conductivity of the
membrane support to ensure thermalization at $4.2\text{\,}\mathrm{K}$.
However, the most straightforward improvement would consist in using a
mechanical resonator with a higher quality factor [53].

The beauty of coherent feedback lies in its potential for processing non-
commuting observables [24]. In the unresolved sideband regime, coherent
feedback opens up the possibility to generate self-interactions and mechanical
squeezing [35], photon-phonon entanglement [22], or to enhance optical
nonlinearities at the single-photon level [19]. Our scheme could also be
exploited in the sideband resolved regime, where optomechanical couplings of
the form $\hat{c}_{j}\hat{B}+\hat{B}^{\dagger}\hat{c}_{j}^{\dagger}$ can be
designed, with $\hat{B}$ being a raising or lowering operator of the
mechanical oscillator. Such couplings are sensitive to both quadratures and
potentially allow for creating non-classical mechanical states using coherent
feedback [3].

In contrast to measurement-based control, coherent feedback avoids the
incoherent addition of feedback and measurement noise, making it a key
technique in the low phonon-number regime [24]. In particular, the
interference of optical input noise can be tuned to realize backaction
cancellation [35, 8]. This makes coherent feedback a promising candidate for
sensing applications, where such backaction cancellation is highly desirable.

Coherent feedback thus opens up new approaches for engineering the dynamics of
quantum systems with potential applications for quantum technology,
measurement and control as well as quantum thermodynamics.

###### Acknowledgements.

We thank Chun Tat Ngai for discussions. This work was supported by the project
“Modular mechanical-atomic quantum systems” (MODULAR) of the European Research
Council (ERC) and by the Swiss Nanoscience Institute (SNI). M.B.A.
acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement
N°101023088. P.P.P. and M.B. acknowledge funding from the Swiss National
Science Foundation (Eccellenza Professorial Fellowship PCEFP2_194268).

##  Appendix A Phonon number calibrations

###  A.1 Phonon number via homodyne detection

In this section, we detail the calibration of the phonon number, which is
determined by performing a homodyne measurement on a leak of the first beam
interacting with the membrane [see Fig. 1(a)]. This is done at room
temperature, where we know the membrane is thermalized to the environment.

In the frequency domain, we can write the field $\hat{c}_{1}$ inside the
cavity as [47]

$\hat{c}_{1}(\omega)=-\chi_{c}(\omega)\left[\sqrt{\kappa\eta_{1}}\hat{a}^{\mathrm{in}}_{1}(\omega)+i\sqrt{2}g_{1}\hat{X}_{m}(\omega)\right],$
(20)

where $\eta_{1}$ is the cavity incoupling efficiency and
$\chi_{c}(\omega)^{-1}=\kappa/2-i(\Delta+\omega)$ the cavity susceptibility.

The output phase quadrature of the light $\hat{P}_{L}$ is related to the one
inside the cavity $\hat{p}_{1}$ via the input-output relation
$\hat{P}_{L}=\hat{P}_{1}^{\mathrm{in}}+\sqrt{\kappa\eta_{1}}\hat{p}_{1}$.

In the thermal-noise-dominated regime, we can neglect the term
$\hat{P}_{1}^{\mathrm{in}}$. Setting
$g_{1}=g_{0}\,\chi_{c}(0)\sqrt{\kappa\eta_{1}}\alpha^{\mathrm{in}}_{1}$ we can
express $\hat{P}_{L}$ in terms of experimentally measurable quantities:

$\hat{P}_{L}(\omega)=-\eta_{1}g_{0}\alpha^{\mathrm{in}}_{1}\,\mathcal{R}(\omega)\hat{X}_{m}(\omega),$
(21)

with the cavity transduction factor

$\mathcal{R}(\omega)=\kappa\,\big{[}\chi_{c}(0)\chi_{c}(\omega)+\chi^{\ast}_{c}(0)\chi^{\ast}_{c}(-\omega)\big{]}\overset{\Delta=0}{=}\frac{8}{\kappa}.$
(22)

The thermal occupation number $\bar{n}_{m}$ is then calibrated by measuring
this light quadrature. For this, we beat the output beam with a strong local
oscillator with power
$P_{\mathrm{LO}}=\hbar\omega_{L}\lvert\alpha_{\mathrm{LO}}\rvert^{2}\gg P_{1}$
and use a Mach-Zehnder interferometer to perform homodyne detection. By means
of a piezoelectric mirror in the local oscillator arm, we can scan the
homodyne angle $\theta$, and the recorded output voltage is given by:

$\hat{D}(\omega)=\sqrt{2}\alpha_{\mathrm{LO}}\big{[}\cos{(\theta)}\hat{X}_{L}(\omega)+\sin{(\theta)}\hat{P}_{L}(\omega)\big{]}.$
(23)

The DC signal of the interference is given by
$D(\omega=0)=2\alpha_{\mathrm{LO}}\alpha^{\mathrm{in}}_{1}\cos\theta$, whose
amplitude $D_{0}=2\alpha_{\mathrm{LO}}\alpha^{\mathrm{in}}_{1}$ can now be
used to calibrate the membrane signal. Locking the interferometer at
$\theta=\pi/2$ we are sensitive to the phase quadrature $\hat{P}_{L}$ encoding
the membrane signal $\hat{X}_{m}$

$\hat{D}_{\theta=\frac{\pi}{2}}(\omega)=\sqrt{2}\alpha_{\mathrm{LO}}\alpha^{\mathrm{in}}_{1}\eta_{1}g_{0}\mathcal{R}(\omega)\hat{X}_{m}(\omega).$
(24)

From this signal $\hat{D}_{\theta=\frac{\pi}{2}}(\omega)$, we can compute the
detected power spectral density (PSD) $S_{DD}(\omega)$

$S_{DD}(\omega)=\frac{1}{2}D_{0}^{2}\left[\eta_{1}g_{0}\,|\mathcal{R}(\omega)|\right]^{2}S_{XX}(\omega),$
(25)

with the membrane displacement power spectral density $S_{XX}(\omega)$. On the
other hand, the average number of phonons is related to the variance of the
membrane displacement as

$\displaystyle\bar{n}_{m}+\frac{1}{2}=\langle\hat{X}^{2}_{m}(t)\rangle=2\int^{\infty}_{0}\bar{S}_{XX}(\omega)\,\frac{d\omega}{2\pi},$
(26)

with the symmetrized PSD $\bar{S}_{XX}(\omega)$. Therefore, combining Eqs.
(25) and (26) the number of phonons can be directly obtained from the recorded
power spectral density as

$\bar{n}_{m}=\frac{4}{D_{0}^{2}\left[\eta
g_{0}\,|\mathcal{R}(\omega)|\right]^{2}}\int_{0}^{\infty}\bar{S}_{DD}(\omega)\,\frac{d\omega}{2\pi}-\frac{1}{2}.$
(27)

###  A.2 Phonon number via area ratios

An alternative way to determine the thermal occupation number of the
mechanical oscillator consists in comparing the measured power spectral
density area in the presence of feedback with the area obtained if only a
single interaction takes place.

In a single-pass interaction, the phonon occupation can be estimated from the
procedure outlined in the previous section or from the measured linewidth
$\Gamma_{m}$ and knowledge of the environment temperature by
$\bar{n}_{\mathrm{calib}}=n_{\mathrm{th}}(T)\gamma_{m}/(\gamma_{m}+\Gamma_{m})$.
We can now associate the area of the measured spectrum $A_{\mathrm{calib}}$ to
an occupation $\bar{n}_{\mathrm{calib}}$.

For this, we first obtain the membrane occupation in the presence of moderate
cooling due to the optical field following standard optomechanical cooling
theory measured as a change in linewidth to its voltage transduction in our
PSD measurement $\bar{S}^{\mathrm{calib}}_{DD}(\omega)$. We can then use the
ratio between the corresponding calibration area $A_{\mathrm{calib}}$ and the
computed occupation number $\bar{n}_{\mathrm{calib}}$ to determine the
occupation number $\bar{n}_{m}$ due to our coherent feedback loop.

In the presence of additional cooling due to the feedback loop, the phonon
number is then given by the ratio of areas

$\displaystyle\bar{n}_{m}=\frac{\bar{n}_{\mathrm{calib}}}{A_{\mathrm{calib}}}A_{DD},$
(28)

with $A_{DD}$ the area of the measured PSD. For this to be accurate, it is
essential to know the temperature $T$ at which the membrane is thermalized. To
determine the actual thermalization temperature at different cryostat
temperatures we use standard optomechanical cooling experiments as shown in
Fig. 6. For each cryostat temperature, we increase the power of a red-detuned
beam and measure the displacement PSD with a resonant probe beam in a homodyne
detection scheme. The ratio of the areas at different temperatures can be used
to infer the temperature of the environment. If we compare the reduction in
areas due to the optomechanical cooling to the theoretical expectation at
those temperatures we find an excellent agreement. Furthermore, this agreement
allows us to exclude excess backaction through classical laser noise for the
powers that we are employing. We find a satisfactory agreement between the two
calibration procedures described in this and the previous section.

Figure 6: Standard dynamical backaction cooling. For different cryostat
temperatures, the plot shows the phonon occupation as a function of cooling
beam power. The shaded area marks the cavity dynamical backaction limit.

##  Appendix B Theoretical description

###  B.1 Langevin equations including losses

In this section, we derive the Langevin equations provided in Eqs. (1), (3),
and (6) from the non-linear optomechanical equations of motion including
additional losses. We start by considering the Langevin equations for the
mechanical oscillator

$\displaystyle\partial_{t}\hat{X}^{\prime}_{m}(t)=$
$\displaystyle\Omega_{m}\hat{P}_{m}(t),$ (29)
$\displaystyle\partial_{t}\hat{P}_{m}(t)=$
$\displaystyle-\Omega_{m}\hat{X}^{\prime}_{m}(t)-\gamma_{m}\hat{P}_{m}(t)-\sqrt{2}g_{0}\sum_{j=1}^{2}\hat{C}_{j}^{\dagger}(t)\hat{C}_{j}(t)$
$\displaystyle-\sqrt{2}\,\hat{\xi}_{\rm th}(t).$

Here $\hat{C}_{j}$ denotes the annihilation operator in the cavity mode $j$
and the radiation pressure force originates from the nonlinear optomechanical
Hamiltonian $\hat{H}_{\mathrm{om},j}=\sqrt{2}\hbar
g_{0}\hat{C}_{j}^{\dagger}\hat{C}_{j}\hat{X}^{\prime}_{m}$ [20]. We now write

$\hat{C}_{j}=\,e^{i\phi_{j}}\left(|\alpha_{j}|+\hat{c}_{j}\right),$ (30)

where $\alpha_{j}=\langle\hat{C}_{j}\rangle$. In the optomechanical coupling,
we drop the non-linear term that is independent of $\alpha_{j}$, as it is
negligible for large average displacements. This results in the equation for
the momentum

$\displaystyle\partial_{t}\hat{P}_{m}(t)=$
$\displaystyle-\Omega_{m}\left[\hat{X}^{\prime}_{m}(t)+\delta
X\right]-\gamma_{m}\hat{P}_{m}(t)$ (31)
$\displaystyle-2\sum_{j=1}^{2}g_{j}\hat{x}_{j}(t)-\sqrt{2}\hat{\xi}_{\rm
th}(t),$ $\displaystyle\delta X=$
$\displaystyle\sqrt{2}\frac{g_{0}}{\Omega_{m}}(|\alpha_{1}|^{2}+|\alpha_{2}|^{2}),$

where $\hat{x}_{j}=(\hat{c}_{j}^{\dagger}+\hat{c}_{j})/\sqrt{2}$ as in the
main text. To recover Eqs. (1) in the main text, we identify
$\hat{X}_{m}=\hat{X}^{\prime}_{m}+\delta X$, which also obeys
$\partial_{t}\hat{X}_{m}=\Omega_{m}\hat{P}_{m}$. We note that $\delta X$ is of
order one for the parameters in the experiment, implying that the mechanical
oscillator exhibits an average displacement of the order of the zero-point
fluctuations.

The Langevin equation for the cavity modes read

$\displaystyle\partial_{t}\hat{C}_{j}(t)=$
$\displaystyle\left(i\Delta-\frac{\kappa}{2}\right)\hat{C}_{j}(t)-i\sqrt{2}g_{0}\hat{C}_{j}\hat{X}^{\prime}_{m}(t)$
(32) $\displaystyle-\sqrt{\kappa}\hat{A}_{j}^{\rm in}(t),$

where $\hat{A}_{j}^{\rm in}=\alpha_{j}^{\rm
in}+\,e^{i\phi_{j}}\hat{a}_{j}^{\rm in}$ denotes the input field for the
respective mode, with $\langle\hat{A}_{j}^{\rm in}\rangle=\alpha_{j}^{\rm
in}$. Using Eq. (30) and dropping the non-linear term in the optomechanical
coupling, we find equations of motion for both the average displacements
$\alpha_{j}$ and the operators describing fluctuations around those averages
$\hat{c}_{j}$. For the averages, we find

$\partial_{t}\alpha_{j}(t)=\left(i\Delta+i\sqrt{2}g_{0}\delta
X-\frac{\kappa}{2}\right)\alpha_{j}(t)-\sqrt{\kappa}\alpha_{j}^{\rm in}.$ (33)

We note that these equations are non-linear because $\delta X$ depends on
$\alpha_{j}$. This non-linear term acts like a displacement-dependent
detuning. However, since $\delta X$ is of order one and $g_{0}\ll\kappa$, this
can safely be ignored and we find the steady-state values

$\alpha_{j}=-\frac{\sqrt{\kappa}\alpha_{j}^{\rm in}}{\kappa/2-i\Delta}.$ (34)

For the fluctuations, we find the Langevin equations

$\displaystyle\partial_{t}\hat{c}_{j}(t)=$
$\displaystyle\left(i\Delta+i\sqrt{2}g_{0}\delta
X-\frac{\kappa}{2}\right)\hat{c}_{j}(t)-i\sqrt{2}g_{j}\hat{X}_{m}(t)$ (35)
$\displaystyle-\sqrt{\kappa}\hat{a}_{j}^{\rm in}(t).$

The contribution to the detuning due to the displacement of the mechanical
oscillator can again safely be ignored. For the first cavity mode, we directly
recover Eq. (3). For the second cavity mode, we first need to determine the
input field operator.

In the case of finite coupling efficiency to the cavity mode, the input
operator for the first cavity mode reads

$\hat{a}_{1}^{\rm in}(t)=\sqrt{\eta_{1}}\hat{a}_{1}^{\prime\,\rm
in}(t)+\sqrt{1-\eta_{1}}\hat{\nu}_{1}(t),$ (36)

where $\hat{a}_{1}^{\prime\,\rm in}$ denotes the input mode driven by the
local oscillator and $\hat{\nu}_{1}$ denotes an additional source of vacuum
noise. Note that $\alpha_{1}^{\rm in}=\sqrt{\eta_{1}}\alpha_{1}^{\prime\rm
in}$, i.e., the cavity mode is driven by a fraction $\eta_{1}$ of the physical
input power. The output mode of the first cavity mode that is used for
feedback is given by

$\hat{a}^{\prime\,\rm out}_{1}(t)=\hat{a}_{1}^{\prime\,\rm
in}(t)+\sqrt{\eta_{1}\kappa}\,\hat{c}_{1}(t).$ (37)

This mode is then combined with an auxiliary local oscillator $\hat{a}_{\rm
aux}^{\prime}$ at a beamsplitter with splitting ratio $\eta_{\rm aux}$, before
being delayed by the time $\tau$. Including further losses arising from the
delay fiber, additional optics as well as coupling to the second cavity mode,
the input mode for the second cavity mode reads

$\displaystyle\hat{a}_{2}^{\rm
in}(t)=\sqrt{1-\eta_{2}\eta_{T}}\hat{\nu}_{2}(t)$ (38)
$\displaystyle+\sqrt{\eta_{2}\eta_{T}}\left[\sqrt{\eta_{\rm
aux}}\,e^{i\varphi}\hat{a}_{1}^{\prime\rm out}(t-\tau)+\sqrt{1-\eta_{\rm
aux}}\,\hat{a}_{\rm aux}^{\prime}(t-\tau)\right]$ $\displaystyle\hskip
14.22636pt=\sqrt{1-\eta_{2}\eta_{T}}\hat{\nu}_{2}(t)+\sqrt{\eta_{1}\eta_{2}\eta_{T}\eta_{\rm
aux}\kappa}\,e^{i\varphi}\hat{c}_{1}(t-\tau)$
$\displaystyle+\sqrt{\eta_{2}\eta_{T}\eta_{\rm
aux}}\,e^{i\varphi}\hat{a}_{1}^{\prime\rm
in}(t-\tau)+\sqrt{\eta_{2}\eta_{T}(1-\eta_{\rm aux})}\,\hat{a}_{\rm
aux}^{\prime}(t-\tau).$

Here the phase $\varphi=\phi_{1}-\phi_{2}$ arises because of the different
phases that enter the definitions of $\hat{c}_{1}$ and $\hat{c}_{2}$, see Eq.
(30). We may now introduce the mode

$\displaystyle\hat{a}_{\rm aux}(t)=\sqrt{\frac{\eta_{2}\eta_{T}(1-\eta_{\rm
aux})}{1-\eta}}\hat{a}^{\prime}_{\rm
aux}(t-\tau)+\sqrt{\frac{1-\eta_{2}\eta_{T}}{1-\eta}}\hat{\nu}_{2}(t)$ (39)
$\displaystyle+\sqrt{\frac{(1-\eta_{1})\eta_{2}\eta_{T}\eta_{\rm
aux}}{1-\eta}}\,e^{i\varphi}\left[\sqrt{1-\eta_{1}}\hat{a}_{1}^{\prime\rm
in}(t-\tau)-\sqrt{\eta_{1}}\hat{\nu}_{1}(t-\tau)\right],$

where we introduced the total efficiency

$\eta=\eta_{1}\eta_{2}\eta_{T}\eta_{\rm aux}.$ (40)

We note that in the ideal limit where $\eta_{1}=\eta_{2}=\eta_{T}=1$, we have
$\hat{a}_{\rm aux}(t)=\hat{a}^{\prime}_{\rm aux}(t-\tau)$, i.e., we shifted
the time-argument. With the help of Eq. (39), we find

$\hat{a}_{2}^{\rm
in}(t)=\sqrt{\eta}\,e^{i\varphi}\left[\sqrt{\kappa}\hat{c}_{1}(t-\tau)+\hat{a}_{1}^{\rm
in}(t-\tau)\right]+\sqrt{1-\eta}\,\hat{a}_{\rm aux}(t),$ (41)

recovering Eq. (6). Importantly, the mode $\hat{a}_{\rm aux}$ is orthogonal to
the mode $\hat{a}_{1}^{\rm in}$ for all values of $\tau$, such that we
preserve the commutation relations from the ideal scenario. However, the
average of the auxiliary mode reads

$\alpha_{\rm aux}=(1-\eta_{1})\sqrt{\frac{\eta_{2}\eta_{T}\eta_{\rm
aux}}{1-{\eta}}}\alpha_{1}^{\prime\rm
in}+\sqrt{\frac{\eta_{2}\eta_{T}(1-\eta_{\rm aux})}{1-\eta}}\alpha_{\rm
aux}^{\prime}.$ (42)

For $\eta_{1}=\eta_{2}=\eta_{T}=1$, we obtain $\alpha_{\rm
aux}^{\prime}=\alpha_{\rm aux}$ as expected. However, for $\eta_{1}\neq 1$,
$\alpha_{\rm aux}$ is non-zero even in the absence of an auxiliary local
oscillator.

With these input modes, we find the average displacements of the cavity modes
in terms of $\alpha_{1}^{\rm in}=\sqrt{\eta_{1}}\,\alpha_{1}^{\prime\rm in}$
and $\alpha_{\rm aux}$

$\displaystyle\alpha_{1}=-\frac{\sqrt{\kappa}\alpha_{1}^{\rm
in}}{\kappa/2-i\Delta},$ (43)
$\displaystyle\alpha_{2}=\frac{1}{\kappa/2-i\Delta}\left[\sqrt{\eta\kappa}\,\frac{{\kappa}/{2}+i\Delta}{{\kappa}/{2}-i\Delta}\alpha_{1}^{\rm
in}-\sqrt{(1-\eta)\kappa}\,\alpha_{\rm aux}\right].$

The Langevin equations in the main text thus fully capture additional loss
channels. However, the amplitudes of the physical inputs
$\alpha_{1}^{\prime\rm in}$ and $\alpha_{\rm aux}^{\prime}$ have to be re-
scaled in order to take into account all the loss channels that are present.

###  B.2 Frequency space

The Langevin equations in Eqs. (1), (3), and (6) can conveniently be written
as a matrix equation in frequency space

$-i\omega\hat{\bm{r}}(\omega)=A(\omega)\hat{\bm{r}}(\omega)+B(\omega)\hat{\bm{r}}_{\rm
in}(\omega),$ (44)

where we introduced the vectors

$\hat{\bm{r}}(\omega)=\begin{pmatrix}\hat{X}_{m}(\omega)\\\
\hat{P}_{m}(\omega)\\\ \hat{x}_{1}(\omega)\\\ \hat{p}_{1}(\omega)\\\
\hat{x}_{2}(\omega)\\\ \hat{p}_{2}(\omega)\end{pmatrix},\quad\hat{\bm{r}}_{\rm
in}(\omega)=\begin{pmatrix}\hat{\xi}_{\rm th}(\omega)\\\ \hat{x}_{1}^{\rm
in}(\omega)\\\ \hat{p}_{1}^{\rm in}(\omega)\\\ \hat{x}_{\rm aux}(\omega)\\\
\hat{p}_{\rm aux}(\omega)\end{pmatrix}.$ (45)

Here, operators in frequency space are given by

$\hat{O}(\omega)=\frac{1}{\sqrt{2\pi}}\int\hat{O}(t)\,\,e^{i\omega t}\,dt.$
(46)

The matrices in Eq. (44) read

$A(\omega)=\begin{pmatrix}0&\Omega_{m}&0&0&0&0\\\
-\Omega_{m}&-\gamma_{m}&-2g_{1}&0&-2g_{2}&0\\\
0&0&-{\kappa}/{2}&-\Delta&0&0\\\ -2g_{1}&0&\Delta&-{\kappa}/{2}&0&0\\\
0&0&-\sqrt{\eta}\kappa\cos(\varphi)\,e^{i\omega\tau}&\sqrt{\eta}\kappa\sin(\varphi)\,e^{i\omega\tau}&-{\kappa}/{2}&-\Delta\\\
-2g_{2}&0&-\sqrt{\eta}\kappa\sin(\varphi)\,e^{i\omega\tau}&-\sqrt{\eta}\kappa\cos(\varphi)\,e^{i\omega\tau}&\Delta&-{\kappa}/{2}\end{pmatrix},$
(47)

and

$B(\omega)=\begin{pmatrix}0&0&0&0&0\\\ -\sqrt{2}&0&0&0&0\\\
0&-\sqrt{\kappa}&0&0&0\\\ 0&0&-\sqrt{\kappa}&0&0\\\
0&-\sqrt{\eta\kappa}\cos(\varphi)\,e^{i\omega\tau}&\sqrt{\eta\kappa}\sin(\varphi)\,e^{i\omega\tau}&-\sqrt{(1-\eta)\kappa}&0\\\
0&-\sqrt{\eta\kappa}\sin(\varphi)\,e^{i\omega\tau}&-\sqrt{\eta\kappa}\cos(\varphi)\,e^{i\omega\tau}&0&-\sqrt{(1-\eta)\kappa}\end{pmatrix}.$
(48)

From Eq. (44), we find that any power spectral density can be written using

$\langle\hat{\bm{r}}(\omega)\hat{\bm{r}}^{T}(\omega^{\prime})\rangle=C(\omega)\langle\hat{\bm{r}}_{\rm
in}(\omega)\hat{\bm{r}}_{\rm in}^{T}(\omega^{\prime})\rangle
C^{T}(\omega^{\prime}),$ (49)

where $C(\omega)=[A(\omega)+i\omega]^{-1}B(\omega)$ and the input spectral
density matrix reads

$\langle\hat{\bm{r}}_{\rm in}(\omega)\hat{\bm{r}}_{\rm
in}^{T}(\omega^{\prime})\rangle=\delta(\omega+\omega^{\prime})\begin{pmatrix}S_{\rm
th}(\omega)&0&0&0&0\\\ 0&1/2&i/2&0&0\\\ 0&-i/2&1/2&0&0\\\ 0&0&0&1/2&i/2\\\
0&0&0&-i/2&1/2\end{pmatrix},$ (50)

with $S_{\rm th}(\omega)$ given in Eq. (2).

Before solving these equations, it is instructive to consider the simplified
scenario where $\Delta=0$ and $\varphi=\pi/2$. From the equation for the first
cavity mode, we find

$\left(\frac{\kappa}{2}-i\omega\right)\hat{p}_{1}(\omega)=-2g_{1}\hat{X}_{m}(\omega)-\sqrt{\kappa}\hat{p}_{1}^{\rm
in}(\omega),$ (51)

illustrating how the momentum quadrature contains information on the position
of the mechanical resonator; $\hat{p}_{1}^{\rm in}$ denotes the input (vacuum)
noise. The momentum quadrature of the first cavity mode is then fed to the
second cavity mode and we find

$\left(\frac{\kappa}{2}-i\omega\right)\hat{x}_{2}(\omega)=\sqrt{\eta\kappa}e^{i\omega\tau}\left[\sqrt{\kappa}\hat{p}_{1}(\omega)+\hat{p}_{1}^{\rm
in}\right]-\sqrt{(1-\eta)\kappa}\hat{x}_{\rm aux}.$ (52)

Finally, the position quadrature of the second cavity mode couples to the
mechanical oscillator, completing the feedback loop. We find for the momentum
quadrature of the mechanical oscillator

$-i\omega\hat{P}_{m}(\omega)=-\Omega_{m}\hat{X}_{m}(\omega)-\gamma_{m}\hat{P}_{m}(\omega)-4\,e^{i\omega\tau}\frac{g_{1}g_{2}\sqrt{\eta}\kappa}{\left({\kappa}/{2}-i\omega\right)^{2}}\hat{X}_{m}(\omega)+\sqrt{2}\hat{\xi}_{\rm
th}(\omega)+\sqrt{2}\hat{\xi}_{\rm fb}(\omega).$ (53)

Using the relation
$-i\omega\hat{X}_{m}(\omega)=\Omega_{m}\hat{P}_{m}(\omega)$, we find that the
real and imaginary parts of the pre-factor of $\hat{X}_{m}(\omega)$ in Eq.
(53) correspond to frequency shift and damping respectively. Since the delay
$\tau$ determines the phase of the pre-factor, it can be used to tune between
a frequency shift and damping, in complete agreement with our analysis in the
main text. Finally, to recover Eqs. (8) in the main text, we neglect the
frequency dependence of the frequency shift and damping and take the
unresolved sideband limit $\kappa\gg\Omega_{m}$.

###  B.3 Eliminating the cavity

Since we are dealing with a linear set of equations, the cavity modes may be
eliminated from Eq. (44). A tedious but straightforward calculation results in

$-i\omega\hat{P}_{m}(\omega)=-[\Omega_{m}+2\delta\Omega_{m}(\omega)]\hat{X}_{m}(\omega)-[\Gamma_{m}(\omega)+\gamma_{m}]\hat{P}_{m}(\omega)+\sqrt{2}\hat{\xi}_{\rm
th}(\omega)+\sqrt{2}\hat{\xi}_{\rm fb}(\omega),$ (54)

where we introduced the frequency-dependent frequency shift

$\delta\Omega_{m}(\omega)={\rm
Re}\left\\{\frac{2\Delta(g_{1}^{2}+g_{2}^{2})}{\Delta^{2}+\left({\kappa}/{2}-i\omega\right)^{2}}-2\,\,e^{i\omega\tau}g_{1}g_{2}\sqrt{\eta}\kappa\frac{2\Delta\left({\kappa}/{2}-i\omega\right)\cos(\varphi)-\left[\Delta^{2}-\left({\kappa}/{2}-i\omega\right)^{2}\right]\sin(\varphi)}{\left[\Delta^{2}+\left({\kappa}/{2}-i\omega\right)^{2}\right]^{2}}\right\\},$
(55)

and the optomechanical damping rate

$\Gamma_{m}(\omega)=\frac{\Omega_{m}}{\omega}{\rm
Im}\left\\{-\frac{4\Delta(g_{1}^{2}+g_{2}^{2})}{\Delta^{2}+\left({\kappa}/{2}-i\omega\right)^{2}}+4\,e^{i\omega\tau}g_{1}g_{2}\sqrt{\eta}\kappa\frac{2\Delta\left({\kappa}/{2}-i\omega\right)\cos(\varphi)-\left[\Delta^{2}-\left({\kappa}/{2}-i\omega\right)^{2}\right]\sin(\varphi)}{\left[\Delta^{2}+\left({\kappa}/{2}-i\omega\right)^{2}\right]^{2}}\right\\}.$
(56)

The feedback noise is described by the spectral density $\langle\hat{\xi}_{\rm
fb}(\omega)\hat{\xi}_{\rm
fb}(\omega^{\prime})\rangle=\delta(\omega+\omega^{\prime})\,S_{\rm
fb}(\omega)$ with

$S_{\rm
fb}(\omega)=\frac{\kappa(g_{1}^{2}+g_{2}^{2})}{(\kappa/2)^{2}+(\Delta+\omega)^{2}}+2\kappa
g_{1}g_{2}\sqrt{\eta}\frac{[(\Delta+\omega)^{2}-(\kappa/2)^{2}]\cos(\varphi+\omega\tau)+\kappa(\Delta+\omega)\sin(\varphi+\omega\tau)}{[(\kappa/2)^{2}+(\Delta+\omega)^{2}]^{2}}.$
(57)

We stress that Eqs. (54-57) involve no further approximations after the
linearization of the optomechanical coupling.

For a high-quality oscillator, we may replace $\delta\Omega_{m}(\omega)$ with
$\delta\Omega_{m}\equiv\delta\Omega_{m}(\Omega_{m})$ and $\Gamma_{m}(\omega)$
with $\Gamma_{m}$ in Eq. (54). This results in a second-order differential
equation describing a damped harmonic oscillator

$\displaystyle\partial_{t}^{2}\hat{X}_{m}(t)=$
$\displaystyle-(\Omega_{m}+\delta\Omega_{m})^{2}\hat{X}_{m}(t)-(\Gamma_{m}+\gamma_{m})\,\partial_{t}\hat{X}_{m}(t)$
(58) $\displaystyle+\Omega_{m}\sqrt{2}\,\hat{\xi}_{\rm
th}(t)+\Omega_{m}\sqrt{2}\,\hat{\xi}_{\rm fb}(t).$

From this equation, we may derive the spectral density

$\displaystyle S_{XX}(\omega)=$ $\displaystyle\frac{1}{2}\frac{S_{\rm
th}(\Omega_{m})+S_{\rm
fb}(\Omega_{m})}{(\Omega_{m}+\delta\Omega_{m}-\omega)^{2}+[({\gamma_{m}+\Gamma_{m}})/{2}]^{2}}$
(59) $\displaystyle+\frac{1}{2}\frac{S_{\rm th}(-\Omega_{m})+S_{\rm
fb}(-\Omega_{m})}{(\Omega_{m}+\delta\Omega_{m}+\omega)^{2}+[({\gamma_{m}+\Gamma_{m}})/{2}]^{2}},$

where we again invoked the high quality factor of the oscillator.

The number of phonons can be obtained from

$\displaystyle 2\bar{n}_{m}+1=$
$\displaystyle\int_{-\infty}^{\infty}[S_{XX}(\omega)+S_{PP}(\omega)]\,\frac{d\omega}{2\pi}$
(60)
$\displaystyle=\int_{-\infty}^{\infty}S_{XX}(\omega)[1+(\omega/\Omega_{m})^{2}]\,\frac{d\omega}{2\pi}$
$\displaystyle\simeq
2\int_{-\infty}^{\infty}S_{XX}(\omega)\,\frac{d\omega}{2\pi},$

which yields Eq. (12) upon using $\Gamma_{m}={S}_{\rm fb}(\Omega_{m})-{S}_{\rm
fb}(-\Omega_{m})$.

###  B.4 The unresolved sideband limit

Here we provide simplified expressions for the unresolved sideband limit,
$\Omega_{m}\ll\kappa$. In contrast to the expressions given in the main text,
we consider a finite detuning $\Delta$. We first consider dynamical backaction
cooling without coherent feedback by setting $\eta=0$. In this case, the
frequency shift and the optomechanical damping rate reduce to

$\delta\Omega_{\rm
dyn}=\frac{2\Delta(g_{1}^{2}+g_{2}^{2})}{\Delta^{2}+\left({\kappa}/{2}\right)^{2}},$
(61)

and

$\Gamma_{\rm
dyn}=-4\frac{\Delta\kappa\Omega_{m}(g_{1}^{2}+g_{2}^{2})}{\left[\Delta^{2}+\left({\kappa}/{2}\right)^{2}\right]^{2}}.$
(62)

For $\Gamma_{m}\gg\gamma_{m}$, the phonon occupation number in Eq. (12)
reduces to

$\bar{n}_{\rm
dyn}=\frac{\Delta^{2}+\left({\kappa}/{2}\right)^{2}}{4|\Delta|\Omega_{m}}-\frac{1}{2},$
(63)

which is minimized at $\Delta=-\kappa/2$ where it reads

$\bar{n}_{\rm dyn}=\frac{\kappa}{4\Omega_{m}}\gg 1.$ (64)

Here, we dropped the term $1/2$ as it becomes negligible. The last equation is
the well-known cooling limit for cavity dynamical backaction cooling in the
unresolved sideband regime [20].

In the presence of coherent feedback ($\eta\neq 0$), we find the following
expressions to lowest-order in $\Omega_{m}/\kappa$

$\delta\Omega_{m}=\frac{2\Delta(g_{1}^{2}+g_{2}^{2})}{\Delta^{2}+\left({\kappa}/{2}\right)^{2}}-2\cos(\Omega_{m}\tau)g_{1}g_{2}\sqrt{\eta}\kappa\frac{\Delta\kappa\cos(\varphi)-\left[\Delta^{2}-\left({\kappa}/{2}\right)^{2}\right]\sin(\varphi)}{\left[\Delta^{2}+\left({\kappa}/{2}\right)^{2}\right]^{2}},$
(65)

and

$\Gamma_{m}=4\sin(\Omega_{m}\tau)g_{1}g_{2}\sqrt{\eta}\kappa\frac{\Delta\kappa\cos(\varphi)-\left[\Delta^{2}-\left({\kappa}/{2}\right)^{2}\right]\sin(\varphi)}{\left[\Delta^{2}+\left({\kappa}/{2}\right)^{2}\right]^{2}}.$
(66)

We note that the coherent feedback allows for a $\Gamma_{m}$ that is
independent of $\Omega_{m}/\kappa$ to lowest-order in this parameter. This
stands in contrast to cavity dynamical backaction cooling, where $\Gamma_{\rm
dyn}\propto\Omega_{m}/\kappa$ to lowest-order [cf. Eq. (62)]. In the limit
$\Gamma_{m}\gg\gamma_{m}$, the number of phonons is then given by

$\bar{n}_{m}=\frac{\kappa}{\Gamma_{m}}\frac{(g_{1}^{2}+g_{2}^{2})}{(\kappa/2)^{2}+\Delta^{2}}+\frac{1}{2}\frac{\cos(\Omega_{m}\tau)}{\sin(\Omega_{m}\tau)}\frac{\kappa\Delta\sin(\varphi)+[\Delta^{2}-(\kappa/2)^{2}]\cos(\varphi)}{\kappa\Delta\cos(\varphi)-[\Delta^{2}-(\kappa/2)^{2}]\sin(\varphi)}-\frac{1}{2}.$
(67)

For $\Delta=0$, Eqs. (65), (66), and (67) reduce to Eqs. (8) and (14).

##  Appendix C Equivalence with measurement-based feedback cooling

In this section, we extend the comparison between coherent feedback and
measurement-based feedback of Sec. VI by considering a generic spectral filter
function $\Xi_{\mathrm{mf}}(\omega)$, which implements an arbitrary feedback
response taking place after the measurement. As for the treatment of coherent
feedback, we can obtain a reduced description for the mechanical variables by
eliminating the cavity mode

$-i\omega\hat{P}_{m}(\omega)=-[\Omega_{m}+2\delta\Omega_{\mathrm{mf}}(\omega)]\hat{X}_{m}(\omega)-[\Gamma_{\mathrm{mf}}(\omega)+\gamma_{m}]\hat{P}_{m}(\omega)+\sqrt{2}\,\hat{\xi}_{\rm
th}(\omega)+\sqrt{2}\,\hat{\xi}_{\mathrm{mf}}(\omega),$ (68)

where we introduced the frequency shift and optomechanical damping rate,
respectively given by

$\displaystyle\delta\Omega_{\mathrm{mf}}(\omega)$
$\displaystyle=\frac{1}{2}{\rm
Re}\left\\{\frac{2g_{1}\Xi_{\mathrm{mf}}(\omega)}{{\kappa}/{2}-i\omega}\right\\}\,,$
(69) $\displaystyle\Gamma_{\mathrm{mf}}(\omega)$
$\displaystyle=-\frac{\Omega_{m}}{\omega}{\rm
Im}\left\\{\frac{2g_{1}\Xi_{\mathrm{mf}}(\omega)}{{\kappa}/{2}-i\omega}\right\\}\,,$
(70)

with $\Gamma_{\mathrm{mf}}=\Gamma_{\mathrm{mf}}(\Omega_{m})$, and collected
the noise terms in the expression

$\hat{\xi}_{\mathrm{mf}}(\omega)=\frac{g_{1}\sqrt{2\kappa}}{{\kappa}/{2}-i\omega}\hat{x}_{\mathrm{in}}(\omega)-\frac{\Xi_{\mathrm{mf}}(\omega)}{\sqrt{2\kappa}}\left(\frac{{\kappa}/{2}+i\omega}{{\kappa}/{2}-i\omega}\right)\hat{p}_{\mathrm{in}}(\omega)-\frac{\Xi_{\mathrm{mf}}(\omega)}{\sqrt{2\kappa}}\sqrt{\frac{1-\eta_{\mathrm{det}}}{\eta_{\mathrm{det}}}}\hat{p}_{0}(\omega)\,.$
(71)

The first term in Eq. (71) describes shot noise due to radiation pressure
interaction with the first cavity, while the second and third terms describe
the noise associated with the feedback process. The spectral density
characterizing the noise $\hat{\xi}_{\mathrm{mf}}$ reads

$\displaystyle S_{\mathrm{mf}}(\omega)$ $\displaystyle=\frac{\kappa
g_{1}^{2}}{\omega^{2}+\left({\kappa}/{2}\right)^{2}}+\frac{|\Xi_{\mathrm{mf}}(\omega)|^{2}}{4\kappa\eta_{\mathrm{det}}}-g_{1}{\rm
Im}\left\\{\frac{\Xi_{\mathrm{mf}}(\omega)}{{\kappa}/{2}-i\omega}\right\\}$
(72) $\displaystyle=\frac{\kappa
g_{1}^{2}}{\omega^{2}+\left({\kappa}/{2}\right)^{2}}+\frac{|\Xi_{\mathrm{mf}}(\omega)|^{2}}{4\kappa\eta_{\mathrm{det}}}+\frac{1}{2}\frac{\omega}{\Omega_{m}}\Gamma_{\mathrm{mf}}(\omega)\,.$

We can now compare the quantities $\delta\Omega_{\mathrm{mf}}(\omega)$,
$\Gamma_{\mathrm{mf}}(\omega)$ and $S_{\mathrm{mf}}(\omega)$ with the
corresponding expressions derived for the case of coherent feedback,
respectively given by Eqs. (55), (56) and (57). We further focus on the case
$\Delta=0$ and $\varphi=\pi/2$, which is relevant for cooling.

By requiring the measurement-based feedback to induce the same effective
damping and broadening as coherent feedback at all frequencies, i.e., by
enforcing $\delta\Omega_{\mathrm{mf}}(\omega)=\delta\Omega_{m}(\omega)$ and
$\Gamma_{\mathrm{mf}}(\omega)=\Gamma_{m}(\omega)$, we get the following filter
function

$\widetilde{\Xi}_{\mathrm{mf}}(\omega)=-4g_{2}\sqrt{\eta}\frac{\,e^{i\omega\tau}}{1-2i{\omega}/{\kappa}}\,.$
(73)

In the above expression, the exponential term describes the effect of delay,
i.e., in order to reproduce the effect of coherent feedback a delay has to be
incorporated in the measurement-based feedback loop. If we compare this
expression to the case of cold damping in Eq. (15), we notice that $\kappa/2$
plays the role of the feedback bandwidth, $4g_{2}\sqrt{\eta}$ the strength of
the feedback and that there is no frequency dependence in the numerator for
$\tau=0$.

We then plug this filter in Eq. (72) and compare the corresponding noise
spectral density with that of coherent feedback Eq. (57). It is
straightforward to show that whenever $\eta=\eta_{\mathrm{det}}$, the ensuing
noise spectral densities $S_{\mathrm{fb}}(\omega)$ and
$S_{\mathrm{mf}}(\omega)$ match for all frequency values and for arbitrary
values of delay. This shows the equivalence of measurement-based feedback
cooling and coherent feedback cooling in more general terms than the
particular case of cold damping.

##  Appendix D Coherent feedback cooling performance in different cavity
regimes

Here, we investigate the comparative performance of coherent feedback cooling
versus standard cavity dynamical backaction cooling in both the resolved and
unresolved sideband regimes. To quantify performance, we examine the lowest
achievable phonon number under different scenarios: For cavity cooling, we
consider the well-known cases of $\Delta=-\Omega_{m}$ for
$\kappa\ll\Omega_{m}$ and $\Delta=-\kappa/2$ for $\kappa\gg\Omega_{m}$, where
the minimum phonon number asymptotically approaches $\kappa/(4\Omega_{m})$. In
Fig. 7 we compare these cooling strategies to coherent feedback cooling, for
$\Delta=-\Omega_{m}$ and $\Delta=0$, which are the optimal detunings in the
two different cavity regimes. Our results demonstrate that, in the resolved
sideband regime, coherent feedback cooling provides no significant advantage
over cavity cooling when operating at $\Delta=-\Omega_{m}$ and using the
optimal parameters $\varphi=\Omega_{m}\tau=\pi/2$ mentioned in the main text.
However, in the unresolved sideband regime, we find that coherent feedback
cooling (which performs best for $\Delta=0$) outperforms standard
optomechanical cooling (which leads to minimal occupation at
$\Delta=-\kappa/2$). Intriguingly, we observe that coherent feedback cooling
performs better deep in the unresolved sideband regime, where the cavity can
actually be ignored and the first interaction is performing an almost
instantaneous readout of the membrane’s motion. The plot is shown for
$\eta\simeq 0.98$. For more realistic scenarios, the minimal phonon number
becomes independent of $\kappa$ early in the unresolved sideband regime,
saturating to the value expressed in the main text in Eq. (14).

Figure 7:  Cooling performance for different cavity regimes for coherent
feedback and cavity dynamical backaction cooling. Blue lines show the
performance for coherent feedback for $\Delta=-\Omega_{m}$ (solid) and
$\Delta=0$ (dashed), in both cases $\varphi=\Omega_{m}\tau=\pi/2$, as
described in the theory section of the manuscript. Red lines show the
performance of cavity cooling for $\Delta=-\Omega_{m}$ (solid) and
$\Delta=-\kappa/2$ (dashed). The black dotted line shows the cavity cooling
limit in the unresolved sideband regime. The coherent feedback curves are
shown for $\eta\simeq 0.98$.

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