# Coherent feedback cooling of a nanomechanical membrane with atomic spins

Gian-Luca Schmid  These authors contributed equally to this work  Department
of Physics and Swiss Nanoscience Institute, University of Basel, 4056 Basel,
Switzerland  Chun Tat Ngai  These authors contributed equally to this work
Department of Physics and Swiss Nanoscience Institute, University of Basel,
4056 Basel, Switzerland  Maryse Ernzer  Department of Physics and Swiss
Nanoscience Institute, University of Basel, 4056 Basel, Switzerland  Manel
Bosch Aguilera  Department of Physics and Swiss Nanoscience Institute,
University of Basel, 4056 Basel, Switzerland  Thomas M. Karg  Current address:
IBM Research Europe, Zurich, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland
Department of Physics and Swiss Nanoscience Institute, University of Basel,
4056 Basel, Switzerland  Philipp Treutlein<EMAIL_ADDRESS>Department of Physics and Swiss Nanoscience Institute, University of Basel,
4056 Basel, Switzerland

###### Abstract

Coherent feedback stabilises a system towards a target state without the need
of a measurement, thus avoiding the quantum backaction inherent to
measurements. Here, we employ optical coherent feedback to remotely cool a
nanomechanical membrane using atomic spins as a controller. Direct
manipulation of the atoms allows us to tune from strong-coupling to an
overdamped regime. Making use of the full coherent control offered by our
system, we perform spin-membrane state swaps combined with stroboscopic spin
pumping to cool the membrane in a room-temperature environment to
${T}=$216\text{\,}\mathrm{mK}$$ ($\bar{n}_{m}=2.3\times 10^{3}$ phonons) in
$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$. We furthermore observe and study
the effects of delayed feedback on the cooling performance. Starting from a
cryogenically pre-cooled membrane, this method would enable cooling of the
mechanical oscillator close to its quantum mechanical ground state and the
preparation of nonclassical states.

##  I Introduction

Hybrid quantum systems in which a mechanical oscillator is coupled to a spin
are a promising platform for fundamental quantum science as well as for
quantum sensing Treutlein  _et al._ (2014); Kurizki  _et al._ (2015); Chu and
Gröblacher (2020). The interest in such systems derives from the fact that the
spin – a genuinely quantum-mechanical object – can be used to control, read-
out, and lend new functionality to the much more macroscopic mechanical
device. Recently, different spin-mechanics interfaces have been realized,
involving the coupling of a mechanical oscillator to (pseudo-)spin systems
such as atomic ensembles Camerer  _et al._ (2011); Jöckel  _et al._ (2015);
Christoph  _et al._ (2018); Møller  _et al._ (2017); Karg  _et al._ (2020);
Thomas  _et al._ (2021), quantum dots Yeo  _et al._ (2014); Montinaro  _et
al._ (2014), superconducting qubits O’Connell  _et al._ (2010); Arrangoiz-
Arriola  _et al._ (2019); Clerk  _et al._ (2020), or impurity spins in solids
Rugar  _et al._ (2004); Arcizet  _et al._ (2011); Barfuss  _et al._ (2015);
Lee  _et al._ (2017), using light-, strain-, or magnetically-mediated
interactions.

Coherent feedback is an intriguing concept that can be studied with such
systems Lloyd (2000); Zhang  _et al._ (2017). In coherent feedback, a quantum
system is controlled through its interaction with another one, in such a way
that quantum coherence is preserved. In contrast to measurement-based feedback
Wiseman and Milburn (2009), coherent feedback does not rely on measurements,
thus avoiding the associated backaction and decoherence. Coherent feedback can
under certain conditions outperform measurement-based feedback in tasks such
as cooling of resonators Hamerly and Mabuchi (2012); Bennett  _et al._ (2014),
and it has been implemented in solid state systems to enhance the coherence
time of a qubit Hirose and Cappellaro (2016). In optomechanical systems, it
has been theoretically studied as a way to generate large nonlinearities at
the single photon level Zhang  _et al._ (2012); Wang and Safavi-Naeini (2017),
to enhance optomechanical cooling and state transfer Harwood  _et al._ (2021),
as well as for entanglement generation Woolley and Clerk (2014); Li  _et al._
(2017); Harwood  _et al._ (2021).

In the context of spin-mechanics interfaces, the mechanical oscillator can act
as the system to be controlled, i.e. the plant, which is coupled to a noisy
thermal bath, and the spin system as the controller, coupled to a zero-
temperature bath. Coherent feedback is achieved by coupling the two systems,
thus reducing the noise in the mechanical system by transferring it to the
spin, where it is dissipated. Additional coherent control of the spin enhances
the cooling performance.

Hybrid systems combining atomic ensembles and mechanical oscillators have been
used for sympathetic cooling by coupling the mechanical vibrations of a
membrane to the center-of-mass oscillation of cold atoms in an optical lattice
Jöckel  _et al._ (2015); Christoph  _et al._ (2018). In these systems the
atomic motion was strongly damped and did not offer the possibility for
coherent control. Furthermore, optical traps for atoms cannot reach MHz
trapping frequencies without introducing substantial photon scattering and
dissipation, restricting this cooling scheme to low-frequency mechanical
oscillators. In contrast, collective spin states of atomic ensembles offer
long coherence times and wide magnetic tuning of the spin precession frequency
across the MHz range. Crucially, a versatile quantum toolbox exists that
provides sophisticated techniques for ground-state cooling and quantum control
Hammerer  _et al._ (2010); Pezzè  _et al._ (2018). This makes it possible to
use the atomic spin as a coherent feedback controller, which can be employed
to efficiently cool and control the mechanical oscillator Vogell  _et al._
(2015), e.g., via a state-swap Wallquist  _et al._ (2010).

Here, we demonstrate coherent feedback control of a nanomechanical membrane
oscillator with the collective spin of an atomic ensemble and employ it to
cool the membrane. For this, we exploit the coherent control offered by our
recently demonstrated spin-membrane interface, where light mediates strong
coupling between the two systems Karg  _et al._ (2020). Using optical pumping
on an internal atomic transition we can modify the spin damping rate and study
the membrane cooling performance in different regimes. We show that coherent
state swaps alternated with spin pumping pulses allow us to extract the noise
from the mechanical system in an efficient way, providing the largest cooling
rate and reaching the phonon steady-state faster than for continuous cooling.
Finally, we study the effect of feedback delay onto the steady-state
temperature of the membrane in the light-mediated coupling between the
mechanical and spin systems. Our observations agree well with a theoretical
model.

##  II Setup

Figure 1:  Sketch of the light-mediated spin-membrane coupling. Light
interacts first with the spin, then with the membrane, and then again with the
spin. The propagation of the light leads to a feedback delay $\tau$. On the
way back from the membrane to the spin, a $\pi$-phase is imprinted on the
light, rendering the spin-membrane interaction effectively Hamiltonian for
zero-delay $\tau=0$. The systems can be approximated by harmonic oscillators
of frequencies $\Omega_{m}$ and $\Omega_{s}$ with damping rates $\gamma_{m}$
and $\gamma_{s}$ coupling them to a bath with $\bar{n}_{m,\mathrm{bath}}$ and
$\bar{n}_{s,\mathrm{bath}}$ phonons, respectively. The oscillators are coupled
at a rate $g$. The spin damping rate can be increased by applying a
$\sigma_{-}$-polarized pumping laser. Figure 2: (a): Time traces of the
membrane occupation number after turning on the coupling to the atoms. The
different traces show measurements with different spin damping rates
$\gamma_{s}$. The dashed lines correspond to the simulation described in the
text based on Eqs. (1) and (2). The dotted line shows the membrane dynamics
without atoms but with the coupling beam turned on. (b): Power spectral
density of the membrane displacement. The dashed lines show a global fit to
the data with the initial phonon occupation $\langle
b^{\dagger}_{i}b_{i}\rangle(t=0)$, $\Omega_{m}$, $\tau$, $g$, and the detector
shot noise level as global fit parameters and $\Omega_{s}$ and $\gamma_{s}$ as
individual fit parameters. All other parameters were taken from independent
calibrations. In (a) and (b), solid lines correspond to the mean and shaded
areas to the standard deviation of 355 measurements.

Our hybrid system consists of a mechanical oscillator and a collective atomic
spin coupled by laser light over a distance of 1 meter in a loop geometry
(Fig. 1). The mechanical oscillator is the (2, 2) square drum mode of a
silicon-nitride membrane Thompson  _et al._ (2008), which has a vibrational
frequency $\Omega_{m}=2\pi\times$1.957\text{\,}\mathrm{MHz}$$ and an intrinsic
quality factor $Q=1.4\times 10^{6}$. The membrane is placed in a single-sided
optical cavity of linewidth $\kappa=2\pi\times$77\text{\,}\mathrm{MHz}$$,
which enhances the optomechanical coupling to external fields. The cavity is
driven by an auxiliary laser beam (not shown in Fig. 1) that is red-detuned
from the cavity resonance, providing some initial cavity optomechanical
cooling of the membrane to $2\times 10^{5}$ phonons Aspelmeyer  _et al._
(2014). The reflection of this beam is used to stabilize the cavity length and
read out the membrane displacement via homodyne detection (detailed in
Appendix A.3).

The collective spin is realised with an ensemble of $1.3\times 10^{7}$ cold
87Rb atoms confined in an optical dipole trap. Strong coupling of the atomic
ensemble to the light is ensured by its large optical depth $d_{0}\approx
300$. The atomic spins are optically pumped into the hyperfine ground state
$|{F}=2,{m_{F}}=-2\rangle$ with respect to a static magnetic field
$B_{0}=$2.8\text{\,}\mathrm{G}$$ perpendicular to the propagation direction of
the coupling laser. The Larmor frequency $\Omega_{s}\propto B_{0}$ is tuned
into resonance with the membrane frequency $\Omega_{m}$. The spin precession
is measured after the first interaction with the coupling laser by picking up
a small fraction of the light (calibration shown in Appendix A.1). The small-
amplitude dynamics of the transverse spin components can be described by a
harmonic oscillator of frequency $\Omega_{s}$ using the Holstein-Primakoff
approximation Hammerer  _et al._ (2010).

A coupling laser beam interacts first with the spin, then with the membrane,
and once again with the spin, as sketched in Fig. 1 and detailed in Karg  _et
al._ (2020). The coupling beam with $1\text{\,}\mathrm{mW}$ optical power is
slightly red-detuned with respect to the membrane cavity and
$-2\pi\times$40\text{\,}\mathrm{GHz}$$ red-detuned from the 87Rb
$\mathrm{D}_{2}$-line. It cools the membrane further to
$\bar{n}_{m\mathrm{,bath}}=2.0\times 10^{4}$ phonons, which broadens its
linewidth to $\gamma_{m}=2\pi\times$262\text{\,}\mathrm{Hz}$$. In presence of
the coupling beam, the spin linewidth is
$\gamma_{s}=2\pi\times$2.2\text{\,}\mathrm{kHz}$$. In the first spin-light
interaction, the $X_{s}$ quadrature of the atomic spin is imprinted onto the
coupling beam via the Faraday interaction Hammerer  _et al._ (2010), resulting
in a modulation of the radiation-pressure force on the membrane. Likewise, the
membrane displacement $X_{m}$ modulates the light reflected from the cavity
Aspelmeyer  _et al._ (2014) which then creates a torque on the spin in the
second interaction. On the way back from the membrane to the spin, the optical
field carrying the spin and membrane signals is phase-shifted by $\pi$ such
that the effective spin-membrane interaction is predominantly Hamiltonian and
the backaction of the light on the spin is suppressed Karg  _et al._ (2019).
Tracing out the light field and neglecting the propagation delay for the
moment, the resonant part of the effective spin-membrane interaction is
described by a beam splitter Hamiltonian $H_{\mathrm{BS}}=\hbar
g(b_{s}^{\dagger}b_{m}+b_{m}^{\dagger}b_{s})$, where $b_{m}$ ($b_{s}$) is the
annihilation operator of a membrane (spin) excitation and $g$ is the effective
spin-membrane coupling rate Karg  _et al._ (2020).

##  III Continuous Cooling

Recently, we demonstrated strong coupling with this spin-membrane interface,
i.e. $2g>(\gamma_{s}+\gamma_{m})\approx\gamma_{s}$ Karg  _et al._ (2020).
Strong coupling is manifested by the hybridization of the membrane and spin
modes which leads to a normal mode splitting of
$2g=2\pi\times$6.8\text{\,}\mathrm{kHz}$$ in the spectrum as shown in Fig.
2(b). In the time domain, strong coupling gives rise to state swaps between
the spin and the membrane at the coupling rate $g$. In Fig. 2(a) we show the
time evolution of the membrane occupation number after switching on the
coupling beam. For $2g>\gamma_{s}$, the thermally excited membrane swaps its
state with the spin, which is initially prepared close to its ground-state, in
half a period $T_{\mathrm{\pi}}=\pi/g$ of the energy exchange oscillations.
After another half period, the thermal state is swapped back onto the membrane
but the phonon number is reduced due to the damping that occurred in the spin
system, whose linewidth is larger than that of the membrane. The oscillations
dephase after approximately $1\text{\,}\mathrm{ms}$ and a steady state with a
membrane occupation of $\bar{n}_{m,\mathrm{ss}}\approx 2.3\times 10^{3}$
phonons is reached, corresponding to a temperature decrease by two orders of
magnitude compared to the initial state. In this process the membrane is
predominantly cooled via its coupling to the cold and damped spin, reaching a
temperature one order of magnitude lower than in the presence of the
optomechanical cooling beams alone.

We now study the effect of increasing the spin damping rate $\gamma_{s}$ on
the coupled dynamics. To increase $\gamma_{s}$ we apply a
$\sigma_{-}$-polarized pump laser along the polarization axis of the spin
(calibration in Appendix A.2). As can be seen in Fig. 2(a), increasing
$\gamma_{s}$ first enhances the membrane cooling, until the overdamped regime
$\gamma_{s}\gg 2g$ is reached where the membrane couples incoherently to a
quasi-continuum of cold spin fluctuations. The membrane decay is then governed
by Fermi’s golden rule, with the occupation number decreasing at the
sympathetic cooling rate $\gamma_{\mathrm{sym}}\approx 4g^{2}/\gamma_{s}$,
i.e. the cooling becomes less effective as $\gamma_{s}$ is increased further.
In this weak-coupling regime, the modes decouple and the membrane spectrum
shows a single Lorentzian peak, broadened by the interaction with the spin,
see Fig. 2(b).

##  IV Stroboscopic Cooling

Previous experiments, which coupled a membrane to the motion of cold atoms
Jöckel  _et al._ (2015); Christoph  _et al._ (2018), lacked both strong
coupling and coherent control over the atoms. In contrast, our strongly
coupled spin-membrane system allows us to implement more elaborate coherent
control schemes. In particular, we can combine strong coupling and strong spin
damping in a stroboscopic fashion in order to cool the membrane much faster
than in the continuous cooling case discussed above. In Fig. 3 we show a
comparison between stroboscopic and continuous cooling, where time traces for
(a) the membrane occupation number and (b) the spin occupation number are
shown. In the stroboscopic sequence we perform a coherent $\pi$-pulse
($T_{\mathrm{pulse}}=$100\text{\,}\mathrm{\SIUnitSymbolMicro s}$$,
$\gamma_{s}=0.6g$) to swap membrane and spin states. Afterwards, we apply an
optical pumping pulse of duration
$T_{\mathrm{pump}}=$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ which increases
the spin damping rate to $\gamma_{s}\approx 60g$ and depletes the spin
occupation on a timescale much shorter than the state swap (gray pulses in
Fig. 3(b)). During the pumping pulse the coupling is kept on. Since the spin
is reinitialised close to the ground state, the next coherent state swap does
not transfer thermal energy back to the membrane but only cools it further. It
takes two to three such iterations of a coherent $\pi$-pulse followed by a
spin pumping pulse to reach the steady state (see Fig. 3). Using this simple
sequence, we can reach the membrane steady state temperature of
$216\text{\,}\mathrm{mK}$ ($\bar{n}_{m,\mathrm{ss}}=2.3\times 10^{3}$ phonons)
in around $200\text{\,}\mathrm{\SIUnitSymbolMicro s}$, approximately a factor
of two faster than for continuous cooling. This exemplarily shows the
advantage of a coherent feedback controller, which enables faster cooling than
if the membrane is coupled with a similar rate to an incoherent, overdamped
system.

Figure 3: (a) Membrane and (b) spin occupation numbers for continuous cooling
at $\gamma_{s}=2g$ and stroboscopic cooling at $\gamma_{s}=0.6g$. The gray
shaded areas indicate the spin pumping pulses (where $\gamma_{s}\approx 60g$).
Solid lines and shaded areas correspond to the mean and standard deviation of
70 measurements and dashed lines correspond to a simulation. Figure 4: Steady
state occupation of the membrane as a function of (a) spin damping rate
$\gamma_{s}$ (at resonance, $\delta=0$) and (b) spin-membrane detuning
$\delta=\Omega_{s}-\Omega_{m}$ at $\gamma_{s}=0.6g$. The solid (dashed) blue
line shows the result of the simulation with (without) delay. In (a), the red
dashed-dotted line indicates the steady-state number given by the rate in Eq.
(3) with $\tau=$15\text{\,}\mathrm{ns}$$. The red shaded area shows the region
for which the dynamics is found to be unstable using the Routh-Hurwitz
criterion. For this measurement, $\bar{n}_{m,\mathrm{bath}}\approx 4.0\times
10^{4}$ phonons and $\gamma_{m}=2\pi\times$94\text{\,}\mathrm{H}\mathrm{z}$$
(independently calibrated without atoms). The data points with error bars
correspond to the mean and the standard deviation of steady state occupations
of 20 (3) experimental realisations in (a) ((b)).

##  V Theoretical Model

Further insight into the dynamics is gained by solving the equations of motion
for the coupled spin-membrane system Karg  _et al._ (2020),

$\displaystyle\ddot{X}_{m}+\gamma_{m}\dot{X}_{m}+\Omega_{m}^{2}X_{m}=-2g\Omega_{m}X_{s}(t-\tau)+\mathcal{F}_{m},$
(1)
$\displaystyle\ddot{X}_{s}+\gamma_{s}\dot{X}_{s}+\Omega_{s}^{2}X_{s}=-2g\Omega_{s}X_{m}(t-\tau)+\mathcal{F}_{s},$
(2)

where terms on the left-hand-side describe the internal dynamics of the damped
oscillators and the first term on the right-hand-side describes the state swap
dynamics including a propagation delay $\tau$ between the spin and the
membrane. We included the generalized Langevin forces $\mathcal{F}_{m}$ and
$\mathcal{F}_{s}$ that capture stochastic force terms due to quantum
fluctuations, thermal and measurement backaction noise (detailed in Appendix
B.1).

We used the following procedures to simulate our experimental results: for the
continuous cooling measurements, we first fitted the spectra for different
$\gamma_{s}$ in Fig. 2(b) globally using a coupled-mode model (fit function
given in Appendix B.2). From this fit, the extracted $\tau$ and $\Omega_{m}$
were used as the input parameters for the simulation. We adapted the technique
described in Nørrelykke and Flyvbjerg (2011) to numerically solve the
equations of motion (1) and (2) and compare the solution to our data (more
details are given in Appendix B.1). To generate each time trace in Fig. 2(a)
(dashed lines) we fitted the numerical solution to our data with only
$\gamma_{s}$ and $\Omega_{s}$ as free parameters. The fit results show a
systematic shift of $\Omega_{s}$ with increasing spin pumping power, likely
due to the light shift induced by the circularly polarised pumping laser (Fig.
7), and $\gamma_{s}$ was observed to be larger than in the independent
calibration of Appendix A.2.

For the stroboscopic cooling measurements, we took the fit parameters from the
continuous cooling measurement and ran the simulation with a time dependent
spin damping rate which was taken to be $\gamma_{s}=0.6g$ during the state
swaps and $\gamma_{s}=60g$ during the pumping pulses. The fit is shown for
membrane and spin in Fig. 3 as a dashed line. The good agreement between fit
and data shows that our model includes all the relevant factors which govern
the coupled dynamics.

##  VI Delayed Feedback

Our hybrid spin-membrane system constitutes a coherent feedback network
Bennett  _et al._ (2014), in which delayed feedback can give rise to
instabilities Ramana Reddy  _et al._ (1998); Reddy  _et al._ (2000); Vochezer
_et al._ (2018). In our experiment, such instabilities show up as a
spontaneous coupled oscillation of spin and membrane, which we observe for
certain values of the spin-membrane detuning $\delta=\Omega_{s}-\Omega_{m}$.
Even at resonance, we have to include the feedback delay to predict the
experimentally measured steady state occupation of the membrane accurately. In
Fig. 4 we plot the measured and simulated occupation numbers of the membrane
in steady state as a function of $\gamma_{s}$ [Fig. 4(a)] and $\delta$ [Fig.
4(b)]. At resonance and for $\Omega_{m}\tau\ll 1$ (as in our system), the
effect of the feedback delay is most apparent in the limit of small
$\gamma_{s}$. The model without delay (light-blue dashed line) predicts a
significantly smaller occupation number compared to both what we observe in
experiments and what is predicted by our model including the feedback delay
(blue solid line). In the large $\gamma_{s}$ limit, the sympathetic cooling
rate is modified to

$\gamma_{\mathrm{sym}}\approx\frac{4g^{2}}{4\delta^{2}+\gamma_{s}^{2}}\left[\gamma_{s}\cos(2\Omega_{m}\tau)+2\delta\sin(2\Omega_{m}\tau)\right]$
(3)

(see Appendix B.3 for derivation). In this limit, the steady state occupation
is given asymptotically by $\langle
b^{\dagger}_{m}b_{m}\rangle_{\mathrm{ss}}=\bar{n}_{m,\mathrm{bath}}\gamma_{m}/(\gamma_{m}+\gamma_{\mathrm{sym}})$,
shown as the red dashed-dotted line in Fig. 4(a). The theory of coupled
oscillators without delay predicts optimal sympathetic cooling at the critical
damping of $\gamma_{s}=2g$ (faded vertical dotted line in Fig. 4). Including
the feedback delay in the model, the minimal occupation number shifts to
larger $\gamma_{s}$ (dark vertical dotted line), because the self-oscillations
have to be compensated by a higher spin damping rate. The experimental data
confirms this theoretical prediction.

Furthermore, we find that the presence of delay lifts the symmetry in
$\delta$, as inferred theoretically from Eq. (3) for large $\gamma_{s}$ and
shown both experimentally and theoretically in Fig. 4(b) for small
$\gamma_{s}=0.6g$. We see that the minimal steady state occupation of the
membrane is obtained for positive detuning $\delta$, i.e.
$\Omega_{s}>\Omega_{m}$, which is true in general for a feedback system with a
delay of $\tau<\pi/(2\Omega_{m})$. For large enough negative $\delta$, we
observe that the coupling drives the system into limit cycle oscillations, see
Fig. 4(b). With our model we can attribute these self-oscillations to the
feedback delay. In this self-driven regime, the resulting membrane occupation
of $6.8\times 10^{7}$ exceeds the spin length by around a factor of three. The
emergence of such instabilities can be characterised using the Routh-Hurwitz
stability criterion Hofer (2008), which indicates whether the real part of one
of the normal modes of the system reverses its sign (shown in Appendix B.4).
In Fig. 4 we indicate such unstable regions for our coupled system by a shaded
area. Our calculations show that the precise value of $\delta$ at which the
driving due to the loop delay exceeds the damping of the coupled system
depends on $\gamma_{s}$. Even at resonance [Fig. 4(a)] self-oscillations are
predicted for small enough $\gamma_{s}$.

The propagation delay is an interesting tuning knob for coherent feedback
experiments, which gives access to Hamiltonian and dissipative dynamics: We
can induce self-oscillations of the system, tune the dependence of the steady
state on system parameters such as damping rate and detuning, or even render
the delay negligible by tuning $2\Omega_{m}\tau$ to a multiple of $2\pi$.

Figure 5: Simulated steady state occupation of membranes for varying cryostat
temperature and different mechanical $Q$ factors. Here, $\gamma_{s}=2g$,
$\delta=0$ and $\tau=$15\text{\,}\mathrm{ns}$$. The insets show the current
membrane with phononic shield used in these experiments and a soft-clamped
membrane for which $Q\approx 5\times 10^{7}$.

##  VII Discussion

In our experiment, the cooling rate of the membrane due to its coupling to the
spin exceeds the cavity-optomechanical cooling rate by more than one order of
magnitude. The lowest achievable phonon occupation of the membrane is thus
given by the competition of cooling the membrane with the spin and heating due
to its coupling to the room-temperature environment. In Fig. 5 we show the
expected membrane steady state occupation for varying environment temperature
and two different membrane designs. In this calculation we include the cavity-
optomechanical cooling of the membrane (which has a negligible effect), the
light-mediated coupling to the spin including backaction of the light, as well
as thermal and quantum mechanical ground state fluctuations of both systems.
The higher quality factors $Q>5\times 10^{7}$ of soft-clamped membranes
Tsaturyan  _et al._ (2017); Reetz  _et al._ (2019) would reduce the thermal
decoherence rate by a factor 25 and allow us to prepare the mechanical
oscillator close to its ground state in a $4\text{\,}\mathrm{K}$ environment.
These technical improvements would realize a mechanical oscillator whose
phonon occupation is limited by quantum backaction instead of thermal noise.
While in the current coupling scheme the double pass eliminates backaction on
the atomic spin, a large membrane quantum cooperativity $C_{m}>1$ would favor
a double pass scheme with coherent cancellation of quantum backaction on the
membrane. This would lead to a higher quantum cooperativity for the spin-
membrane coupling Karg  _et al._ (2019). Further, the feedback control of the
membrane could be improved by increasing the quantum cooperativity of the spin
system. This involves gaining a better understanding of the spin decoherence
sources and achieving a larger spin-light coupling rate.

In this work we implemented a relatively simple coherent feedback sequence
based on coherent state swaps of pulse area $\pi$ interleaved with short spin
pumping pulses. In the future, it would be interesting to explore more
elaborate feedback sequences to optimize the cooling in a specific situation.
For example, the duty cycle of the stroboscopic cooling sequence could be
changed over time to cool a mechanical oscillator with a high initial
occupation that exceeds the spin length. Initially, short coupling pulses of
pulse area $\ll\pi$ could remove excitations without saturating the spin, and
once the phonon number is sufficiently reduced, the pulse area could be
increased to minimize the final temperature.

Our coherent feedback cooling scheme is a rather general technique that can be
applied to any physical system with a strong light-matter interface. This
includes cavity optomechanical systems or mechanical oscillators without an
optical cavity. Moreover, similar cooling schemes could be implemented in the
microwave domain with electromechanical oscillators Clerk  _et al._ (2020)
coupled to solid-state spin systems. The macroscopic distance between the
feedback controller and the target system enables modular control schemes in
analogy to classical feedback in electrical engineering. This opens up the new
possibility to use coherent feedback control in quantum networks.

The coherent control and bidirectional Hamiltonian coupling employed in this
work pave the way towards more elaborate quantum protocols such as the
generation of non-classical mechanical states via state swaps Wallquist  _et
al._ (2010) as well as further studies of coherent feedback in the quantum
regime Lloyd (2000); Wiseman and Milburn (2009); Hamerly and Mabuchi (2012);
Zhang  _et al._ (2017).

###### Acknowledgements.

We thank Christoph Bruder for a careful reading of the manuscript. 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). MBA acknowledges funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement N°101023088.

##  Appendix A Calibrations

###  A.1 Calibration of the spin signal

The spin occupation was calibrated by using the off-resonant Faraday
interaction Hammerer  _et al._ (2010) between the atoms and the light. In
contrast to the looped coupling scheme in which light interacts twice with the
spin, the calibration is performed by measuring the light directly after the
first interaction with the spin. Hence, the light does not interact with the
membrane nor with the atoms a second time.

For a single spin-light interaction, the Hamiltonian of the Faraday
interaction is given by Geremia  _et al._ (2006)

$H_{\mathrm{int}}=\hbar\alpha_{1}S_{z}F_{z}$ (4)

where $\alpha_{1}$ is the vector polarizability of the atoms, $S_{z}$ is the
circularly polarized component of the Stokes vector of the light, and $F_{z}$
is the collective spin component along the propagation direction of the probe
laser. The input-output relation for the $S_{y}$ Stokes vector component of
the probe light yields

$S_{y}^{(\mathrm{out})}=S_{y}^{(\mathrm{in})}+\alpha_{1}S_{x}^{(\mathrm{in})}F_{z}.$
(5)

For the experiments in this paper, the probe laser was linearly polarized with
an angle of $55^{\circ}$ with respect to the magnetic field in order to
minimise frequency shifts due to the tensor interaction with the light Geremia
_et al._ (2006), which otherwise would give rise to inhomogeneous broadening
of the spin. In the following we define $S_{x}$ as the difference between the
flux of light linearly polarized at $55^{\circ}$ and the flux in the
orthogonal linear polarization, such that we can approximate the $S_{x}$
operator as a classical quantity $S_{x}^{(\mathrm{in})}\approx
S_{x}^{(\mathrm{out})}\approx\langle S_{x}\rangle$ and define the Faraday
angle $\theta_{F}=S_{y}^{(\mathrm{out})}/(2\langle
S_{x}\rangle)=\alpha_{1}F_{z}/2$. For the calibration measurement, we slowly
rotate the spin to align it to the propagation direction of light. Thus, the
collective atomic spin points along the $z$-axis, and the $F_{z}$ component of
the spin can be approximated by $F_{z}\approx\langle
F_{z}\rangle={F}N_{a}=2N_{a}$ where the number of atoms in the dipole trap
$N_{a}$ was measured independently by absorption imaging. The
$S_{y}^{(\mathrm{out})}$ Stokes vector component of the out-going field was
determined by a polarization homodyne measurement. Measuring $\langle
S_{x}\rangle$ independently allows us to determine the Faraday angle (shown in
Fig. 6 for different atom numbers). By knowing the number of atoms from
absorption imaging and the Faraday angle from polarization homodyne
measurement, the ensemble-averaged vector polarizability
$\alpha_{1}=\theta_{F}/N_{a}$ can be calculated.

Figure 6: DC Faraday rotation signal of the atomic ensemble spin-polarized
along the optical propagation axis at a probe laser detuning of
$-2\pi\times$40\text{\,}\mathrm{GHz}$$. In the experiment, first the Faraday
rotation angle $\theta_{F}$ was measured with a weak, far-detuned probe pulse
and then the number of atoms $N_{a}$ was determined by absorption imaging.
Each data point with error bar corresponds to the mean and the standard
deviation of five experimental runs at the same MOT loading time.

With this calibrated value for the vector polarizability $\alpha_{1}$, a
measurement of the Faraday angle $\theta_{F}$ yields directly the $F_{z}$
component of an arbitrary spin state. If the collective spin is aligned along
a magnetic field perpendicular to the propagation direction of the light
field, the $F_{z}$ component oscillates at the Larmor frequency. In this case
of an oscillating signal, the polarization homodyne measurement of the out-
going field is demodulated by a lock-in amplifier which returns the root mean
square (rms) amplitude
$V_{S_{y},\mathrm{rms}}^{\mathrm{$50\text{\,}\mathrm{\SIUnitSymbolOhm}$}}\propto\bar{S}_{y}$
(where $\bar{S}_{y}$ is the slowly varying amplitude of $S_{y}$). In order to
determine the Faraday angle from this rms amplitude and the DC measurement of
$V_{S_{x}}\propto\langle S_{x}\rangle$ using an oscilloscope, one has to first
multiply the rms amplitude by a factor $\sqrt{2}$ to get a peak amplitude
voltage and further by a factor of $2$ to compensate for the impedance
mismatch between the $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ input impedance of
the lock-in amplifier and the high input impedance of the oscilloscope.
Including these factors, we get the slowly varying amplitude of the Faraday
angle
$\bar{\theta}_{F}=\sqrt{2}V_{S_{y},\mathrm{rms}}^{\mathrm{$50\text{\,}\mathrm{\SIUnitSymbolOhm}$}}/V_{S_{x}}$.
By normalising the spin signal by the square-root of the total spin length we
obtain the slowly varying amplitude of the $X_{s}$-quadrature of the spin

$\bar{X}_{s}=\frac{\bar{F}_{z}}{\sqrt{\langle
F_{x}\rangle}}=\frac{2\bar{\theta}_{F}}{\alpha_{1}\sqrt{2N_{a}}}=\frac{2V_{S_{y},\mathrm{rms}}^{\mathrm{$50\text{\,}\mathrm{\SIUnitSymbolOhm}$}}}{\alpha_{1}V_{S_{x}}\sqrt{N_{a}}}.$
(6)

From this result, we can use the equipartition theorem to calculate the number
of excitations of the spin oscillator:

$\bar{n}_{s}+\frac{1}{2}=\frac{\langle
X_{s}(t)^{2}+P_{s}(t)^{2}\rangle}{2}=\bar{X}_{s}^{2}=\frac{2\bar{\theta}_{F}^{2}}{\alpha_{1}^{2}N_{a}}.$
(7)

where $X_{s}(t)$ and $P_{s}(t)$ are the fast rotating quadratures of the spin
oscillator. In the looped experiment, only a small fraction of the light was
measured in between the first interaction of the spin and the interaction of
the light with the membrane. This in-loop measurement was calibrated using a
coherent spin excitation and comparing it to the polarization homodyne
measurement presented in this section.

###  A.2 Calibration of the spin damping rate

One of the main parameters in the experiments is the spin damping rate
$\gamma_{s}$. In order to measure the spin damping rate in the presence of all
lasers but without coupling to the membrane, we detuned the coupling laser
from the cavity resonance ($|\Delta|\gg\kappa$). The laser thus is reflected
from the incoupling mirror of the cavity and only the spin is probed. For the
calibration measurements, the spin was coherently excited by a weak RF-pulse.
The spin signal was measured by detecting the remaining signal on the light
after the second pass. It is normalised to occupation numbers [shown in Fig.
7(a)]. The damping rate $\gamma_{s}$ is extracted from the exponential fit to
the temporal dynamics [Fig. 7(b)] and the frequency $\Omega_{s}$ is extracted
from a Lorentzian fit to the spectrum [Fig. 7(c)]. For optical pumping power
larger than $P_{\mathrm{pump}}>$0.7\text{\,}\mathrm{\SIUnitSymbolMicro W}$$,
the spectra were too broad to provide reasonable fit results [and are
therefore not shown in Fig. 7(c)]. In Fig. 7(b) and (c), fit parameters for
the coupled dynamics are shown.

Figure 7: Measurement of the spin in the absence of coupling to the membrane
after it is excited by a weak RF-pulse: (a) Time trace of double pass
measurement of the spin with different pumping powers (range from $0$ to
$10\text{\,}\mathrm{\SIUnitSymbolMicro W}$). The dashed lines show fits with
an exponential decay. The spin linewidth (b) and spin frequency (c) are
plotted as a function of the pump power. The dashed lines in (b) and (c) show
a linear fit to the spin linewidth and resonance frequency. The crosses show
the fit parameters extracted from Fig. 2(a) which were used as input for the
simulations. The data shown in (a) is an average over seven experimental
realisations and was used to fit the exponential decay [for (b)] and the
Lorentzian peak [for (c)]. The error bars in (b) and (c) show the fit-error of
the corresponding quantity.

###  A.3 Calibration of the membrane signal

The vibrations of the membrane are detected via their effect on the phase of
the beam reflected from the cavity (Aspelmeyer  _et al._ , 2014). In
particular, the membrane vibrations modulate the cavity resonance frequency
$\omega_{c}$. For small membrane displacements, we can write
$\omega_{c}(x_{m})\approx\omega_{c}+Gx_{m}$, where
$G=-\mathrm{d}\omega_{c}/\mathrm{d}x_{m}$ is the cavity frequency shift per
membrane displacement. For a single-sided cavity, the phase $\phi_{c}$ of the
beam reflected from the cavity with respect to the incoming beam is related to
the cavity detuning $\Delta=\omega_{L}-\omega_{c}$ by (Aspelmeyer  _et al._ ,
2014)

$\phi_{c}=\arctan\left[\frac{\kappa\Delta}{(\kappa/2)^{2}+\Delta^{2}}\right],$
(8)

where $\kappa$ is the cavity linewidth. A change in the cavity frequency
$\delta\omega_{c}=Gx_{m}$ thus leads to a change in the phase of the reflected
beam by an amount
$\delta\phi_{c}=-(\mathrm{d}\phi_{c}/\mathrm{d}\Delta)\delta\omega_{c}\approx-{4}Gx_{m}/{\kappa}$,
where the approximation holds for small detunings $|\Delta|\ll\kappa$. We can
write the previous expression in terms of the vacuum optomechanical coupling
strength $g_{0}=Gx_{\mathrm{ZPF}}$ as

$\delta\phi_{c}=-\frac{4g_{0}}{\kappa}\frac{x_{m}}{x_{\mathrm{ZPF}}},$ (9)

where we have introduced the zero-point fluctuation amplitude of the membrane
$x_{\mathrm{ZPF}}=(\hbar/2m_{\mathrm{eff}}\Omega_{m})^{1/2}$, with
$m_{\mathrm{eff}}$ the effective mass of the vibration mode. These phase
variations $\delta\phi_{c}$ can now be read interferometrically by means of
balanced homodyne detection. For this, the beam reflected from the cavity is
combined with a strong local oscillator in a 50:50 beam splitter. The output
beams are subsequently photodetected and the output signals subtracted. The
recorded balanced voltage can be written as

$V=V_{0}\cos(\Delta\phi),$ (10)

with $V_{0}$ the modulation amplitude, proportional to the square-root of the
power of the beam reflected from the cavity and of the local oscillator beam
and with $\Delta\phi=\phi_{c}-\phi_{\mathrm{LO}}$ where $\phi_{\mathrm{LO}}$
is the phase of the local oscillator.

The modulation amplitude $V_{0}$ is inferred by modulating
$\phi_{\mathrm{LO}}$, thanks to a movable mirror in the local oscillator path
which allows to generate path differences of a few wavelengths. $V_{0}$ can be
extracted from the contrast of the observed interference fringes. In order to
detect the phase fluctuations $\delta\phi_{c}$ of $\phi_{c}$ induced by the
membrane motion, we lock the relative phase $\Delta\phi$ to $\pi/2$, i.e., the
point where the slope of the fringes is maximal. For small shifts
$\delta\phi_{c}\ll\pi/2$, the recorded voltage variation $\delta V(t)$ is
directly proportional to $\delta\phi_{c}(t)$, and thus to $x_{m}(t)$. In
practice, $\delta V(t)$ is effectively increased by a factor $1/\eta_{c}$ due
to imperfect cavity coupling, such that

$x_{m}(t)=\frac{\delta
V(t)}{\eta_{c}V_{0}}\frac{x_{\mathrm{ZPF}}\kappa}{4g_{0}}.$ (11)

In order to determine membrane phonon occupation $\bar{n}_{m}(t)$ we first
define the dimensionless membrane quadrature operators
$X_{m}=x_{m}/(\sqrt{2}x_{\mathrm{ZPF}})$ and
$P_{m}=\sqrt{2}x_{\mathrm{ZPF}}p_{m}/\hbar$, defined so that
$[X_{m},P_{m}]=\mathrm{i}$. We can now write

$\displaystyle\langle H(t)\rangle$ $\displaystyle=\hbar\Omega_{m}\frac{\langle
X_{m}(t)^{2}+P_{m}(t)^{2}\rangle}{2}$ (12)
$\displaystyle=\hbar\Omega_{m}\left(\bar{n}_{m}(t)+\frac{1}{2}\right).$

By means of the equipartition theorem, we can write $\langle
X_{m}(t)^{2}\rangle_{t}=\langle P_{m}(t)^{2}\rangle_{t}$ and thus relate the
measured voltage variations to the membrane phonon occupation number. In
practice, we do not measure the voltage $\delta V(t)$ but its rms value
$\delta V(t)^{50\,\Omega}_{\mathrm{rms}}$, which we further need to multiply
by a factor of 2 due to impedance mismatch of our measuring instrument. To
convert the measured rms value to amplitude variations, we thus need an
overall $2\sqrt{2}$ factor. This finally yields

$\displaystyle\bar{n}_{m}(t)+\frac{1}{2}$ $\displaystyle=\langle
X_{m}(t)^{2}\rangle_{t}$ (13) $\displaystyle=\left(\frac{\delta
V(t)^{50\,\Omega}_{\mathrm{rms}}}{\eta_{c}V_{0}}\right)^{2}\left(\frac{\kappa}{2g_{0}}\right)^{2}.$

The values of $\kappa=2\pi\times$77\text{\,}\mathrm{MHz}$$ and
$g_{0}=2\pi\times$224\text{\,}\mathrm{Hz}$$, have been independently
calibrated from the width of the Pound-Drever-Hall signal and by measuring the
optomechanical response to an optical amplitude modulation tone, respectively.

##  Appendix B Theoretical model for the membrane-spin coupling

We modelled our membrane-spin coupling by two coupled harmonic oscillators as
shown in Eqs. (1) and (2). In the following we show how we characterised,
simulated, and approximated the system starting from these equations. In
section B.1 the stochasic simulation of the system is presented. In section
B.2 we show the derivation of the fit function for the spectra. From the
spectrum, we calculate the sympathetic cooling rate and the resonance
frequency shift in the weak coupling limit in section B.3. Finally, we show
the Routh-Hurwitz stability analysis of the coupled dynamics with delay in
section B.4.

###  B.1  Simulation of the spin-membrane dynamics

In this section, we provide some details on the simulation method we used to
solve the stochastic equations of motion Eqs. (1) and (2) for the spin-
membrane system. This simulation follows closely the algorithm presented in
Nørrelykke and Flyvbjerg (2011). For the simulation, we rewrite the equations
of motion as four coupled first-order differential equations for
$\tilde{X}_{j}$ and $\tilde{P}_{j}$, with $j\in(m,s)$ in a frame rotating at
the membrane frequency $\Omega_{m}$ (operators in the rotating frame are
denoted with a tilde) and apply the rotating wave approximation (RWA). In the
limit where the propagation delay is small compared to other timescales
involved in the coupled dynamics (i.e.
$\tau\ll\gamma_{j}^{-1},g^{-1},\delta^{-1}$), the change of the oscillator
quadratures during the time $\tau$ can be neglected in the rotating frame i.e.
$\tilde{X}_{j}(t)\approx\tilde{X}_{j}(t-\tau)$ and
$\tilde{P}_{j}(t)\approx\tilde{P}_{j}(t-\tau)$. The equations of motion then
read

$\frac{\mathrm{d}}{\mathrm{d}t}\begin{pmatrix}\tilde{X}_{m}(t)\\\
\tilde{P}_{m}(t)\\\ \tilde{X}_{s}(t)\\\ \tilde{P}_{s}(t)\\\
\end{pmatrix}=-\mathbf{M}\begin{pmatrix}\tilde{X}_{m}(t)\\\
\tilde{P}_{m}(t)\\\ \tilde{X}_{s}(t)\\\ \tilde{P}_{s}(t)\\\
\end{pmatrix}+\begin{pmatrix}-\sin{(\Omega_{m}t)}\mathcal{F}_{m}(t)\\\
\cos{(\Omega_{m}t)}\mathcal{F}_{m}(t)\\\
-\sin{(\Omega_{m}t)}\mathcal{F}_{s}(t)\\\
\cos{(\Omega_{m}t)}\mathcal{F}_{s}(t)\end{pmatrix},$ (14)

where we have split the dynamics into the $4\times 4$ dynamical matrix

$\mathbf{M}=\begin{pmatrix}{\gamma_{m}}/{2}&0&-g\sin\left(\Omega_{m}\tau\right)&-g\cos\left(\Omega_{m}\tau\right)\\\
0&{\gamma_{m}}/{2}&g\cos\left(\Omega_{m}\tau\right)&-g\sin\left(\Omega_{m}\tau\right)\\\
-g\sin\left(\Omega_{m}\tau\right)&-g\cos\left(\Omega_{m}\tau\right)&{\gamma_{s}}/{2}&-\delta\\\
g\cos\left(\Omega_{m}\tau\right)&-g\sin\left(\Omega_{m}\tau\right)&\delta&{\gamma_{s}}/{2}\end{pmatrix},$
(15)

and a stochastic part, given by the generalized noise forces
$\mathcal{F}_{j}(t)=\sqrt{2\gamma_{j}}F^{\mathrm{(tot)}}_{j}(t)$. The total
force noise $\mathcal{F}_{j}^{\mathrm{(tot)}}(t)$ includes the thermal noise
$F_{j}^{\mathrm{(th)}}(t)$ and the backaction noise $F^{\mathrm{(ba)}}_{j}(t)$
which itself depends on the optical vacuum noise $F^{\mathrm{(in)}}_{j}(t)$.
Thus, it is given by

$\displaystyle F_{j}^{\mathrm{(tot)}}(t)$
$\displaystyle=F_{j}^{\mathrm{(th)}}(t)+F_{j}^{\mathrm{(ba)}}(t)$ (16)
$\displaystyle=F_{j}^{\mathrm{(th)}}(t)+\sqrt{\frac{2\Gamma_{j}}{\gamma_{j}}}F_{j}^{\mathrm{(in)}}(t),$

where $\Gamma_{j}$ is the measurement rate of the individual system. The noise
terms $F^{(\nu)}_{j}(t)$, $\nu\in\mathrm{(th,in)}$ can be expressed explicitly
in terms of the product of a noise amplitude and a zero mean, delta correlated
noise $f_{j}^{(\nu)}(t)$:

$\displaystyle
F_{j}^{\mathrm{(th)}}(t)=\sqrt{\bar{n}_{j,\mathrm{bath}}+\frac{1}{2}}f^{\mathrm{(th)}}_{j}(t),$
$\displaystyle
F_{m}^{\mathrm{(in)}}(t)=\sqrt{\frac{\eta^{2}}{2}}f^{\mathrm{(in)}}_{m}(t),$
(17) $\displaystyle
F_{s}^{\mathrm{(in)}}(t)=\sqrt{\frac{1-\eta^{4}}{2}}f^{\mathrm{(in)}}_{s}(t),$
(18)

where $\eta^{2}\approx 0.8$ is the power transmission coefficient of the light
between the spin and the membrane and $\bar{n}_{j,\mathrm{bath}}$ is the
number of thermal phonons in the individual system. The thermal noise
amplitude is calculated from the fluctuation dissipation theorem while for the
derivation of the backaction noise we refer to Karg  _et al._ (2019). The
number of thermal phonons of the membrane $\bar{n}_{m,\mathrm{bath}}$ was
measured by homodyne detection in presence of all laser beams but without
loading the atoms. This calibrated value agrees very well with an estimation
from comparing the spectral linewidth in presence of the cooling and coupling
beams with the spectral linewidth of the uncooled membrane and the calculated
room temperature occupation of the membrane. We assumed the spin pumping to be
perfect such that the spin oscillator environment is in its quantum mechanical
ground state (i.e. $\bar{n}_{s,\mathrm{bath}}=0$).

The approach given in Nørrelykke and Flyvbjerg (2011) allows for an exact
simulation of the stochastic dynamics for a single oscillator for arbitrary
time steps, which we extend to the case of two coupled oscillators with delay.
This is done by calculating for each time step the coherent evolution and the
noise separately:

$\begin{pmatrix}\tilde{X}_{m}(t_{i+1})\\\ \tilde{P}_{m}(t_{i+1})\\\
\tilde{X}_{s}(t_{i+1})\\\
\tilde{P}_{s}(t_{i+1})\end{pmatrix}=\mathrm{e}^{-\mathbf{M}\Delta
t}\begin{pmatrix}\tilde{X}_{m}(t_{i})\\\ \tilde{P}_{m}(t_{i})\\\
\tilde{X}_{s}(t_{i})\\\
\tilde{P}_{s}(t_{i})\end{pmatrix}+\begin{pmatrix}\Delta\tilde{X}_{m}^{t_{i}\rightarrow
t_{i+1}}\\\ \Delta\tilde{P}_{m}^{t_{i}\rightarrow t_{i+1}}\\\
\Delta\tilde{X}_{s}^{t_{i}\rightarrow t_{i+1}}\\\
\Delta\tilde{P}_{s}^{t_{i}\rightarrow t_{i+1}}\end{pmatrix},$ (19)

where $\Delta t=t_{i+1}-t_{i}$ is one simulation time step, and
$\Delta\tilde{X}_{j}^{t_{i}\rightarrow t_{i+1}}$,
$\Delta\tilde{P}_{j}^{t_{i}\rightarrow t_{i+1}}$ are terms for the stochastic
noise which enters the system in between time $t_{i}$ and $t_{i+1}$. We
performed the simulation at time steps comparable to the oscillation period
$\Omega_{m}^{-1}$. Thus, the noise terms
$\Delta\tilde{X}_{j}^{t_{i}\rightarrow t_{i+1}}$ and
$\Delta\tilde{P}_{j}^{t_{i}\rightarrow t_{i+1}}$ are correlated which is taken
into account by following the calculation of noise variances and covariances
in Nørrelykke and Flyvbjerg (2011). Because the coupling between the two
oscillators is much slower than the simulation time step
$g\ll\Omega_{m}\approx\Delta t^{-1}$ we neglect the correlation of noise
building up between the oscillators during one simulation step. Thus, we can
treat the noise of both oscillators separately. In order to simulate the
system more efficiently, we perform the simulation in time steps of multiples
of one frame rotation $\Delta t=k\cdot 2\pi/\Omega_{m}$, $k=1,2,3...$ such
that the noise amplitudes [proportional to
$\sin(\Omega_{m}t),\cos(\Omega_{m}t)$, see Eq. (14)] are the same for each
step of the simulation.

###  B.2 Fit function for the power spectral density of mechanical
displacement

In this section, we provide some details on the coupled-mode model used for
fitting the power spectral density of the mechanical displacement shown in
Fig. 2(b). For this, we first Fourier transform the equations of motion Eqs.
(1) and (2), which allows us to derive the following effective
susceptibilities

$\displaystyle\chi_{m,0}(\omega)^{-1}X_{m}(\omega)+2g\,\mathrm{e}^{\mathrm{i}\omega\tau}X_{s}(\omega)=-\sqrt{2\gamma_{m}}F_{m}^{(\mathrm{tot})}(\omega),$
(20)
$\displaystyle\chi_{s,0}(\omega)^{-1}X_{s}(\omega)+2g\,\mathrm{e}^{\mathrm{i}\omega\tau}X_{m}(\omega)=-\sqrt{2\gamma_{s}}F_{s}^{(\mathrm{tot})}(\omega),$
(21)

where we have defined the individual oscillator susceptiblities as

$\chi_{i,0}(\omega)=\frac{\Omega_{i}}{\Omega_{i}^{2}-\omega^{2}-\mathrm{i}\omega\gamma_{i}}.$
(22)

Solving for $X_{m}$ and $X_{s}$ yields

$\displaystyle X_{m}(\omega)=$
$\displaystyle\chi_{m,\mathrm{eff}}(\omega)\big{[}-\sqrt{2\gamma_{m}}F_{m}^{(\mathrm{tot})}(\omega)$
$\displaystyle+2g\,\mathrm{e}^{\mathrm{i}\omega\tau}\sqrt{2\gamma_{s}}\chi_{s,0}(\omega)F_{s}^{(\mathrm{tot})}(\omega)\big{]},$
(23) $\displaystyle X_{s}(\omega)=$
$\displaystyle\chi_{s,\mathrm{eff}}(\omega)\big{[}-\sqrt{2\gamma_{s}}F_{s}^{(\mathrm{tot})}(\omega)$
$\displaystyle+2g\,\mathrm{e}^{\mathrm{i}\omega\tau}\sqrt{2\gamma_{m}}\chi_{m,0}(\omega)F_{m}^{(\mathrm{tot})}(\omega)\big{]},$
(24)

where we have introduced the effective susceptibilities of the membrane and
spin oscillators as

$\displaystyle\chi_{m,\mathrm{eff}}(\omega)^{-1}=\chi_{m,0}(\omega)^{-1}-4g^{2}\mathrm{e}^{\mathrm{i}2\omega\tau}\chi_{s,0}(\omega),$
(25)
$\displaystyle\chi_{s,\mathrm{eff}}(\omega)^{-1}=\chi_{s,0}(\omega)^{-1}-4g^{2}\mathrm{e}^{\mathrm{i}2\omega\tau}\chi_{m,0}(\omega).$
(26)

We used this model to fit the power spectral densities of the mechanical
displacement spectra [see Fig. 2(b)] using as fit function
$a^{2}|\chi_{m,\mathrm{eff}}(\omega)|^{2}$ where $a$ is a global scaling
factor accounting for the noise terms driving the system.

###  B.3 Derivation of the sympathetic cooling rate

Here, we derive the sympathetic cooling rate for the mechanical oscillator
given in Eq. (3). For this, let us first write Eq. (25) explicitly

$\displaystyle\chi_{m,\mathrm{eff}}(\omega)^{-1}=$
$\displaystyle\frac{1}{\Omega_{m}}\Bigg{(}\Omega_{m}^{2}-\omega^{2}-\mathrm{i}\omega\gamma_{m}$
$\displaystyle-4g^{2}\mathrm{e}^{\mathrm{i}2\omega\tau}\frac{\Omega_{m}\Omega_{s}\left(\Omega_{s}^{2}-\omega^{2}+\mathrm{i}\omega\gamma_{s}\right)}{\left(\Omega_{s}^{2}-\omega^{2}\right)^{2}+\left(\omega\gamma_{s}\right)^{2}}\Bigg{)},$
(27)

which can be written in the form of

$\chi_{m,\mathrm{eff}}(\omega)^{-1}=\frac{1}{\Omega_{m}}\left[\Omega_{m}^{2}-\delta\Omega_{\rm{shift}}^{2}-\omega^{2}-\mathrm{i}\omega\left(\gamma_{m}+\gamma_{\mathrm{sym}}\right)\right].$
(28)

Here, we have defined an effective frequency shift
$\delta\Omega_{\mathrm{shift}}$ and the sympathetic cooling rate
$\gamma_{\mathrm{sym}}$, which for $\omega=\Omega_{m}$ read

$\displaystyle\delta\Omega_{\mathrm{shift}}^{2}=$
$\displaystyle\frac{4g^{2}\Omega_{m}\Omega_{s}}{\left(\Omega_{s}^{2}-\Omega_{m}^{2}\right)^{2}+\left(\Omega_{m}\gamma_{s}\right)^{2}}$
$\displaystyle\times\big{[}\left(\Omega_{s}^{2}-\Omega_{m}^{2}\right)\cos\left(2\Omega_{m}\tau\right)-\Omega_{m}\gamma_{s}\sin\left(2\Omega_{m}\tau\right)\big{]},$
(29)

and

$\displaystyle\gamma_{\mathrm{sym}}=$
$\displaystyle\frac{4g^{2}\Omega_{m}\Omega_{s}}{\left(\Omega_{s}^{2}-\Omega_{m}^{2}\right)^{2}+\left(\Omega_{m}\gamma_{s}\right)^{2}}$
$\displaystyle\times\left[\gamma_{s}\cos\left(2\Omega_{m}\tau\right)+\frac{\Omega_{s}^{2}-\Omega_{m}^{2}}{\Omega_{m}}\sin\left(2\Omega_{m}\tau\right)\right].$
(30)

For $\Omega_{s}\approx\Omega_{m}$ and large spin damping $\gamma_{s}>g$, we
get a simplified expression for the frequency shift and sympathetic cooling
rate [Eq. (3)]

$\displaystyle\delta\Omega_{\text{shift}}^{2}\approx\frac{4g^{2}\Omega_{m}}{4\delta^{2}+\gamma_{s}^{2}}\left[2\delta\cos\left(2\Omega_{m}\tau\right)-\gamma_{s}\sin\left(2\Omega_{m}\tau\right)\right],$
(31)
$\displaystyle\gamma_{\mathrm{sym}}\approx\frac{4g^{2}}{4\delta^{2}+\gamma_{s}^{2}}\left[\gamma_{s}\cos\left(2\Omega_{m}\tau\right)+2\delta\sin\left(2\Omega_{m}\tau\right)\right],$
(32)

where $\delta=\Omega_{s}-\Omega_{m}$.

###  B.4 Routh-Hurwitz stability criterion of the coupled system

Figure 8: Evaluation of the stability of the coupled system using the Routh-
Hurwitz criterion: The colored regions (i.e. region above each solid line)
show the sets of parameters for which the coupled dynamics is stable for a
given value of the feedback delay. Without propagation delay, every set of
detunings and spin damping leads to stable dynamics. For
$\tau=$80\text{\,}\mathrm{ns}$$ we have $\Omega_{m}\tau\approx 1$ thus the
validity of the Taylor expansion of the exponential function in presence of
small delays reaches its limit. For the stability estimations shown here we
used $2g=2\pi\times$6.8\text{\,}\mathrm{kHz}$$,
$\gamma_{m}=2\pi\times$262\text{\,}\mathrm{Hz}$$, and
$\Omega_{m}=2\pi\times$1.957\text{\,}\mathrm{MHz}$$

In this section we present a stability analysis in which the Routh-Hurwitz
criterion Hofer (2008) from control theory is applied to our linearly coupled
spin-membrane oscillators. The criterion provides a convenient means to assess
the stability of our linear systems without solving the equations of motion.
In this treatment, we exclude the Langevin noise, as we are interested to see
if the delayed coupled oscillator dynamics is stable by itself. We then
explore the experimental parameter space to see under which conditions the
coupled system becomes unstable. We take the equations of motion for the
delayed coupled system Eqs. (1) and (2) neglecting the noise terms

$\displaystyle\ddot{X}_{m}+\gamma_{m}\dot{X}_{m}+\Omega_{m}^{2}X_{m}=-2g\Omega_{m}X_{s}(t-\tau),$
(33)
$\displaystyle\ddot{X}_{s}+\gamma_{s}\dot{X}_{s}+\Omega_{s}^{2}X_{s}=-2g\Omega_{s}X_{m}(t-\tau).$
(34)

Substituting the ansatz $X_{j}(t)=X_{{j}}(s)\,\mathrm{e}^{st}$ where
$s\in\mathbb{C}$ yields

$\displaystyle\left(s^{2}+s\gamma_{m}+\Omega_{m}^{2}\right)X_{m}(s)=-2g\Omega_{m}\,\mathrm{e}^{-s\tau}X_{s}(s),$
(35)
$\displaystyle\left(s^{2}+s\gamma_{s}+\Omega_{s}^{2}\right)X_{s}(s)=-2g\Omega_{s}\,\mathrm{e}^{-s\tau}X_{m}(s).$
(36)

Solving the simultaneous equations Eqs. (35) and (36), we obtain the
characteristic equation for non-trivial solutions $X_{m}\neq 0$,

$\left(s^{2}+s\gamma_{m}+\Omega_{m}^{2}\right)\left(s^{2}+s\gamma_{s}+\Omega_{s}^{2}\right)-4g^{2}\Omega_{m}\Omega_{s}\,\mathrm{e}^{-2s\tau}=0.$
(37)

For clarity, we consider here small propagation delays
$\tau\ll{1}/{\Omega_{j}}$ and apply a first order Taylor expansion
$\exp({-2s\tau})\approx(1-2s\tau)$ (in the actual simulation we keep terms up
to 4th order). We then obtain

$\displaystyle 0=$ $\displaystyle
s^{4}+(\gamma_{s}+\gamma_{m})s^{3}+(\Omega_{m}^{2}+\Omega_{s}^{2}+\gamma_{m}\gamma_{s})s^{2}$
$\displaystyle+(\Omega_{s}^{2}\gamma_{m}+\Omega_{m}^{2}\gamma_{s}+8g^{2}\Omega_{m}\Omega_{s}\tau)s$
$\displaystyle+\Omega_{m}\Omega_{s}(\Omega_{m}\Omega_{s}-4g^{2}).$ (38)

Having our dynamics in this polynomial form, we can define the polynomial
coefficients of a fourth order polynomial by

$p(s)=a_{4}s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}=0,\>a_{4}>0.$ (39)

In order to apply the Routh-Hurwitz criterion, the so-called Hurwitz matrix
containing the polynomial coefficients has to be defined. For a fourth order
polynomial this matrix reads

$H_{4}=\begin{pmatrix}a_{3}&a_{1}&0&0\\\ a_{4}&a_{2}&a_{0}&0\\\
0&a_{3}&a_{1}&0\\\ 0&a_{4}&a_{2}&a_{0}\end{pmatrix}.$ (40)

According to the Routh-Hurwitz criterion, the system dynamics is
asymptotically stable if all the principal minors of the Hurwitz matrix are
non-zero and positive. Application of the Hurwitz criterion leads to the
following stability criteria for a fourth order polynomial system:

$\displaystyle\Delta_{1}=\left|a_{3}\right|>0,$ (41)
$\displaystyle\Delta_{2}=\begin{vmatrix}a_{3}&a_{1}\\\
a_{4}&a_{2}\end{vmatrix}=a_{2}a_{3}-a_{4}a_{1}>0,$ (42)
$\displaystyle\Delta_{3}=\begin{vmatrix}a_{3}&a_{1}&0\\\ a_{4}&a_{2}&a_{0}\\\
0&a_{3}&a_{1}\end{vmatrix}=a_{1}\Delta_{2}-a^{2}_{3}a_{0}>0,$ (43)
$\displaystyle\Delta_{4}=\det(H_{4})=a_{0}\cdot\Delta_{3}>0.$ (44)

In our system, the coefficients are given explicitly by

$\displaystyle a_{4}=1,$ (45) $\displaystyle a_{3}=\gamma_{s}+\gamma_{m},$
(46) $\displaystyle a_{2}=\Omega_{s}^{2}+\Omega_{m}^{2}+\gamma_{s}\gamma_{m},$
(47) $\displaystyle
a_{1}=\gamma_{m}\Omega_{s}^{2}+\gamma_{s}\Omega_{m}^{2}+8g^{2}\Omega_{m}\Omega_{s}\tau,$
(48) $\displaystyle
a_{0}=\Omega_{s}\Omega_{m}\left(\Omega_{s}\Omega_{m}-4g^{2}\right).$ (49)

Since $\Omega_{s}\Omega_{m}\gg 4g^{2}$, all coefficients are positive. Thus,
the criterion $\Delta_{1}$ is fulfilled and the criterion $\Delta_{4}$ depends
directly on the criterion $\Delta_{3}$. Therefore, only $\Delta_{2}$ and
$\Delta_{3}$ are left to be checked. In order to get an intuition on the
stability for different parameters, Fig. 8 shows the stable regions as a
function of spin damping, detuning and delay.

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