# A macroscopic object passively cooled into its quantum ground state of
motion: beyond single-mode cooling

D. Cattiaux∗, I. Golokolenov∗, S. Kumar∗, M. Sillanpää∗∗, L. Mercier de
Lépinay∗∗, R. R. Gazizulin∗, X. Zhou∗∗∗, A. D. Armour∗∗∗∗, O. Bourgeois∗, A.
Fefferman∗ and E. Collin∗,† (*) Univ. Grenoble Alpes, Institut Néel - CNRS
UPR2940,
25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France
(**) Departement of Applied Physics, Aalto University, FI-00076 Aalto, Finland
(***) IEMN, Univ. Lille - CNRS UMR8520, Av. Henri Poincaré, Villeneuve d’Ascq
59650, France
(****) Centre for the Mathematics and Theoretical Physics of Quantum Non-
Equilibrium Systems
and School of Physics and Astronomy, University of Nottingham, Nottingham NG7
2RD, United Kingdom

###### Abstract

† Corresponding Author<EMAIL_ADDRESS>
##  I ABSTRACT

The building blocks of Nature, namely atoms and elementary particles, are
described by quantum mechanics. This fundamental theory is the ground on which
physicists have built their major mathematical models RefFeynman . Today, the
unique features of quantum objects have led to the advent of promising quantum
technologies Qubits ; Qualgo . However, the macroscopic world is manifestly
classical, and the nature of the quantum-to-classical crossover remains one of
the most challenging open question of Science to date. In this respect, moving
objects play a specific role Legett1 ; Leggett2 . Pioneering experiments over
the last few years have begun exploring quantum behaviour of micron-sized
mechanical systems, either by passively cooling single GHz modes, or by
adapting laser cooling techniques developed in atomic physics to cool specific
modes far below the temperature of their surroundings Teufel ; Painter ;
Kippen ; Aspel ; SillanpaaEnt ; Groblacher . Here instead we describe a very
different approach, passive cooling of a whole micromechanical system down to
500$~{}\mu$K, reducing the average number of quanta in the fundamental
vibrational mode at 15 MHz to just 0.3 (with even lower values expected for
higher harmonics); the challenge being to be still able to detect the motion
without disturbing the system noticeably. With such an approach higher
harmonics and the surrounding environment are also cooled, leading to
potentially much longer mechanical coherence times, and enabling experiments
questioning mechanical wave-function collapse BassiCollapse , potentially from
the gravitational background BouwmeesterPRL ; Penrose , and quantum
thermodynamics RefQthermo . Beyond the average behaviour, here we also report
on the fluctuations of the fundamental vibrational mode of the device in-
equilibrium with the cryostat. These reveal a surprisingly complex interplay
with the local environment and allow characteristics of two distinct
thermodynamic baths to be probed.

##  II INTRODUCTION

Centre-of-mass motion stands out in quantum mechanics Legett1 ; Leggett2 . It
plays a central role in quantum models of gravity Penrose ; Stamp  and is at
the core of continuous spontaneous localisation (or collapse) models (CSL)
BassiCollapse . Understanding the quantum behaviour of macroscopic moving
objects could thus be the key to unification of quantum mechanics and general
relativity, solving at the same time the long-standing issue of the wave-
packet reduction postulate interpretation BookJoos . In practice CSL models
are most effectively challenged by probing lengthscales of order
$10-100~{}$nm: within the mesoscopic range BassiCollapse . Besides, motion is
at the core of the basic definition of heat: phonons arise in quantum
mechanics as the quasi-particle describing how atoms move. It is therefore
important to explore practical ways in which thermodynamics could be probed at
the quantum level beyond electromagnetic degrees of freedom
JukkaQuantumPhotons ; JukkaQuantumThermo .

Very few setups can detect motion near the Heisenberg limit, that is with the
minimum back-action allowed by quantum mechanics ClerkPRA ; SchwabSET ;
VanDerZant . In practice, this can be conveniently realized by coupling the
mechanical degree of freedom to an optical cavity BraginskyManukin ; LKB . The
tremendous capabilities of opto-mechanics for force sensing have led recently
to gravitational wave detection LIGO . One extremely promising technology is
the microwave version of opto-mechanics, where the mechanical element
modulates the resonance frequency of an $RLC$ circuit RegalNatPhys2008 ;
Teufel . It inherits the properties of conventional opto-mechanics, is
directly compatible with quantum electronics technologies
ClerkDevoretSchoelkopf ; SillanpaaNEMSQubit , with the great advantage of low
energy photons being more compatible with cryogenic setups Teufel ;
SillanpaaEnt ; PRXSchwab .

Conventionally, a mechanical mode can be said in its quantum ground state when
its thermal population $n(T)<1$, which can be achieved by lowering the
temperature $T$. Alternatively, the high degree of control reached within
optomechanical systems enables the use of back-action cooling Braginsky ;
Teufel . This comes at a cost: the mode is strongly damped by the light field,
while the surrounding bath remains warm Aspel ; steele . GHz acoustics has
been passively cooled to the ground state using conventional dilution
cryostats Oconnell ; SAW ; PainterPhonoCryst . However, these systems are
limited to extremely small centre-of-mass displacements (zero in the case of
breathing modes Oconnell ; PainterPhonoCryst ), even though they do contain a
very large number of individual atoms. Hence, they are not suitable for tests
of quantum decoherence and collapse theories Legett1 .

Mechanical modes within micro/nano-mechanical systems with macrocopic masses
and which can tolerate large motional amplitudes have resonance frequencies in
the MHz range. Their passive cooling therefore necessitates sub-milliKelvin
temperatures, which is the range attained by ultimate cryogenics: nuclear
adiabatic demagnetisation. The ability to measure the system without
disturbing it, and the demonstration of its thermal equilibrium with the
cryostat are both significant challenges pickett .

Here we report on in-equilibrium ground-state cooling of a whole 15-micron
diameter aluminium mechanical device resonating at $\omega_{m}=2\pi\times
15.1~{}$MHz in its first flexure and coupled to a $\omega_{c}=2\pi\times
5.7~{}$GHz microwave cavity, installed on a nuclear demagnetization cryostat
reaching about 500$~{}\mu$K. Remarkably, no signs of thermal saturation are
detected in the mechanical properties as shall be discussed below. The
mechanical device consists of a 50 nm vacuum-gap capacitor (see Fig. 1) which
forms a drum-head SillanpaaEnt , and the cryogenic setup consists of a home-
made laminar copper nuclear stage mounted on a wet dilution cryostat (see Ref.
ZhouPRAppl  for details). The single-phonon opto-mechanical coupling $g_{0}$
for these modes is measured to be about $2\pi\times 230~{}$Hz. Single-tone
microwave opto-mechanics is used here only for motion detection AKMreview . A
pump tone (of power $P_{in}$) is applied at an angular frequency
$\omega_{c}+\omega_{m}$ (blue-detuned scheme), or conversely at
$\omega_{c}-\omega_{m}$ (red-detuned scheme). The mechanical motion generates
a sideband peak in the output spectrum at the cavity frequency $\omega_{c}$.
Energy is transferred between the mechanical mode and the microwave field,
resulting in either enhanced damping of the motion as we increase the drive
power (red-detuning), or amplification (blue-detuning). Measurements are
performed at the lowest possible powers ($n_{cav}\propto P_{in}$ about 300 -
600 drive photons confined in the cavity) in order to limit the impact of
damping/amplifying. However, driving the system up to the blue-detuned
instability (when the damping vanishes), we can show that the mechanical mode
exhibits self-sustained motion in the nanometer range CiteCattiauxPRR , which
means that these devices are potentially very well adapted for exploring CSL
physics. Furthermore, no physical heating of the device can be detected in
this range of injected powers, down to $500~{}\mu$K, which is remarkable.
Optical systems are usually limited by their ability to feed-in energy from
the photon field, making pulsed experiments mandatory PainterPhonoCryst ;
davisTLS . Details on the setup and measurements can be found in Supplementary
Information.

##  III RESULTS

Figure 1: Mechanical properties versus temperature. Main graph: mechanical
mode frequency shift $(\omega_{m}[T]-\omega_{0})/(2\pi)$ [with respect to
$\omega_{0}/(2\pi)\approx 15.1~{}$MHz] as a function of cryostat temperature.
The black line is a simple Logarithmic fit, while the dashed line is the full
theoretical fit function (see text). An optical picture of a device from the
same batch is also shown. Bottom inset: corresponding mechanical damping
$\Gamma_{m}/(2\pi)$. The black line is a linear fit (see text). Top inset:
measured temperatures at beginning of cooling in a demagnetization process
(starting at the dashed vertical). The red curve corresponds to the cryostat
$T_{cryo}$, while the two blue curves are mechanical mode temperatures
$T_{mode}$ calculated from the spectrum (from two distinct runs, see text).

Formally, three distinct temperatures have to be considered here: the cryostat
$T_{cryo}$, the fundamental mechanical mode $T_{mode}$, and the baths
$T_{baths}$ directly in contact with it ZhouPRAppl . Demonstrating that
thermal equilibrium is maintained across these different subsystems is the key
challenge for passive cooling, which we shall address first. Furthermore,
characterizing the properties of the baths that couple to the mechanical mode
is of itself a significant issue. This is our second topic, demonstrating the
capabilities of our microwave/microkelvin platform.

In Fig. 1 we present the mechanical frequency shift $\omega_{m}(T)-\omega_{0}$
(with respect to the high-temperature value $\omega_{0}$, main graph) and the
damping $\Gamma_{m}$ (bottom inset) of the first flexural mode, as a function
of cryostat temperature $T_{cryo}$. These parameters are obtained from
Lorentzian fits of the measured sideband peak (position and linewidth,
respectively). Below 100 mK, the mechanical damping saturates which is a
signature of clamping losses RefClampLoss ; RefClampLoss2 . This means that
“phonon tunneling” dominates energy relaxation, and therefore one of the baths
to be considered is the phonons surrounding the drum device. On the other
hand, the frequency shift follows a Logarithmic temperature dependence in the
whole range, as is commonly observed with low temperature nanomechanics
cite_Mohanty . This is interpreted as the signature of Two-Level Systems (TLS)
defects present in the structure, that couple to the macroscopic mode via the
strain field RefGlassULT2 ; cite_Kunal . The exact fit expression from theory
is shown as the dashed line in Fig. 1 (see Supplementary Information for
details) ReviewPhilips . This leads us to identify a second bath coupled to
the mechanics, namely the TLSs. The absence of thermal saturation on
$\omega_{m}(T)$ proves that this bath does cool to the lowest temperatures
ZhouPRAppl . Since it thermalises through the bulk phonons themselves
ReviewPhilips , the surrounding phonon bath has to be cold as well. We can
thus infer $T_{TLS}\approx T_{phonons}\approx T_{baths}\simeq T_{cryo}$.

Figure 2:  Passive ground state cooling. a, Main graph: the peak area of blue
and red sideband pumping measurements leads to two different
$T_{blue},T_{red}$ estimates, $\propto n+1$ and $\propto n$ (blue and red
lines) respectively, a phenomenon known as sideband asymmetry PRXSchwab .
Right insets: sideband peaks (blue and red colour for Stokes and anti-Stokes)
measured for four different temperatures, with their Lorentzian fits
(vertically shifted to 2-photon baseline for readability). The slight shift in
position (with respect to the actual $\omega_{m}[T]$) is due to a small
remnant optical spring due to imperfect tuning of the pump frequency, while
the difference in peak heights at high temperature is due to the small
amplifying/de-amplifying occurring at the finite input power we use (see text
and Supplementary Information). The smallest Anti-Stokes peak required 4 days
of continuous averaging. Left top inset: ratio of anti-Stokes and Stokes peak
areas enabling sideband asymmetry thermometry (the black line is the
theoretical expectation). b, Measurement of in-equilibrium population
fluctuations at 220, 13, 1.4 and 0.65 mK (top to bottom) obtained with 20
minutes averaging time per point. Middle: time traces (9 h span) demonstrating
very slow (and large, see also inset Fig. 1) fluctuations. The shaded areas
delimit the regions below 1 phonon. Left: spectrum (FFT transform of
autocorrelation) showing a $1/f^{2}$ type dependence (full line fit) with a
low-frequency cutoff $1/t_{c}$ (dashed vertical). Right: corresponding
histograms from which average $n$ and standard deviation $\sigma_{ph}$ can be
defined (with Gaussian fit displayed; the slight negativity for the two lowest
graphs is due to the finite precision of the analysis procedure, see text and
Supplementary Information). Error bars mostly arise from the finite stability
of mechanical parameters (frequency, damping, see Fig. 1) and are explained in
Supplementary Information. Note the arrow on the central lowest time-trace
that points to a time slot where the measured population drops below our
resolution for about 5 minutes (see text).

Cooling from about 10 mK to the lowest temperature takes about 2 to 3 days;
the beginning of the process is shown in Fig. 1 top inset. The mechanical mode
temperature $T_{mode}$ is inferred from the area of the measured sideband peak
using a blue-detuned pumping, as explained below. We clearly see that, on
average, the drum-head mode and the cryostat temperatures follow each other,
in good equilibrium. Furthermore, we can also see that the mechanics
reproducibly displays both very large, and very slow temperature fluctuations.
In the following we will start by discussing the averaged behaviour in detail
before going on to describe the properties of the fluctuations.

The sideband signal encodes the position-fluctuations spectrum of the
mechanical mode. At low drive powers, the area of the measured peaks with blue
and red pumping (Stokes and Anti-Stokes sidebands) are proportional to
$P_{in}\times(n+1)$ and $P_{in}\times(n)$ respectively, with
$n(T_{mode})=1/(Exp[(\hbar\omega_{m})/(k_{B}T_{mode})]-1)$ the Bose-Einstein
population AKMreview ; PRXSchwab . It is therefore convenient for the
experimentalist to define from these areas two “effective mode temperatures”:

$T_{blue}=(n+1)(\hbar\omega_{m})/k_{B}$ for blue-detuned pumping, measuring
Stokes peak,

$T_{red}=(n)(\hbar\omega_{m})/k_{B}$ for red-detuned pumping, measuring anti-
Stokes peak.

We will in the following demonstrate thermalisation of $T_{mode}$ by studying
$T_{blue},T_{red}$ and their ratio as a function of $T_{cryo}$. In Fig. 2 a
(right insets) we show sideband spectra measured at different cryostat
temperatures and the same drive power (about 300 photons), with their
Lorentzian fits. For this microwave input, a slight amplifying/de-amplifying
exists that is visible from the peak height difference of the high temperature
spectra (blue and red colour for Stokes and anti-Stokes). In order to extract
the genuine small-drive limit, we measure the power dependence for each
temperature and pumping scheme. This enables us to plot in Fig. 2 a (main
graph) the $T_{blue},T_{red}$ dependencies with respect to $T_{cryo}$. Lines
are theoretical calculations, with no free parameters. At high temperature,
$T_{blue}\approx T_{red}\approx T_{cryo}$, as it should; this is essentially
the limit of conventional experiments using commercial dilution cryostats
ZhouPRAppl , and justifies the wording “effective temperature”.

However below typically 10 mK, sideband asymmetry is visible: for blue
pumping, $T_{blue}$ saturates while for red $T_{red}$ vanishes (see fits in
Fig. 2 a main graph). This effect is a signature of zero-point-fluctuations in
the microwave cavity PRXSchwab . While $T_{blue},T_{red}$ are no longer simply
proportional to $T_{cryo}$, the asymmetry leads to an in-built primary
thermometry by plotting the ratio of Anti-Stokes over Stokes peak areas
AssymThermo . Indeed, in the ideal case of low powers this reduces to
$n/(n+1)$; for the finite microwave drive amplitude used for the peaks
displayed, the function is renormalized by the ratio of the amplified/de-
amplified peak linewidths. This is shown in the top left inset, Fig. 2 a; with
the black line being theory with no free parameters. Technical details are
given in Supplementary Information. The experiment therefore demonstrates
cooling down to an average population for the first flexure of 0.3 quanta
(c.f. the lowest Stokes data point in Fig. 2 a). A discussion on the thermal
modeling of the device can be found in Supplementary Information.

Although the long-time average mode temperature always matches that of the
cryostat to within measurement uncertainties, we observe strong fluctuations
that occur over a surprisingly long timescale, as mentioned earlier. Landauer
famously remarked ”the noise is the signal” landauer , and in this case the
fluctuations provide important information about the complexity of the thermal
environment surrounding the mechanics. To characterize them, we acquire
continuously (at fixed drive power, 600 photons, and $T_{cryo}$) sideband
spectra using the blue-detuned pumping scheme at a reasonably high repetition
rate (typically 1 second per file), and then post-process these data with a
sliding average of window 20 minutes that produces a fitable peak ZhouPRAppl .
From the fit we can extract peak position, width and area as a function of
time. The area can then be converted (from the calibrated power-dependence)
into mode temperature (top inset, Fig. 1), or population (Fig. 2 b centre)
ZhouPRAppl . Width and position shall be discussed later in this article.
While a true reconstruction of the phonon tunnelling statistics is out of the
scope of the present paper (the measurement is not at the single phonon
level), what we obtain is a sort of “smoothed” estimator of the mode
occupation number. Furthermore, since the finite measuring power disturbs the
system we must always correct for this to obtain the unpumped behaviour.
Errors associated with the correction process explain the slight negativity of
the lowest histograms in Fig. 2 b. These issues are discussed explicitly below
and in Supplementary Information.

Figure 3:  Frequency and population standard deviations. Main: standard
deviation of the phonon population extracted from Fig. 2 b as a function of
average population. The black line is a $\propto\sqrt{n}$ guide to the eye.
Top inset: frequency shift standard deviation calculated over 10 h averaging
as a function of cryostat temperature. The line is a power law fit. The
magenta point is inferred from data taken in the self-oscillating regime; in
both graphs the orange dot is obtained with a measuring input power twice
smaller (see text).

From time-traces we compute the fluctuation’s spectra (Fig. 2 b left). These
can be fit by a Lorentzian shape: producing a $1/f^{2}$ dependence (Brown
noise) above a cutoff frequency $1/t_{c}$ (see full lines). Such a spectrum is
reminiscent of an Ornstein-Uhlenbeck process OUprocess . The question that
arises then is: from what thermodynamic bath? We expect fluctuations due to
coupling to the phonon bath to have a characteristic timescale
$\tau_{m}=2/\Gamma_{m}\approx 1~{}$ms, the mechanical relaxation time. This
timescale is clearly outside of our bandwidth (which is at best about 1 Hz);
besides, we observe that the time $t_{c}\gg\tau_{m}$ is of the order of $5(\pm
2)$ hours, and not specifically dependent on temperature (see Fig. 2 b left).
Therefore, our experiment does not directly probe phonon fluctuations. It is
thus natural to consider the second identified bath: TLSs. As shall be
discussed below, from the frequency (and damping) fluctuations we infer that
TLSs are certainly involved, in a complex and specific way. And indeed, ultra-
low temperature experiments that studied their dynamics report on extremely
slow timescales, which are consistent with our $t_{c}$ parpiaSpikes . A
detailed discussion on the statistical treatment is given in Supplementary
Information.

At low temperatures the fluctuations can be especially interesting. For
example, note the arrow in Fig. 2 b centre that points to a time slot where
the measured mode population stays at essentially zero over about 5 minutes.
Knowing that high frequency modes do cool down to dilution temperatures with
no sign of thermal saturation Oconnell ; Painter , higher frequency modes of
the structure have to be empty with a high probability. Making an analogy with
electronic transport, this result is reminiscent of the one of Ref.
Jukka_NO_QP  that demonstrated the absence of any quasi-particles in a
superconductor over seconds.

The statistical distributions computed from the time-traces look reasonably
Gaussian (Fig. 2 b right). We can then extract the average population $n$ and
its standard deviation $\sigma_{ph}$. This is shown in Fig. 3 (main).
Remarkably, we find that the behaviour of $\sigma_{ph}$ is well-described by
$0.5\sqrt{n}$, implying a sub-Poissonian mechanism. Note that this fit extends
from the truly quantum range up to the classical one, without showing any
crossover behaviour. This level of fluctuations is smaller than what we would
expect in terms of the fluctuations in the average energy of a system coupled
to a simple thermal bath thermoADA ; RefHO . However, this is not surprising
since fluctuations associated with the phonon bath are effectively averaged
over in the measurement. The observed fluctuations are thus most likely a
signature of the (potentially rather complex burnett ; TLS_report ) mode-TLS
interaction.

The behaviour of $\sigma_{ph}$ is completely different from frequency and
damping noises measured at the same time. They both display true $1/f^{2}$
spectra (with no low frequency cutoff observed), and shall thus be
characterized for a given time span (so-called Allan deviation) MailletACSNano
. Their temperature dependence is similar, and reversed from the one of phonon
population, see inset of Fig. 3 presenting the frequency standard deviation
$\sigma_{f}$: it grows as we cool. This resembles what has been reported on
superconducting cavities, and attributed to TLSs present in the constitutive
materials burnett ; here, it would be the TLSs generating frequency shifts
from strain-coupling cite_Kunal ; RefGlassULT ; RefGlassULT2 . Details on the
statistical analysis can be consulted in Supplementary Information.

Before concluding, we should point out that, as for any measurement, even if
the detection power is kept very low it is nonetheless non-zero. There is
therefore a finite back-action from the detection scheme onto the mechanical
mode: the amplifying/de-amplifying seen in the spectra of Fig. 2 a, plus a
small cavity noise. It is rather straightforward to correct for these, and
recalculate average phonon populations ZhouPRAppl ; AKMreview . It is however
perfectly natural to wonder how the finite measurement drive affects the
fluctuations themselves, in magnitude, spectral properties and distribution
shape. In order to verify experimentally that recalculated fluctuation
characteristics do not depend on pump power, we measured a time-trace at half
the drive and processed it in the same way: this leads to the orange dot in
Fig. 3, which is in agreement with all other data. A thorough discussion on
the impact of the measurement scheme can be found in Supplementary
Information.

##  IV DISCUSSION

In conclusion, we have demonstrated ground-state cooling of a whole drum-head
device (of mass $5\times 10^{-14}~{}$kg), an object that can move its centre-
of-mass substantially (typically tens of nm in self-sustained motion of first
flexure), from in-equilibrium properties of the lowest frequency mode.
Fluctuations in the occupation of the mechanical mode reveal the complexity of
the baths to which it is coupled. The microwave/microkelvin platform developed
for this experiment opens the path to a unique new regime for
experimentalists: quantum thermodynamics can be addressed with phonons through
mesoscopic equilibrium properties near the ground state of motion, instead of
electrons (or photons) JukkaQuantumPhotons ; JukkaQuantumThermo . The
stochastic nature of quantum heat transport can be studied from one of the
collective, macroscopic mechanical degrees of freedom of the device towards
the continuum formed by the substrate, through the constriction made by the
clamping region. “Conventional” baths (bulk phonons, Two Level Systems) are
thermalized; this is mandatory for thermodynamics, but also a unique advantage
for the unraveling of new contributions postulated in CSL models
BouwmeesterPRL ; BassiCollapse . This is a feature which is absent for other
ground-state cooling platforms focused on macroscopic motion Teufel ; Aspel .
Besides, modal-coupling can be controlled with all modes being cold, down to
the lowest frequency one: mechanical decoherence from nearby “hot modes” (i.e.
with a large thermal population) is thus avoided OLIVEModeC ; DykmanModeC .
This is a unique advantage of having a whole object ground-state cooled, in
comparison with a single working mode as is realized in all other systems
Teufel ; Oconnell ; SAW ; PainterPhonoCryst ; Aspel ; steele .

Passive cooling allows the ground state to be reached while preserving the
mechanical $Q$. This potentially provides much longer mechanical coherence
times, enabling unique sensitivities for force sensing to be attained Caves ;
the figure of merit being $\frac{\Gamma_{m}}{2\pi}\times n\approx 100$ for us
at 500$~{}\mathrm{\SIUnitSymbolMicro}$K, with plenty of scope for improvements
in $\Gamma_{m}$ (see e.g. Refs. Teufel ; Schwab7mK ). The unexpected
properties of the fluctuations reported here call for theoretical input on
both thermodynamics and the constitutive matter RefQthermo ; RefGlassULT ;
RefGlassULT2 ; parpiaSpikes . As for superconducting mesoscopic electronic
devices TLS_report , Two Level Systems are certainly the key to the
understanding of the complex microscopic environment interacting with the
mechanical mode.

The present work demonstrates the compatibility between microwave
optomechanics and ultra-low temperatures. The next generation of experiments
will incorporate a TWPA (Travelling Wave Parametric Amplifier) in order to
open the detection bandwidth, potentially down to the phonon relaxation time,
while reaching the quantum limit JPALehnert . This would enable the detection
of single phonon jumps in-and-out of the mechanical mode, similarly to
electrons in an SET (Single Electron Transistor), which represents the ”holy-
grail” of quantum thermal transport experiments. Besides, the microwave
circuitry is fully compatible with standard quantum electronics; which means
that future developments will also incorporate a quantum bit
SillanpaaNEMSQubit . This would enable experiments directly focused on the
study of quantum mechanical decoherence, as proposed e.g. in Ref.
ArmourBlencoweNJP . Such proposals that rely on macroscopic motion can only be
implemented on low frequency devices bouwmeesterDM , as opposed to GHz modes.

## ACKNOWLEDGEMENTS

We wish to thank O. Maillet, A. Heidmann, P. Verlot and F. Marquardt for very
useful discussions. We acknowledge support from the ERC CoG grant ULT-NEMS No.
647917 (E.C.), StG grant UNIGLASS No. 714692 (A.F.), the STaRS-MOC project
from Région Hauts-de-France and ISITE-MOST project (X.Z.). A.D.A. was
supported through a Leverhulme Trust Research Project Grant (RPG-2018-213),
and M.S. was supported by the Academy of Finland (contracts 308290, 307757,
312057), by the European Research Council (615755-CAVITYQPD), and by the Aalto
Centre for Quantum Engineering. The work was performed as part of the Academy
of Finland Centre of Excellence program (project 312057). We acknowledge
funding from the European Union’s Horizon 2020 research and innovation program
under grant agreement No. 732894 (FETPRO HOT). We acknowledge the use of the
Néel Cryogenics facility. The research leading to these results has received
funding from the European Union’s Horizon 2020 Research and Innovation
Programme, under grant agreement No. 824109, the European Microkelvin Platform
(EMP).

## AUTHOR CONTRIBUTIONS

D.C. ran the experiment and made all measurements. M.S. and L. M. de l’E.
designed and fabricated the sample. X.Z. installed and calibrated the whole
microwave platform, and R.G. took care of the cryogenics. I.G. and S.K. made
preliminary microwave experiments. O.B. analysed and modelled the thermal
aspects of the experiment. E.C. designed the experiment. A.D.A., A.F. and E.C.
supervised the experiment and the data analysis. All authors participated in
the writing of the manuscript.

## COMPETING INTERESTS

The authors declare no competing interests.

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SUPPLEMENTARY INFORMATION for
A macroscopic object passively cooled into its quantum ground state of motion:
beyond single-mode cooling

##  Appendix A Setup

###  A.1 Microwave wiring

Figure 4:  Microwave circuit schematic. Simplified circuitry of the microwave
setup (note attenuation, circulators with 50$~{}\Omega$ load, $R-\phi$
opposition settings); see Ref. ZhouPRAppl  for details.

The microwave setup is a completely conventional one. Two platforms have been
equipped and calibrated: a commercial dry Bluefors® cryostat for preliminary
measurements (not presented in the core of the manuscript), and the home-made
wet nuclear demagnetization cryostat. On the Bluefors®, a LNF® cryo-HEMT is
present on the 4 K stage with a power combiner mounted in front of it allowing
to cancel the strong pump tone (avoiding amplifier saturation). On the nuclear
demag. it is a Caltech® cryo-HEMT bolted on the 4 K stage. Both have a second
room temperature LNF® HEMT amplifier. On the nuclear demag. cryostat the
“opposition line” made with the power combiner is at room temperature; the
Caltech amplifier is linear enough so that it never saturates with the pump
powers we use (from an Agilent® 20 GHz synthesiser). After mixing up the
signal with a local tone (LO) shifted by $\pm\Omega_{m}+2~{}$MHz, the
measurement is performed with a Zurich Instrument® lock-in that is demodulated
at the shift frequency (used in spectrum mode). A basic drawing is provided in
Fig. 4. The coaxial lines from room temperature to 4 K are stainless steel,
and all the cryogenic routing below is NbTi apart from the mixing chamber
level which is copper. Details with complete schematics can be found in Ref.
ZhouPRAppl  and Ref. CattiauxThese . The noise background (brought back at
input of the cryogenic HEMT) is on the nuclear demag. setup about 100 photons
in this experiment. Note that this is clearly not single-shot, but the circuit
has the valuable advantage of being simple.

Figure 5:  Tuning fork 3He thermometry. Main: temperature dependence of the
damping (in Hz) measured on the tuning fork. Insets: Lorentzian resonance
peaks of the tuning fork corresponding to different working points. See Ref.
ZhouPRAppl  for details.

###  A.2 Cryogenics

The nuclear demagnetization cryostat is made of two stages: a (conventional)
home-made dilution unit that reaches base temperature of 10 mK, followed by a
copper laminar nuclear stage on which the experimental cell is mounted
paperYuriSatge . The two stages are connected to each other through a
Lancaster-made aluminium heat switch lancHeatSwtch . On the dilution unit, the
temperature is inferred from a calibrated carbon resistor and a commercial
magnetic field fluctuation thermometer (MFFT) from Magnicon®. On the nuclear
stage we use a home-made 3He thermometer: a quartz tuning fork immersed in the
fluid that measures its viscosity RefFork . This (almost) primary thermometry
also provides a fixed temperature point: the superfluid transition $T_{c}$ at
0.95 mK (for 0 bar pressure). Calibration of the thermometer is shown in Fig.
5. Note that near $T_{c}$ the fork is very strongly damped, and the
sensitivity is rather poor; however above and below, we can infer temperature
very accurately. Details can be looked at in Ref. ZhouPRAppl  and Ref.
CattiauxThese .

The ultra-low temperature cooling process is single shot, and starts with a
precooling period of about a week: the heat switch is closed (a small magnetic
field is applied on the aluminium heat switch to make it normal), and the
copper is cooled to (almost) the mixing chamber temperature while a 7 T field
is applied to it. The coil producing this field is well compensated, and the
cell is surrounded by a superconducting lead shield. After that, the heat
switch is opened (the aluminium is made superconducting by removing the small
field in the switch), and the large field of 7 T is slowly decreased down to a
final value $B_{fin}$ (always larger than 100 mT); this makes the nuclear
spins of the copper atoms cool down, which in turn cool down the conduction
electrons. The cryostat can stay cold for about a week at an almost fixed
temperature (see e.g. Ref. PobellBook  for details on the technique). The
beginning of the cooling process is shown in Fig. 1 top inset of the paper.

Figure 6:  Nuclear demag. cryostat and copper cell. Left: picture of the
dilution unit stage of the nuclear demagnetisation cryostat. Right: bottom of
the nuclear stage (outside of the large demag. field zone) with the copper
cell mounted (back: “weakly-coupled” drum run, front: this run). The 3He
thermometer is also shown.

###  A.3 Multiple runs

3 different chips have been measured on the 2 platforms. They are mounted in
copper cells with SMA connectors. On the nuclear demag. cryostat, the cell is
bolted on a copper clamp that is located at the bottom of the laminar copper
stage; the 3He thermometer is just underneath it. The copper clamp is
connected to the stage and the thermometer on either side with 1 mm diameter
annealed silver wires. Pictures of the demag. cryostat are shown in Fig. 6.

Of the 3 samples measured, we have: a beam NEMS made in Grenoble which was
used in Ref. ZhouPRAppl , a drum-head NEMS made in Aalto and used in Ref.
CiteCattiauxPRR  that we shall call “weakly-coupled”, and another drum-head
from Aalto (the one of this experiment), that we shall call “strongly-
coupled”. The beam device displayed in the first flexure large amplitude
fluctuations (nicknamed “spikes”) at low temperatures, described in Ref.
ZhouPRAppl . They come from a stochastic driving force of unknown origin, that
has been observed in different laboratories over the world. The first drum
(“weakly-coupled”) showed below about 30 mK an apparent thermal decoupling of
the fundamental mode that seemed to share some of the “spikes” characteristics
CattiauxThese . It was therefore impossible to measure in thermal equilibrium
the mechanical modes of these structures down to (and below) 10 mK. For both
devices, the microwave power required for the measurements is quite large,
$n_{cav}$ in the range $10^{4}-10^{6}~{}$photons confined in the cavity (see
Section B below for the definition of $n_{cav}$).

However, the second drum could be measured down to very low powers, typically
$300-600~{}$photons (using the so-called blue and red-detuned schemes, see
below), thanks to its “strongly-coupled” nature. In these conditions, none of
the above problems was seen (consistently with results from other labs, e.g.
Ref. Schwab7mK ); the remaining issues are fluctuations in the device
properties themselves, which are always seen in nanomechanical experiments (as
far as they are looked for), as discussed in the core of the paper (and see
below, section C). The link between fluctuations, “spikes”, and microwave
power remains mysterious, and one can only speculate on that (see Ref.
ZhouPRAppl  and Section C below).

##  Appendix B Single-tone microwaves

###  B.1 Chip properties

The device studied in this experiment is an aluminium drum-head (see Fig. 1
picture) of diameter 15$~{}\mu$m, 100 nm thickness suspended about 50 nm above
a gate electrode coupled to a $\omega_{c}=2\pi\times 5.7~{}$GHz meander
microwave cavity. It is made on sapphire. The total damping of the cavity
resonance is $\kappa_{tot}=2\pi\times 500~{}$kHz, for an external coupling of
$\kappa_{ext}=2\pi\times 240~{}$kHz. The measurement is performed in
reflection, with a single coupling port; we are in the resolved sideband
regime $\omega_{m}\gg\kappa_{tot}$.

The two lowest modes of the mechanical structure have been measured; the
strongly coupled $[0,0]$ fundamental mode at $\omega_{m}=2\pi\times
15.1~{}$MHz (for a low temperature damping of $\Gamma_{m}=2\pi\times
420~{}$Hz), and the first non-axisymmetric mode $[0,1]$ at 25.9 MHz (for a
damping of about 100 Hz) with a very small optomechanical coupling (non-zero
because of the slight imperfect mode shape due certainly to the exact clamp
structure). The next axisymmetric mode $[1,0]$ should be around 36 MHz but has
not been characterized. From simulations, the device stores about 240 MPa
stress. Mechanical damping is dominated by (rather large) clamping losses
below 100 mK, which is certainly linked to the large amount of stress present
in the device ignacio . In our experiment, the relaxation time of the mode is
thus $\tau_{m}=2/\Gamma_{m}\approx 1~{}$ms, independent of temperature below
100 mK.

###  B.2 Basic calibrations

Figure 7:  Optomechanics schemes. Output spectra, with the cavity response
depicted as dashed (not to scale). The pump tone is the vertical arrow (Dirac
peak). Top: red-detuned pumping scheme. Bottom: blue-detuned pumping scheme.
Only the enhanced sideband at $\omega_{c}$ is measured.

Only strict single-tone optomechanics has been used, such that the basic
theory does strictly apply. This means that blue (microwave drive frequency
$\omega_{c}+\omega_{m}$) and red (microwave drive frequency
$\omega_{c}-\omega_{m}$) detuned pumping experiments have been performed in
distinct runs. Only these two schemes have been used; see Fig. 7 for a
schematic of the techniques. The “green” pumping (pump tone applied at the
cavity frequency $\omega_{c}$) could not be used; below typically 50 mK, the
areas of the two measured sidebands were not consistent anymore with the
cryostat temperature, a feature that is observed with the “spike” problem
ZhouPRAppl . We suspect that this is again due to the large amount of
microwave power applied (which for this scheme is again on the order of
$10^{5}$ photons). This feature, which seems to affect all microwave
optomechanical devices, deserves to be studied but is outside of the scope of
this work.

We first characterized the optomechanical coupling $g_{0}$ by measuring at
fixed temperature $T_{cryo}=100~{}$mK the Brownian sideband peak as a function
of applied microwave power. The peak areas and the peak widths are shown in
Figs. 8 and 9 respectively.

The lineshapes are Lorentzians (see fits Fig. 2 a insets). Basic formulas
allow to link the measured peak area $A$ (in photons/s) and linewidth $\Delta
W$ (in rad/s) to the effective NEMS mode population detected $n_{ef\\!f}$ and
applied microwave power $P_{in}$ AKMreview ; ZhouPRAppl ; CattiauxThese :

$\displaystyle A$ $\displaystyle=$
$\displaystyle\Gamma_{opt}\,n_{ef\\!f}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for
red},$ (1) $\displaystyle A$ $\displaystyle=$
$\displaystyle\Gamma_{opt}\left(n_{ef\\!f}+1\right)\,\,\,\,\mbox{for blue},$
(2) $\displaystyle\Delta W$ $\displaystyle=$
$\displaystyle\Gamma_{m}\pm\Gamma_{opt},$ (3) $\displaystyle\Gamma_{opt}$
$\displaystyle=$ $\displaystyle 4\frac{g^{2}}{\kappa_{tot}},$ (4)
$\displaystyle g^{2}$ $\displaystyle=$ $\displaystyle g_{0}^{2}\,n_{cav},$ (5)
$\displaystyle n_{cav}$ $\displaystyle=$
$\displaystyle\frac{P_{in}\,\kappa_{ext}}{\hbar\omega_{c}\,\omega_{m}^{2}},$
(6) $\displaystyle n_{ef\\!f}$ $\displaystyle=$
$\displaystyle\frac{n(T)\,\Gamma_{m}+N_{noise}\Gamma_{opt}}{\Gamma_{m}\pm\Gamma_{opt}},$
(7) $\displaystyle n(T)$ $\displaystyle=$
$\displaystyle\frac{1}{Exp[(\hbar\omega_{m})/(k_{B}T)]-1},$ (8)

with the sign referring to red (+) or blue (-) detuning schemes, using the
resolved-sideband condition $\kappa_{tot}\ll\omega_{m}$. $n(T)$ is the Bose-
Einstein thermal distribution, where $T=T_{mode}$ by definition. $n_{cav}$ is
the number of drive photons confined in the cavity for the measurement. From
fits (black lines) we extract the single phonon coupling $g_{0}=2\pi\times
230~{}$Hz.

Figure 8:  Area of sideband versus power. Blue and red detuned measurements
performed at 100 mK. Lines are fits (a slight out-of-equilibrium contribution
is visible at large powers for red detuned pumping, which is discussed later
in the text); the dashed vertical marks the threshold towards self-sustained
oscillations (see text).

Let us discuss these simple expressions and what they actually mean. At high
temperature when $n_{ef\\!f}\gg 1$, Eqs. (1,2) tell us that the areas obtained
with blue and red pumping schemes are essentially equivalent; they are just
proportional to the mode population $n_{ef\\!f}$. However, when $n_{ef\\!f}$
becomes very small, they start to be different: this effect known as sideband
asymmetry PRXSchwab , is in itself a proof that the system is very cold. In
Eq. (7) which defines the effective mode population, we see that an extra term
appears in addition to the Bose population: $N_{noise}$. This term is linked
to the photon noise $N_{cav}$ present in the microwave cavity (not to be
confused with $n_{cav}$) through:

$\displaystyle N_{noise}$ $\displaystyle=$ $\displaystyle
N_{cav}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for red},$ (9)
$\displaystyle N_{noise}$ $\displaystyle=$ $\displaystyle
N_{cav}+1\,\,\,\,\,\,\,\,\,\,\mbox{for blue}.$ (10)

For simplicity in our discussion, we assumed that the cavity photon noise and
the one arising from the microwave port are the same. In Eq. (9), we neglected
a term $(\kappa_{tot}/4\omega_{m})^{2}$ which is small in the limit of
sideband resolved setups; this is actually the lower bound for red detuned
active cooling fixed by quantum mechanics AKMreview . A generic discussion can
be found in Ref. CattiauxThese . The “$+1$” appearing in Eq. (10) is a
signature of zero point fluctuations in the cavity field AKMreview . If the
noise $N_{cav}$ was limited only by the thermal noise of the cavity, and if we
take the expected electronic temperature of 40 mK to evaluate it (see Section
D), we would obtain $10^{-3}$. This is clearly negligible, but unfortunately
not the whole story, as shall be discussed in Section B.3.

The red detuned scheme de-amplifies/cools the mode (population decreased,
linewidth increased) while the blue scheme amplifies/heats (population
increased, linewidth decreased). When the linewidth becomes zero (at
$P_{thr}$, Figs. 8 and 9), the mechanical mode enters self-sustained
oscillations CiteCattiauxPRR . This is shown in Fig. 10. The amplitude of
motion becomes then very large, and the mechanical frequency shifts because of
the Duffing nonlinear effect (stretching of the membrane, Duffing coefficient
about 20 Hz/nm2, see inset) CattiauxDuffing . From simulations CiteCattiauxPRR
, we infer that the motion is in the nanometer range.

Figure 9:  Width of sideband versus power. Blue and red detuned measurements
performed at 100 mK. Lines are fits (pointing at $\Gamma_{m}$ for zero power
injected); the dashed vertical marks the threshold towards self-sustained
oscillations (see text).  Figure 10:  Self-sustained oscillations. Amplitude
of sideband signal as a function of power at 18 mK, in the blue detuned
scheme. The dashed line marks the threshold were the linewidth becomes zero
(see Figs. 8 and 9). Inset: measured mechanical frequency shift versus power.

When the pump tone is not perfectly tuned, the position of the measured
sideband moves linearly with applied power; an effect known as optical spring
AKMreview . This is shown in Fig. 11 for our experimental settings, and
explains the slight difference in the actual peak positions in Fig. 2 a (but
has a marginal impact on other measured parameters). The pump position
$\omega_{c}\pm\omega_{0}$ has been kept constant over the whole experiment
(with $\omega_{0}$ the high temperature value of $\omega_{m}$). As such, the
optical spring measured was temperature-independent; the $T$-dependent
mechanical frequency shift (Fig. 1) being far too small compared to
$\kappa_{tot}$ to have any effect on this setting. But precisely because the
mechanical frequency shifts with temperature, any microwave-induced physical
heating of the device would add-up and create an additional (and temperature-
dependent) contribution to the effect presented in Fig. 11 (see e.g. Ref.
ZhouPRAppl ). This is not observed in our case, and we therefore conclude that
no relevant microwave-induced physical heating of the device is present, at
any temperature, in the range that we explored.

Figure 11:  Optical spring. Mechanical frequency shift as a function of
microwave power at 15 mK for both red and blue detuned schemes. Lines are fits
(see text).

The point of the experiment is to use optomechanics as a probe, as little
invasive as possible: ideally for extremely small injected power
$n_{cav}\rightarrow 0$, the amplification/de-amplification is negligible.
Ideally as well, the stochastic back-action arising from the (very small, but)
finite cavity/microwave port population $\Gamma_{opt}\,N_{noise}$ should also
be essentially zero. In this limit, one would just measure a sideband signal
proportional to injected power, subject only to sideband asymmetry, i.e.
obtaining an area from Eqs. (1-2) proportional to $n(T)$ in the red detuned
scheme, and $n(T)+1$ for blue. This leads us to define the “effective
temperatures” $T_{blue},T_{red}$ from the area $A$ quoted in the core of the
paper:

$\displaystyle T_{blue}$ $\displaystyle=$
$\displaystyle(n+1)\frac{\hbar\omega_{m}}{k_{B}}\,\,\,\mbox{blue-detuned },$
(11) $\displaystyle T_{red}$ $\displaystyle=$
$\displaystyle(n)\frac{\hbar\omega_{m}}{k_{B}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{red-
detuned}.$ (12)

These are practical parameters for the experimentalist: above typically 10 mK,
they both equalise to the mode temperature $T_{mode}$. Having them follow
$T_{cryo}$ is thus a proof of thermalisation.

Figure 12:  Normalised areas. Areas normalized to injected power and linewidth
as a function of $P_{in}$. Top: blue-detuned pumping. Bottom: red-detuned
pumping. Lines are fits explained in the text.

We therefore keep the drive power in the range 300 to 600 photons for
$n_{cav}$, the lowest we can afford for practical reasons. Even if small there
is still a measurable amplification/de-amplification in this range; but this
is easily corrected for since the ratio $\Gamma_{opt}/\Gamma_{m}$ is known
(see however the discussion in Section C about damping fluctuations). This is
what is applied to create the main graph of Fig. 2 a. However, the sideband
peaks displayed in insets have not been corrected; this is why an asymmetry is
visible at high temperature between the Stokes and anti-Stokes peaks. As such,
the ratio of their areas is bound at high temperature by
$(\Gamma_{m}-\Gamma_{opt})/(\Gamma_{m}+\Gamma_{opt})\approx 0.5$, as can be
seen in the top inset, Fig. 2 a (sideband asymmetry thermometry). The line in
the same inset graph is theory; in principle an independent proof of
thermalisation down to the lowest temperatures.

###  B.3 Technical heating

Unfortunately, the second assumption discussed in the previous paragraph
(ideally cold microwave mode, $N_{cav}=0$) is also not fully verified: as we
increase the injected microwave power $P_{in}$, there is technical heating of
the microwave mode (to be distinguished from physical heating of the whole
structure; see Fig. 8 the red detuned data). This effect is known in the
community, and not always easy to circumvent. It is presumably due to out-of-
equilibrium photons that populate the cavity and act (through back-action)
onto the mechanical degree of freedom. These out-of-equilibrium photons are
believed to originate in the phase noise of the microwave source; an ultimate
limitation of any setup. The foot of the Dirac peak of the pump “leaks” into
the cavity, so to speak. But the actual mechanism behind this is not fully
understood, and the magnitude of this effect does depend on sample and cool
down: it has to be carefully characterized in each run.

To demonstrate this for our experiment, we plot in Fig. 12 the area $A$
normalised to power $P_{in}$ and linewidth $\Delta W$ as a function of
$P_{in}$, for different temperatures $T_{cryo}$. And indeed, we see at the
largest power a deviation, characteristic of this effect (without which it
should remain flat, and $\propto T_{blue},T_{red}$). It can be fit by an out-
of-equilibrium contribution $\propto(P_{in})^{a}$ (a power law dependence with
exponent $a$) ZhouPRAppl , see lines in Fig. 12. This term is independent of
the scheme, and of temperature. It is of the order of 1 out-of-equilibrium
photon at the lowest drive used (300 photons), and can be easily corrected
for; it is negligible in all conditions presented in the core of the paper,
except for two measurements performed around 700$~{}\mu$K (the time-trace
taken with the blue scheme, Fig. 2 b, 600 photons drive, and the anti-Stokes
peak Fig. 2 a, 300 photons), and the time-trace taken at 1.4 mK (blue scheme,
Fig. 2 b, 600 photons drive). In these extreme cases, the correction
nonetheless remains smaller than a factor two. In order to certify that there
is no flaw in the analysis, a time trace with 300 photons has also been taken
and processed at 14 mK; and indeed results are identical for 300 and 600 drive
photons (see Section C below).

As a concluding remark on the mode populations, we can compute the $n(T)$ of
the nearest modes at the lowest temperature of 500$~{}\mu$K assuming perfect
thermalization, which as discussed in the core of the paper is essentially
guaranteed by our results obtained on the lowest mode (see also the discussion
on thermal properties in Section D below). At this temperature the fundamental
mode $[0,0]$ stores 0.3 quanta on average. The nearest non-axisymmetric mode
$[0,1]$ stores about 0.1 quanta while the next axisymmetric mode $[1,0]$ has
0.03 quanta. All higher modes verify $n(T)\leq 1~{}\%$.

##  Appendix C Statistical analysis

Beyond thermalisation, a striking result of the experiment concerns the very
slow, and rather large fluctuations that are seen in mechanical properties as
a function of time. Obviously our measurement is not single-shot (detection
background about 100 photons at 5 GHz), and it is not a QND measurement of the
population ($\hat{n}$ operator); what we measure is the spectrum of the
$\hat{x}$ operator, the position. We need between 5 and 20 minutes at our
highest drive level in blue-detuned pumping (600 photons) in order to have an
acquisition data averaged enough to be fitable (depending on temperature).
This means that what we measure is a “smoothed” estimator of the time-
dependent mechanical properties, but not the real-time ones. But it is an
invaluable, and unique, information about whatever processes are at stake in
the thermodynamical equilibrium of the device.

The setup and measurement scheme has been kept basic, so that it can be easily
modelled by “conventional single-tone optomechanics”. This makes the
interpretation of measured data rather simple, and the only experimental
parameter linked to the detection that we will have to consider for potential
biases in the analysis is the (even if small) power $P_{in}$ (or number of
photons $n_{cav}$). This has been already discussed in the previous Section.
And of course, because of the nature of the data presented we have to consider
carefully the potential flaws that the statistical analysis may imply. These
aspects are the subject of this Appendix.

Figure 13:  Standard deviation versus acquisition time. Phonon standard
deviation measured at different $T_{cryo}$ normalised to 108 mK data, with an
averaging window of 20 minutes (from 600 photon drive in blue-detuned scheme).
Inset: frequency standard deviation at 108 mK. The stars are measured with 300
photons (see text).

###  C.1 Potential statistical flaws

The first statistical parameter we can consider is the full acquisition time.
In Fig. 13 (main graph) we show the phonon standard deviation $\sigma_{ph}$
measured at different $T_{cryo}$ scaled on the 108 mK values, as a function of
the length of the acquisition time. In inset, we give the frequency standard
deviation $\sigma_{f}$ measured at 108 mK (damping noise is discussed in the
next Section). For these data, all fits have been performed on time-traces
where single points are obtained by 20 minutes averages. As explained in the
core of the paper, this averaging window is sliding through the acquired set
of data (which has a time step of 1 second). Apart from the highest
temperatures, no peak can be seen on single files. $\sigma_{ph}$ and
$\sigma_{f}$ have rather different behaviours, which are characteristic of the
nature of their fluctuation spectra. For full acquisition times smaller than
$t_{c}=5$ hours, $\sigma_{ph}$ grows essentially linearly, similarly to the
whole behaviour of $\sigma_{f}$. This comes from the $1/f^{2}$ nature of the
spectra. However, above 5 hours it becomes flat: this is a signature of the
low-frequency cutoff $1/t_{c}$ seen in the fluctuation spectrum. The standard
deviation of the phonon noise is bound, and measuring with an acquisition time
of 10 hours guarantees to have a proper estimate of $\sigma_{ph}$. On the
other hand, $\sigma_{f}$ grows continuously: the spectrum is not bound at low
frequency, this noise is actually non-stationary. We therefore have to quote
for what acquisition length we compute $\sigma_{f}$; this is what people refer
to in the literature with the Allan deviation NatNanotechHentz . Here again,
we shall compute the standard deviation with a time slot of 10 hours.

Figure 14:  Standard deviation versus averaging window. Phonon standard
deviation at different $T_{cryo}$ for acquisition times longer than 10 hours
scaled on 108 mK data (line is linear guide). Inset: frequency standard
deviation versus acquisition time for 3 different averaging window, at 108 mK.
Lines are fits, and the dashed one is the extrapolation for zero averaging
(see text). All data taken at 600 photons in blue-detuned scheme.

More importantly, we have to consider the effect of the averaging window on
the quoted standard deviations. In Fig. 14 we show the effect of the size of
the window on the $\sigma_{ph}$ measured in the plateau region of Fig. 13
(beyond 10 hours acquisition, scaled on the value at 108 mK). As expected, we
see a rather smooth decrease of $\sigma_{ph}$ when we increase the averaging
time. At first order, this looks like a linear dependence; at most, when
averaging 20 minutes we lose about $20~{}\%$ of the total phonon fluctuation
amplitude. This has been corrected for in Fig. 3 of the core of the paper. In
inset on Fig. 14, we demonstrate the impact of the averaging procedure on
$\sigma_{f}$; we plot it as a function of acquisition time for 3 different
averaging windows. As expected, the more we average the smaller the
$\sigma_{f}$ value. In all cases, the dependence seems linear with respect to
acquisition length. Increasing the averaging time seems to impact much more
the short acquisitions (the 0-time intercept goes down very quickly by
increasing the averaging), while it is less pronounced at large times. For 10
hours acquisitions, a 20 minutes averaging window decreases the impact of
frequency fluctuations on the data by about $20~{}\%$ (see dashed line in Fig.
14). This has been corrected for in the $\sigma_{f}(T)$ plots.

One important aspect of the measurement which also has to be discussed within
the statistical analysis is the finite power at which (in blue-detuned
pumping) the measurement is performed. For technical reasons, this amplifies
the fluctuations: even when the average population is below one, the
acquisition is performed around about 3 phonons. It is thus expected than,
within our resolution all distributions shown in Fig. 2 b look Gaussian (in
accord with the central limit theorem). Knowing the gain
$\Gamma_{m}/(\Gamma_{m}-\Gamma_{opt})$, one can easily recalculate the
original average value. However, it is perfectly reasonable to wonder if the
procedure could modify the amplitude of fluctuations (like e.g. squeeze them).
We therefore performed a time-trace measurement at half the drive (300
photons), at a temperature high enough such that the averaging settings we use
would still be enough to process the data over decent timescales (we took 14
mK; see stars in Fig. 13). This leads to the orange dot in the graphs (Fig. 3
of the paper, Fig. 15 below and stars in Fig. 13). As one can clearly see, the
obtained standard deviations are equivalent to the ones extracted with 600
photons. The overall shape of the spectra reported are also independent of
microwave power and statistical analysis: the $1/f^{2}$ nature is absolutely
robust (for both phonons and frequency/damping noises), as well as the $t_{c}$
cutoff time seen for phonon fluctuations. We thus conclude that there is no
flaw in the magnitude of the fluctuations reported. Also: if there was an
influence from other sources of noise (especially electric noise for what
concerns the optical field, damping noise impacting the area measurement or
fluctuations in cryostat temperature), we would expect an effective saturation
at the lowest temperatures. This is not observed, and $\sigma_{ph}$ seems to
follow $n$ monotonically in Fig. 3.

In order to conclude our discussion of the statistical analysis, we shall
comment on the relevant timescales involved in the experiment. The physical
filtering is about 1 Hz from the lock-in, and the averaging extrapolation is
performed from a typical time of a minute. Therefore, our analysis is a proper
treatment of the data that reproduces genuinely only the slow component of the
population fluctuations. To be concrete, the inset of Fig. 1 suggests that the
mechanical damping writes $\Gamma_{m}=\Gamma_{ph}+\Gamma_{TLS}$, with
$\Gamma_{ph}\gg\Gamma_{TLS}$ (and thus $\Gamma_{m}\approx\Gamma_{ph}$). The
phonon contribution is by far too fast compared to our measurement; its
contribution to the statistics is suppressed by a factor of the order of 10
s$/$1 ms. On the other hand, the TLS contribution $\Gamma_{TLS}$ (presumably
of order $1/t_{c}$) is slow enough to be perfectly reproduced by our analysis:
this is the genuine mechanism that has to underlie our results. Also by
construction, our data actually contain all possible sources of slow noise
integrated together (in particular the impact of $1/f$-like electronic noise,
the $1/f^{2}$ damping noise discussed below, and even the fridge long-term
temperature instability). It is also true for the frequency fluctuations. One
could therefore guess that our $\sigma_{ph}$, $\sigma_{f}$ could only possibly
over-estimate genuine TLS- fluctuations, if any imperfection is to be claimed.
For this reason, it is rather surprising to measure a sub-Poissonian
statistics; it would have been naively more natural to expect a result above
the $\sigma_{ph}=\sqrt{n}$ dependence. We thus conclude that the prefactor
$0.5$ in the $\sigma_{ph}=0.5\sqrt{n}$ law provides important insight into the
TLS bath and the way in which it interacts with the mechanics.

###  C.2 Frequency and damping fluctuations

Besides frequency and area fluctuations, we also report on damping
fluctuations. These are usually not looked at, but are always present in
nanomechanical systems MailletACSNano . Actually, all the essential
characteristics of the frequency noise ($1/f^{2}$ spectrum, Gaussian-like
distribution, rare telegraph-like jumps, and $1/T^{a}$ temperature dependence
below 200 mK) appear to be shared by damping noise. However, these two noises
seem to be not (at least perfectly) correlated. This is demonstrated in Fig.
15.

Figure 15:  Damping noise. Comparing frequency (top) and damping (bottom)
fluctuations as a function of temperature (standard deviation
$\sigma_{\Gamma}$ computed for 10 h acquisition, as for $\sigma_{f}$). The
magenta point is inferred from the self-oscillating state measurements (see
Fig. 10 above), in agreement with the Brownian motion result (also on Fig. 3
of the paper at 18 mK). Insets: Gaussian like statistics and $1/f^{2}$
spectra. In the last inset, we report on a damping sudden jump, while nothing
is seen on the frequency. The black lines in the main graphs are power laws
$1/(T_{cryo})^{a}$ guides to the eye. The orange dots, as in other graphs, are
obtained at half the drive (see text).

Even if not strictly equivalent, the frequency noise looks quite similar to
what is found in superconductors: $1/f^{a}$-like, growing with decreasing
temperature burnett . Also, the damping noise seems to dominate over other
sources of errors (namely the amplifier noise, and then the fit error) when
evaluating mechanical parameters, and is thus responsible for the scatter in
the data. We can only speculate on the origin of all material-dependent
fluctuations, namely “spikes” ZhouPRAppl , frequency noise, and damping noise.
These may be related to the same underlying mechanisms, but it will require
further work to be proven. Two-Level-Systems (TLS) are often invoked when
dealing with these issues, since they explain the frequency shifts
$\omega_{m}(t)-\omega_{0}$ as well as the high temperature dependence of the
damping $\Gamma_{m}$. The exact fit function (dashed line in Fig. 1) for the
frequency shift reads
$\omega_{m}(T)-\omega_{0}\propto(Re[\Psi(1/2+(\hbar\omega_{0})/(2\pi
ik_{B}T))]-Ln[(\hbar\omega_{0})/(k_{B}T)])$ with $\Psi$ the Digamma function
ReviewPhilips . This leads to the observed logarithmic-like dependence (full
line in Fig. 1). The damping happens to be linear with temperature at high
temperature, a feature also observed for other NEMS devices in the same
temperature range OlivePRBTLS . An intriguing effect has been noticed in the
mechanical frequency shift at the lowest temperatures (below 10 mK): even
though the equilibrium temperature follows the Log (or equivalently Digamma)
law, we discovered temperature hysteresis when ramping up the temperature
after cooling. This could be the signature of some sort of “glassy” behaviour
within the TLS bath (subject to TLS-TLS interactions? See e.g. Ref.
AndrewParpiaTLS ; burnett ), but we are again limited to speculations and one
would need further investigations to understand this. More discussions can be
found in Ref. CattiauxThese .

##  Appendix D Thermal modeling

The experimental work presented in the paper is about the thermalisation of a
mesoscopic mechanical object down to 500$~{}\mu$K. We present in this Appendix
the basic thermal modeling of the device, that consistently supports our
experimental result. First, we will review the fundamental models to calculate
the specific heat (Debye model) and the thermal transport (Kinetic equation
and Casimir model). Basic material-dependent parameters will be reminded.

Nomenclature:

  * •

$k$ thermal conductivity W.m-1.K-1

  * •

$K$ thermal conductance W.K-1

  * •

$c_{p}$ specific heat in J.m-3.K-1

  * •

$C$ heat capacity J.K-1

  * •

$\Lambda$ phonon mean-free-path

The modeling is kept basic, since only orders of magnitudes are sought. As
such, we should point out already that the specific heat of materials is not
affected by small dimensions at the 100-nanometer scale (we can thus keep
using bulk values) Cleland . On the other hand, for the thermal conductance we
need  to take into account the small dimensions of the heat conductor (and
thus renormalize the bulk values and adapt them to our 2D-membrane sample).

###  D.1 Thermal properties of superconducting aluminium membrane

The heat carriers that will matter for calculating the characteristic
thermalization time (and other parameters) are the phonons. Indeed, thermal
transport in Al at low enough temperature is dominated by phonons and not any
more by electrons. The experiment is performed below 200 mK: the temperature
is then far lower than the superconducting critical temperature $T_{c}$ of
aluminium ($T_{c}\approx 1.2$ K); we can thus consider that all the conducting
electrons are BCS-condensed, and consequently the thermal transport will
happen through the lattice only (phonons), because the condensed electrons
cannot carry entropy (then do not contribute to thermal transport) BCS . This
can be seen in the thermal transport measurements done by Zavaritskii zav
presented in Fig. 16 where the thermal conductivity dramatically changes
between $T_{c}$ and $T_{c}/5$: below 0.2 K one sees that only the phonon
contribution remains as it varies like $T^{3}$.

As well, the contribution of electrons to the specific heat $c_{p}$ is
decreasing exponentially below $T_{c}$; this is a standard result that can
also be calculated in BCS-theory BCS . We can thus also neglect electrons in
the specific heat, and consider only lattice vibrations for the heat capacity
of the object. We shall come back at the end of this Appendix to electronic
properties.

Figure 16: Thermal conductivity. Measurement done by Zavaritskii zav  in 1958 of thermal conductivity in aluminium, on bulk materials (few centimeter large). The extrapolation of that curve and the corresponding fit are given in Fig. 18 (and Tab. 1). Image adapted from Ref. zav .  Material | $v_{s}$ | $\theta_{D}$ | $\rho$ | $c_{p}$ | $k_{bulk}$
---|---|---|---|---|---
Al | 6700 | 468 | 2 700 | 0.41 $\times T^{3}$ | 23.4 $\times T^{3}$
Table 1: Aluminium (Al) data for thermal properties: the speed of sound
$v_{s}$ in m/s, the mass density $\rho$ in kg/m3, the Debye temperature
$\theta_{D}$ (in K). Specific heat in J.m-3.K-1 and (bulk) thermal
conductivity in W.m-1.K-1.

###  D.2 Specific heat of the Al membrane

We will use the Debye model to calculate the phonon specific heat of aluminium
associated to the crystal lattice, in other terms the phonon specific heat.
For $T\ll\theta_{D}$ (the Debye temperature), $c_{p}$ is given by the
following equation using the averaged speed of sound $v_{s}$:

$c_{p}=\frac{2\pi^{2}}{5}\frac{k_{B}^{4}}{\hbar^{3}v_{s}^{3}}T^{3}.$ (13)

The specific heat is in J.m-3.K-1. To convert it to J.kg-1.K-1, if necessary,
one has to divide Eq. (13) by the density $\rho$ in kg.m-3 (see e.g. Ref.
Cleland ). Parameters are given in Tab. 1 with the thermal conductance fit on
Ref. zav  (see Fig. 18). We find a specific heat for the aluminium crystal
lattice of $c_{p}=0.41\times T^{3}$ J.m-3.K-1.

###  D.3 Thermal conductivity, Kinetic equation and phonon mean-free-path in
the structure

The second important calculation is about estimating the thermal conductance
of the structure in order to evaluate its thermalisation time (and potential
thermal gradients). The simplified model we consider is shown in Fig. 17.

Figure 17: Thermal model. Left: schematics (not to scale) of the suspended
aluminium disk (at temperature $T_{1}$) and the (cut) torus connecting the
disk to the heat bath (the sapphire substrate, at $T_{0}$), see image in Fig.
1. Right: thermal model of the suspended disk membrane of heat capacity
$C\approx C_{membrane}$ and thermal conductance $K\approx K_{torus}$ of the
torus (we neglect then the small cuts). “heat leak” stands for all the heat
sources acting on the membrane (see discussion in text).

The phonon thermal conductance is related to specific heat and phonon mean-
free-path through the kinetic equation (coming from the Boltzmann transport
equation at low temperature, see e.g. Ref. Cleland ):

$k=\frac{1}{3}c_{p}\Lambda v_{s}.$ (14)

From this equation, we can calculate the bulk phonon mean-free-path $\Lambda$
if we know the bulk (experimental) thermal conductivity $k$ and the specific
heat $c_{p}$. The measured bulk thermal conductivity is taken from Zavaritskii
zav  (Fig. 16) by fitting the lower part of the curve below 0.2 K (see Tab. 1;
and the power law extrapolation in Fig. 18). This gives a mean-free-path for
bulk phonons of $\Lambda=2.5\times 10^{-2}$ m (2.5 cm, and independent of
temperature since both $k$ and $c_{p}$ have the same $T^{3}$ dependence). From
this result, we shall simplify the problem by considering the membrane as
isothermal at temperature $T_{1}$, and assume that the temperature gradient
(if any) happens at the torus region. We should now define the thermal
conductance of this zone.

When we build a system where one dimension is smaller than this mean-free-
path, the phonons will scatter on the rough surfaces that confine them; then
$\Lambda$ will be reduced. The new mean-free-path can be obtained following
the Casimir model Ziman ; Casimir1938 ; Heron , a rather simple model, where
we assume that the phonons are fully scattered on surfaces (a sort of “phonon
black body”). This is indeed what happens in the torus region, where the
thickness $e_{p}$ is only about 100 nm. Then the new (“confined”)
$\Lambda_{torus}$ is essentially given by the smallest dimension of the
thermal conduction channel (a “worse estimate”, here the thickness already
quoted, much smaller than the radii $r_{1}$ and $r_{2}$). Taking as an order-
of-magnitude approximation $\Lambda_{torus}\approx e_{p}$ (then $10^{4}$ times
smaller than the bulk one), the “confined” thermal conductivity writes:

$k_{Al\,nano}=\frac{1}{3}c_{p}\Lambda_{torus}v_{s},$ (15)

where $c_{p}$ is the bulk specific heat in J.m-3.K-1, and $\Lambda_{torus}$
the new phonon mean-free-path in the Al torus. This is giving a thermal
conductivity of $k_{Al\,nano}=10^{-4}\,T^{3}$ W.m-1.K-1.

Figure 18: Thermal conductivity fit. Bulk thermal conductivity of aluminium as
a function of $T^{3}$ extrapolated from Zavaritskii zav  up to high
temperatures enabling to calculate the power law $k\approx$ 23.4 $T^{3}$
W.m-1.K-1 (Tab. 1).

###  D.4 Characteristic thermalisation time

The characteristic thermalisation time $\tau_{th}$ is given by
$\tau_{th}=C_{membrane}/K_{torus}$. We define the membrane heat capacity:

$C_{membrane}=c_{p}V,$ (16)

with $V=e_{p}\pi r_{1}^{2}$ the volume of the suspended part in Fig. 17 (with
$r_{1}=7~{}\mu$m). The numerical value is then $C_{membrane}=6\times
10^{-18}\,T^{3}$ J.K-1. In order to estimate the thermal conductance
$K_{torus}$ of the Al membrane to the heat bath, we make a simple “worse
estimate” calculation assuming that the thermal gradient develops on the full
width of the torus, from $r_{1}$ to $r_{2}$ (see Fig. 17). Since
$r_{1},r_{2}\gg\Lambda_{torus}$, we shall assume that the conduction is
diffusive in the torus. Noting that the dominant phonon wavelength
$\lambda_{dom.}\approx 2.23\,\hbar v_{s}/(k_{B}T)$ is of the order of $e_{p}$
around 1 K, we can treat the problem as pure 2D transport, leading to the
simple result Maasilta :

$K_{torus}=\frac{2\pi e_{p}}{ln(r_{2}/r_{1})}k_{Al\,nano},$ (17)

with the two radii $r_{1}=$7 $\mu$m and $r_{2}=$10 $\mu$m. We thus get as an
estimate $K_{torus}=1.6\times 10^{-10}\,T^{3}~{}$W.K-1.

We obtain from our estimates $\tau_{th}=4\times 10^{-8}~{}$s (that is 40
nanoseconds), independent of temperature. This is obviously a rough estimate,
giving however the correct order of magnitude. It tells us that the bath
itself can accept/give thermal energy at a very fast rate, up to around a GHz.

###  D.5 Kapitza thermal resistance and heat dissipation within the aluminium

The simple model assumes that the sapphire substrate and the copper cell
(which are macroscopic) are well anchored at the cryostat temperature $T_{0}$.
But there is an extra thermal boundary resistance that we did not include yet:
the Kapitza resistance between the aluminum torus and the sapphire, which is
due to the acoustic mismatch between the two solids PobellBook . We take from
Swartz et al. swartz  the thermal conductance per surface given by:
$k_{ac.\,mis.}=0.1\,T^{3}~{}$W.cm-2.K-1, then $K_{Kapitza}=k_{ac.\,mis.}\times
S_{torus}$ with $S_{torus}$ the area of the torus. From our dimensions, we
calculate a contact surface $S_{torus}=1.6\times 10^{-6}~{}$ cm2, which leads
to a Kapitza thermal conductance of $1.6\times 10^{-7}\,T^{3}$ W.K-1. This
value is much larger than $K_{torus}$, which means that we can indeed safely
neglect it in our discussion.

At the lowest temperatures, a thermal gradient might develop between the bulk
(at temperature $T_{0}$) and the aluminum drum-head (at $T_{1}$):

$T_{1}-T_{0}=P_{heat}/K_{torus},$ (18)

with $P_{heat}$ the total power that is absorbed by the suspended membrane
(Fig. 17). The question that naturally arises is: what mechanisms eventually
limit $P_{heat}$? Talking only about known (and somehow steady, thus not
considering the “spike” issue here) mechanisms, one immediately thinks of the
microwave power injected, or the imperfect r.f. filtering that reaches the
sample. The former is measured to be negligible (see Section B), while the
latter couples essentially to the electrons (see following subsection), not to
the phonons. Black-body radiation is also negligible, since the whole chip is
enclosed in a copper cell thermalised to the nuclear stage, which acts as a
shield. Heating due to cosmic rays or natural radioactivity (from the building
itself) is also an eligible mechanism ParpiaRad , although almost impossible
to estimate. Since the cross section of the device (and its thickness) is
extremely small, we shall also neglect it.

Beyond radiation, one can also think about adsorption of gas particles: using
4He as exchange gas in the 4 K pre-cooling process, if the remaining vacuum in
the can is not low enough at cryogenic temperatures, atoms may desorb from hot
parts and condense on the cold ones (i.e. the chip), creating thus an almost
permanent heat leak PobellBook . This phenomenon is well known, and has been
taken care of. Finally, the last external mechanism that people take seriously
in consideration is mechanical vibrations. This is known to be detrimental to
low frequency devices, generating huge heat loads if not properly handled by
complex suspension systems oosterkamp . For high frequency devices, this seems
to be much less of a problem: they do thermalise to tens of milliKevin with
conventional cryogenics Teufel . In our case, the whole cryostat is suspended
on air-mounts with about a ton of concrete anchoring, which is required for
cryogenic purposes (the copper stage shall not vibrate in the 7 T field,
otherwise eddy currents would create a huge heat leak) PobellBook .

We are therefore left with internal sources of heat within the Al membrane,
which can become (very) relevant at ultra-low temperatures PobellBook . These
are time-dependent (but with rather long time-scales, so that they always
contribute to some extent), and originate in H2 inclusions, structural defects
(e.g. grain boundaries in the metal layer, or TLSs in glasses) that can relax,
or even radioactive impurities (citing documented effects). It is impossible
to estimate quantitatively this contribution, apart from fitting results from
an actual experiment. We shall thus simply consider a reasonable upper bound
that supports our finding: about 0.1 nW/kg (after a few weeks of experiments
below 1 K) PobellBook . This number corresponds to a negligible heating of the
mechanical element above 1 mK, and about 100$~{}\mu$K temperature increase
only around 600$~{}\mu$K.

###  D.6 Conduction electrons

We shall conclude our discussion on thermal properties by a quick look at the
electron bath in the (superconducting) aluminium. The electron bath decouples
from the phonon bath in the membrane following the equation Roukes :

$P_{e-}=Vg_{e-ph}\left(T^{5}_{e-}-T^{5}_{ph}\right),$ (19)

where $P_{e-}$ is the power arriving on the electrons, $T_{e-}$ and $T_{ph}$
are respectively the electronic and phononic temperatures, and $V$ the volume
of the drum. $g_{e-ph}$ is the electron-phonon coupling constant, taken here
to be $0.4\times 10^{9}~{}$W.K-5m-3 Wellstood . From the discussion of the
previous section, we can take $T_{ph}=T_{1}\approx T_{0}$.

Since these electrons hold a vanishingly small specific heat BCS , any small
source of heat $P_{e-}$ easily keeps them hot: this is a major problem in
quantum electronics, which required a lot of work over the years to prevent
superconducting circuits to be poisoned by quasi-particles (i.e. non-paired
electrons) Jukka_NO_QP . As an illustration here, a rather correct filtering
of r.f. noise leading to a heat leak of 1 fW would create a base temperature
for the electrons of 40 mK, while phonons do cool below 1 mK. An extremely
good filtering producing $P_{e-}\approx 1~{}$aW would still keep $T_{e-}$ at
about 10 mK, among the best electronic temperatures reported in
nanostructures.

## DATA AVAILABILITY

The data that support the findings of this study are openly available at the
following URL/DOI: https://cloud.neel.cnrs.fr/index.php/s/CnnYPKn8XHYZgXa.