Introduction

Centre-of-mass motion stands out in quantum mechanics1,2. It has a central role in quantum models of gravity3,4 and is at the core of continuous spontaneous localisation (or collapse) models (CSL)5. Understanding the quantum behaviour of macroscopic moving objects could thus be the key to the unification of quantum mechanics and general relativity, solving at the same time the long-standing issue of the wave-packet reduction postulate interpretation6. In practice, CSL models are most effectively challenged by probing length scales of order 10−100 nm: within the mesoscopic range5. 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 freedom7,8.

Very few setups can detect motion near the Heisenberg limit, that is with the minimum back-action allowed by quantum mechanics9,10,11. In practice, this can be conveniently realised by coupling the mechanical degree of freedom to an optical cavity12,13. The tremendous capabilities of optomechanics for force sensing have led recently to gravitational wave detection14. One extremely promising technology is the microwave version of optomechanics, where the mechanical element modulates the resonance frequency of an RLC circuit15,16. It inherits the properties of conventional optomechanics, is directly compatible with quantum electronics technologies17,18, with the great advantage of low energy photons being more compatible with cryogenic setups16,19,20.

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 cooling16,21. This comes at a cost: the mode is strongly damped by the light field, while the surrounding bath remains warm22,23,25. GHz acoustics has been passively cooled to the ground state using conventional dilution cryostats26,27,28. However, these systems are limited to extremely small centre-of-mass displacements (zero in the case of breathing modes26,28), even though they do contain a very large number of individual atoms. Hence, they are not suitable for tests of quantum decoherence and collapse theories1.

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 challenges29.

Here, we report on in-equilibrium ground-state cooling of a whole 15-micron diameter aluminium mechanical device resonating at ωm = 2π × 15.1 MHz in its first flexure and coupled to a ωc = 2π × 5.7 GHz microwave cavity, installed on a nuclear demagnetisation cryostat reaching ~500 μK. 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-head19, and the cryogenic setup consists of a home-made laminar copper nuclear stage mounted on a wet dilution cryostat (see ref. 30 for details). The single-phonon optomechanical coupling g0 for these modes is measured to be ~2π × 230 Hz. Single-tone microwave optomechanics is used here only for motion detection31. A pump tone (of power Pin) is applied at an angular frequency ωc + ωm (blue-detuned scheme), or conversely at ωc − ωm (red-detuned scheme). The mechanical motion generates a sideband peak in the output spectrum at the cavity frequency ωc. 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 (ncav ∝ Pin ~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 nanometre range32, 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 μK, which is remarkable. Optical systems are usually limited by their ability to feed-in energy from the photon field, making pulsed experiments mandatory28,33. Details on the setup and measurements can be found in Supplementary Notes 1 and 2, respectively.

Fig. 1: Mechanical properties versus temperature.
figure 1

Main graph: mechanical mode frequency shift (ωm[T] − ω0)/(2π) (with respect to ω0/(2π) ≈ 15.1 MHz) as a function of cryostat temperature. The black line is a simple Logarithmic fit, whereas 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 Γm/(2π). The black line is a linear fit (see text). Top inset: measured temperatures at beginning of cooling in a demagnetisation process (starting at the dashed vertical). The red curve corresponds to the cryostat Tcryo, whereas the two blue curves are mechanical mode temperatures Tmode calculated from the spectrum (from two distinct runs, see text). Error bars correspond to type A uncertainties extracted from the fitting process (Supplementary Note 3).

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Results

Brute-force cooling

Formally, three distinct temperatures have to be considered here: the cryostat Tcryo, the fundamental mechanical mode Tmode, and the baths Tbaths directly in contact with it30. Demonstrating that thermal equilibrium is maintained across these different subsystems is the key challenge for passive cooling, which we shall address first. Furthermore, characterising 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 ωm(T)−ω0 (with respect to the high-temperature value ω0, main graph) and the damping Γm (bottom inset) of the first flexural mode, as a function of cryostat temperature Tcryo. 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 losses34,35. This means that ‘phonon tunnelling’ 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 nanomechanics36. 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 field37,38. The exact fit expression from theory is shown as the dashed line in Fig. 1 (see Supplementary Note 3 for details)39. This leads us to identify a second bath coupled to the mechanics, namely the TLSs. The absence of thermal saturation on ωm(T) proves that this bath does cool to the lowest temperatures30. As it thermalises through the bulk phonons themselves39, the surrounding phonon bath has to be cold as well. We can thus infer TTLS ≈ Tphonons ≈ Tbaths ≃ Tcryo.

Cooling from ~10 mK to the lowest temperature takes ~2–3 days; the beginning of the process is shown in Fig. 1 top inset. The mechanical mode temperature Tmode 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 Pin × (n + 1) and Pin × (n), respectively, with n(Tmode) = 1/(Exp[(ℏωm)/(kBTmode)] − 1) the Bose–Einstein population20,31. It is therefore convenient for the experimentalist to define from these areas two ‘effective mode temperatures’:

Tblue = (n + 1)(ℏωm)/kB for blue-detuned pumping, measuring Stokes peak,

Tred = (n)(ℏωm)/kB for red-detuned pumping, measuring anti-Stokes peak.

We will in the following demonstrate thermalisation of Tmode by studying Tblue, Tred and their ratio as a function of Tcryo. In Fig. 2a (right insets), we show sideband spectra measured at different cryostat temperatures and the same drive power (~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. 2a (main graph) the Tblue, Tred dependencies with respect to Tcryo. Lines are theoretical calculations, with no free parameters. At high temperature, Tblue ≈ Tred ≈ Tcryo, as it should; this is essentially the limit of conventional experiments using commercial dilution cryostats30, and justifies the wording ‘effective temperature’.

Fig. 2: Passive ground state cooling.
figure 2

a Main graph: the peak area of blue and red sideband pumping measurements leads to two different Tblue, Tred estimates, ∝ n + 1 and ∝ n (blue and red lines) respectively, a phenomenon known as sideband asymmetry20. 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 two-photon baseline for readability). The slight shift in position (with respect to the actual ωm[T]) is due to a small remnant optical spring due to imperfect tuning of the pump frequency, whereas the difference in peak heights at high temperature is owing to the small amplifying/de-amplifying occurring at the finite input power we use (see text and Supplementary Note 2). 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/f2 type dependence (full line fit) with a low-frequency cutoff 1/tc (dashed vertical). Right: corresponding histograms from which average n and standard deviation σph 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 Note 3). Error bars (type A uncertainty) mostly arise from the finite stability of mechanical parameters (frequency, damping, see Fig. 1) and are explained in Supplementary Note 3. Note the arrow on the central lowest time trace that points to a time slot where the measured population drops below our resolution for ~5 mins (see text).

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Quantum ground state

However, below typically 10 mK, sideband asymmetry is visible: for blue pumping, Tblue saturates while for red Tred vanishes (see fits in Fig. 2 the main graph). This effect is a signature of zero-point fluctuations in the microwave cavity20. Although Tblue, Tred are no longer simply proportional to Tcryo, the asymmetry leads to in-built primary thermometry by plotting the ratio of Anti-Stokes over Stokes peak areas40. 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. 2a; with the black line being theory with no free parameters. Technical details are given in Supplementary Note 2. 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. 2a). A discussion on the thermal modelling of the device can be found in Supplementary Note 4.

Mode fluctuations

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’41, and in this case, the fluctuations provide important information about the complexity of the thermal environment surrounding the mechanics. To characterise them, we acquire continuously (at fixed drive power, 600 photons, and Tcryo) sideband spectra using the blue-detuned pumping scheme at a reasonably high repetition rate (typically 1 s per file), and then post-process these data with a sliding average of window 20 mins that produces a fittable peak30. 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. 2b centre)30. Width and position shall be discussed later in this article. Although 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. 2b. These issues are discussed explicitly below and in Supplementary Note 3.

From time traces we compute the fluctuation’s spectra (Fig. 2b left). These can be fit by a Lorentzian shape: producing a 1/f2 dependence (Brown noise) above a cutoff frequency 1/tc (see full lines). Such a spectrum is reminiscent of an Ornstein-Uhlenbeck process42. 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 τm = 2/Γm ≈ 1ms, 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 tc ≫ τm is of the order of 5(±2) hours, and not specifically dependent on temperature (see Fig. 2b 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 tc43. A detailed discussion on the statistical treatment is given in Supplementary Note 3.

At low temperatures, the fluctuations can be especially interesting. For example, note the arrow in Fig. 2b centre that points to a time slot where the measured mode population stays at essentially zero over ~5 mins. Knowing that high-frequency modes do cool down to dilution temperatures with no sign of thermal saturation26,44, 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. 45 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. 2b right). We can then extract the average population n and its standard deviation σph. This is shown in Fig. 3 (main). Remarkably, we find that the behaviour of σph 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 bath46,47. 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 complex48,49) mode-TLS interaction.

Fig. 3: Frequency and population standard deviations.
figure 3

Main: standard deviation of the phonon population extracted from Fig. 2b as a function of the 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). Error bars correspond to type A uncertainties extracted from the fitting process (Supplementary Note 3).

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The behaviour of σph is completely different from frequency and damping noises measured at the same time. They both display true 1/f2 spectra (with no low-frequency cutoff observed), and shall thus be characterised for a given time span (so-called Allan deviation)50. Their temperature dependence is similar and reversed from the one of phonon population, see inset of Fig. 3 presenting the frequency standard deviation σf: it grows as we cool. This resembles what has been reported on superconducting cavities, and attributed to TLSs present in the constitutive materials48; here, it would be the TLSs-generating frequency shifts from strain-coupling37,38,51. Details on the statistical analysis can be consulted in Supplementary Note 3.

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. 2a, plus a small cavity noise. It is rather straightforward to correct for these and recalculate average phonon populations30,31. 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 Notes 2 and 3.

Discussion

In conclusion, we have demonstrated ground-state cooling of a whole drum-head device (of mass 5 × 10−14 kg), an object that can move its centre-of-mass substantially (typically tens of nm in self-sustained motion of the 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 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)7,8. 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, TLS) are thermalized; this is mandatory for thermodynamics, but also a unique advantage for the unravelling of new contributions postulated in CSL models5,52. This is a feature that is absent for other ground-state cooling platforms focused on macroscopic motion16,23. 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 avoided53,54. This is a unique advantage of having a whole object ground-state cooled, in comparison with a single working mode as is realised in all other systems16,23,24,26,27,28.

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 attained55; the figure of merit being \(\frac{{{{\Gamma }}}_{{{\mbox{m}}}}}{2\pi }\times n\approx 100\) for us at 500 μK, with plenty of scopes for improvements in Γm (see e.g., refs. 16,56). The unexpected properties of the fluctuations reported here call for theoretical input on both thermodynamics and the constitutive matter37,43,51,57. As for superconducting mesoscopic electronic devices49, TLS 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 limit58. This would enable the detection of single-phonon jumps in-and-out of the mechanical mode, similarly to electrons in a 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 bit18. This would enable experiments directly focused on the study of quantum mechanical decoherence, as proposed, e.g., in ref. 59. Such proposals that rely on macroscopic motion can only be implemented on low-frequency devices60, as opposed to GHz modes.