On-chip thermometry for microwave optomechanics implemented in a nuclear demagnetization cryostat

TL;DR

This paper presents a novel microwave optomechanics technique for on-chip thermometry in a nuclear demagnetization cryostat, covering temperatures from below 500 μK to about 1 K, revealing an unexplained instability below 100 mK.

Contribution

It introduces an on-chip thermometry method using microwave optomechanics in ultra-low temperature environments, highlighting an unexplained instability phenomenon at millikelvin temperatures.

Findings

Demonstrated temperature measurement from below 500 μK to 1 K.

Identified an unstable intrinsic driving force below 100 mK.

Proposed mitigation strategies for the instability.

Abstract

We report on microwave optomechanics measurements performed on a nuclear adiabatic demagnetization cryostat, whose temperature is determined by accurate thermometry from below 500$~\mu$K to about 1$~$Kelvin. We describe a method for accessing the on-chip temperature, building on the blue-detuned parametric instability and a standard microwave setup. The capabilities and sensitivity of both the experimental arrangement and the developed technique are demonstrated with a very weakly coupled silicon-nitride doubly-clamped beam mode of about 4$~$MHz and a niobium on-chip cavity resonating around 6$~$GHz. We report on an unstable intrinsic driving force in the coupled microwave-mechanical system acting on the mechanics that appears below typically 100$~$mK. The origin of this phenomenon remains unknown, and deserves theoretical input. It prevents us from performing reliable experiments below typically 10-30$~$mK; however no evidence of thermal decoupling is observed, and we propose that the same features should be present in all devices sharing the microwave technology, at different levels of strengths. We further demonstrate empirically how most of the unstable feature can be annihilated, and speculate how the mechanism could be linked to atomic-scale two level systems. The described microwave/microkelvin facility is part of the EMP platform, and shall be used for further experiments within and below the millikelvin range.

Equations

$${\mathcal{A}}_0=\mathcal{M}{P}_{in}{T}_{mode}.$$

$$n_{cav}\left(P_{in}\right)=\frac{P_{in}{\kappa }_{ext}/2}{\hslash \left({\omega }_c+\Delta \right)\left({\Delta }^2+{\kappa }_{tot}^2/4\right)}.$$

$${\gamma }_{eff}\left(P_{in}\right)$$

$$\mathcal{A}$$

$${\left(S_{12}\left[\omega \right]\right)}_{dB}=20\log \left|1-\frac{Q_{tot}{\left(Q_{ext}+I{Q}_{im}\right)}^{-1}}{1+2I{Q}_{tot}\frac{\omega -{\omega }_c}{{\omega }_c}}\right|,$$

Full text

Mechanics, Condensed Matter Physics, Quantum Physics

I Introduction

Advances in clean-room technologies within the last decades make it possible to create mechanical elements with one (or more) dimensions below a micron, namely NEMS (Nano-Electro-Mechanical Systems) roukescleland . These objects can be embedded in electronic circuits, and today even within quantum electronic circuits cleland2010 ; quantelecsimmonds ; quantelec2 . Indeed a breakthrough has been made with microwave optomechanics, essentially shifting the concepts of optomechanics into the microwave domain lehnert2008 . These quantum-mechanical devices can then be thought of as a new resource for quantum electronics and quantum information processing, with the realization of e.g. quantum-limited optical-photon/microwave-photon converters and non-reciprocal microwave circuits convOskar ; cindy ; clerknonrec ; simmondsamplifnonrecip ; kippenbergamplifnonrecip ; jfink . Profound quantum concepts are also under study, with for instance mechanical motion squeezing schwavsciencesqueeze and recently mechanical entanglement of separate objects sillanpaaintrique .

In order to operate at the quantum limit, the mechanical mode in use has to be initially in its quantum ground state; for experiments on microwave optomechanics based on megahertz motion, this shall rely on a strong red-detuned pump signal that actively cools the mode, with an environment remaining “hot” kippenbergamplifnonrecip ; sillanpaamultimode ; schwabtones . For most applied issues this is not a problem, even if it brings an additional complexity with a microwave tone that has to be kept on, which could lead to heating of the circuit and mixing with other signals involved. However, for all experiments aiming at a careful characterization of the mechanical decoherence due to the thermal bath, this is impractical: the whole system is required to be in thermodynamic equilibrium. As such, theoretical proposals testing the grounds of quantum mechanics have been released in the literature, both on microwave setups using quantum circuits ArmourBlencowe and conventional optomechanics (i.e. with lasers) bouwmeesterNJP .

Indeed, the first attempt to use nuclear adiabatic demagnetization for optomechanics is due to Dirk Bouwmeester, building on the technology of Leiden Cryogenics KlecknerThesis . This ambitious setup was aimed at cooling a macroscopic moving mirror actuated/detected by optical means. On the other hand, a microwave-based optomechanical system hosting a NEMS is much less demanding; the heat loads from both the photon field and the internal heat release of materials are much less. As such, these setups have already proven to be compatible with dilution (millikelvin) technology cleland2010 ; quantelecsimmonds ; quantelec2 ; lehnert2008 ; kippenbergamplifnonrecip ; schwavsciencesqueeze ; sillanpaaintrique .

We have thus built a microwave platfrom for optomechanics on a nuclear adiabatic demagnetization cryostat. The microwave circuitry has been kept as basic as possible so far for demonstration purposes; no JPA (Josephson Parametric Amplifier) has been used, and we rely only on intrinsic properties of optomechanics for the measurement AKMreview . The cryostat reaches temperatures below 500$~{}\mu$K, and is equipped with accurate thermometry from the lowest temperatures up to about 1 K: using a noise SQUID-based (Superconducting QUantum Interference Device) thermometer noiseJohn plus a 3He-fork thermometer Rob3Hefork .

We report on measurements realized on this platform with a very weakly coupled doubly-clamped beam flexural mode embedded in an on-chip cavity. Beam-based devices are indeed a tool of choice for applications like ultimate sensing (e.g. single molecule mass spectrometry roukesmass ), while the smallness of the photon-phonon coupling enables demonstrating the sensitivity of the methods used: we build on the optomechanical blue-detuned pumping instability to extract the on-chip temperature. On the other hand, popular drumhead devices achieve routinely much higher couplings schwavsciencesqueeze ; quantelecsimmonds (from $\times 10$ to $\times 100$); the very same methods can then be applied with much less power, therefore limiting heating problems. We report not only on the thermalization of the mechanical mode itself, but also on the bulk of the mechanical element with the temperature of its constitutive TLSs (Two-Level Systems).

Below about 100 mK, the mechanical device starts to be self-driven out-of-equilibrium by a strong stochastic force of unknown origin. We report on this phenomenon with a detailed account of the observed features, which have been confirmed in different laboratories but never documented to our knowledge sillanpaPrivCom ; schwabPHD ; contactLehnert . This thorough description is calling for theoretical understanding, and we speculate on the possible ingredients that could be underlying this effect. We demonstrate that the strongest events can be canceled by applying a DC voltage onto the transmission line. However, for beam-based devices the remaining (small) events seem to limit the measurement capabilities to about 10 mK, roughly the lowest achievable temperature for dilution cryostats. This is actually an order of magnitude better than what has been reported so far for doubly-clamped beams sillanpaamultimode ; schwabrocheleau ; TeufelBEAM . Looking also at the spread in the reported thermalization temperatures of drum-like devices, obtained in completely similar conditions, we suspect that this genuine phenomenon should be present in all devices sharing the same microwave readout scheme and constitutive materials, at different levels of expression. In this respect, the present description is extremely relevant for the community, and calls for further experiments on other types of devices below 10 mK.

By carefully characterizing the self-heating due to the continuously injected microwave power, we infer that for the extremely weakly coupled mode used here the technique is suitable down to temperatures of the order of $1-2~{}$milliKelvin; for much larger couplings (like in drum structures) where less power is needed, in the absence of any uncontrolled intrinsic drive the technique should be functional down to the lowest achievable temperatures. As a comparison, similar experiments in the millikelvin range using optics could only be performed in pulsed mode DavisTLS .

II Experiment

The microwave setup that we have chosen for this demonstration is relatively basic, and resembles the one described in Ref. lehnert2008 . Details on the microwave wiring can be found in Appendix A. The measurement is performed in transmission, with a coplanar transmission line coupled on-chip to microfabricated microwave cavities (see Fig. 1 left). The practical aspect of this design is multiplexing: in Fig. 1 one cavity hosts our on-chip thermometer while the others can be used for more complex experiments.

The microwave cavities are realized through laser lithography and RIE (Reactive Ion Etching) of a 120 nm thick layer of niobium (Nb). A typical resonance curve at about $\omega_{c}/(2\pi)\approx 6~{}$GHz is shown in Fig. 1 center, displaying a (bi-directional) coupling rate of $\kappa_{ext}/(2\pi)\approx 100~{}$kHz and a total damping rate of about $\kappa_{tot}/(2\pi)\approx 150~{}$kHz. The NEMS mechanical element we use as on-chip thermometer is made from 80 nm thick high-stress silicon-nitride (SiN, 0.9 GPa), grown on top of silicon. It is a 50$~{}\mu$m long doubly-clamped beam of width 300 nm. It is covered by a 30 nm layer of aluminum (Al), capacitively coupled to the cavity through a 100 nm gap. The aluminum part has been patterned using standard e-beam lithography and lift-off, while the beam was released through RIE etching of the silicon-nitride followed by a selective XeF2 silicon etching KunalsJLTP . The silicon-nitride has not been removed below the niobium layer. The mechanical resonance of the first flexural mode we use is shown in Fig. 1 right, around $\Omega_{m}/(2\pi)\approx 4~{}$MHz, with damping rate $\gamma_{m}/(2\pi)$ of order 10 Hz. Such a mechanical mode still hosts about $n_{thermal}\approx 6~{}$phonons around 1 mK, large enough to be considered in the classical limit and thus well adapted to mode thermometry.

The microwave setup has been carefully calibrated by electronic means; we therefore quote injected powers in Watts applied on-chip, and detected spectra in Watts/Hz (expressed in number of 6 GHz photons) at the output port of the chip. These units are adapted to both issues of heat leaks (cryogenics) and signal-to-noise (electonics) discussed in this Article. Two setups have been used: a dry commercial BlueFors${}^{\textregistered}\,$ (BF) machine for preliminary experiments (base temperature 7.5 mK), and then the home-made Grenoble nuclear adiabatic demagnetization cryostat (operated down to 400$~{}\mu$K) paperYuriSatge . Particular care has been taken in the construction of the two cryostats’ temperature scales, and details are given in Appendix A.

The opto-mechanical interaction arises from the force exerted by light onto movable objects AKMreview . It corresponds to the transfer of momentum carried by light (i.e. photon particles) to the surfaces on which it reflects. In a cavity design with one movable mirror, the retarded nature of this so-called radiation pressure force (due to the finite lifetime of light inside the cavity) leads to damping or anti-damping of the motion, depending on the frequency detuning of the input light with respect to the cavity AKMreview .

Our moving end mirror is the NEMS beam capacitively coupled to the microwave resonator of Fig. 1 (the cavity) lehnert2008 . The standard schemes we use are illustrated in Fig. 2, with an input wave of power $P_{in}$ at frequency $\omega_{c}+\Delta$. We are in the so-called resolved-sideband regime, with $\gamma_{m},\kappa_{tot}\ll\Omega_{m}$ AKMreview . When we pump power in the system exactly at the frequency of the cavity $\omega_{c}$ (detuning $\Delta=0$), the mechanical motion leads to a phase shift of the light AKMreview . The Brownian motion of the mechanical element thus imprints two equivalent sidebands in the spectrum that we can measure. For a red-detuned pump (frequency detuning $\Delta=-\Omega_{m}$, Fig. 2), energy quanta from the mechanics (phonons) can be transferred to the optical field (photons). This leads to the well-known sideband cooling technique AKMreview ; firstopticsSBcool ; microwaveSBCoolLehnertPRL . The so-called anti-Stokes sideband peak in the spectrum is favored, and the mechanical mode is damped by light. On the other hand for a blue-detuned pump ($\Delta=+\Omega_{m}$, Fig. 2), the optical field generates a parametric instability in the mechanics AKMreview ; clerkmarquardtharris2010 . The Stokes sideband peak is favored, and the mechanics is amplified through anti-damping. Eventually one can reach at high pumping powers the self-oscillation regime kippenbergOscillVahalaPRL2005 ; delftsteeleOscill .

We start by calibrating the optomechanical interaction. The optical damping and anti-damping are linear in applied power $P_{in}$, see Fig. 3. From a fit [Eq. (3) below], we can infer the so-called single photon coupling strength $g_{0}=\frac{1}{2}\omega_{c}\frac{1}{C}\frac{dC}{dx}x_{zpf}$ with $x_{zpf}$ the zero-point-motion AKMreview . This is essentially a geometrical parameter, arising from the modulation $\frac{dC}{dx}$ of the microwave mode capacitance $C$ by the beam motion lehnert2008 . We find $g_{0}/(2\pi)\approx 0.55\pm 0.1~{}$Hz, corresponding to the out-of-plane flexure. This coupling is particularly small, the idea being to take advantage of that to demonstrate the sensitivity of our method. The magnitude of the output power is fit to theory AKMreview , leading to a calibration of the measured phonon mode population/temperature [performed at 210 mK, parameter $\cal M$ in Eq. (1) and Fig. 14]. More details on the optomechanics measurements can be found in Appendix B.

III Method

The method we propose builds on the parametric instability of the blue-detuned pumping scheme. When the pump tone is applied at $\omega_{c}$, the size of the two equivalent sideband peaks (their measured area ${\cal A}_{0}$, in photons/s) is simply proportional to injected power $P_{in}$ and mode temperature $T_{mode}$ AKMreview :

[TABLE]

This optomechanics scheme alters neither the measured position of the sideband peaks (detuned by $\pm\Omega_{m}[T_{beam}]$), nor their linewidth $\gamma_{m}(T_{beam})$: both are determined by mechanical properties, which depend on the beam temperature $T_{beam}$. The lineshapes are Lorentzian. We introduce the number of stored photons in the cavity $n_{cav}$, function of both $P_{in}$ and $\Delta$:

[TABLE]

On the other hand for blue-detuned pumping, as we increase the injected power $P_{in}$ (but keep it below the instability threshold), the area $\cal A$ of the Stokes peak is amplified. The blue/red-detuned pumping expressions write AKMreview :

[TABLE]

in the limit of negligible cavity thermal population. For $\Delta>0$, the last term in Eq. (4) after the $\times$ sign is a gain, illustrated in Fig. 4. It arises from the anti-damping, with $\gamma_{ef\!f}$ the linewidth of the Lorentzian peak. Controlling the applied power $P_{in}$, from the knowledge of system parameters one can straightforwardly recalculate the value of ${\cal A}_{0}$, and thus of $T_{mode}$ (i.e. the temperature of the mode in absence of optomechanical pumping). In Fig. 4 we demonstrate 18.5 dB gain, which is greater than the previously reported maximum for a similar setup using a graphene device delftsteeleOscill . Essentially only $\gamma_{m}$ depends on temperature, and has to be known to apply Eq. (4). It can be obtained easily from a measurement of the mechanical effective damping (the linewidth of the Lorentzian Stokes peak, Fig. 3), by either extrapolating to $P_{in}\rightarrow 0$ or defining the position of the threshold $P_{thr}\propto\gamma_{m}$ [with Eq. (3) at $\gamma_{ef\!f}=0$, see Figs. 3, 4 and 15]. More details are given in Appendix B. Obviously, the main requirement for this $T_{mode}$ estimate is the stability of experimental parameters. The mechanical mode itself happens to be the limiting element (see Fig. 3 and Fig. 5 below), leading to finite error bars at large gains in Fig. 4; fluctuations are further discussed in Sections IV and V, see Fig. 8.

The measured mechanical damping rates $\gamma_{m}$ and resonance frequencies $\Omega_{m}$ are shown in Fig. 5. The displayed dependencies are characteristic of NEMS devices in the millikelvin range: a damping $\gamma_{m}\propto T^{\alpha}$ with $0.3<\alpha<2.5$ and a logarithmic frequency shift $\propto\ln\left(T/T^{\ast}\right)$ with $T^{\ast}$ a characteristic temperature (see fits in Fig. 5). For all materials (from monocrystalline to amorphous) this behavior is understood as a signature of TLSs (Two-Level Systems) TLSMohanty ; TLSPashkin ; KunalPRB ; KunalPRL ; FaveroTLSPRL ; DavisTLS ; Painter : either defects (e.g. for monocrystalline Si, or polycrystalline Al), or constitutive of the atomic arrangement (for amorphous SiN). Direct coupling of the first flexural mode to the phonon bath (i.e. clamping losses) RoukesLifshitzClamp is negligible for these structures at millikelvin temperatures.

Within the TLS model the mechanical mode is coupled to the two level systems, which are themselves coupled to the external bath: the electrons and the (thermal) phonons present in the moving structure KunalPRL . For superconducting materials, the electronic contribution is negligible and the TLSs temperature should reflect the phononic temperature in the beam (i.e. the temperature of high frequency modes well-coupled to the clamping ends), which we simply define as $T_{beam}$. By inverting the fits in Fig. 5 it is thus straightforward to extract $T_{beam}$.

The aim of our work is thus to compare the temperature of the cryostat $T_{cryo}$ to $T_{beam}$ and $T_{mode}$. These results are analyzed in Section IV; however it is mandatory to quantify beforehand the impact of the microwave pump power on the defined temperatures. For this purpose we use the “in-cavity” pumping scheme (Fig. 2). We measure, at a given temperature $T_{cryo}$, the mechanical characteristics $\gamma_{m}$, $\Omega_{m}$ and the area ${\cal A}_{0}$ of the two sideband peaks as a function of injected microwave power $P_{in}$. Using respectively the fits of Fig. 5 and Eq. (1), we can recalculate the expected temperatures $T_{beam}$ and $T_{mode}$ under microwave irradiation. Since the local heating should be proportional to the local electric field squared confined onto the NEMS, we discuss these results as a function of $n_{cav}$. The properties of the cavity itself as a function of the pump settings are discussed in Appendix B.

A typical result obtained at 210 mK is shown in Fig. 6 (main graph). Both $T_{beam}$ (obtained equivalently from damping and frequency shift) and $T_{mode}$ display the same linear dependence on $n_{cav}$, and the two sidebands are equivalent: this demonstrates that the effect is indeed thermal. Defining the slope of the fit as $\sigma$, we can extract this coefficient as a function of $T_{cryo}$ (Fig. 6 inset). This temperature-dependence is non-trivial, and no heating model is provided here: such a model should take into account the microwave absorption in the materials, the energy flow in the beam plus the clamping zone slab (suspended by the fabrication undercut), and finally the anchoring to the bulk of the chip. Nonetheless, we can use this graph to estimate the NEMS heating for a given $T_{cryo}$ and $n_{cav}$ in the blue-detuned pumped scheme. As a result, we extrapolate that applying a power of order $P_{thr}$ at 1 mK would heat the beam by about 1 mK; above 10 mK, the heating is essentially negligible (see error bars on $T$ axis of Fig. 15). Knowing the smallness of the coupling $g_{0}$ employed here, this demonstrates the capabilities of the method. Furthermore, because of this microwave-heating it is obviously meaningless to report experiments below about $T_{cryo}\approx 1~{}$mK for this first “ultimate” cooling attempt.

IV In-equilibrium Results

From fits to Eq. (4) of the power-dependent Stokes peak area, we thus extract $T_{mode}$. Reversing the fits of the mechanical parameters $\gamma_{m}$, $\Omega_{m}$ (Fig. 5) we obtain $T_{beam}$. Both are displayed as a function of $T_{cryo}$ in Fig. 7, for our two experimental setups. We demonstrate a thermalization from about 10 mK to 1 K of the mode and of the whole beam; the device is in thermal equilibrium over 2 orders of magnitude in $T_{cryo}$. Reported lowest thermodynamic temperatures in the literature lie all within the range 10 - 30 mK cleland2010 ; quantelecsimmonds ; quantelec2 ; lehnert2008 ; kippenbergamplifnonrecip ; schwavsciencesqueeze ; sillanpaaintrique ; microwaveSBCoolLehnertPRL ; delftsteeleOscill ; jfink ; however one work reports a potential mode temperature for an Al-drum of order 7 mK, consistent with base temperature of dry dilution cryostats schwab7mK . Similarly, a lowest temperature of 7 mK is reported for a gigahertz phononic crystal Painter ; but obviously such a mode cannot be used for phonon thermometry at millikelvin temperatures.

As far as thermalization between measured $T_{cryo}$ and $T_{mode},T_{beam}$ is concerned, Fig. 7 reproduces the state-of-the-art, but does not go yet beyond even though the cryostat cools well below 7 mK. The reason for this is discussed in Section V: below typically 100 mK, the system displays huge amplitude fluctuations which hinder the measurements (Fig. 8). These features have been seen by other groups for beam-based microwave optomechanical devices containing an Aluminum layer, but never reported so far sillanpaPrivCom ; schwabPHD ; contactLehnert ; jfinkpriv . Until recently, this essentially prevented experiments from being performed on these types of devices below physical temperatures of order 100 mK sillanpaamultimode ; schwabrocheleau ; TeufelBEAM ; remarkably however, (Al covered) ladder-type Si beams jfink ; jfinkpriv seem to be less susceptible to this problem than simple doubly-clamped beams. On the other hand, large signal fluctuations have not been observed to date for (Al) drum-like structures sillanpaPrivCom ; contactLehnert , and do not show up in schemes which do not involve microwaves (e.g. magnetomotive measurements of SiN and Al beams OlivierPhD ; TLSPashkin , or laser-based measurements of Si beams DavisTLS ). This is what enabled nano-mechanical experiments to be conducted at base temperature of dilution cryostats.

Up to now, the only possibility to deal with these large events was post-selection, which is extremely time-consuming and even stops being useable at all at the lowest temperatures. The origin of this phenomenon remains unknown, and we can only speculate on it in Section V hereafter. Note that there is no evidence of thermal decoupling in Fig. 7.

More conventional frequency $\Omega_{m}$ and damping $\gamma_{m}$ fluctuations ACSnanoUs are also present (see e.g. Fig. 8). These features have been reported for essentially all micro/nano mechanical devices, as soon as they were looked for; their nature also remains unexplained, and their experimental magnitude is much greater than all theoretical expectations RoukesHentzNatNanotech . Frequency noise essentially leads to inhomogeneous broadening OliveNJP . It does not alter the area $\cal A$ measurement, but does corrupt both frequency and linewidth estimates. This noise comes in with a $1/f$-type component RoukesHentzNatNanotech ; ACSnanoUs , plus telegraph-like jumps MartialPhD . It leads to the finite error bars in Fig. 5 inset; below 10 mK, the mechanical parameters $\Omega_{m}$ and $\gamma_{m}$ cannot be measured accurately. Similar damping fluctuations ACSnanoUs are more problematic, since the amplification gain Eq. (4) depends on $\gamma_{m}$. The error bars of Fig. 5 (main graph) are essentially due to this; they translate into a finite error for the estimate of the gain, Fig. 4, which itself limits the resolution on $T_{mode}$ (Fig. 7, top).

V unstable drive force features

In Fig. 8 we show a typical series of spectral acquisitions as a function of time, around 1 mK. We see very large amplitude fluctuations which start to appear around 100 mK, and get worse for lower temperatures (regardless of the scheme used): the “spikes” grow even larger, but more importantly their occurrence increases. We studied these events in the whole temperature range accessible to our experiment. Their statistics seems to be rather complex, and shall be the subject of another Article. Key features are summarized in this Section. For blue-detuned pumping the spikes worsen as pump power increases, while for red-detuned pumping it is the opposite, suggesting that the effective damping of the mode plays an important role. With in-cavity pumping, spiky features are also present at very low powers, but not at high powers when the NEMS physical temperature exceeds about 100 mK. The recorded heights can be as large as equivalent mode temperatures in the Kelvin range. Around $10-30~{}$mK, post-selection becomes impossible.

With the aim of searching for the origin of this effect, we have characterized it in various situations. We first realized that cycling the system from the lowest temperatures to above 100 mK was producing a sort of “reset”. But very quickly (a matter of hours), after cooling down again the large spikes happen to dominate the signal again. We then tried to apply a small magnetic field to the system; this was not very conclusive. However, applying a DC voltage had a drastic effect on these random features. This is illustrated in Fig. 9: with a few volts on the chip’s coplanar transmission line all the large features disappear. The averaged signal (deeply buried into the noise) recovers a reasonable Lorentzian lineshape (see fit in Fig. 9), while the shape of the spikes is not resolved (Fig. 8). Details on the DC voltage biasing are given in Appendices A and B.

In discussing the source of this feature, a few comments have to be made. What is shown in Fig. 8 is primarily fluctuations of the output optical field. These are detected only on the Stokes and Anti-Stokes peaks, for any of the schemes shown in Fig. 2. Furthermore, the threshold to self-oscillation in the blue-detuned pumping scheme displays a large hysteresis (certainly due to nonlinearities in the system, see Fig. 15, Appendix B). We have noticed that when the microwave power applied (at frequency $\omega_{c}+\Omega_{m}$) lies within this hysteresis, the spiky events seem to be able to trigger the self-oscillation. This would not be possible if the amplitude fluctuations measured were only in the detected signal, at the level of the HEMT. We thus have to conclude that we see genuine mechanical amplitude fluctuations. However, these cannot be due to damping fluctuations alone that could trigger self-oscillations, since we do see the same type of features when pumping red-detuned or in-cavity.

If these fluctuations were due to the input field itself, from Fig. 6 we would reasonably conclude that the NEMS beam would be heated to rather high temperatures, leading to broad and very shifted in frequency (see Fig. 5) Stokes/Anti-Stokes peaks. This is not compatible with the measurement of Fig. 8. The only reasonable conclusion seems thus to be that we do suffer from a genuine extra stochastic force acting on the mechanical element. This is consistent with a stronger sensitivity to the phenomenon when the effective damping of the mode is small (bue-detuned scheme). Since a DC voltage applied only to the cell can drastically modify the measured features, the source has to be on-chip. Noting furthermore than with an in-cavity pumping, it disappears when the beam temperature exceeds 100 mK, we conclude that it should even be within the mechanical element. But the mechanism remains mysterious: citing only documented effects in other areas of research, is it linked to vortex motion in the superconductor vortexNoise , trapped charges chargeNoise , adsorbed molecules adsorbate or to atomic-size Two-Level-Systems in dielectrics (beyond the standard friction model) dielecTLS ?

The low temperature properties of NEMS are described within the tunneling model of Two-Level-Systems (TLSs): for damping, frequency shifts, and phase fluctuations DavisTLS ; fongPRB ; ACSnanoUs . It is thus natural to consider strongly coupled individual TLSs as the most probable source of our problems. Besides, while the actual nature of these microscopic defects remains elusive in most systems, they could be generated in many ways beyond the standard atomic configuration argument ustinov ; an electron tunneling between nearby traps would be a TLS strongly coupled to its electromagnetic environment, among other possibilities adsorbate . For Al-based NEMS, these would create (only a few) defects present in (or on) the Al layer; they should carry a dipole moment, which couples them to the microwave drive as well as to the electric field generated by the applied DC voltage. This field distorts their potentials, such that they could get locked in one state and “freeze”. Furthermore, our results seem to be very similar to those of Ref. ParpiaGamma obtained with a macroscopic mechanical glass sample, where “spiky” events were demonstrated to be originating in the interaction with low-level radioactivity (gamma rays). These results suggest a parametric coupling to TLS at GigaHertz frequencies mediated by the microwave drive, but were the energy corresponding to the large peaks would be provided by the external radiation.

The reason why the mechanism should be dependent on the low phononic dimensionality or size of the device (typ. width about 100 nm, much smaller than the phonon wavelength at 10 mK) is nontrivial. One simple argument could be that the spring constant of the modes under consideration are very different: about 1 N/m for megahertz beams and 100 N/m for drumheads. This could justify why beam-based structures are more reactive to external force fluctuations; an immediate consequence of this argument is then that membrane-based Al devices are not truly immune to force fluctuations, but are just less sensitive: if that should be true, cooling them to low enough temperatures would eventually revive the same features as for beam-based NEMS.

To conclude, let us concentrate on the measurements performed at ultra-low temperatures with a DC voltage bias (of the type of Fig. 9, bottom). Even with the help of the in-built parametric amplification, the signal is very small and requires decent averaging, typically here about 30 minutes for reasonable error bars. Even if the resonance peak is Lorentzian, below typically 20 mK the measured area $\cal A$ does not correspond to the actual cryostat’s temperature $T_{cryo}$: it is always bigger, but the actual value presents large fluctuations in magnitude. This is demonstrated in Fig. 10, with identical measurements performed at 23 mK and 7 mK. What is shown is how the measured area of the Stokes peak evolves over time, performing a sliding average over the whole set of acquired data. In the former case, we see that the fluctuations of the measured area are not more than about $+60~{}\%$; they are much smaller for higher temperatures, leading to proper estimates of $T_{mode}$. However, for the latter these are greater than $300~{}\%$. Besides, fluctuations happen to have an extremely slow dynamics: while spikes switch on/off faster than our acquisition time, their overall occurrence fluctuates over a day (Fig. 10). By no means could this behavior be explained by a thermal decoupling of the device from the cryostat. As a consequence, even the calm zones in Fig. 8 are corrupted by the phenomenon shown in Fig. 10. This is essentially why no reliable data could be acquired below 10 mK; but from the DC biasing and the continuous monitoring of the Stokes peak, thermal equilibrium has been demonstrated at about ten times lower temperature than previously reported in microwave doubly-clamped NEMS experiments sillanpaamultimode ; schwabrocheleau ; TeufelBEAM .

VI Conclusion

We presented measurements of a microwave optomechanical system performed on a nuclear adiabatic demagnetization cryostat, able to reach temperatures well below the 10 mK limit of conventional dilution machines. Relying on a fairly standard microwave wiring and the in-built parametric amplification provided by a blue-detuned pumping, we devised a method providing accurate thermometry of both the mechanical mode and its on-chip environment (the Two-Level Systems to which it couples). The experiment was conducted on a beam Nano-Electro-Mechanical System embedded in an on-chip microwave cavity. The efficiency of the method is demonstrated with a very low opto-mechanical coupling. Thermalization is shown from 10 mK to 1 K with no sign of thermal decoupling.

However at very low temperatures we report strong fluctuations in the signal amplitude which prevented any experiments to be conducted at ultra-low temperatures. These features appear around 100 mK and have been observed in different laboratories, but had never been studied in details so far. We demonstrate the characteristics of these fluctuations, and argue that they are due to an extra stochastic driving force of unknown origin. Microwave irradiation seems to trigger the phenomenon. Applying a DC voltage of a few Volts on-chip cancels the large spiky events, but a small component of this extra random drive persists, with variations over a typical timescale of about a day.

It is unclear if all the fluctuations characteristics present in the devices (amplitude, frequency, damping) are linked to the same underlying mechanism. One could even imagine that temperature-dependent non-linear effects could impact the phonon-photon coupling, beyond the lowest (geometrical) order $g_{0}$. However, it is likely that these effects are present in all experimental systems at different levels of expressions, since all NEMS/MEMS share the same overall characteristics (especially damping, frequency shifts and phase noise typical of Two-Level Systems physics). It is thus tempting to relate this stochastic force to a mechanism mediated by some kind of microscopic TLSs, driven by microwaves but blocked under DC voltage biasing. This stochastic driving force can mimic to some extent a thermal decoupling, and could explain why some drumhead devices in the literature refuse to cool down below typically 20-30 mK. While being a limitation for experimentalists, this phenomenon definitely deserves theoretical investigations. The present work also calls for further experiments at lower temperatures, using other types of devices (e.g. drums). This shall be performed in the framework of the European Microkelvin Platform (EMP) EnssPickettEMP .

Note added in proof: following the present work, a collaboration between Aalto University and Institut Néel has started with the aim of cooling down a drumhead Al device as low as possible on our adiabatic nuclear demagnetization platform. We have evidence that the same features as for beams are present, at a different level of expression. This shall be published elsewhere.

(†) Corresponding Author: eddy.collin@neel.cnrs.fr

Acknowledgements.

We acknowledge the use of the Néel facility Nanofab for the device fabrication, and the Néel Cryogenics facility with especially Anne Gerardin for realization of mechanical elements of the demagnetization cryostat. X.Z. and E.C. would like to thank Olivier Arcizet and Benjamin Pigeau for help in room-temperature characterization of the NEMS devices using optics; E.C. would like also to thank O. Arcizet, M. Sillanpää, K. Lehnert, J. Teufel, S. Barzanjeh, K. Schwab, J. Parpia and M. Dykman for very useful discussions. We acknowledge support from the ERC CoG grant ULT-NEMS No. 647917, StG grant UNIGLASS No. 714692 and the STaRS-MOC project from Région Hauts-de-France. 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).

Appendix A setup

Two similar microwave setups have been used in these experiments. Their common features are described in Fig. 11. This wiring is basic and can also be found within the literature lehnert2008 . Essentially, they are built around a cryogenic HEMT (High Electron Mobility Transistor) placed at about 4 K and two circulators mounted on the mixing chamber of the dilution units. On the BlueFors${}^{\textregistered}\,$ (BF) machine, the HEMT is a Low Noise Factory${}^{\textregistered}\,$ 4-8 GHz bandwidth, with a measured noise of about 3 K (10 photons) at 6 GHz. On the nuclear adiabatic demagnetization cryostat, it is a Caltech 1-12 GHz bandwidth with a measured noise of about 15 K (50 photons at 6 GHz). The (dashed green) boxed component below the HEMT in Fig. 11 represents a power combiner used to realize an opposition line. On the BF setup, this is mandatory to avoid saturation of the cryogenic HEMT from the strong blue-detuned pump tone. On the demag. cryostat, the cryogenic HEMT is linear enough so this protection is not necessary. This choice has been made because of space constrains: feeding an extra microwave opposition line in the nuclear adiabatic demagnetization cryostat would be very demanding. The filtering of the injection lines (DC and microwave) is also described in Fig. 11.

Gains and noise levels of the full chain have been carefully checked with respect to HEMT working point. Besides, each component has been tested at 4 K prior to mounting. The whole setup has then been calibrated, using an Agilent${}^{\textregistered}\,$ microwave generator EXG N5173B and an Agilent${}^{\textregistered}\,$ spectrum analyser MXA N9020A. The measurements presented in the core of the paper have been realized using a Zurich Instruments${}^{\textregistered}\,$ UHFLI lock-in detector operating in spectrum mode. The signal is mixed down with a Local Oscillator (LO) and detected at frequency $\pm\Omega_{m}+2~{}$MHz (the shift avoiding overlap of Stokes/Anti-Stokes signals). The generators used were from Agilent${}^{\textregistered}\,$ or Keysight${}^{\textregistered}\,$ brands leading to equivalent data quality. The absolute error in the calibrations is estimated at about $\pm 2~{}$dB over the whole set of realized runs; within these error bars, the two cryogenic platforms gave the same quantitative results.

Particular care has been taken in thermalization issues. The experimental cell is thus made of annealed high-purity copper. It is mounted on a cold finger either bolted onto the mixing chamber plate of the BF machine, or connected through silver wires to the bottom of the nuclear stage of the demag. cryostat. This stage is of laminar type, about 1 kg of high-quality copper paperYuriSatge . On the top it is connected through a Lancaster-made Al heat switch ULANCHSpaper to the mixing chamber of the home-made dilution unit. Both dilution cryostats reach base temperatures of order 7-10 mK depending on cooling power settings. The PCB board mounted inside the cell is hollow in its center; the chip is pressed there directly onto the copper block, by means of a copper/indium spring. At very large microwave powers, we do see heating at the intermediate stages where attenuators are anchored. However, the nuclear stage thermometers do not show any heating, even at the highest powers used.

Thermometry below typically 30 mK is no easy task; below 10 mK it requires the expertise of ultra-low temperature laboratories (see e.g. EMP partners EnssPickettEMP ). In the graphs of Fig. 7, the $x$ axis is as important as the $y$ one; this point is not always emphasized in the literature. We thus paid particular care in providing an almost primary temperature scale for our two microwave platforms. The use of the word almost here should be understood as follows: we also used resistive thermometers (RuO2, carbon Speer-type) calibrated against primary devices, and our primary thermometers (which do follow known temperature dependencies) are calibrated at a single point in temperature for practical reasons. Our laboratory has a long history of working for the construction of the ultra-low temperature scale RefHenriULTS . In our case, the BF cryostat has been equipped with a Magnicon${}^{\textregistered}\,$ MFFT noise thermometer and a CMN paramagnetic salt (Cerium Magnesium Nitrate) thermometer. The MFFT was bolted at mixing chamber level while the CMN was directly mounted on the cold finger. On the demag. cryostat, an MFFT is mounted at the top of the nuclear stage while a 3He thermometer is connected at the bottom; the exerimental cell is actually between the nuclear stage and this thermometer. When their working range overlap, the thermometers agree within typically 2 - 5 $~{}\%$; the error bars on $x$ axes of the temperature plots also take into account thermal gradients and slow drifts.

The nuclear adiabatic demagnetization process requires large magnetic fields to be used (here, 8 T). One usually starts at $T_{ini}\approx 10~{}$mK with the superconducting coil surrounding the stage fully magnetized at $B_{ini}$ (heat switch open), and then demagnetizes to about $B_{fin}\approx 100~{}$mT to reach the lowest possible temperatures (in the present runs, we worked down to 400$~{}\mu$K, Fig. 12). The cryostat can then stay cold over a week (heat leak $<100~{}$pW); another cycle needs then to be initiated with a pre-cooling of the system. The process being adiabatic, the nuclear spins of copper are cooled according to $B_{ini}/B_{fin}=T_{ini}/T_{fin}$ PobellBook . Our magnet is compensated on both sides, but nonetheless small stray fields are present when it is magnetized (represented in Fig. 11 by the small boxed coil). In order to make sure that these fields do not introduce any slant in the data, we performed measurements above 10 mK but within a demagnetization cycle: we started at 20 mK insead of 10 mK, and reduced the field by 1/2. This can then be compared to experiments performed without any field, at base temperature of the dilution unit. The microwave cavity does shift in frequency with applied field (see Appendix B), but no other difference could be found. All data taken below 10 mK have been obtained exclusively with the nuclear adiabatic demagnetization technique.

The lowest part of our temperature scale is obtained from a 3He thermometer (see Fig. 12). It is based on the viscosity measured in the fluid with an immersed probe, a mechanical resonator. In the past, vibrating wires were the best choice for this task GuenaultViW ; ClemensViW . Today, people use quartz tuning forks which are more practical Rob3Hefork ; ULANCFork ; ULANCFork2 . Our thermometer is thus a nested cell containing two tuning forks (one in the outside cell, the other in the inside), filled with silver sinters connected to the nuclear stage by silver wires. The outside cell serves as a thermal shield for the inner one, and does not cool down below typically 1 mK. The inner cell is directly connected to the cold finger that hosts the microwave cell.

3He is a Fermi liquid above about 1 mK, and becomes superfluid below. At $0~{}$bar pressure, $0~{}$T field this state is called 3He-B, and the properties of this amazing fluid have been extensively studied over the years Book3HeVollhardt . For instance, the viscosity of 3He is well known for both normal and superfluid states, see e.g. HookHall1 ; HookHall2 . In principle, from the knowledge of the geometry of the immersed object and its surface rugosity, one can calculate the friction and thus the broadening of the resonance. In practice, it is much more efficient to calibrate the device by scaling its properties on known measurements ULANCFork ; ULANCFork2 .

In order to perform this scaling, we measured the damping of the quartz tuning fork thermometer (inner cell) as a function of the final field of the demagnetization process $B_{fin}$. The protocol was to demagnetize by steps, and then remagnetize to $B_{ini}$ in order to verify that the process was indeed adiabatic (by recovering the initial temperature $T_{ini}$). It is then straightforward to calculate the temperature of the nuclear spins of copper for each step, which should be in equilibrium with electrons (and finally 3He) if one waits long enough. The result is shown in Fig. 12. The line is the expected behavior from published results. Note that the superfluid transition $T_{c}$ also acts as a fixed point in the temperature scale.

Appendix B Microwave optomechanics

The microwave cavity is the central element of the technique. It happens to be very sensitive to external conditions: cell temperature, pump settings (power, detuning), DC voltage bias and magnetic field. In order to avoid any systematic error, the protocol is thus to measure the cavity for each setting. On the BF cryostat, in order to mimic stray fields from the demagnetization protocol and to study the impact of (small) magnetic fields on the unstable features reported in Section V, a small coil has been installed on top of the cell (boxed element in Fig. 11). It can create perpendicular fields up to about 10$~{}\mu$T. The voltage DC bias is applied only to the cell which is placed within a Bias Tee and a DC Block (see boxed elements in Fig. 11). Above about 4 V, the chip starts to heat (certainly because of currents flowing within the silicon). All reported measurements obtained with a DC bias have thus been acquired at 3 V DC, ramping slowly periodically (every 1/2 hour approximately) the voltage to almost 4 V and back again. At zero bias, after some time the “spikes” eventually reappear. They did not seem to react particularly to the small magnetic fields used.

The cavity parameters are extracted from a transmission response measurement (amplitude of $S_{21}$ component of $\left[S\right]$ matrix). This is measured by applying a very weak probe tone (see boxed element in Fig. 11) while keeping the strong pump on. We verified that the probe power was small enough not to alter the measurement. The data (see Fig. 1) is fit to the expression RefS12fitAlessandro :

[TABLE]

with $I$ the pure imaginary, $Q_{tot}=\omega_{c}/\kappa_{tot}$ and $Q_{ext}=\omega_{c}/\kappa_{ext}$. $Q_{im}$ corresponds to an imaginary component of the external impedance, usually attributed to the inductive bonding wires (leading to the asymmetry of the fit in Fig. 1 center).

The fits are always of good quality. $\kappa_{ext}$ is very stable and reproducible, and does not vary by more than about $\pm 5~{}\%$. On the other hand $\kappa_{tot}=\kappa_{ext}+\kappa_{in}$ varies between typically $120~{}$kHz and $190~{}$kHz from one cool-down to the other. $\kappa_{in}$ represents the internal decay processes, taking into account all microscopic mechanisms. At low pump powers, $\kappa_{tot}$ happens to be worse (i.e. larger) than at high powers. This is true up to some maximum above which we start to substantially degrade the cavity’s properties; in terms of $n_{cav}$ this limit lies in the range of $10^{8}-10^{9}$ photons. In the range studied the temperature dependence of $\kappa_{tot}$ is rather small. It is immune to the magnetic fields and voltages applied within the parameter range studied. However, the resonance position of the cavity is very sensitive to all parameters: so the main issue of the fit is to define $\omega_{c}$ as accurately as possible.

The properties of the cavity are too complex to allow an analysis of the type of Fig. 6 (performed for the NEMS), which could tell us by how much the microwave mode (and/or the chip) heats at a given pump power. We therefore measured directly the thermal population of the cavity with respect to $P_{in}$. We present these data in Fig. 13 as a function of $\sqrt{n_{cav}}$: on pure phenomenological grounds, the dependence seems then to be linear (see guides for the eyes in Fig. 13). We could not perform this measurement below typically 400 mK because the signal was too weak. The power-dependence is rather different from Fig. 6 and does not seem to be a true heating of the chip itself; it has thus to be noise fed into the cavity by the pump generator. Indeed, similar features but with different levels of cavity populations have been measured using different brands of generators. The relevant outcome of this graph is the extra (out-of-equilibrium) cavity population induced by the strong pump; this is the number quoted in magenta in Fig. 13 for each temperature, at the largest powers displayed. This number is obtained by dividing the photon flux by $\kappa_{ext}/2$ (bi-directional coupling). Injecting it in the theoretical expressions AKMreview , we find out that this effect remains always negligible with respect to other problems creating the finite error bars of Fig. 7.

From the careful measurement of the cavity at each point, the knowledge of the heating effects in Figs. 6 and 13 we can guarantee that the experiment is performed in the best possible conditions. To be then quantitative, we finally need to be able to convert photon population in the Stokes/Anti-Stokes peaks into phonon populations. To do so, we need to know the parameter $\cal M$ introduced in Eq. (1). In principle, one could calculate it from theory, knowing all experimental details of the setup AKMreview . However, since all of these parameters are known within some experimental error, the final value obtained for $\cal M$ would be of poor precision. It is thus much more efficient to calibrate it at a given temperature: this is performed in Fig. 14, at sufficiently high $T_{cryo}$ to guarantee a good thermal coupling and enough resolution in signal (210 mK). In this plot we show the signal (area) obtained for the 3 schemes presented in Fig. 2, as a function of $P_{in}$ (they thus all overlap at low powers). The lines are fit to theory Eqs. (3,4) AKMreview , with the dashed one taking into account the heating produced by the large $n_{cav}$ reached with the “in-cavity” scheme. We thus obtain ${\cal M}\approx 1.8\times 10^{12}~{}$photons/s/W/K for our device. Note that at the highest powers used, in the red-detuned pumping scheme the mode cooled from $210~{}$mK down to approximately $20~{}$mK.

Our aim was to demonstrate the capabilities of the method for temperatures as low as possible and a coupling $g_{0}$ which is not especially large. We therefore relied on the out-of-plane flexure of our NEMS beam; in-plane and out-of-plane modes were both characterized first at room temperature using optics ArcizetThanks . The out-of-plane flexural mode was found at 3.660 MHz, shifted about 280 kHz from the in-plane one. Both had a quality factor of about $6\,500$. At low temperatures, the method then relies on the blue-detuned pump instability: below the threshold it enables amplification of the weak signal, and the position of the threshold itself gives access to the parameter $\gamma_{m}$ needed for quantitative fits. It is thus in principle not necessary to be able to resolve the signal at low powers; in particular, in these conditions sideband-assymetry thermometry would be impractical.

The definition of $\gamma_{m}$ from the threshold position is illustrated in Fig. 15. Ramping up the power, from Eq. (3) we can recalculate the damping rate from the position of the threshold $P_{thr}$ (full symbols and full line). Furthermore, it happens that this threshold towards self-sustained oscillations is hysteretic (Fig. 15 empty symbols). This effect is not yet modeled, but it clearly arises from non-linearities in the system. For microwave optomechanics, this are genuine geometrical nonlinearites, as opposed to thermal nonlinearities seen in optics faveroOscill . The temperature-dependence of the down-sweep threshold (large amplitude motion) is even stronger than for the up-sweep one (small amplitude). Again, no thermal decoupling can be seen down to 10 mK; the heating effect (Fig. 6) has been taken into account, and has only a small impact on the data.