Force sensitivity of multilayer graphene optomechanical devices

Mechanical resonators based on low-dimensional materials are promising for force and mass sensing experiments. The force sensitivity in these ultra-light resonators is often limited by the imprecision in the measurement of the vibrations, the fluctuations of the mechanical resonant frequency and the heating induced by the measurement. Here, we strongly couple multilayer graphene resonators to superconducting cavities in order to achieve a displacement sensitivity of 1.3 fm Hz−1/2. This coupling also allows us to damp the resonator to an average phonon occupation of 7.2. Our best force sensitivity, 390 zN Hz−1/2 with a bandwidth of 200 Hz, is achieved by balancing measurement imprecision, optomechanical damping, and measurement-induced heating. Our results hold promise for studying the quantum capacitance of graphene, its magnetization, and the electron and nuclear spins of molecules adsorbed on its surface.

A powerful method to obtain efficient electrical readout of small resonators is to amplify the interaction between mechanical vibrations and the readout field using a superconducting microwave cavity [27][28][29] . Increasing the field in the cavity improves the readout sensitivity and eventually leads to dynamical back-action on the thermo-mechanical noise. This effect has been studied intensively on comparatively large micro-fabricated resonators, resulting for instance in enhanced optomechanical damping 38,39 , ground-state cooling of mechanical vibrations 40,41 and displacement imprecision below the standard quantum limit 42,43 . Another phenomenon often observed when detecting and manipulating the motion of mechanical resonators is the induced heating that can occur through Joule dissipation and optical adsorption 28,44 . Heating is especially prominent in tiny mechanical resonators because of their small heat capacity. An additional difficulty in characterizing mechanical vibrations is related to the fluctuations of the mechanical resonant frequency, also called frequency noise, which are particularly sizable in small resonators endowed with high-quality factors Q 10 .
Here we study the force sensitivity of multilayer graphene mechanical resonators coupled to superconducting cavities. In particular, we quantify how the force sensitivity is affected by dynamical back-action, Joule heating and frequency noise upon increasing the number of pump photons inside the cavity. We demonstrate a force sensitivity of S tot F À Á 1=2 ¼ 390 AE 30 zN Hz À 1=2 , of which E50% arises from thermo-mechanical noise and E50% from measurement imprecision. The force sensitivity tends to be limited by measurement imprecision and frequency noise at low pump power, and by optomechanical damping and Joule heating at high pump power.

Results
Thermal force noise and imprecision force noise. A fundamental limit of force sensing is set by the thermo-mechanical noise of the eigenmode that is measured. According to the fluctuation-dissipation theorem, the associated thermal force noise is white and is quantified by where T mode is the temperature of the mechanical eigenmode, and m eff is its effective mass 8,45 . This force noise is transduced into a mechanical resonance with line width G spectral eff and height S mode z in the displacement spectrum (Fig. 1). Importantly, equation (1) shows that the low mass of graphene decreases the size of the thermo-mechanical force noise. However, a drawback of tiny resonators with high Q-factors is their tendency to feature sizable frequency noise that broadens the resonance and, therefore, increases the size of the force noise 10,46 .
Measuring mechanical vibrations with high accuracy is key to resolving small forces, since the imprecision in the measurement contributes to the force sensitivity. The force sensitivity S tot F is given by the sum of the thermal force noise S mode F and the imprecision force noise S imp F , where the latter is the result of the white noise background with strength S imp z in the displacement spectrum (Fig. 1a). The challenge with mechanical resonators based on low-dimensional systems is to reach the limit S imp F oS mode F . When detecting the motion of graphene resonators with microwave cavities, one typically operates in the resolved sideband limit [27][28][29] , where the cavity decay rate k is significantly smaller than the mechanical resonance frequency o m . This is interesting for force sensing, because pumping on the red sideband allows to enhance the mechanical damping rate by G opt , and therefore to reduce the harmful effect of frequency noise, as we will discuss below. In addition, this allows to increase the measurement bandwidth, as is often done in magnetic resonance force microscopy experiments 33 while keeping S mode F constant. The drawback of red sideband pumping compared to pumping at the cavity resonant frequency is an increased imprecision force noise at high pump powers. In the red-detuned pump regime,  (1) and (2), respectively. The quantum back-action noise is neglected for simplicity. For the plots most of the parameters are those of device B, but we estimate the mass assuming that the graphene flake is a single layer. Further we choose n add ¼ 0.5, T bath ¼ 0.015 K, and n p ¼ 2 Á 10 5 in a (see text).

Red-detuned pump
the measurement imprecision contributes to the force sensitivity by the amount with k ext the external coupling rate of the cavity, n add the noise added by the amplifier chain at the output of the device, G m spectral the intrinsic line width of the resonator, n p the number of pump photons in the cavity, and g 0 the single-photon optomechanical coupling. Figure 1b shows the pump power dependence of the force sensitivity S tot F expected in the absence of Joule heating and frequency noise. The  increase of S tot F at high n p is due to the dynamical back-action, which enhances the mechanical line width by G opt ¼ 4n p g 0 2 /k.
Device characterization. Our devices consist of a suspended graphene mechanical resonator capacitively coupled to a superconducting niobium (Nb) cavity ( Fig. 2a-c). The graphene resonators are circular with a radius of RE1.6 mm. Here we present data of two devices. The graphene resonator of device A has a thickness of approximately 25 layers, and the one of device B 5-6 layers. This corresponds respectively to an effective mass of m eff ¼ (4.1 ± 0.8) Á 10 À 17 kg and (9.6 ± 0.8) Á 10 À 18 kg. The uncertainty results from extracting the mass with different methods including optical contrast measurements, thickness measurements with atomic force microscopy and the measured electrostatic softening of the mechanical resonators (see Supplementary Note 1 and Supplementary Equation 2). The fundamental mode of devices A and B vibrates at o m /2p ¼ 67 Here V g is the constant voltage applied between the graphene flake and the superconducting cavity. In order to improve the attachment of the graphene flake to its support, we clamp it between crosslinked poly(methyl metracylate) and the contact electrodes; the detailed fabrication is described elsewhere 29 . The separation between the graphene resonator and the cavity counter electrode at V g ¼ 0 V is assumed to be equal to the hole depth, which is typically d 0 E85 nm in our devices as measured with atomic force microscopy. Varying V g allows us to tune the separation between the graphene resonator and the cavity counter electrode 24,29,[47][48][49] , modifying the graphene-cavity capacitance, the cavity frequency o c and o m (Fig. 2g,h). The superconducting cavity is a coplanar waveguide resonating at about o c /2p ¼ 7.4 GHz. We choose a single-port, quarter wavelength, reflection geometry, so that the cavity is connected to ground on one end, allowing to apply a well-defined constant voltage between the cavity and the graphene flake. The other end of the cavity is coupled to a transmission line via a capacitor C ext with a coupling rate k ext ¼ 2p Â 850 kHz for device A; the total cavity decay rate is Here k int accounts for the internal energy loss. We detect the vibrations of the graphene resonator with high precision by pumping the cavity with an electromagnetic field, and probing its mechanical sideband. This sideband is generated by the capacitive modulation of the pump field at frequency o p /2p by the graphene vibrations at o m /2p. We usually set o p ¼ o c À o m and probe the electromagnetic field that exits the cavity at o c . We measure the device at the cryostat base temperature of 15 mK if not stated otherwise. The cavity output field is amplified with a high electron-mobility-transistor mounted at the 3 K stage of the cryostat. Mechanical noise spectra are detected with a spectrum analyser at room temperature. For a detailed description of the measurement setup, see Supplementary Fig. 1 and Supplementary Note 2. In addition, we perform ring-down measurements to determine the mechanical dissipation rate G decay eff of the graphene resonator. Spectral measurements are not suitable for quantifying reliably G decay eff because of the potentially substantial frequency noise of graphene resonators.
We characterize the single-photon optomechanical coupling and show that the coupling can be significantly enhanced by deflecting the membrane towards the cavity electrode. For this, we quantify the optomechanical scattering rate G opt using ring-down measurements at V g ¼ 0 V and V g ¼ 3.002 V for device A. Figure 3a,b shows the measured dissipation rate G decay eff as a function of cavity pump photon number n p for blue and red detuned pumping. The measurements are well described by G decay eff ¼ G decay m AE G opt where G decay m corresponds to the intrinsic mechanical dissipation rate, and ± to red and blue detuned pumping at o p ¼ o c 8o m , respectively. By increasing V g from 0 to 3.002 V we obtain a strong increase of the optomechanical coupling from g 0 ¼ 2p Â 9.7 Hz to g 0 ¼ 2p Â 42.6 Hz. We estimate that the separation d between the membrane and the cavity counter electrode is reduced from 88 to 33 nm when varying V g from 0 to 3.002 V. The calibration of both g 0 and n p is robust, while the quantification of the reduction of d is approximative; AE G opt (red and blue lines) using g 0 /2p ¼ 9.7 Hz in a and g 0 /2p ¼ 42.6 Hz in b. The inset in b shows a ring-down measurement for n p ¼ 1.4 Á 10 6 . We plot the normalized vibration amplitude as a function of time t. The resonator is driven with a capacitive driving force for tot 0 . At t 0 the drive is switched off and the vibration amplitude decays freely (t4t 0 ). We fit the data with an exponential decay (black line) using z 2 ðtÞ ¼ z 2 max expð À t À t0 t Þ with a decay rate G decay eff ¼ 1=t ¼ 2p Á 8:4 kHz. The vibration amplitude in ring-down measurements is larger than that in undriven displacement spectra, so that the motion in ring-down measurements can be resolved with lower n p . Thermal calibration and sideband cooling. In order to calibrate the mechanical phonon occupation and the mode temperature T mode , we measure the mechanical thermal motion spectrum while varying the cryostat temperature 40 . This is done by pumping the cavity with a weak pump tone on the red sideband. The integrated area of the thermal resonance is proportional to the mode temperature according to the equipartition theorem. For temperatures above 100 mK the area is linearly proportional to the cryostat temperature, showing that the mode is in thermal equilibrium with the cryostat (Fig. 4b). This linear dependence serves as a precise calibration to relate the resonance area to the averaged phonon occupation n m and the mode temperature T mode . Below 100 mK the mechanical mode does not thermalize well with the cryostat. The origin of this poor thermalization at low temperature may be related to the heating induced by the pump field (see below) 28 , and a non-thermal force noise 50 such as the electrostatic force noise related to the voltage noise in the device. As a next characterization step, we investigate the mechanical phonon occupation when increasing the power of the pump tone on the red sideband and keeping the temperature of the cryostat constant at T cryo ¼ 15 mK. The measured resonance gets broader and its area smaller (Fig. 4c), showing that the mechanical mode is damped and cooled 38,39 . At the largest available pump power, the phonon occupation reaches n m ¼ 7.2 ± 0.2 (Fig. 4e). This is the lowest phonon occupation reached in a mechanical resonator based on graphene 27,28,51 . The error in the estimation of n m is given by the standard error obtained from five successive spectral measurements.
Displacement sensitivity and force sensitivity. The improved coupling allows us to achieve also an excellent displacement sensitivity S imp z (Fig. 4d). At the largest pump power, we obtain ðS imp F Þ 1=2 ¼ 1:3 AE 0:2 fm Hz À 1=2 , which compares favourably to previous works 27,51,52 . The error in S imp z is given by the uncertainty in the estimation of m eff . We obtain S imp z from the noise floor of the measured power spectral the zero-point motion amplitude 27 . The displacement sensitivity scales as 1/n p (Fig. 4d). By comparing the measurement to the expected displacement sensitivity we obtain that the equivalent noise added by the amplifier chain is n add ¼ 32. This is a reasonable value for a high electronmobility-transistor amplifier mounted at 3 K 42,53 . We now quantify the force sensitivity as a function of the microwave pump power (Fig. 5a,e). Since the mechanical resonances in the measured displacement spectra are well described by Lorentzian line shapes, the thermal force noise is quantified using S mode The best force sensitivity we achieve for device A is S tot F À Á 1=2 ¼ 5:8 aN Hz À 1=2 with a mechanical bandwidth of 20 kHz (Fig. 5a,d). In device B we reach a force sensitivity of S tot F À Á 1=2 ¼ 390 AE 30 zN Hz À 1=2 with a mechanical bandwidth of 0.2 kHz (see Fig. 5e,h). The error in the estimation of the force sensitivity is obtained from both the uncertainty in the mass and the fluctuations in the measurement of S tot F , which we evaluate by calculating the standard error of 10 measurements. This force sensitivity compares favourably with the best sensitivities obtained with micro-fabricated resonators ( S tot F À Á 1=2 ¼ 510 zN Hz À 1=2 ) 42,45 , albeit it is not as good as that of resonators based on carbon nanotubes 8,10 . Compared to previous devices, the mechanical bandwidth of graphene resonators is much higher, which enables faster detection of sudden force changes.

Discussion
We plot both S mode F and S imp F as a function of cavity pump photon population in Fig. 5b. As expected, the imprecision force noise decreases at low n p and increases at high n p due to the enhanced damping caused by the optomechanical back-action. The thermal force noise S mode F appears roughly constant when varying n p as a result of the competing effects of Joule heating and frequency noise. Joule heating is caused by the microwave current in the graphene flake induced by the pump field. This results in the increase of the temperature T bath of the thermal bath coupled to the mechanical mode as well as the mechanical dissipation rate 26,28 . We can infer the product T bath Á G decay m from the measurements of n m and G decay eff in Figs 3b and 4e using When increasing the pump power, Joule heating significantly increases the product T bath G decay m (Fig. 5c), and therefore the size of the thermal force noise (equation (1)). We see next that the effect of frequency noise leads to the opposite dependence of the thermal force noise on pump power. Frequency noise enhances the spectral line width by the amount dG noise , when the fluctuations of the resonant frequency are described by a white noise 8 . The measurements of G spectral eff and G decay eff as a function of pump power can be well described by equation (5) with dG noise /2p ¼ 8.7 kHz (Fig. 5d). Importantly, Fig. 5d shows that G spectral eff is comparable to G decay eff at large pump power, showing that the relative contribution of dG noise to G spectral eff gets negligible upon increasing n p . As the cooling efficiency described by equation (4)  frequency noise (see chapter 7 in 54 ), the thermal force noise is quantified by Taking into account the measured effects of Joule heating and frequency noise in equation (6), the thermal force noise S mode F is expected to remain roughly constant as a function of n p (dark yellow line in Fig. 5b), in agreement with the measurements. Overall, the best force sensitivity we achieve in this device is S tot F À Á 1=2 ¼ 5:8 aN Hz À 1=2 at n p E4 Á 10 6 (Fig. 5a). While the force sensitivity in this device is primarily limited by the measurement imprecision, the thermal force noise is affected to a large extent by frequency noise at low n p and by Joule heating at high n p .
In device B, the graphene resonator has a lower mass and a narrower mechanical line width, two assets for high force sensitivity ( Fig. 5e-h). The spectral line width corresponds to a mechanical quality factor of QE200,000. In this device we reach a force sensitivity of S tot F À Á 1=2 ¼ 390 AE 30 zN Hz À 1=2 at n p E4 Á 10 5 (see Fig. 5e). In an attempt to improve the thermal anchoring of device B compared to device A, the graphene contact electrodes contain an additional Au layer between the graphene and the Nb layer 28,55 . The normal metal layer is expected to increase the thermal conductance between the graphene flake and the contact electrodes through electron diffusion, which allows for better heat dissipation into the contacts. However, device B is still strongly affected by Joule heating, which substantially increases the value of S mode F when increasing the pump power (Fig. 5f,g). The heating is so strong that we are not able to reduce the phonon occupation n m with sideband cooling. We attribute the strong heating to the fact that the resonator is significantly thinner than the one of device A and therefore has a smaller heat capacity. The effect of frequency noise on the spectral line width is negligible for pump powers above n p E4 Á 10 5 . We do not know the origin of the frequency noise but it might be related to charged two-level fluctuators in the device. The force sensitivity is here primarily limited by the measurement imprecision at low n p , and by the thermomechanical force noise and Joule heating at high n p .
In the future, the force sensitivity of graphene optomechanical devices can be further improved using a quantum-limited Josephson parametric amplifier 53 . This readout will improve the measurement imprecision, by lowering n add in S imp F . In addition, it will be possible to resolve the thermal vibrations with lower pump power, which is crucial to reduce Joule heating while working with low-mass graphene resonators. A quantum-limited amplifier with n add ¼ 0.5 may allow to achieve 47 zN Hz À 1/2 force sensitivity at 15 mK taking the mass of a single-layer graphene resonator with the diameter and the quality factor of device B (Fig. 1b). With only modest device improvements, it may be possible to probe the fundamental limit of continuous displacement detection imposed by quantum mechanics, since the force noise associated to quantum backaction ðS qba F Þ 1=2 ¼ ð2' o m m eff G decay eff Þ 1=2 ¼ 1:1 aN Hz À 1=2 is approaching S mode F À Á 1=2 ¼ 4:3 aN Hz À 1=2 measured at n p ¼ 1.4 Â 10 7 for device A. Force sensing with resonators based on two-dimensional materials hold promise for detecting electron and nuclear spins 33 using superconducting cavities compatible with relatively large magnetic fields 56 , and studying the thermodynamic properties of twodimensional materials, such as the quantum capacitance and the magnetization 35 .

Methods
Cavity characterization. In Fig. 2d,e we plot the coefficient |S 11 | 2 and the phase of the reflected signal when sweeping the frequency over the cavity resonance at o c /2p ¼ 7.416 GHz. To extract the external coupling rate k ext and the internal loss rate k int we fit the measurement with the line shape expected for a one-port reflection cavity 57 which yields k int /2p ¼ 950 kHz and k ext /2p ¼ 850 kHz at V g ¼ 3.002 V for device A.