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 fm Hz. This coupling also allows us to damp the resonator to an average phonon occupation of . Our best force sensitivity, zN Hz with a bandwidth of 200 Hz, is achieved by balancing measurement imprecision, optomechanical damping, and 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.
Considerable effort has been devoted to developing mechanical resonators based on low-dimensional materials, such as carbon nanotubes Sazonova et al. (2004); Jensen et al. (2008); Chiu et al. (2008); Lassagne et al. (2009); Steele et al. (2009); Gouttenoire et al. (2010); Chaste et al. (2012); Moser et al. (2013); Ganzhorn et al. (2013); Moser et al. (2014); Benyamini et al. (2014); Häkkinen et al. (2015), semiconducting nanowires Ayari et al. (2007); Gil-Santos et al. (2010); Arcizet et al. (2011); Nichol et al. (2012, 2013); Sansa et al. (2014); Gloppe et al. (2014); Montinaro et al. (2014); Mathew et al. (2015); Nigues et al. (2015), graphene Bunch et al. (2007); Chen et al. (2009); Eichler et al. (2011); Miao et al. (2014); Singh et al. (2014); Song et al. (2014); Weber et al. (2014), and monolayer semiconductors Lee et al. (2013); van Leeuwen et al. (2014); Wang et al. (2014). The specificity of these resonators is their small size and their ultra-low mass, which enables sensing of force and mass with unprecedented sensitivities Moser et al. (2014); Chaste et al. (2012). Such high-precision sensing capabilities hold promise for studying physical phenomena in new regimes that have not been explored thus far, for instance, in spin physics Rugar et al. (2004), quantum electron transport Bleszynski-Jayich et al. (2009); Chen et al. (2016), light-matter interaction Gloppe et al. (2014) and surface science Wang et al. (2010); Tavernarakis et al. (2014). However, the transduction of the mechanical vibrations of nanoscale mechanical systems into a measurable electrical or optical output signal is challenging. As a result, force and mass sensing is often limited by the imprecision in the measurement of the vibrations, and cannot reach the fundamental limit imposed by thermo-mechanical noise.
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 Song et al. (2014); Weber et al. (2014); Singh et al. (2014). 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 Arcizet et al. (2006); Gigan et al. (2006), ground-state cooling of mechanical vibrations Teufel et al. (2011); Chan et al. (2011), and displacement imprecision below the standard quantum limit Teufel et al. (2009); Anetsberger et al. (2010). 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 Song et al. (2014); Meenehan et al. (2014). 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 Moser et al. (2014).
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 zN Hz, of which arises from thermo-mechanical noise and 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.
ii.1 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 is the temperature of the mechanical eigenmode, and is its effective mass Mamin and Rugar (2001); Moser et al. (2013). This force noise is transduced into a mechanical resonance with line width and height in the displacement spectrum (Fig. 1). Importantly, Eq. 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 -factors is their tendency to feature sizable frequency noise that broadens the resonance and, therefore, increases the size of the force noise Moser et al. (2014); Zhang et al. (2014).
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 is given by the sum of the thermal force noise and the imprecision force noise , where the latter is the result of the white noise background with strength in the displacement spectrum (Fig. 1a). The challenge with mechanical resonators based on low-dimensional systems is to reach the limit . When detecting the motion of graphene resonators with microwave cavities, one typically operates in the resolved sideband limit Weber et al. (2014); Song et al. (2014); Singh et al. (2014), where the cavity decay rate is significantly smaller than the mechanical resonance frequency . This is interesting for force sensing, because pumping on the red sideband allows to enhance the mechanical damping rate by , 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 Rugar et al. (2004) while keeping 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, the measurement imprecision contributes to the force sensitivity by the amount
with the external coupling rate of the cavity, the noise added by the amplifier chain at the output of the device, the intrinsic line width of the resonator, the number of pump photons in the cavity, and the single-photon optomechanical coupling. Figure 1b shows the pump power dependence of the force sensitivity expected in the absence of Joule heating and frequency noise. The increase of at high is due to the dynamical back-action, which enhances the mechanical line width by .
ii.2 Device characterization
Our devices consist of a suspended graphene mechanical resonator capacitively coupled to a superconducting niobium cavity (Fig. 2a-c). The graphene resonators are circular with a radius of . Here we present data of 2 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 kg and kg. The uncertainty results from extracting the mass with different methods including optical contrast measurements, thickness measurements with atomic force microscopy (AFM) 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 MHz and MHz at V, respectively. Here 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 cross-linked poly(methyl metracylate)(PMMA) and the contact electrodes; the detailed fabrication is described elsewhere Weber et al. (2014). The separation between the graphene resonator and the cavity counter electrode at V is assumed to be equal to the hole depth, which is typically nm in our devices as measured with AFM. Varying allows us to tune the separation between the graphene resonator and the cavity counter electrode Chen et al. (2009); Singh et al. (2010); Bao et al. (2012); Chen et al. (2013); Weber et al. (2014), modifying the graphene-cavity capacitance, the cavity frequency , and (Figs. 2g,h). The superconducting cavity is a coplanar waveguide resonating at about 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 with a coupling rate kHz for device A; the total cavity decay rate is MHz (see Methods). Here 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 by the graphene vibrations at . We usually set and probe the electromagnetic field that exits the cavity at . We measure the device at the cryostat base temperature of mK if not stated otherwise. The cavity output field is amplified with a high electron-mobility-transistor (HEMT) mounted at the 3 K stage of the cryostat. Mechanical noise spectra are detected with a spectrum analyzer 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 of the graphene resonator. Spectral measurements are not suitable for quantifying reliably 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 using ring-down measurements at V and V for device A. Figures 3a,b show the measured dissipation rate as a function of cavity pump photon number for blue and red detuned pumping. The measurements are well described by where corresponds to the intrinsic mechanical dissipation rate, and to red and blue detuned pumping at , respectively. By increasing from to V we obtain a strong increase of the optomechanical coupling from Hz to Hz. We estimate that the separation between the membrane and the cavity counter electrode is reduced from nm to nm when varying from to V. The calibration of both and is robust, while the quantification of the reduction of is approximative; see Supplementary Notes 1 and 3, Supplementary Fig. 2 and Supplementary Equations 1, 3-5.
ii.3 Thermal calibration and sideband cooling
In order to calibrate the mechanical phonon occupation and the mode temperature , we measure the mechanical thermal motion spectrum while varying the cryostat temperature Teufel et al. (2011). 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 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 and the mode temperature . Below 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) Song et al. (2014), and a non-thermal force noise Rocheleau et al. (2010) 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 mK. The measured resonance gets broader and its area smaller (Fig. 4c), showing that the mechanical mode is damped and cooled Arcizet et al. (2006); Gigan et al. (2006). At the largest available pump power, the phonon occupation reaches (Fig. 4e). This is the lowest phonon occupation reached in a mechanical resonator based on graphene Song et al. (2014); Barton et al. (2012); Singh et al. (2014). The error in the estimation of is given by the standard error obtained from 5 successive spectral measurements.
ii.4 Displacement sensitivity and force sensitivity
The improved coupling allows us to achieve also an excellent
displacement sensitivity (Fig. 4d). At the largest pump power, we obtain fm Hz, which compares favorably to previous works Barton et al. (2012); Singh et al. (2014); Cole et al. (2015). The
error in is given by the uncertainty in the
estimation of . We obtain from the
noise floor of the measured power spectral density
using with the zero-point motion amplitude Singh et al. (2014). The displacement sensitivity scales as (Fig. 4d) . By comparing the measurement to the expected displacement sensitivity
We now quantify the force sensitivity as a function of the microwave pump power (Figs. 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 with the effective mechanical susceptibility . Similarly, we obtain the imprecision force noise with . The best force sensitivity we achieve for device A is aN Hz with a mechanical bandwidth of kHz (Fig. 5a,d). In device B we reach a force sensitivity of zN Hz with a mechanical bandwidth of kHz (see Figs. 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 , which we evaluate by calculating the standard error of 10 measurements. This force sensitivity compares favorably with the best sensitivities obtained with micro-fabricated resonators (zN Hz) Mamin and Rugar (2001); Teufel et al. (2009), albeit it is not as good as that of resonators based on carbon nanotubes Moser et al. (2013, 2014). Compared to previous devices, the mechanical bandwidth of graphene resonators is much higher, which enables faster detection of sudden force changes.
We plot both and as a function of cavity pump photon population in Fig. 5b. As expected, the imprecision force noise decreases at low and increases at high due to the enhanced damping caused by the optomechanical back-action. The thermal force noise appears roughly constant when varying 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 of the thermal bath coupled to the mechanical mode as well as the mechanical dissipation rate Song et al. (2014); Miao et al. (2014). We can infer the product from the measurements of and in Figs. 3b, 4e using
When increasing the pump power, Joule heating significantly increases the product (Fig. 5c), and therefore the size of the thermal force noise (Eq. 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 ,
when the fluctuations of the resonant frequency are described by a white noise Moser et al. (2013). The measurements of and as a function of pump power can be well described by Eq. 5 with kHz (Fig. 5d). Importantly, Fig. 5d shows that is comparable to at large pump power, showing that the relative contribution of to gets negligible upon increasing . As the cooling efficiency described by Eq. 4 remains unaltered by frequency noise (see chapter 7 in Dykman (2012)), the thermal force noise is quantified by
Taking into account the measured effects of Joule heating and frequency noise in Eq. 6, the thermal force noise is expected to remain roughly constant as a function of (dark yellow line in Fig. 5b), in agreement with the measurements. Overall, the best force sensitivity we achieve in this device is aN Hz at (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 and by Joule heating at high .
In device B, the graphene resonator has a lower mass and a narrower mechanical line width, two assets for high force sensitivity (Figs. 5e-h). The spectral line width corresponds to a mechanical quality factor of . In this device we reach a force sensitivity of zN Hz at (see Figs. 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 Fong et al. (2013); Song et al. (2014). 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 when increasing the pump power (Figs. 5f,g). The heating is so strong that we are not able to reduce the phonon occupation 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 . 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 , and by the thermo-mechanical force noise and Joule heating at high .
In the future, the force sensitivity of graphene optomechanical devices can be further improved using a quantum-limited Josephson parametric amplifier Castellanos-Beltran et al. (2008). This readout will improve the measurement imprecision, by lowering in . 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 may allow to achieve zN Hz 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 aN Hz is approaching aN Hz measured at for device A. Force sensing with resonators based on two-dimensional materials hold promise for detecting electron and nuclear spins Rugar et al. (2004) using superconducting cavities compatible with relatively large magnetic fields Samkharadze et al. (2016), and studying the thermodynamic properties of two-dimensional materials, such as the quantum capacitance and the magnetization Chen et al. (2016).
iv.1 Cavity characterization
In Figs. 2d,e we plot the coefficient and the phase of the reflected signal when sweeping the frequency over the cavity resonance at GHz. To extract the external coupling rate and the internal loss rate we fit the measurement with the line shape expected for a one-port reflection cavity Aspelmeyer
et al. (2014)
which yields kHz and kHz at V for device A. The rates of Device B are kHz and kHz at V.
Appendix A Acknowledgements
We thank P. Verlot and M. Dykman for discussions. We acknowledge financial support by the ERC starting grant 279278 (CarbonNEMS), the EE Graphene Flagship (contact no. 604391), from MINECO and the Fondo Europeo de Desarrollo Regional (FEDER) through grant MAT2012-31338 and FIS2015-69831-P, the Fundació Privada Cellex, the Severo Ochoa Excellence Grant, and the Generalitat through AGAUR.
Appendix B Author contributions
P.W. fabricated the devices, the process being developed by P.W. and J.G. P.W., J.G. and A.N. carried out the experiment with support from J.V.C. The data analysis was done by P.W. and J.G. with inputs from A.B. The experimental setup was built by J.G. with support from P.W. P.W. and A.B. wrote the manuscript with comments from J.G. and A.N. A.B. and J.G. conceived the experiment and supervised the work.
Competing financial interests: The authors declare no competing financial interests.
- Corresponding author. E-mail: firstname.lastname@example.org.
- Sazonova, V., Yaish, Y., Ustunel, H., Roundy, D., Arias, T. A., and McEuen, P. L. A tunable carbon nanotube electromechanical oscillator, Nature 431, 284-287 (2004).
- Jensen, K., Kim, K., and Zettl, A. An atomic-resolution nanomechanical mass sensor, Nat. Nanotech. 3, 533-537 (2008).
- Chiu, H. Y., Hung, P., Postma, H. W. C., and Bockrath, M. Atomic-scale mass sensing using carbon nanotube resonators, Nano Letters 8, 4342-4346 (2008).
- Lassagne, B., Tarakanov, Y., Kinaret, J., Garcia-Sanchez, D., and Bachtold, A. Coupling mechanics to charge transport in carbon nanotube mechanical resonators, Science 325, 1107-1110 (2009).
- Steele, G. A. et al. Strong coupling between single-electron tunneling and nanomechanical motion, Science 325, 1103-1107 (2009).
- Gouttenoire, V. et al. Digital and FM demodulation of a doubly clamped single-walled carbon-nanotube oscillator: Towards a nanotube cell phone, Small 6, 1060-1065 (2010).
- Chaste, J., Eichler, A., Moser, J., Ceballos, G., Rurali, R., and Bachtold, A. A nanomechanical mass sensor with yoctogram resolution, Nat. Nanotech. 7, 301-304 (2012).
- Moser, J. et al. Ultrasensitive force detection with a nanotube mechanical resonator, Nat. Nanotech. 8, 493-496 (2013).
- Ganzhorn, M., Klyatskaya, S., Ruben, M., and Wernsdorfer, W. Strong spin-phonon coupling between a single-molecule magnet and a carbon nanotube nanoelectromechanical system, Nat. Nanotech. 8, 165-169 (2013).
- Moser, J., Eichler, A., Güttinger, J., Dykman, M. I., and Bachtold, A. Nanotube mechanical resonators with quality factors of up to 5 million, Nat. Nanotech. 9, 1007-1011 (2014).
- Benyamini, A., Hamo, A., Kusminskiy, S. V., von Oppen, F., and Ilani, S. Real-space tailoring of the electron-phonon coupling in ultraclean nanotube mechanical resonators, Nat. Phys. 10, 151-156 (2014).
- Häkkinen, P., Isacsson, A., Savin, A., Sulkko, J., and Hakonen, P. Charge sensitivity enhancement via mechanical oscillation in suspended carbon nanotube devices, Nano Letters 15, 1667-1672 (2015).
- Ayari, A. et al. Self-oscillations in field emission nanowire mechanical resonators:â A nanometric dcac conversion, Nano Letters 7, 2252-2257 (2007).
- Gil-Santos, E. et al. Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires, Nat. Nanotech. 5, 641-645 (2010).
- Arcizet, O., Jacques, V., Siria, A., Poncharal, P., Vincent, P., and Seidelin, S. A single nitrogen-vacancy defect coupled to a nanomechanical oscillator, Nat. Phys. 7, 879-883 (2011).
- Nichol, J. M., Hemesath, E. R., Lauhon, L. J., and Budakian, R. Nanomechanical detection of nuclear magnetic resonance using a silicon nanowire oscillator, Phys. Rev. B 85, 054414 (2012).
- Nichol, J. M., Naibert, T. R., Hemesath, E. R., Lauhon, L. J., and Budakian, R. Nanoscale fourier-transform magnetic resonance imaging, Phys. Rev. X 3, 031016 (2013).
- Sansa, M., Fernández-Regúlez, M., Llobet, L., San Paulo, A., and Pérez-Murano, F. High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators, Nat. Commun. 5, 4313, (2014).
- Gloppe, A. et al. Bidimensional nano-optomechanics and topological backaction in a non-conservative radiation force field, Nat. Nanotech. 9, 920-926 (2014).
- Montinaro, M. et al. Quantum dot opto-mechanics in a fully self-assembled nanowire, Nano Letters 14, 4454-4460 (2014).
- Mathew, J. P., Patel, R., Borah, A., Maliakkal, C. B., Abhilash, T. S., and Deshmukh, M. M. Nanoscale electromechanics to measure thermal conductivity, expansion, and interfacial losses, Nano Letters 15, 7621-7626 (2015).
- Nigues, A., Siria, A., and Verlot, P. Dynamical backaction cooling with free electrons, Nat. Commun. 6, 8104, (2015).
- Bunch, J. S. et al. Electromechanical resonators from graphene sheets, Science 315, 490-493 (2007).
- Chen, C. et al. Performance of monolayer graphene nanomechanical resonators with electrical readout, Nat. Nanotech. 4, 861-867 (2009).
- Eichler, A., Moser, J., Chaste, J., Zdrojek, M., Wilson-Rae, I., and Bachtold, A. Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene, Nat. Nanotech. 6, 339-342 (2011).
- Miao, T., Yeom, S., Wang, P., Standley, B., and Bockrath, M. Graphene nanoelectromechanical systems as stochastic-frequency oscillators, Nano Letters 14, 2982-2987 (2014).
- Singh, V., Bosman, S. J., Schneider, B. H., Blanter, Y. M., Castellanos-Gomez, A., and Steele, G. A. Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity, Nat. Nanotech. 9, 820-824 (2014).
- Song, X., Oksanen, M., Li, J., Hakonen, P. J., and Sillanpää, M. A. Graphene optomechanics realized at microwave frequencies, Phys. Rev. Lett. 113, 027404 (2014).
- Weber, P., Güttinger, J., Tsioutsios, I., Chang, D. E., and Bachtold, A. Coupling graphene mechanical resonators to superconducting microwave cavities, Nano Letters 14, 2854-2860 (2014).
- Lee, J., Wang, Z., He, K., Shan, J., and Feng, P. X. L. High frequency MoS2 nanomechanical resonators, ACS Nano 7, 6086-6091 (2013).
- van Leeuwen, R., Castellanos-Gomez, A., Steele, G. A., van der Zant, H. S. J., and Venstra, W. J. Time-domain response of atomically thin MoS2 nanomechanical resonators, Appl. Phys. Lett. 105, 041911 (2014).
- Z. Wang, and Feng, P. X. L. Dynamic range of atomically thin vibrating nanomechanical resonators, Appl. Phys. Lett. 104, 103109 (2014).
- Rugar, D., Budakian, R., Mamin, H. J., and Chui, B. W. Single spin detection by magnetic resonance force microscopy, Nature 430, 329-332 (2004).
- Bleszynski-Jayich, A. C. et al. Persistent currents in normal metal rings, Science 326, 272-275 (2009).
- Chen, C. et al. Modulation of mechanical resonance by chemical potential oscillation in graphene, Nat. Phys. 12, 240-244, (2016).
- Wang, Z., Wei, J., Morse, P., Dash, J. G., Vilches, O. E., and Cobden, D. H. Phase transitions of adsorbed atoms on the surface of a carbon nanotube, Science 327, 552-555 (2010).
- Tavernarakis, A. et al. Atomic monolayer deposition on the surface of nanotube mechanical resonators, Phys. Rev. Lett. 112, 196103 (2014).
- Arcizet, O., Cohadon, P. F., Briant, T., Pinard, M., and Heidmann, A. Radiation-pressure cooling and optomechanical instability of a micromirror, Nature 444, 71-74 (2006).
- Gigan, S. et al. Self-cooling of a micromirror by radiation pressure, Nature 444, 67-70 (2006).
- Teufel, J. et al. Sideband cooling of micromechanical motion to the quantum ground state, Nature 475, 359-363 (2011).
- Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum ground state, Nature 478, 89-92 (2011).
- Teufel, J. D., Donner, T., Castellanos-Beltran, M. A., Harlow, W. J., and Lehnert, K. W. Nanomechanical motion measured with an imprecision below that at the standard quantum limit, Nat. Nanotech. 4, 820-823 (2009).
- Anetsberger G. et al. Measuring nanomechanical motion with an imprecision below the standard quantum limit, Phys. Rev. A 82, 061804 (2010).
- Meenehan, S. M. et al. Silicon optomechanical crystal resonator at millikelvin temperatures, Phys. Rev. A 90, 011803 (2014).
- Mamin, H. J. and Rugar, D. Sub-attonewton force detection at millikelvin temperatures, Appl. Phys. Lett. 79, 3358 (2001).
- Zhang, Y., Moser, J., Güttinger, J., Bachtold, A., and Dykman, M. I. Interplay of Driving and Frequency Noise in the Spectra of Vibrational Systems, Phys. Rev. Lett. 113, 255502 (2014).
- Singh, V. et al. Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators, Nanotechnology 21, 165204 (2010).
- Bao, W. et al. In situ observation of electrostatic and thermal manipulation of suspended graphene membranes, Nano Letters 12, 5470-5474 (2012).
- Chen, C. et al. Graphene mechanical oscillators with tunable frequency, Nat. Nanotech. 8, 923-927 (2013).
- Rocheleau, T., Ndukum, T., Macklin, C., Hertzberg, J. B., Clerk, A. A., and Schwab, K. C. Preparation and detection of a mechanical resonator near the ground state of motion, Nature 463, 72-75 (2009).
- Barton, R. A. et al. Photothermal self-oscillation and laser cooling of graphene optomechanical systems, Nano Letters 12, 4681-4686 (2012).
- Cole, R. M. et al. Evanescent-field optical readout of graphene mechanical motion at room temperature, Phys. Rev. Applied 3, 024004 (2015).
- Castellanos-Beltran, M., Irwin, K., Hilton, G., Vale, L., and Lehnert, K. Amplification and squeezing of quantum noise with a tunable Josephson metamaterial, Nat. Phys. 4, 929-931 (2008).
- Dykman, M. I., ed., Fluctuating Nonlinear Oscillators (Oxford, 2012).
- Fong, K. C. et al. Measurement of the electronic thermal conductance channels and heat capacity of graphene at low temperature, Phys. Rev. X 3 (4), 041008 (2013).
- Samkharadze, N. et al. High-Kinetic-Inductance Superconducting Nanowire Resonators for Circuit QED in a Magnetic Field, Phys. Rev. Applied 5, 044004 (2016).
- Aspelmeyer, M., Kippenberg, T. J., and Marquardt, F. Cavity Optomechanics, Rev. Mod. Phys 86, 1391 (2014).