A few-layer graphene nanomechanical resonator driven by multifrequency digital signals

Domain: physics
Published: 2025-12-04
Editorial outcome: major_revision

Abstract
Nanomechanical resonators driven by multifrequency signals merge mesoscopic vibration physics with radiocommunication technologies. Their fundamental property is resonant filtering: they respond only to driving signals with frequency components within their mechanical bandwidth. Compared to single-tone drives, multifrequency drives produce richer dynamics, with all frequency components transduced simultaneously into vibrations of varying amplitudes and phases that superimpose and interfere. Here, we employ a few-layer graphene nanomechanical resonator as a filter for broadband, digitally modulated video signals. We transduce the modulated signals into modulated vibrations and subsequently demodulate these into a nanomechanical video. This approach shows vibrational features that are absent in single-tone or noise-driven responses.

Introduction
Nanomechanical resonators enable the study of mesoscopic vibrations within individual systems that can be tuned and controlled1–4. Owing to their small masses, they transduce weak driving forces near resonance into vibrations of sizeable amplitude. Their frequency response resembles that of resonant tuning circuits in analogue radio receivers, with the force acting as the driving voltage and the mechanical displacement acting as the circuit charge. Just as the resonant frequency of an electrical circuit can be tuned using a variable capacitor or inductor, the resonant frequencies of the mechanical response, which are the frequencies of the vibrational modes, can be adjusted in situ using capacitive5,6 and dielectric forces7, or by straining the resonator either mechanically8 or through its thermal expansion9,10. As in the electrical case, the frequency bandwidth of the mechanical response is governed by both dissipation and resonant frequency fluctuations3. A narrower electrical response enhances frequency selectivity, while a narrower mechanical response facilitates the sensing of smaller driving forces. A broader response can also be advantageous. For instance, a wide-bandwidth tuning circuit is required to isolate a modulated radio signal and capture its full spectral content while filtering out signals in adjacent bands. Similarly, nanomechanical resonators with a large bandwidth W can respond to nearly resonant forces whose broad spectra are contained within the mechanical response11–14. Such large-bandwidth resonators also enable the study of resonant frequency fluctuations induced far below resonance by fluctuators with correlation times as short as 1/W 15–20. In essence, nanomechanical resonators function as tunable bandpass filters for nearly resonant forces and as detectors for nonresonant fluctuators.

While the nanomechanical response to broadband force noise and low-frequency fluctuators has been thoroughly studied, the response to multifrequency coherent drives has received comparatively less attention. In many cases where multifrequency signals are applied, only a single frequency component is used to measure the response, while the other components serve different purposes. For example, amplitude-modulated6,21,22 and frequency-modulated signals23 have been employed to drive the resonator at the single frequency of the carrier, with the modulation sidebands used to detect the vibrations. Two-tone signals, consisting of one tone near resonance and a second tone outside the response, have been employed to explore various vibrational phenomena. Examples abound and include pump-probe spectroscopy for measuring intermode coupling24, parametrically induced mode coupling, used to heat or cool vibrations25–28 and to engineer nonlinear friction29, and symmetry-breaking parametric resonators for binary information encoding30–32. By contrast, applications involving multifrequency signals where the entire spectrum directly drives vibrations remain relatively limited. Two-tone drives have been used to demonstrate signal mixing with nonlinear nanomechanical resonators33,34. The interplay between a near-resonant two-tone drive and parametric pumping has enabled the realization of mechanical logic gates35. A combination of strong and weak near-resonant tones has been shown to squeeze the thermomechanical noise36. Resonators driven nonlinearly by two strong tones near resonance have exhibited unconventional dynamics, where the beating between the tones acts as a slow modulation of the bifurcation points37,38. Overall, studying nanomechanical resonators driven by multifrequency signals sheds light on the physics of mesoscopic vibrations and leads to proof-of-concept demonstrations for potential applications.

Appealing demonstrations of the physics of multifrequency vibrations are provided by nanomechanical radio receivers and transmitters for frequency modulated (FM) signals39,40. Nanomechanical FM receivers have been realized using suspended nanotubes23,41–43 and nanomachined silicon tuning forks44. The FM voltage waveform is capacitively coupled to the receiving resonator, creating a modulated driving force that the resonator passively responds to. By contrast, nanomechanical FM transmitters output a modulated radio-frequency signal. These have been demonstrated using suspended graphene45 and nanomachined polysilicon disks46,47. In these devices, the vibrating element forms part of an electrical positive feedback loop that sustains self-oscillations. The oscillation frequency is modulated by a stream of low-frequency voltage pulses or by an audio waveform, which induces a slowly varying capacitive force between the oscillator and a nearby electrode, thereby slowly modulating the spring constant. This approach, akin to “keying” the baseband components of the vibrations, is reminiscent of Morse code telegraphy. In summary, nanomechanical radio receivers and transmitters illustrate the richness of mesoscopic vibrations driven by multifrequency signals. By producing an audible copy of these vibrations, they also appeal to science and technology enthusiasts.

Here, we demonstrate that nanomechanical resonators based on few-layer graphene (FLG) can transduce a multifrequency, digital video signal into modulated vibrations that encode an accurate copy of the video. We quantify the transduction using standard signal quality metrics for digital communications. We have chosen FLG resonators for their large resonant frequency tunability within the high-frequency (HF) and the very-high-frequency (VHF) bands of the radio spectrum, and for the compromise they offer between broad bandwidths and moderate force sensitivities. A key point in our study is that the signal bandwidth is larger than or comparable to that of the mechanical response. This contrasts with nanomechanical radio receivers and transmitters discussed above, which process audio baseband signals with bandwidths much smaller than the mechanical bandwidth. A wide-bandwidth coherent drive serves as an effective probe for exploring the physics of the mechanical response. Because the resonator behaves as a filter, it shapes the multifrequency driving signal and transduces it into multifrequency vibrations with different phases and amplitudes that superimpose and interfere. We find that these interferences have a profound impact on the signal-to-noise ratio and on the bit error ratio of the transduction. We theoretically model the response to our multifrequency drive and successfully reproduce our experimental data. We establish a relation between the mechanical bandwidth and the maximum signal bandwidth that can be transduced without information loss in the baseband components of the vibrations. Our work contributes to the recent study of resonators driven nonlinearly by multiple tones. Whereas earlier studies explored the case of two-tone driving37,38, we investigate a previously unexplored regime in which the resonator is coherently driven by a near continuum of frequency components across its full mechanical bandwidth. We also envision experimental applications in the context of renewed interest in HF and VHF band communications48. Even though data rates in these bands are low, HF and VHF remain among the most robust and reliable options for long-distance communication in scenarios where other channels are unavailable. Future HF communication systems may be required to transmit and receive weak signals encoded with compressed images or videos48, while narrowband digital television has already been demonstrated in the VHF band49. Our results suggest that nanomechanical resonators could serve as passive receivers for such signals, just as they do for audio-modulated radio waves.

Results
We begin by briefly describing the principle of our measurements. Our experimental setup is shown in Fig. 1a. The FLG resonator, estimated from its optical contrast to be composed of ~10 layers, is shaped as a 3 μm diameter drum suspended over a local gate electrode (Supplementary Note 1). It is kept at room temperature in a vacuum enclosure and held by a piezopositioner. Two similar devices have been measured, yielding comparable results. Homemade flexible transmission lines, fabricated from copper-clad ribbons of Kapton, deliver radio-frequency signals to the gate without impeding the motion of the positioner50,51. With the resonator grounded, the voltage waveform incident on the gate, s(t), creates a capacitive force Fd(t) ≃ (1+Γ)Cg' Vdc s(t), where Γ ≃ 1 is the reflection coefficient at the gate, Cg' is the derivative of the capacitance between the resonator and the gate with respect to displacement along the out-of-plane z^-direction (Fig. 1a), Vdc is a dc voltage applied to the gate, and t is time. Near resonance, Fd(t) drives flexural vibrations along z^. To measure these vibrations, we place the resonator in an optical standing wave formed between the gate and a quarter-wave plate52, a standard configuration for studying two-dimensional resonators1,53–56. The amount of optical energy absorbed by the resonator from the standing wave is modulated at the frequency of the flexural vibrations, resulting in oscillations in the output voltage of a photodetector placed in the path of the reflected light54.

The originality of our setup lies in our technique to create a modulated drive and demodulate the nanomechanical response. The baseband signal of the drive is built from a video in MP4 format, which we convert into a video Transport Stream (TS). The TS data stream is split into two bit streams that are used to associate each symbol in the baseband signal with 2 bits. We implement a 4-quadrature amplitude modulation scheme (4-QAM, equivalent to quadrature phase shift keying). In this scheme, each symbol is represented by two nearly simultaneous voltage pulses, Iin = Vp and Qin = Vq, with V a voltage and p = ±1, q = ±1 (vertical bars in Fig. 1b). These pulses are filtered using a root raised cosine (RRC) filter, producing two smoothly varying signals, ~Iin(t) and ~Qin(t). This filter both minimizes intersymbol interference (when paired with a matched RRC filter at the receiver) and ensures that the resonator is driven by a slowly modulated waveform. Without this filtering, the resonator would attenuate much of the information encoded in the high-frequency components of the pulses. The signals ~Iin(t) and ~Qin(t) modulate the amplitude of the in-phase and quadrature components of the driving waveform, s(t) = ~Iin(t)cos(2πfd t) + ~Qin(t)sin(2πfd t), where fd is the carrier (drive) frequency (Supplementary Note 2). Both baseband modulation (bit-to-symbol mapping) and carrier modulation of s(t) are implemented using GNU Radio, an open-source software development toolkit for signal processing57. The resulting waveform s(t) is converted into an analog signal using a low-cost software-defined radio (SDR) transmitter (HackRF One by Great Scott Gadgets58). The calculated power spectral density S of the driving force Fd(t), averaged over multiple realizations, is shown in Fig. 1c as a function of spectral frequency f. We consider a drive power at the gate Pd = 10log10[(1+Γ)2<s2(t)>/(50×10−3)] ≃ −31 dBm, where <⋅> denotes time averaging. We set Vdc = 13 V and estimate Cg' ≃ 9×10−10 Fm−1 with COMSOL. The spectrum S(f) is centered at fd and spans a full bandwidth W ≃ 1.3Rsym, where Rsym = 106 Hz is the symbol rate and 1.3 accounts for the filter roll-off factor. The area under S(f) corresponds to the variance of Fd(t), which depends only on the drive amplitude and is independent of Rsym. Importantly, Fd(t) is not a delta-correlated noise but rather a coherent waveform (Supplementary Note 2). Figure 1d shows the normalized magnitude and the phase of the linear mechanical response. The bandwidth of the drive W exceeds the resonator’s mechanical bandwidth Wm = fm/Qm ≃ 0.56×106 Hz, where fm ≃ 56×106 Hz is the resonant frequency of the fundamental mode and Qm ≃ 100 is its quality factor.

To measure the voltage v(t) at the output of the photodetector, we use a low-cost SDR receiver (NESDR Smart by Nooelec58) that downconverts the signal to baseband. Using GNU Radio, we correct transmission delays with a polyphase filter bank59,60 and apply matched filtering using an RRC filter. This yields the in-phase and quadrature components of the received signal, expressed as v(t) = ~Iout(t)cos(2πfd t) + ~Qout(t)sin(2πfd t). We decimate ~Iout(t) and ~Qout(t) at the symbol rate Rsym to obtain the discrete symbol components Iout and Qout. These values, acquired for 2 s, are plotted on a constellation diagram in Fig. 1e. This diagram helps visualize the process of converting symbols back into bits, the decision to ascribe a received symbol to a pair of bits depending on which quadrant the symbol appears in61. The GNU Radio flowcharts used for modulation and demodulation are provided in Supplementary Note 3. Finally, Fig. 1f shows a frame from a nanomechanical video obtained by converting the Iout and Qout streams back into an MP4 video using FFmpeg, an open-source multimedia processing tool62. The frame shows an early configuration of our setup, featuring an optical objective outside the vacuum enclosure.

Obtaining a nanomechanical video requires optimizing the drive power Pd, the drive frequency fd, and the symbol rate Rsym. We vary these parameters individually and monitor their impact on the constellation diagram of the demodulated outputs Iout and Qout. As fd is stepped through resonance, the constellation diagram undergoes significant changes that are best visualized as animations (Supplementary Movie 1 shows diagrams measured as fd is increased at various Pd values, with Rsym = 0.5×106 Hz). A simple metric we extract from the constellation diagram is the time-averaged length of the I−Q vector in one quadrant, defined as L = <|Iout + iQout|>. The spread of the symbol clouds (Fig. 1e) also contains useful information. We quantify it within a single quadrant as d = <[(Iout − <Iout>)2 + (Qout − <Qout>)2]1/2>. The logarithmic ratio, ε = 20log10(L/d), can be interpreted as a signal-to-noise ratio, since noise at the output of the receiver and the dephasing of vibrations by the mechanical response both contribute to the spread d. In digital communications, ε is known as the reciprocal of the error vector magnitude of the symbols. Larger values of ε indicate better symbol separation and correspond to smoother video playback.

Measurements of L(fd) and ε(fd) are shown as dots in Fig. 2a–d and Fig. 2e–h, respectively, for several values of Pd at Rsym = 0.5×106 Hz (the minimum values of L and ε required for successful video decoding are given in Supplementary Note 4). Similar measurements at Rsym = 106 Hz are presented in Fig. 3. In all measurements, we set Vdc = 13 V, corresponding to a mechanical resonant frequency fm ≃ 56×106 Hz. For both symbol rates, the lineshapes of L(fd) broaden and become increasingly asymmetric as Pd increases, consistent with the onset of nonlinearities in the mechanical response. While the L(fd) curves resemble broadened resonance peaks, the ε(fd) traces exhibit richer features. At low drive power, ε(fd) peaks near fd ≃ fm and develops two local minima away from resonance (Fig. 2e), the latter being more pronounced at Rsym = 106 Hz (Fig. 3e). At higher Pd, the local minimum in ε(fd) that appears where |∂L/∂fd| is largest deepens significantly, and the maxima in ε(fd) shift further from resonance (Figs. 2h and 3h). These extrema in ε(fd) are important features, as they directly affect the quality of the demodulated video.

To understand these observations, we numerically solve the resonator’s equation of motion in response to the multifrequency driving force Fd(t). The equation reads63,64:

z̈ + (ωm/Qm)ż + ηz2ż + ωm2z + αz2z = Fd(t)/meff, (1)

where z is the resonator displacement, ωm = 2πfm is the angular resonant frequency, η is the nonlinear damping coefficient, and α is the Duffing elastic constant. The values Qm ≃ 100, η ≃ 1.2×105 kg m−2 s−1 and α ≃ −1014 kg m−2 s−2 (arising from the electrical potential acting on the resonator, see ref. 65) are estimated from the response to a single-tone drive (Supplementary Note 5). The effective mass is estimated as meff ≃ 2×10−17 kg based on the resonator shape and the number of graphene layers. As in the experiment, the baseband signals ~Iin(t) and ~Qin(t) that define Fd(t) are computed from two streams of in voltage pulses. The pulse amplitudes randomly take values ±V at a rate Rsym. These pulse streams are processed with a raised cosine (RC) filter (Fig. 1b). The use of this filter is justified, given that most of the measured noise appears at the receiver output. For completeness, we also consider the scenario where white Gaussian force noise is added at the drive stage, in addition to receiver noise. This case requires a cascade of two RRC filters: one at the transmitter and one at the receiver. However, we find that modeling receiver noise alone suffices to explain our measurements.

We demodulate z(t) to obtain ~Iout(t) = 2h(t) * [ξz(t)cos(2πfd t)] and ~Qout(t) = 2h(t) * [ξz(t)sin(2πfd t)], where h(t) is a lowpass filter, ξ is a conversion factor in V m−1, and * denotes convolution. The symbol components Iout and Qout are obtained by decimating ~Iout(t) and ~Qout(t) at the symbol rate Rsym, accounting for the delay introduced by the RC filter. This simple model reproduces our measured L(fd) and ε(fd) traces (solid blue lines in Figs. 2 and 3). The only fit parameter in the analysis of L(fd) is ξ ≃ 7×105 V m−1. Notably, the lineshape of L(fd) is broader than that obtained with an unmodulated drive at the same Pd (Supplementary Note 2). To analyze ε(fd), we model fluctuations δ~Iout and δ~Qout at the receiver output as additive white Gaussian noise with variance σ2. This noise contributes to the spread of the symbol clouds, leading to the modified expression d2 + <δ~Iout2> + <δ~Qout2> = d2 + 2σ2. We use σ2 ≃ 3.9×10−19 m2 as a single fit parameter in the analysis of ε(fd). Measurements of the noise at the receiver output are presented in Supplementary Note 6. Interestingly, the lineshape of ε(fd) near fm is not influenced by the noise at the receiver. The parameter σ2 is only required to capture the decay of ε(fd) away from fm, and it has only a little incidence on the most interesting features that are the minima in ε(fd). To show this, in Fig. 3e–h we display our calculations for ε(fd) with no added noise, that is, without any fit parameter (orange traces). In this case, ε increases as fd is stepped away from fm, while the features near fm remain similar to those in the presence of noise. This indicates that the spread of the symbol clouds, d(fd), near fm is caused by a process other than noise. This process is the loss of coherence in the response, resulting from the superposition of multifrequency vibrations with different amplitudes and phases. It is a hallmark of vibrations driven by a coherent, multifrequency signal and is not seen with single-tone drives or with force noise. It is caused by the resonator acting as a filter in the case where the spectrum of the drive is broader than the response, W ≳ Wm. If the drive were narrowband, W ≪ Wm, the input symbol components (Iin, Qin) would be transformed into (Iout, Qout) by a simple rotation through the phase angle of the response. By contrast, each frequency component in the broadband drive is transduced into a vibrational frequency component with a distinct amplitude and phase. Where fd is far from fm, and provided that receiver noise is low, (Iout, Qout) still result from a simple rotation through a constant angle (≃ 0 or −π), since the magnitude and the phase of the response depend on frequency only weakly. In this case, Iout ∝ ±Iin and Qout ∝ ±Qin. Where fd ≃ fm, W ≃ Wm, and where the response is linear (that is, at low Pd), we find Iout ∝ Qin and Qout ∝ −Iin (Supplementary Note 7), meaning that input and output symbols are simply swapped. This has no adverse effect on video demodulation (Supplementary Note 8). However, the dephasing effect of the response on Iout and Qout is strongest where |fd − fm| ≃ W/2 and W ≳ Wm. This case is depicted in Fig. 1c, d, where the upper frequency edge of the drive spectrum is near resonance and the lower frequency edge is far from it. There, ~Iout(t) contains dephased frequency components from both ~Iin(t) and ~Qin(t), and similarly for ~Qout(t) [Supplementary Note 7]. The fact that the in-phase component of the vibrations depends on both the in-phase and quadrature components of the drive, and that the quadrature response likewise depends on both drive components, can already be seen by solving Eq. (1) in the linear regime, yielding:

~Iout(t) ≃ κ ∫ dt' exp(−Γm t') [~Iin(t−t')sin(Δωt') − ~Qin(t−t')cos(Δωt')]
~Qout(t) ≃ κ ∫ dt' exp(−Γm t') [~Iin(t−t')cos(Δωt') + ~Qin(t−t')sin(Δωt')] (2)

where κ = |ξVdc(1+Γ)Cg'|, ωm = 2πfm, Γm = πWm, and Δω = 2π(fd − fm). As a result, decimating ~Iout(t) and ~Qout(t) yields broad distributions of Iout and Qout going through zero, which reduces ε. Experimentally, the condition of |fd − fm| ≃ W/2 coincides with the minima observed in ε(fd) for Rsym = 0.5×106 Hz and 106 Hz in the linear regime.

In the nonlinear regime (high Pd), our model also captures both the deepening of the local minimum (dip) in ε(fd) near the point of steepest slope in L(fd), and the frequency shift of the dip, Δfdip, as Pd increases. In Supplementary Note 9, we show that Δfdip is approximately proportional to the square of the change in vector length, ΔLdip, that is, Δfdip / (ΔLdip)2. This behavior is consistent with the quartic relationship between the frequency of a vibrational mode and the vibrational amplitude caused by weak conservative nonlinearities in the case where the mode is driven by a single tone drive66–68. It is also consistent with the lineshape broadening of the mechanical response induced by the interplay between conservative nonlinearities and a force noise13,14,69, as further detailed in Supplementary Note 9.

To complement our measurements of ε and directly quantify the effect of the mechanical response on the driving bit stream, we measure the bit error ratio (BER) of the nanomechanical transduction. This ratio represents the number of misread bits in the received stream (output bit stream) relative to the number of bits in the stream sent by the transmitter (input bit stream). Measuring BER is not straightforward because propagation and filtering delays complicate the direct comparison of the input and output bit streams. We solve this problem by inserting a marker at the beginning of the input bit stream. The marker consists of a train of bits ‘0’ followed by a train of bits ‘1’, which are easily identifiable on the receiver side. This enables a bit-by-bit comparison between the input and output streams. As an example, Fig. 4a shows a short input bit stream (hollow squares) and the corresponding received output bit stream (dots), obtained with Rsym = 0.5×106 Hz. Figure 4b–f illustrates BER as a function of fd, measured at Rsym = 0.5×106 Hz (blue dots) and Rsym = 106 Hz (orange dots), spanning from the linear to the nonlinear regime [with increasing Pd from (b) to (f)]. At each fd, BER is computed from a single 15-second measurement of the output bit stream. We find that BER at Rsym = 0.5×106 Hz is consistently lower than at Rsym = 106 Hz, indicating that the minimum bit error ratio, BERmin, depends on Rsym. At the largest Pd and Rsym = 0.5×106 Hz (Fig. 4f), we measure BERmin ≃ 5×10−6, corresponding to 5 misread bits per million sent to the resonator. This limit is set by noise in our setup (Supplementary Notes 4 and 6). Furthermore, we observe that BER(fd) develops a sharp peak as Pd increases. This peak coincides with the emergence of the lower frequency dip in ε(fd) (Supplementary Note 9). At the peak frequency, the symbol clouds on the constellation diagram are strongly distorted, as confirmed by our model (Fig. 4g), leading to decoding failure of the nanomechanical video. In Supplementary Note 10, we present our calculated L(fd), ε(fd), BER(fd), and constellation diagrams as an embedded animation to help visualize how these quantities relate to one another.

With regard to applications, we now examine the dependence of BERmin on Rsym. Our goal is to determine how Rsym*, the highest symbol rate at which BERmin remains low, varies with the mechanical linewidth Wm. We interpret Rsym* as the effective filtering bandwidth of the resonator for the video signal. We focus on BER at resonance in the linear regime, where BER(fd) = BERmin (Fig. 4b). In Fig. 4h, we plot BERmin as a function of Rsym, calculated using Eq. (1) for Wm = 104 Hz, 105 Hz, and 106 Hz (gray traces, from left to right). The dark traces correspond to calculations without noise, while the light traces include additive Gaussian noise with variance σ2, as used in Figs. 2 and 3. Each trace computed without noise exhibits a kink near Rsym ≃ 3.5Wm, which sets an upper bound for Rsym*. When noise is added, Rsym* shifts to lower values, particularly for larger Wm (that is, lower Qm), as BERmin(Rsym) begins to increase at lower symbol rates. The red dots in Fig. 4h represent measured values of BERmin(Rsym), obtained at Pd = −28 dBm, near the onset of the nonlinear regime and in the presence of noise. From these data, we extract an experimental filtering bandwidth of Rsym* ≃ 106 Hz. Additional measurements of BERmin across a range of Pd are shown in Fig. 4i (see also Supplementary Note 11). As expected, these results indicate that nanomechanical information encoding suffers less information loss at higher drive powers.

We now demonstrate the filtering and video encoding capabilities of our resonator. Figure 5a shows that the nanomechanical response (measured here using a single-tone drive) depends on the gate voltage Vdc. Since both ε and BER strongly depend on |fd − fm|, we can transmit multiple videos with well separated fm simultaneously, and selectively demodulate one of them by appropriately tuning fm with Vdc. Our experimental setup for broadcasting two videos in parallel is illustrated in Fig. 5b. Two SDR transmitters are used: one operating at fdA = 57.8×106 Hz = fm, and the other at fdB = 60.8×106 Hz = fm. Both transmitters are configured with Rsym = 106 Hz and Pd = −22 dBm, a power at which the resonator responds nonlinearly. The drive frequencies are chosen to avoid the dip in ε(fd). Figure 5c shows ~Iout(t) and ~Qout(t), with Vdc initially set to 14 V and the receiver’s center frequency fr set to fA at t = 0. Between t = 17 s and t = 19 s, Vdc is ramped from 14 V to 15 V, and fr is set in software to fB. Then, between t = 40 s and t = 42 s, Vdc is ramped back down from 15 V to 14 V, and fr is reset to fA. Figure 5c shows that the desired broadcast, A or B, can be selected by adjusting Vdc.

In the spirit of previous nanomechanical devices used as signal processors39, we envision that our FLG resonators, as well as other resonators based on similar principles, may serve as passive radio receivers operating in the HF and VHF bands of the radio spectrum. With a resonant frequency ranging from ≃ 40×106 Hz to ≃ 60×106 Hz (Fig. 5a), our resonator operates within the 6-meters subband, which is internationally allocated for amateur radio use. In this subband, digital amateur television (DATV) broadcasts have been conducted using signals with a 300 kHz bandwidth49. In the United Kingdom, DATV with a broader bandwidth of up to 500 kHz is permitted in the 2-meter amateur radio band (144–148 MHz)49. Smaller-diameter FLG resonators, or higher-order vibrational modes, can operate in this frequency range. The filtering bandwidth of our resonator, along with the maximum 4-QAM data rate of its transduction, 2Rsym* ≃ 2×106 bps, is fully compatible with DATV.

While our experimental setup may not yet be practical for receiving amateur radio broadcasts, it can be further simplified51. The resonator would serve a dual purpose: as a front-end, tunable bandpass filter, and as a passive optocoupler. It could be placed at the output of the preamplifier that is typically connected to the feed point of a receiving antenna70. As a filter, it would reject jamming and other interfering signals in a manner similar to surface acoustic wave filters, but with the added benefit of a tunable center frequency, and without the fabrication complexity of more advanced tunable front-end filters71. As an optocoupler, it would passively convert the radio signal into modulations of reflected light intensity through the nanomechanical transduction, effectively eliminating the issue of radio frequency interference caused by common-mode currents in the coaxial cable that connects the antenna to the demodulator70. The detection of weaker radio signals may be improved by using more advanced SDR systems with lower receiver noise and by engineering the mirror underneath the resonator to enhance the responsivity of the device72. Admittedly, the idea of using nanomechanical resonators for radio communication remains largely academic due to challenges related to device reproducibility and thermal drift40. Nevertheless, these resonators may eventually enable alternative architectures for shortwave radio receivers, facilitating the decoding of images and videos carried by weak signals in noisy environments61.

Methods
Optical detection of vibrations
The incident light is provided by a Helium-Neon laser operating at a wavelength of λ = 633 nm. The laser output is coupled into a single-mode optical fiber for spatial filtering. A half-wave plate sets the linear polarization of the emerging light, which then passes through a quarter-wave plate to become circularly polarized. The light is focused onto the resonator and the gate electrode, acting as a mirror. After traversing the quarter-wave plate again, the reflected beam emerges linearly polarized, orthogonal to the incident polarization. A polarizing beam splitter separates the incident and reflected beams, directing the reflected light to the photodetector (Thorlabs APD430A2 avalanche photodetector). As a function of time t, the voltage at the output of the photodetector reads:

v(t) ≃ Pinc (∂R/∂z) δz(t) * ψ(t) + δvb, (3)

where Pinc ≃ 3×10−6 W is the optical power incident on the resonator, R is the optical reflectance of the device composed of the resonator, the gate electrode, and the vacuum gap between them, δz is the vibrational amplitude of the resonator (that is, its dynamic displacement), z is the static displacement of the resonator, |∂R/∂z| ≃ 3×106 m−1 is the calculated optical responsivity of the device52, ψ is the response function of the photodetector in units of V W−1 s−1, δvb is a background voltage noise, and * denotes convolution.

Driving voltage and driving force waveforms
To build the driving voltage and the driving force waveforms, we start with two streams of pulses pk = ±1 and qk = ±1, generated at a pulse (symbol) rate Rsym. The streams are filtered using either a raised cosine (RC) filter or a root raised cosine (RRC) filter, yielding waveforms ~p(t) and ~q(t) that vary continuously in time. Where noise is added to the signal after the transmitter and before the receiver, an RRC filter is applied at the transmitter to shape the pulses, and a second RRC filter is applied at the receiver to filter the baseband signal. The cascade of these two RRC filters is equivalent to a single RC filter and serves to minimize intersymbol interference, just as a single RC filter does in a noiseless scenario. The waveforms ~p(t) and ~q(t) can be expressed as:

~p(t) = Σ pk g(t − k Tsym), ~q(t) = Σ qk g(t − k Tsym), (4)

where g is the impulse response of the filter and Tsym is the reciprocal of the symbol rate, Tsym = 1/Rsym. For a RC filter, g reads:

g(t) = [sin(πt/Tsym) / (πt/Tsym)] [cos(πβt/Tsym) / (1 − (2βt/Tsym)2)] (6)

where β is the roll-off factor of the filter. The filtered pulses are used to modulate the amplitude of the in-phase component and that of the quadrature component of the waveform x(t) defined as:

x(t) = ~p(t)cos(ωd t) + ~q(t)sin(ωd t), (7)

where ωd is the angular frequency of the drive. Given a peak voltage amplitude Vin, the driving voltage waveform s(t) is defined as:

s(t) = Vin x(t) = ~Iin(t)cos(ωd t) + ~Qin(t)sin(ωd t), (8)

where ~Iin(t) and ~Qin(t) are the baseband in-phase and quadrature components of the drive, respectively. They are defined as:

~Iin(t) = Vin ~p(t), ~Qin(t) = Vin ~q(t): (9)

The power of s(t) that would be dissipated across a 50 Ohm resistor in units of dBm is:

Pin = 10log10[<s2(t)> / (50×10−3)] = 10log10[Vin2 <x2(t)> / (50×10−3)], (10)

where <⋅> averages over time. Hence, the peak voltage Vin reads:

Vin = [10Pin/10 × (50×10−3) / <x2(t)>]1/2: (11)

The total voltage between the gate and the resonator is the total voltage of the standing wave at the gate:

Vd = (1+Γ)Vin, (12)

where Γ ≃ 1 is the reflection coefficient of s(t) at the gate. The equivalent driving power that would be dissipated across a 50 Ohm resistor in units of dBm is then:

Pd = 10log10[(1+Γ)2<s2(t)> / (50×10−3)] = 20log10(1+Γ) + Pin ≃ 6 + Pin: (13)

The driving force waveform reads:

Fd(t) = |Vdc| (1+Γ) |dCg/dz| s(t), (14)

where Vdc is the dc voltage applied between the resonator and the gate, and dCg/dz is the derivative with respect to the displacement of the resonator of the capacitance between the resonator and the gate.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Review for nature communication: A graphene nanomechanical video receiver\nCe Zhang, Heng Lu, Chen Yang, YuBin Zhang, FengNan Chen, Ying Yan,\nand Joel Moser\nThe author has used 2D material resonator as a building block for mechanical signal processors. These specific resonators\nhave indeed a large frequency tunability compared to more conventional nanoresonator.\nThey discuss about previous reports of similar measurements as radio receiver and transmitter for FM signals (or AM). Ref\n26 and 39 are important examples of real radio applications with a carbon nanotube, similar to the present device. It can be\nnotice that the signal processing is much more complex, by orders of magnitude, in the present video encoding. The main\ndifference with the present result is the wide modulation bandwidth. It allows a larger encoding of information and to\ndetermine the bit error rate of the nanomechanical resonator.\nThe author declare also that their work demonstrates the use of digital bit streams as a probe of nanomechanical resonators.\nThe paper is well written, the discussion is precise and clear. A lot of work is required to obtain this type of result or video.\nThe achievement is technically very impressive. It seems the author also agree that the two videos represent an interesting\ndemonstration and a very huge work but 2d materials resonators are far from the best solution to obtain these videos. They\nfocus on the mechanical response of the resonator to a time series of logical signal. The measurement technic is innovative.\nThe sharp peak and minima in the error bar and the bit rate error are by themselves very interesting for the community. The\ndiscussion is well described. However, at this stage, I’m not fully convinced about the importance of this fundamental issue\nand the future impact of this work.\nI have to admit that the information on GNU radio programming is a part where I'm not an expert but I was able to follow the\nflow of explanations. All the programs seem to be very well details in the SI.\nComment:\n1) Usually, the vibration is measured by down mixing (electrically or optically) and filter at the kHz-10 kHz range, which\nmeans a response to a sinusoidal excitation is average on 1 ms-100 µs. For a ~50 MHz resonator, it means 50000-5000\noscillations. Here the authors are measuring with a pulse rate of 500 kHz which correspond to 100 oscillations. In other\noptomechanical system, real time measurement is possible but what is the limit of detection in optomechanic 2d materials\n2) It can be interesting no measure the input noise of the NESDR smart component in order for readers to compare with the\noutput noise of the photodetector and with their own measurements. We have all technical information on the set-up but\nnothing on the photodetector, it seems?\n3) Figure S5 is very interesting and understandable at first glance compare with Figure 3 (and maybe Figure 2). I propose to\nswitch them or combine the two figures. Maybe I missed some detail here. However, Figure 3e and 3f are also intriguing.\n4) In the SI, the Figure and the text are a mess, please put the good figure with the good Sup note parts. It is for clarity. We\nhave GNU flow chart in sup note part S2 or S3\n5) It is difficult to have a good feeling of the error vector magnitude eps (f): is it related to the signal treatment, to the input\nnoise of the image, to the noise of the vibration? Maybe a better explanation on this term is necessary since it is a very\nimportant in the paper and discussion\n6) I find very interesting the part on the discussion about the minima and its origin. This point is strongly related to the\nprevious one. It is possible to simulate this from the motion equation and it is due to the dephasing. I understand partially the\ndiscussion but the argument about the phase origin and not the noise from only Figure 2e and 2f is difficult to understand in\nthe present form. Is it possible to improve this part?\n7) In Figure 2, especially e and f, I expect the same scale in frequency in order to compare them, I also can expect the same\ndiscussion with figure 2e and 2f in Figure 3 with stronger non linearities and stronger minima.", "reviewer_2": "This paper concerns the continuous transmission of data via transduction of a digital electronic signal into the vibration of a\ngraphene drum. The authors claim that the graphene drum can encode this signal into an optical field by modulation of the\nabsorption, and the signal can be extracted by demodulation with a series of commercial signal processing equipment. Two\nsignals in quadrature appear to provide data redundancy to reduce error. Nonlinear effects were found to be a significant\nfactor in determining the bit error rate (BER) at high drive power. The authors provide a video they claim was transmitted via\nthe graphene. The tunability of the graphene enables the selective transmission of video signals at different frequencies.\nThe paper is well-written and will likely be highly cited, given that it demonstrates an application of graphene NEMS in\ntechnology. Having said that, the scientific and technological significance of the results is quite low. The data transmission\nrate is significantly lower than existing commercial standards and relies on a complex laboratory setup, and the observed\nnonlinear effects have been extensively studied. That said, we recommend publication, provided the following comments\ncan be addressed.\n1. The paper refers to the system as a “video receiver”, however, the authors only demonstrate the use of a graphene\nmembrane to transduce the electrical signal into mechanical vibrations and an optical field, which are arguably less useful\nthan the original electrical signal. The authors should clarify whether the graphene behaves as a receiver and not a\ntransmitter, and on the technological significance of this device.\n2. The authors claim that the broadband video signal is useful as a probe for nonlinear effects. I cannot see any particular\nreason that the video signal is more useful than simple narrowband white noise. Noise-driven nonlinear effects have been\nstudied in depth by many researchers. See, for instance:\nDykman, et al. PRA 1990\nZhang, et al. PRL 2014\nZhang, et al. PRB 2015\nMaillet, et al. PRB 2017\nStambaugh & Chan PRB 2006\nThe authors should comment on the differences between their video signal and generic narrowband white noise.\n3. The nonlinear regime features a critical amplitude and frequency at which two stable oscillation states emerge, as\ndiscussed extensively in Dykman’s “Mesoscopic Physics” review. The authors should determine this amplitude and\nfrequency and discuss with regards to the features in Figures 3, 4, S5 and S7.\n4. The discussion of the two conditions for the BER is confusing, as the authors state that both conditions result in zero error.\nThe authors should clarify how these two “zero-error” conditions result in the observed BER.\n5. The quality factor of the resonator should be mentioned in the main text or captions when discussing the graphene\nparameters, instead of at the end.\n6. The BER will likely be highly dependent on the ratio of the relaxation time of the graphene to the correlation time of the\ndata signal. So far, the authors have shown the BER at signal correlation times below, and roughly equal to, the graphene\nrelaxation time (Rp = 1 MHz and 0.5 MHz, respectively). A plot of the minimum BER versus pulse rate and discussion of the\noscillator response with regards to this ratio would be very helpful.", "reviewer_3": "The paper presents a graphene transducer that converts electrical signals into an optical signal. The authors use the\ngraphene system to transmit a digital video signal, a digitally modulated driving force, and create a \"video receiver\". On the\none hand, the motivation to build a video receiver from a highly complex graphene resonator experiment is highly\nquestionable. Electronic video receivers are a commodity that is cheap and ubiquitous. It is not clear how a graphene\nresonator could contribute to any improvement. The chance, that such a graphene-based video receiver finds its way into a\nconsumer product is quasi-zero. On the other hand, the implementation of the video receiver application with a\nnanoelectromechanical system has resulted in interesting new insights into driving a NEMS resonator with a digitally\nmodulated signal in the linear and nonlinear regime. Furthermore, the authors show the possibility of operating at different\nbase-band frequencies by electrostatic tuning of the resonators' spring constant.\nApart from lacking a clear motivation and potential applications for the NEMS-based video receiver, the authors erroneously\ncompare it to the Nipkow wheel. The only similarity between the two systems is that they are both mechanical devices; no\nfurther analogies hold. While the Nipkow wheel was a pioneering technology eventually supplanted by electronic television,\nthe graphene video receiver does not represent a groundbreaking innovation and is already outdated upon introduction.\nDespite the few interesting observations of driving with a digitally modulated signal, I can't recommend the publication of this\nmanuscript in Nature Communications.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts"}, "author_response": "Response to the comments of the Reviewers\n(Manuscript NCOMMS-24-26392)\nWe are grateful for the time and the care that the Editor and the Reviewers have taken in\nreviewing our manuscript. We have made substantial revisions to our manuscript to address\nthe comments we received in full. Please find below our point-by-point response to these\ncomments.\nResponse to Reviewer #1\nWe thank the Reviewer for taking the time to consider our manuscript and for offering help-\nful comments.\nThe Reviewer writes:\nTheydiscussaboutpreviousreportsofsimilarmeasurementsasradioreceiverand\ntransmitterforFMsignals(orAM).Ref26and39areimportantexamplesofreal\nradioapplicationswithacarbonnanotube,similartothepresentdevice. Itcanbe\nnotice that the signal processing is much more complex, by orders of magnitude,\nin the present video encoding. The main difference with the present result is\nthe wide modulation bandwidth. It allows a larger encoding of information and\nto determine the bit error rate of the nanomechanical resonator. The author\ndeclare also that their work demonstrates the use of digital bit streams as a\nprobe of nanomechanical resonators. The paper is well written, the discussion is\nprecise and clear. A lot of work is required to obtain this type of result or video.\nThe achievement is technically very impressive. It seems the author also agree\nthat the two videos represent an interesting demonstration and a very huge work\nbut 2d materials resonators are far from the best solution to obtain these videos.\nThey focus on the mechanical response of the resonator to a time series of logical\nsignal. The measurement technic is innovative. The sharp peak and minima in\nthe error bar and the bit rate error are by themselves very interesting for the\ncommunity. The discussion is well described.\nWe thank the Reviewer for summarizing our work and for appreciating the novelty of our\nmeasurement technique. Indeed, the wide modulation bandwidth of the signals we employ,\nwhich is on the scale of the bandwidth of the response of our nanomechanical resonator, is\n1\nkey to encoding information in the vibrations and to quantifying the bit error ratio of the\ntransduction.\nThe Reviewer writes:\nHowever, at this stage, I’m not fully convinced about the importance of this\nfundamental issue and the future impact of this work.\nWe thank the Reviewer for this comment. It prompted us to rewrite most of our manuscript\ntoclarifytheimportanceofourworkinthefieldofnanomechanicsandtoaddressitspossible\napplications in radio communications. We have written a new introduction where we present\nour work in the context of the current research on the physics of mesoscopic vibrations.\nIt reads:\n(...) Abroader responseis alsoadvantageous. For example, a large-bandwidth tuningcir-\ncuit is necessary to isolate a modulated radio signal and capture the entirety of its spectrum,\nwhile filtering away signals in adjacent bands. Similarly, nanomechanical resonators with a\nlarge bandwidth ∆f can respond to nearly resonant forces whose spectra are as broad as\nthe mechanical response [11, 12, 13, 14]. They can also be used to study resonant frequency\nfluctuations induced much below resonance by fluctuators with correlation times as short\nas 1/∆f [15, 16, 17, 18, 19, 20]. In essence, nanomechanical resonators behave as tunable\nbandpass filters for nearly resonant forces and as detectors for nonresonant fluctuators.\nWhile the nanomechanical response to broadband force noise and low frequency fluctua-\ntors has been thoroughly studied, the response to multifrequency coherent drives has thus far\nreceived less attention. Where multifrequency signals are applied, oftentimes only a single\nfrequency component is used to measure the response while the other components are used\nfor different tasks. For example, amplitude modulated [6, 21, 22] and frequency modulated\nsignals [23] have been employed to drive the resonator at the single frequency of the carrier,\nwhile the modulation sidebands have been used to detect the vibrations. Two-tone signals,\nwith one tone near resonance and the second tone outside of the response, have been used to\nstudy a variety of vibrational phenomena. Examples abound and include pump-probe spec-\ntroscopy, where intermode coupling is measured [24]; parametrically-induced mode coupling,\nused to heat or cool vibrations [25, 26, 27, 28] and to engineer nonlinear friction [29]; and\nsymmetry-breaking parametric resonators used for binary information encoding [30, 31, 32].\nBy contrast, the scope of applications for multifrequency signals whose entire spectra drive\nvibrations has been more limited. Two-tone drives have been used to demonstrate signal\nmixing with nonlinear nanomechanical resonators [33, 34]. The interplay of a two-tone drive\nnear resonance and parametric pumps has enabled the realization of mechanical logic gates\n[35]. The combination of a strong and a weak driving tone near resonance has been shown\nto squeeze thermomechanical noise [36]. Resonators driven nonlinearly by two large tones\nnear resonance have displayed unconventional dynamics, with the beating between the tones\nacting as a slow modulation of the bifurcation points [37, 38]. Overall, studying nanome-\nchanical resonators using signals composed of multifrequency components sheds light on the\nphysics of mesoscopic vibrations and enables useful applications.\n2\n(...) Here, we demonstrate that nanomechanical resonators based on few-layer graphene\n(FLG) can transduce a multifrequency, digital video signal into modulated vibrations that\nhost an accurate copy of the video. We quantify the transduction using signal quality met-\nrics for digital communications. We have chosen FLG resonators for their large resonant\nfrequency tunabilities within the high frequency (HF) and the very high frequency (VHF)\nbands of the radio spectrum, and for the compromise they offer between broad bandwidths\nand moderate force sensitivities. Of interest in our study is the fact that the signal band-\nwidth is larger than or comparable to that of the mechanical response. This is in contrast\nto nanomechanical radio receivers and transmitters mentioned above, which process audio\nbaseband signals whose bandwidth is necessarily much smaller than the mechanical band-\nwidth. A wide-bandwidth coherent drive acts as a useful probe to study the physics of the\nmechanical response. Because the resonator behaves as a filter, it shapes the multifrequency\ndriving signal and transduces it into multifrequency vibrations with different phases and\namplitudes that superimpose and interfere. We find that these interferences have a profound\nimpact on the signal-to-noise ratio and on the bit error ratio (BER) of the transduction.\nWe model the response to our multifrequency drive theoretically and are able to reproduce\nour data. We establish a relation between the mechanical bandwidth and the largest signal\nbandwidth that can be transduced without loss of information in the baseband components\nof the vibrations. Our work contributes to the recent research effort on the physics of res-\nonators driven nonlinearly by multiple tones. Yet, where only two tones were used earlier\n[37, 38], here we study an unexplored regime where the response is coherently driven by a\nnear continuum of frequency components within the whole mechanical bandwidth.\nWe also address possible applications of our work in the field of radio communications.\nAt the end of our new introduction, we write:\nWe also envision experimental applications at a time of renewed interest in communica-\ntions within the HF and the VHF bands [48]. Even though their data rates are low, HF\nand VHF communications still provide the most robust and reliable means of long-distance\ncommunications in situations where access to other channels is compromised. Future HF\ncommunications will require the ability to occasionally transmit and receive weak signals\nencoded with compressed images and videos [48], while narrowband digital television has al-\nready been experimented with in the VHF band [49]. Our results show that nanomechanical\nresonators may act as passive receivers for such signals, as they do for radiowaves modulated\nat audio frequencies.\nFurthermore, in the section “Discussion”, we write:\nWithregardtoapplications, wenowconsiderthedependenceofBER onR [BER\nmin sym min\nis the minimum of the bit error ratio and R is the symbol rate]. We wish to find out how\nsym\nR∗ , the largest R at which BER is small, changes with the mechanical linewidth ∆f.\nsym sym min\nWe identify R∗ as the filtering bandwidth of the resonator for the video signal. We focus\nsym\non BER at resonance in the linear regime, where BER(f ) = BER (Fig. 4b). In Fig. 4h\nd min\n[Fig. R1h], we plot BER as a function of R calculated using Eq. (1) for ∆f = 104 Hz,\nmin sym\n3\nnoise added, while light traces are calculated with additive Gaussian noise of variance σ2\nused in Figs. 2 and 3. Every trace without noise shows a kink near R ≃ 3.5∆f, which\nsym\nmarks an upper bound for R∗ . Additive noise restricts R∗ to lower values, especially\nsym sym\nfor larger ∆f (lower Q ), as BER starts to increase from a lower R . Red dots in\nm min sym\nFig. 4h [Fig. R1h] represent BER (R ) measured at P = −28 dBm, at the onset of\nmin sym d\nthe nonlinear regime, and in the presence of noise. We extract the experimental filtering\nbandwidth R∗ ≃ 106 Hz. BER data measured over a range of P are displayed in\nsym min d\nFig. 4i [Fig. R1i] (see also Supplementary Note 10). They show, somewhat expectedly, that\nnanomechanical information encoding incurs less information loss at higher drive power.\nh i 100\n20\n15 10-2\n10\n5\n10-4\n0\n10-6\n104 105 106 107 0.5 1 1.5\nR (Hz)\nsym\n)2-01(\nREB\nnim REB\nnim\nR (106 Hz)\nsym\nFigure R1: Fig. 4h-i. Bit error ratio (BER) of the nanomechanical transduction.\n(h) Minimum of BER in the linear regime as a function of R . Grey traces are calculated\nsym\nwithout (dark grey) and with added noise at the receiver (light grey). Up triangle: ∆f =\nat P = −28 dBm. (i) Measured BER as a function of R . From top (red) to bottom\nd min sym\n(blue), P = −28 dBm, −25 dBm, −22 dBm, −19 dBm. Measurements in (h, i) were\nd\nperformed in a noisier environment than measurements shown in (b-g). Vdc = 13 V in all\ng\npanels.\nFinally, at the end of our manuscript, we write:\nIn the spirit of previous nanomechanical devices employed as signal processors [39], we\nenvision that our FLG resonators, and other resonators based on similar principles, may be\nemployed as small footprint, passive radio receivers in the HF and the VHF bands of the\nradio spectrum. With a resonant frequency ranging from ≃ 40×106 Hz to ≃ 60×106 Hz\n(Fig. 5a), our resonator operates in the 6-meter subband that is internationally allocated to\nthe use of amateur radio. In this subband, digital amateur television (DATV) broadcasting\nhas been conducted using signals with a 300 kHz bandwidth [49]. In the United Kingdom,\nDATV with a broader bandwidth of up to 500 kHz is allowed in the 2-meter amateur radio\nband (144–148 MHz) [49]. Smaller diameter FLG resonators, or higher order vibrational\nmodes, can operate in this band. The filtering bandwidth of our resonator, and the maxi-\nmum 4-QAM data rate of its transduction 2R∗ ≃ 2×106 bps, are then fully compatible\nsym\nwith DATV. Arguably, our simple experimental setup may still not be practical enough for\nreceiving amateur radio messages, but it can be simplified further [51]. The resonator would\nserve the dual purpose of a front-end, tunable bandpass filter and of a passive optocoupler.\nIt would be placed at the ouput of the preamplifier that is usually connected to the feedpoint\n4\nof a receiving antenna [69]. As a filter, it would reject jamming and other interfering signals\nthe way surface acoustic wave filters do, but with the additional benefit of a tunable cen-\nter frequency, and without the fabrication complexity of more advanced tunable front-end\nfilters [70]. As an optocoupler, it would passively convert the radio signal into modulations\nof reflected light intensity through the nanomechanical transduction, effectively eliminat-\ning the problem of radio frequency interferences caused by common-mode currents in the\ncoaxial cable that connects the antenna to the demodulator [69]. Admittedly, the idea of\nemploying nanomechanical resonators as devices for radio communcation mostly possesses\nan academic appeal, given the limitations posed by device reproducibility and thermal drifts\n[40]. However, these resonators may ultimately enable alternate architectures for shortwave\nradioreceivers, facilitatingthedecodingofimagesandvideoscarriedbyweaksignalsthrough\na noisy environment [61].\nThe Reviewer writes:\n1) Usually, the vibration is measured by down mixing (electrically or optically)\nand filter at the kHz-10 kHz range, which means a response to a sinusoidal\nexcitation is average on 1 ms-100 (cid:181)s. For a 50 MHz resonator, it means 50000-5000\noscillations. Here the authors are measuring with a pulse rate of 500 kHz\nwhich correspond to 100 oscillations. In other optomechanical system, real time\nmeasurement is possible but what is the limit of detection in optomechanic 2d\nmaterials\nWe thank the Reviewer for raising the interesting issue of real time measurements. To\nour knowledge, measuring the flexural vibrations of 2-D resonators in real time has been\ndemonstrated with graphene oscillators in Prof. Hone’s group [see Fig. R2 below from Chen,\nC., Lee, S., Deshpande, V. V., Lee, G.-H., Lekas, M., Shepard, K. & Hone, J. Graphene me-\nchanical oscillators with tunable frequency. Nat. Nanotechnol. 8, no. 12, 923–927 (2013).\nhttps://doi.org/10.1038/nnano.2013.232]. Other groups, especially those working on\n2-D resonators embedded in superconducting microwave cavities, may have been able to\nperform real time measurements of vibrations as well.\nFigure Redacted\nFigure R2: Fig. 2a in C. Chen, et al., Nat. Nanotechnol. (2013).\nextract bit streams from single-shot measurements of the vibrations. Because we are inter-\nested in encoding information in the vibrations, we cannot average the mechanical signal\nthat we read out. Our way of acquiring signals and extracting information encoded in them\nhas its limits, of course. To address this point, we have created a new Supplementary Note 3\nentitled “I −Q vector length and reciprocal of error vector magnitude at low drive power”.\nIt reads:\nThe averaged length of the I − Q vector in one quadrant of the constellation diagram,\nL = ⟨|I +iQ |⟩, and the reciprocal of the error vector magnitude ε, both at the output\nout out\nof the photodetector, are shown in Fig. R3 as a function of drive frequency f at a symbol\nd\nrate R = 0.5 × 106 Hz. Low values for the drive power P are used (see Eq. (S10) for\nsym d\nthe definition of P ). We find that the smallest ε at which a video signal can be decoded is\nd\nabout 12 dB (peak value of ε(f ) in Fig. R3d).\nd\na 6 b 6 c 6\n-40 dBm -37 dBm -34 dBm\n4 4 4\n2 2 2\n0 0 0\nd e f\n15 15 15\n10 10 10\n5 5 5\n54 55 56 57 58 54 55 56 57 58 54 55 56 57 58\nε\n)Vm(\nL\n)Bd(\nε\n)Vm(\nL\n)Bd(\nε\n)Vm(\nL\n)Bd(\nf (MHz) f (MHz) f (MHz)\nd d d\nFigure R3: Supplementary Fig. 6. L and ε at low drive power. (a-c) Averaged I − Q\nvector L and (d-f) reciprocal of error vector magnitude ε as a function of drive frequency\nf at low drive power P on the gate and at a symbol rate R = 0.5 × 106 Hz. (a, d)\nd d sym\nP = −40 dBm. (b, e) P = −37 dBm. (c, f) P = −34 dBm.\nd d d\nTo understand this lower bound, we express L and the characteristic size d of the symbol\ncloud in one quadrant. L2 reads:\n(cid:32) (cid:33)2 (cid:32) (cid:33)2\n1 (cid:88) 1 (cid:88)\nL2 = I + Q = µ2 +µ2 , (1)\nN n N n I Q\nn n\nwhere N is the number of (I ,Q ) measurements in one quadrant. d2 reads:\nn n\nd2 = 1 (cid:88)(cid:2) (I −µ )2 +(Q −µ )2(cid:3) = σ2 +σ2 , (2)\nN n I n Q I Q\nn\n6\nwhere σ2 is the variance of {I } and σ2 is the variance of {Q }. At low P , the mechanical\nI n Q n d\nresponse is linear. On resonance, d is determined by noise, which we assume to be white\nwith a Gaussian statistics and with a mean µ and a variance σ2. Hence, µ ≃ µ ≡ µ and\nI Q\nσ ≃ σ ≡ σ. Using Eqs. (1) and (2),ε reads:\nI Q\n(cid:18) L2(cid:19)\n(cid:16)µ(cid:17)\nε = 10log ≃ 20log . (3)\n10 d2 10 σ\nIn Fig. R4, we plot a calculated constellation diagram for µ/σ = 4, which corresponds to\nε ≃ 12 dB. The symbol clouds are still well separated, each one residing in its own quadrant.\nUpon reducing µ/σ, hence lowering ε, the symbol clouds spread to the other quadrants and\nthe decoding fails.\nFigure R4: Supplementary Fig. 7. Calculated constellation diagram in the case where\nthe spread of the symbol clouds is determined by white Gaussian noise and µ/σ = 4 (see\ntext for details). Inner circles have a radius of σ, outer circles have a radius of 4σ.\nThe Reviewer writes:\n2) It can be interesting [to] measure the input noise of the NESDR smart compo-\nnent in order for readers to compare with the output noise of the photodetector\nand with their own measurements. We have all technical information on the\nset-up but nothing on the photodetector, it seems?\nWe thank the Reviewer for this question. It prompted us to investigate the source of the\nnoise in our measurements. To address this point, we have created a new Supplementary\nNote 5 entitled “Noise at the output of the receiver”. It reads:\nWe estimate the noise at the output of the receiver from a large collection of 1.18 million\nmeasurements of (I ,Q ) at low drive power P = −40 dBm and at a drive frequency\nout out d\nf = 51 × 106 Hz far away from resonance. Figure R5a shows the power density function\nd\n7\n(PDF) of I , Fig. R5b shows the PDF of Q , and Fig. R5c shows the distribution of\nout out\n(I ,Q ). From these PDFs and distribution, we conclude that the fluctuations of I and\nout out out\nQ follow a white Gaussian statistics. The variance of I and Q is σ2 ≃ 0.19 mV2.\nout out out\n1\n0.8\n0.6\n0.4\n0.2\n0\n-2 -1 0 1 2\nI (mV)\nout\n)1-Vm(\nFDP\n1\n0.8\n0.6\n0.4\n0.2\n0\n-2 -1 0 1 2\nQ (mV)\nout\n)1-Vm(\nFDP\n2\n1\n0\n-1\n-2\n-2 -1 0 1 2\nI (mV)\nout\n)Vm(\nQ\ntuo\na b c\nFigure R5: Supplementary Fig. 9. Noise at the output of the receiver. (a) Power density\nfunction (PDF) of I . (b) PDF of Q . (c) Distribution of (I ,Q ). Traces in (a) and\nout out out out\n(b) are Gaussian curve fits using the measured variance of I and Q .\nout out\nData in all panels\nare measured at low drive power P = −40 dBm and drive frequency f = 51×106 Hz far\nd d\naway from resonance.\nTo narrow down the source of this noise, we separately measure the power spectral den-\nsity of voltage fluctuations (PSD) at the output of the photodetector and the noise at the\noutput of the software-defined radio receiver. Figure R6 shows the PSD at the output of\nthe photodetector measured with a spectrum analyzer at a frequency of 56 MHz (near the\nmechanical resonant frequency) as a function of optical power P directly incident on the\ninc\nphotodetector. The orange square shows the noise equivalent power of the photodetector\nspecified by the supplier (APD 430A2 by Thorlabs). The dots show PSD averaged between\nto the maximum and the minimum PSD within this frequency range. Given that P ranges\ninc\nfrom 2 µW to 4 µW in our nanomechanical experiment, and given that our nanomechan-\nical signals are measured within a ≃ 2 MHz-wide frequency band (the bandwidth of our\nsoftware-defined radio receiver), we estimate that the variance of voltage fluctuations that\noriginate from the output of the photodetector and that appear at the input of the SDR\nreceiver ranges from ≃ 0.2 mV2 to ≃ 0.5 mV2. Our value of σ2 ≃ 0.19 mV2 measured above\nis at the lower bound of this range. We surmise that digital filtering in our GNU Radio\ncodes, including the root raised cosine filter at the receiver, contributes to lowering the noise\nwe measure in our nanomechanical experiment.\nFinally, we estimate the measurement background of our SDR receiver. As an input\nsignal, we use a voltage noise with a Gaussian statistics and a standard deviation of 1 mV\nwithin a 100 MHz bandwidth. The variance is calibrated for a 50 Ohm impedance load,\nwhich is the input impedance of our SDR receiver. Figure R6b shows the PDF of the in-\nphase components of the signal at the output of the SDR receiver acquired for 30 s using a\nreceiver bandwidth of 2.4 MHz. Using the same measurement settings, Fig. R6c shows the\ncorresponding PDF where the only noise present comes from the 50 Ohm output impedance\n8\nof the noise source (the output of the source is nominally set to ‘off’). Using the mea-\nsurements in Fig. R6b as a calibration, we estimate that the smallest amplitude of voltage\nfluctuations the SDR can measure is larger than 0.1 mV and is limited by digitization error.\nOverall, noise in our measurements mostly corresponds to voltage fluctuations at the\noutput of the photodetector. Lowering the intensity of this noise would enable higher order\ndigital modulation (m-QAM with m > 4) at lower drive power.\n10-12\n10-13\n10-14\n10-15\n0 2 4 6 8\nP ( W)\ninc\n)zH/2V(\nzHM\n65\nta\nS\nvv\n6\n4\n2\n0\n-4 -2 0 2 4\nI (mV)\nout\n)1-Vm(\nFDP\n0.4\n0.2\n0\n-4 -2 0 2 4\nI (mV)\nout\n)1-Vm(\nFDP\na b\nc\nFigure R6: Supplementary Fig. 10. Identifying the main source of noise in our mea-\nsurements. (a) Power spectral density (PSD) of voltage fluctuations at the output of the\nphotodetector at a frequency of 56 MHz. Dots: PSD averaged between 55 MHz and 57 MHz.\nThe upper and lower bounds of the gray shaded area correspond to the maximum and the\nminimum PSD within this frequency range. (b) Power density function (PDF) of a cali-\nbrated white Gaussian noise with a standard deviation of 1 mV measured at the output of\nthe software-defined radio receiver. (c) PDF of the signal at the output of the SDR with the\ncalibration noise source switched off.\nThe Reviewer writes:\n3) Figure S5 is very interesting and understandable at first glance compare with\nFigure 3 (and maybe Figure 2). I propose to switch them or combine the two\nfigures. Maybe I missed some detail here. However, Figure 3e and 3f are also\nintriguing.\nWe thank the Reviewer for this comment. To address it, we have created two new figures\nin the main text, namely Fig. 2 [Fig. R7] and Fig. 3 [Fig. R8], which appear below.\n9\n20\n15\n10\n5\n18\n14\n10\n52 54 56 58 52 54 56 58 52 54 56 58 52 54 56 58\nf (MHz) f (MHz) f (MHz) f (MHz)\nd d d d\n)Vm(\nL\na b c d\n-31 dBm -28 dBm -25 dBm -22 dBm\ne f g h\nε\n)Bd(\n0\nsignal at R = 0.5×106 Hz. (a-d) Vector length L of I and Q as a function of drive\nsym out out\nfrequency f . (e-h) Signal-to-noise ratio ε(f ). Drive power at the gate P = −31 dBm (a,\nd d d\ne), −28 dBm (b, f), −25 dBm (c, g), and −22 dBm (d, h). Dots are measured data.\nSolid\nlines are calculated traces. Vdc = 13 V in all panels.\ng\n10\n5\n54 56 58 54 56 58 54 56 58 54 56 58\nf (MHz) f (MHz) f (MHz) f (MHz)\nd d d d\n)Vm(\nL\na b c d\n-31 dBm -28 dBm -25 dBm -22 dBm\ne f g h\nε\n)Bd(\n0\n18\n14\n10\nsignal at R = 106 Hz. (a-d) Vector length L of I and Q as a function of drive\nsym out out\nfrequency f . (a-d) Vector length L of I and Q as a function of drive\nsym out out\nfrequency f . (e-h) Signal-to-noise ratio ε(f ). Drive power at the gate P = −31 dBm\nd d d\n(a, e), −28 dBm (b, f), −25 dBm (c, g), and −22 dBm (d, h). Dots are measured data\n[measurement noise in (f) is a little higher than in (e, g, h)]. Blue (orange) solid lines are\ncalculated traces with noise on (off). Vdc = 13 V in all panels.\ng\n4) In the SI, the Figure and the text are a mess, please put the good figure with\nthe good Sup note parts. It is for clarity. We have GNU flow chart in sup note\npart S2 or S3\nWe thank the Reviewer for this comment. We have improved the arrangement of the\nfigures and of the text related to them in Supplementary Information.\nThe Reviewer writes:\n5) It is difficult to have a good feeling of the error vector magnitude eps (f): is it\nrelated to the signal treatment, to the input noise of the image, to the noise of\nthe vibration? Maybe a better explanation on this term is necessary since it is\na very important in the paper and discussion\nWe thank the Reviewer for addressing the origin of the spread of the symbol clouds in\nthe reciprocal of the error vector magnitude ε, and the dependence of this spread on drive\nfrequency f . This point is indeed essential. To address it, in section “Results” we write:\nd\nThe simplest quantity we extract from the constellation diagram is the time-averaged\nlength of the I − Q vector in one quadrant, L = ⟨|I + iQ |⟩. The spread of the\nout out\nsymbol clouds (see Fig. 1e) is informative as well. We define it within one quadrant as\nd = ⟨(I − ⟨I ⟩)2 + (Q − ⟨Q ⟩)2⟩1/2. One may interpret the logarithm of the ratio,\nout out out out\nε = 20log (L/d), as a signal-to-noise ratio, because noise at the output of our receiver and\n10\nthe dephasing of vibrations by the mechanical response both contribute to the spread of\nthe clouds (see below). In digital communications, ε is known as the reciprocal of the error\nvector magnitude of the symbols. The larger ε is, the more separated the symbol clouds are\nand the smoother the video playback will be.\nWe elaborate on the meaning of ε in the following response.\nThe Reviewer writes:\n6) I find very interesting the part on the discussion about the minima and its\norigin. Thispointisstronglyrelatedtothepreviousone. Itispossibletosimulate\nthis from the motion equation and it is due to the dephasing. I understand\npartially the discussion but the argument about the phase origin and not the\nnoise from only Figure 2e and 2f is difficult to understand in the present form.\nIs it possible to improve this part?\nWe thank the Reviewer for this comment. It prompted us to calculate ε(f ) without\nd\nadditive white Gaussian noise in the drive or at the receiver. To our surprise, we found\nthat we could reproduce the most important features in our measured ε(f ), which are its\nd\nmaximum at resonance and its minima off resonance, without any fit parameter. In the\nsection “Discussion”, we write:\nd m\nσ2 [the variance of additive white Gaussian noise at the receiver, a fit parameter confirmed\nby subsequent measurements] is only needed to capture the decay of ε(f ) away from f ,\nd m\nbut it has only little incidence on the most interesting features that are the minima in ε(f ).\nd\nTo show this, in Fig. 3e-h [Figs. R8e-h] we display our calculations for ε(f ) with no noise\nd\nadded, that is, without any fit parameter (orange traces). In this case, ε increases as f is\nd\nstepped away from f , while the features near f are similar to those with noise on. This\nm m\nobservation indicates that the spread of the symbol clouds d(f ) near f is caused by a\nd m\nprocess other than noise. This process is the loss of coherence of the response caused by\nthe superposition of multifrequency vibrations with different amplitudes and phases. It is\na distinctive feature in vibrations driven by a coherent, multifrequency signal that is unseen\nin vibrations actuated by single tone drives or by a force noise. It is a result of the resonator\nacting as a filter in the case where the spectrum of the drive is broader than the response,\nW ≳ ∆f. If the drive were narrowband, W ≪ ∆f, then the symbol components (I ,Q )\nin in\nwould simply be transformed into (I ,Q ) by a rotation through the phase angle of the\nout out\nresponse. By contrast, every frequency component in the broadband drive gets transduced\ninto a vibrational frequency component with a different amplitude and a different phase.\nWhere f is far away from f , and provided that receiver noise is low, (I ,Q )\nd m out out\nstill result from a simple rotation through a single angle (≃ 0 or −π), since the magnitude\nand the phase of the response depend on frequency only weakly. In this case, I ∝ ±I\nout in\nand Q ∝ ±Q . Where f ≃ f , W ≃ ∆f, and where the response is linear, we find\nout in d m\nI ∝ Q and Q ∝ −I (Supplementary Note 6), so the symbols in the output stream are\nout in out in\nsimply exchanged (with no incidence on the video demodulation, as shown in Supplementary\nNote 7). However, the dephasing effect of the response on I and Q is strongest where\nout out\n|f − f | ≃ W/2 and W ≳ ∆f. This case is depicted in Figs. 1c-d, where the upper fre-\nd m\nquency edge of the drive spectrum is nearly resonant while the lower frequency edge is away\n˜\nfrom resonance. There, we find that I (t) contains dephased frequency components both\nout\n˜ ˜ ˜\nfrom I (t) and from Q (t), and so does Q (t) [Supplementary Note 6]. As a result, the\nin in out\n˜ ˜\ndecimation of I (t) and Q (t) produces broad distributions of I and Q going through\nout out out out\nzero, which decreases ε. Experimentally, |f −f | ≃ W/2 is indeed where we find minima\nd m\nin ε(f ) for R = 0.5×106 Hz and 106 Hz in the linear regime (low P ).\nd sym d\nIn the nonlinear regime (high P ), our model also reproduces the deepening of the local\nd\nminimum (dip) in ε(f ) near the largest slope in L(f ) and the frequency shift of the dip\nd d\n∆f as P is increased. We show in Supplementary Note 8 that ∆f is nearly propor-\ndip d dip\ntional to the square of the change in vector length ∆L , ∆f ∝ ∆L2 . This behavior\ndip dip dip\nis consistent with the quartic relation between the frequency of a vibrational mode and the\nvibrational amplitude caused by weak conservative nonlinearities in the case where the mode\nis driven by a single tone drive [65, 66, 67]. It is also consistent with the lineshape broadening\nof the mechanical response induced by the interplay between conservative nonlinearities and\na force noise [13, 14, 68], as described in Supplementary Note 8.\n12\n60\n50\n40\n30\n20\n10\n0\n)mn(\nL\n25\n20\n15\n10\n5\n)Bd(\n0.4\n0.3\n0.2\n0.1\n0\n52 53 54 55 56 57 58 59\nf (MHz)\nd\nREB\na\nb\nc\n0\n-1\n-2\n-3\n-4\n0 1000 2000\nL2 (nm2)\ndip\n)zHM(\nf\npid\nFigure R9: Supplementary Fig. 12. Dip in ε and peak in BER in the nonlinear regime.\n(a) L(f ). Circles indicate the value of L at f where ε dips. (b) ε(f ). (c) BER(f ). In\nd d d d\n(a-c), P ranges from −31 dBm (lowest trace in blue) to −4 dBm (leftmost trace in red).\nd\nInset to (c): frequency shift ∆f of the dip in ε(f ) as a function of the square of the change\ndip d\nin vector length ∆L2 , where frequency shifts and vector length changes are with reference\ndip\nto calculations at P = −34 dBm. P ranges from −34 dBm (∆L = 0) to 0 dBm. The\nd d dip\neffective mass and the coefficients of the conservative and dissipative forces are taken from\nSupplementary Note 4. No addititive white Gaussian noise is used. R = 106 Hz in all\nsym\npanels.\nnonlinear regime”. It reads:\nThe dip in ε(f ) and the peak in BER(f ) are striking features in our measurements.\nd d\nWe address them theoretically in the case of the linear regime in Supplementary Note 6.\nTo understand how they relate to L(f ) in the nonlinear regime, in Fig. R9 we show our calcula-\nd\ntions for L [Fig. R9a], ε [Fig. R9b], and BER [Fig. R9c] as a function of f at R = 106 Hz,\nd sym\nfor P ranging from −31 dBm to −4 dBm, and without addititive white Gaussian noise.\nd\nThe dip in ε(f ) and the peak in BER(f ) appear concurrently.\nd d\nThe inset to Fig. R9c shows additional calculations with a finer f step where we resolve\nd\nthe dip in ε(f ) and the corresponding I −Q vector length at the dip, L , for P ranging\nd dip d\nfrom −34 dBm to 0 dBm. There, the frequency shift ∆f of the dip is plotted as a function\ndip\nof the square of the change in vector length ∆L , where frequency shifts and vector length\ndip\nchanges are with reference to calculations at P = −34 dBm. Our model shows that ∆f\nd dip\nis proportional to ∆L2 . This behavior is consistent with the quartic relation between the\ndip\nfrequency of a vibrational mode and the vibrational amplitude caused by weak conservative\nnonlinearities in the case where the mode is driven by a single tone drive [65, 66, 67]. It\nis also consistent with the lineshape broadening of the mechanical response induced by the\ninterplay between conservative nonlinearities and a force noise [68]. There, vibrational am-\nplitude fluctuations are converted into resonant frequency fluctuations. If the characteristic\namplitude of frequency fluctuations is larger than the linear decay rate of the resonator, the\nmeasured response is a superposition of instantaneous responses at various resonant frequen-\ncies. As a result, the frequency linewidth of the measured response is proportional to the\nvariance of the amplitude fluctuations [13, 14].\nThe Reviewer writes:\n7) In Figure 2, especially e and f, I expect the same scale in frequency in order\nto compare them, I also can expect the same discussion with figure 2e and 2f in\nFigure 3 with stronger non linearities and stronger minima.\nWe thank the Reviewer for this comment. We have moved Figs. 2e, f in our previ-\nous manuscript to Figs. 1c, d in the current manuscript [Figs. R10c, d]. These panels are\nreproduced below.\nFurthermore, we have entirely rewritten the discussion that used to address Figs. 2e and\n2f in the previous version of the manuscript, and that now addresses Figs. 2 and 3 in the\ncurrent version [Figs. R7 and R8]. In the section “Discussion”, we write:\nTo understand these observations, we numerically solve the equation of motion for the\nresonator in response to the multifrequency driving force F (t). The equation reads [63, 64]:\nd\n(cid:20) (cid:21) (cid:20) (cid:21)\nω η α F (t)\nz¨+ m + z2 z˙ + ω2 + z2 z = d , (4)\nQ m m m m\nm eff eff eff\nwith z the resonator displacement, ω = 2πf the angular resonant frequency, η the non-\nm m\nlinear damping coefficient, and α the Duffing elastic constant. Q ≃ 100, η ≃ 1.2 ×\nm\n14\n)1-zH\n2N\n03-01(\nS\n1 0\n0.5\n0\n55 56 57\nχ )\n(gra\nπ -\n2\nπ -\nχ\n).mron(\n|\n|\nc 104\n102\nf\n100 d\nd\nf m\nf (MHz)\nFigure R10: Figs. 1c-d. (c) Averaged power spectral density of driving force S(f) as a\nfunction of spectral frequency calculated with a raised cosine filter, for a drive power at the\ngate P = −31 dBm, and for R = 106 Hz. S(f) is superimposed on the background of\nd sym\nthermal forces. (d) Magnitude and argument of the linear mechanical response χ calculated\nfor Q = 100.\nm\n105 kg m−2 s−1 and α ≃ −1014 kg m−2 s−2 are estimated from the response to a single\ntone drive (Supplementary Note 4). m ≃ 2×10−17 kg is estimated from the shape of the\neff\n˜ ˜\nresonator and the number of graphene layers. As in the experiment, I (t) and Q (t) that\nin in\ndefine the baseband signal of F (t) are computed from two streams of pulses. The height of\nd\nthe pulses takes on the values ±V randomly at a rate R . Pulse streams are processed\nin sym\nwith a raised cosine filter (Fig. 1b). The use of this filter is justified in the case where most\nof the experimental noise appears at the output of the receiver, as in our measurements. For\ncompleteness, we have also considered the case where, in addition to noise at the receiver, a\nwhite Gaussian force noise is added to the drive. This case requires a cascade of two RRC\nfilters, one at the transmitter and one at the receiver. Yet, we have found that the simpler\nscenario based on receiver noise only is sufficient to explain our measurements. Demodu-\n˜ ˜\nlating z(t) yields I (t) = 2h(t)∗[ξz(t)cos(2πf t)] and Q (t) = 2h(t)∗[ξz(t)sin(2πf t)],\nout d out d\nwhere h is a lowpass filter, ξ is a conversion factor in units of V m−1, and ∗ is the convo-\n˜\nlution operator. We obtain the symbol components I and Q by decimating I (t) and\nout out out\n˜\nQ (t) at the rate R , taking into account the delay between input and output signals\nout sym\ncaused by the raised cosine filter. We find that this simple model reproduces our measured\nL(f ) and ε(f ) data [calculations are shown as solid blue traces in Figs. 2 and 3 (Figs. R7\nd d\nand R8)]. The only fit parameter in our analysis of L(f ) is ξ ≃ 7 × 105 V m−1. The\nd\nlineshape of L(f ) is broader than the response to a drive with unmodulated in-phase and\nd\nquadrature components at the same P (Supplementary Note 1). To analyze ε(f ), we model\nd d\n˜ ˜\nfluctuations δI and δQ at the output of the receiver as additive white Gaussian noise\nout out\nwith variance σ2. This noise contributes to the spread of our calculated symbol clouds as\nd2 → d2+⟨δI ˜2 ⟩+⟨δQ ˜2 ⟩ = d2+2σ2. We use σ2 ≃ 3.9×10−19 m2 as a single fit parameter\nout out\nin our analysis of ε(f ). Measurements of the noise at the output of the receiver are shown\nd\nin Supplementary Note 5.\nOnce again, we thank the Reviewer for carefully reading our manuscript and for offering\nthoughtful comments.\nResponse to Reviewer #2\nWe thank the Reviewer for taking the time to consider our manuscript and for offering help-\nful comments.\nThe Reviewer writes:\nThis paper concerns the continuous transmission of data via transduction of a\ndigital electronic signal into the vibration of a graphene drum. The authors claim\nthat thegraphene drum canencode this signal into an optical fieldby modulation\nof the absorption, and the signal can be extracted by demodulation with a series\nof commercial signal processing equipment. Two signals in quadrature appear\nto provide data redundancy to reduce error. Nonlinear effects were found to be\na significant factor in determining the bit error rate (BER) at high drive power.\nThe authors provide a video they claim was transmitted via the graphene. The\ntunability of the graphene enables the selective transmission of video signals at\ndifferent frequencies.\nThe paper is well-written and will likely be highly cited, given that it demonstrates\nan application of graphene NEMS in technology. Having said that, the scientific\nandtechnologicalsignificanceoftheresultsisquitelow. Thedatatransmission\nrate is significantly lower than existing commercial standards and relies on a\ncomplex laboratory setup, and the observed nonlinear effects have been ex-\ntensively studied. That said, we recommend publication, provided the following\ncomments can be addressed.\nWe thank the Reviewer for accurately summarizing our work. We also thank the Re-\nviewer for commenting on the scientific and technological significance of our results. This\ncomment prompted us to rewrite most of our manuscript to clarify the importance of our\nwork in the field of nanomechanics and to address its possible applications in radio commu-\nnications. We elaborate on the changes we have made to our manuscript in our responses to\nthe Reviewer’s comments that follow.\nThe Reviewer writes:\n1. The paper refers to the system as a “video receiver”, however, the authors\nonly demonstrate the use of a graphene membrane to transduce the electrical\nsignal into mechanical vibrations and an optical field, which are arguably less\nuseful than the original electrical signal. The authors should clarify whether the\ngraphene behaves as a receiver and not a transmitter, and on the technological\nsignificance of this device.\nWe thank the Reviewer for this comment. We clarify the difference between a nanome-\nchanicalreceiverandananomechanicaltransmitterinournewintroduction. There,wewrite:\nAppealing demonstrations of the physics of multifrequency vibrations are provided by\nnanomechanical radio receivers and transmitters for frequency modulated (FM) signals\n16\n[39, 40]. Nanomechanical FM receivers have been realized with suspended nanotubes [23,\n41, 42, 43] and with nanomachined silicon tuning forks [44]. The FM voltage waveform is\ncapacitively coupled to the receiving resonator, creating a modulated driving force that the\nresonatorpassivelyrespondsto. Bycontrast, nanomechanicalFMtransmittersouputamod-\nulated radio frequency signal. They have been demonstrated with suspended graphene [45]\nand with nanomachined polysilicon disks [46, 47]. In these devices, the vibrating element\nis part of an electrical positive feedback loop that drives it to a regime of self-sustaining\noscillations. The frequency of the oscillations is modulated by a stream of low frequency\nvoltage pulses or by an audio waveform, which creates a slowly varying capacitive force\nbetween the oscillator and a nearby electrode that slowly modulates the spring constant.\nSuch a technique of ‘keying’ the slowly modulated components, or baseband signal, of the\nvibrations is reminiscent of the use of a Morse key in telegraphy. In brief, nanomechanical\nradio receivers and transmitters embody the richness of mesoscopic vibrations driven by\nmultifrequency drives. Importantly, they also offer the additional reward of an audible copy\nof the vibrations.\nFurthermore, we address the technological significance of our device at the end of our\nnew introduction and in the concluding part of our manuscript. Our new introduction reads:\nWe also envision experimental applications at a time of renewed interest in communica-\ntions within the HF and the VHF bands [48]. Even though their data rates are low, HF\nand VHF communications still provide the most robust and reliable means of long-distance\ncommunications in situations where access to other channels is compromised. Future HF\ncommunications will require the ability to occasionally transmit and receive weak signals\nencoded with compressed images and videos [48], while narrowband digital television has al-\nready been experimented with in the VHF band [49]. Our results show that nanomechanical\nresonators may act as passive receivers for such signals, as they do for radiowaves modulated\nat audio frequencies.\nIn the concluding part, we write:\nIn the spirit of previous nanomechanical devices employed as signal processors [39], we\nenvision that our FLG resonators, and other resonators based on similar principles, may be\nemployed as small footprint, passive radio receivers in the HF and the VHF bands of the\nradio spectrum. With a resonant frequency ranging from ≃ 40×106 Hz to ≃ 60×106 Hz\n(Fig. 5a), our resonator operates in the 6-meter subband that is internationally allocated to\nthe use of amateur radio. In this subband, digital amateur television (DATV) broadcasting\nhas been conducted using signals with a 300 kHz bandwidth [49]. In the United Kingdom,\nDATV with a broader bandwidth of up to 500 kHz is allowed in the 2-meter amateur radio\nband (144–148 MHz) [49]. Smaller diameter FLG resonators, or higher order vibrational\nmodes, can operate in this band. The filtering bandwidth of our resonator, and the maxi-\nmum 4-QAM data rate of its transduction 2R∗ ≃ 2×106 bps, are then fully compatible\nsym\nwith DATV. Arguably, our simple experimental setup may still not be practical enough for\nreceiving amateur radio messages, but it can be simplified further [51]. The resonator would\nserve the dual purpose of a front-end, tunable bandpass filter and of a passive optocoupler.\nof a receiving antenna [69]. As a filter, it would reject jamming and other interfering signals\nthe way surface acoustic wave filters do, but with the additional benefit of a tunable cen-\nter frequency, and without the fabrication complexity of more advanced tunable front-end\nfilters [70]. As an optocoupler, it would passively convert the radio signal into modulations\nof reflected light intensity through the nanomechanical transduction, effectively eliminat-\ning the problem of radio frequency interferences caused by common-mode currents in the\ncoaxial cable that connects the antenna to the demodulator [69]. Admittedly, the idea of\nemploying nanomechanical resonators as devices for radio communcation mostly possesses\nan academic appeal, given the limitations posed by device reproducibility and thermal drifts\n[40]. However, these resonators may ultimately enable alternate architectures for shortwave\nradioreceivers, facilitatingthedecodingofimagesandvideoscarriedbyweaksignalsthrough\na noisy environment [61].\nThe Reviewer writes:\n2. The authors claim that the broadband video signal is useful as a probe for\nnonlinear effects. I cannot see any particular reason that the video signal is more\nuseful than simple narrowband white noise. Noise-driven nonlinear effects have\nbeen studied in depth by many researchers. See, for instance:\nDykman, et al. PRA 1990; Zhang, et al. PRL 2014; Zhang, et al. PRB 2015;\nMaillet, et al. PRB 2017; Stambaugh & Chan PRB 2006\nThe authors should comment on the differences between their video signal and\ngeneric narrowband white noise.\nWe thank the Reviewer for suggesting we elaborate on the differences between our video\nsignal and narrowband white noise. In our new introduction, we write:\n(...) Abroader responseis alsoadvantageous. For example, a large-bandwidth tuningcir-\ncuit is necessary to isolate a modulated radio signal and capture the entirety of its spectrum,\nwhile filtering away signals in adjacent bands. Similarly, nanomechanical resonators with a\nlarge bandwidth ∆f can respond to nearly resonant forces whose spectra are as broad as\nthe mechanical response [11, 12, 13, 14]. They can also be used to study resonant frequency\nfluctuations induced much below resonance by fluctuators with correlation times as short\nas 1/∆f [15, 16, 17, 18, 19, 20]. In essence, nanomechanical resonators behave as tunable\nbandpass filters for nearly resonant forces and as detectors for nonresonant fluctuators.\nWhile the nanomechanical response to broadband force noise and low frequency fluctua-\ntors has been thoroughly studied, the response to multifrequency coherent drives has thus far\nreceived less attention. Where multifrequency signals are applied, oftentimes only a single\nfrequency component is used to measure the response while the other components are used\nfor different tasks. For example, amplitude modulated [6, 21, 22] and frequency modulated\nsignals [23] have been employed to drive the resonator at the single frequency of the carrier,\nwhile the modulation sidebands have been used to detect the vibrations. Two-tone signals,\nwith one tone near resonance and the second tone outside of the response, have been used to\n18\nstudy a variety of vibrational phenomena. Examples abound and include pump-probe spec-\ntroscopy, where intermode coupling is measured [24]; parametrically-induced mode coupling,\nused to heat or cool vibrations [25, 26, 27, 28] and to engineer nonlinear friction [29]; and\nsymmetry-breaking parametric resonators used for binary information encoding [30, 31, 32].\nBy contrast, the scope of applications for multifrequency signals whose entire spectra drive\nvibrations has been more limited. Two-tone drives have been used to demonstrate signal\nmixing with nonlinear nanomechanical resonators [33, 34]. The interplay of a two-tone drive\nnear resonance and parametric pumps has enabled the realization of mechanical logic gates\n[35]. The combination of a strong and a weak driving tone near resonance has been shown\nto squeeze thermomechanical noise [36]. Resonators driven nonlinearly by two large tones\nnear resonance have displayed unconventional dynamics, with the beating between the tones\nacting as a slow modulation of the bifurcation points [37, 38]. Overall, studying nanome-\nchanical resonators using signals composed of multifrequency components sheds light on the\nphysics of mesoscopic vibrations and enables useful applications.\n(...) Our work contributes to the recent research effort on the physics of resonators driven\nnonlinearly by multiple tones. Yet, where only two tones were used earlier [37, 38], here we\nstudy an unexplored regime where the response is coherently driven by a near continuum of\nfrequency components within the whole mechanical bandwidth.\nFurthermore, in the section “Results”, we write:\nThe calculated power spectral density S of the driving force F (t), averaged over many\nd\nrealizations of F (t), is shown in Fig. 1c [R11c] as a function of spectral frequency f. We\nd\nconsider a drive power at the gate P = 10log [(1+Γ)2⟨s2(t)⟩/(50×10−3)] ≃ −31 dBm,\nd 10\nwhere ⟨·⟩ averages over time. We use Vdc = 13 V and C′ ≃ 9 × 10−10 F m−1 estimated\ng\nwith COMSOL. S(f) is centered at f and its full spectral bandwidth reads W ≃ 1.3R ,\nd sym\nwhere R = 106 Hz is the symbol rate and 1.3 accounts for the roll-off factor of the root\nsym\nraised cosine filter. The area under S(f) is the variance of F (t); it does not depend on\nd\nR , and only depends on the amplitude of the drive. We also emphasize that F (t) is not\nsym d\na delta-correlated noise; it is instead a coherent waveform (Supplementary Note 1).\nIn the section “Discussion”, we write:\nInterestingly, the lineshape of ε(f ) near f is not influenced by the noise at the receiver.\nd m\nσ2 [the variance of additive white Gaussian noise at the receiver, a fit parameter confirmed\nby subsequent measurements] is only needed to capture the decay of ε(f ) away from f ,\nd m\nbut it has only little incidence on the most interesting features that are the minima in ε(f ).\nd\nTo show this, in Fig. 3e-h [Figs. R14e-h] we display our calculations for ε(f ) with no noise\nd\nadded, that is, without any fit parameter (orange traces). In this case, ε increases as f is\nd\nstepped away from f , while the features near f are similar to those with noise on. This\nm m\nobservation indicates that the spread of the symbol clouds d(f ) near f is caused by a\nd m\nprocess other than noise. This process is the loss of coherence of the response caused by\nthe superposition of multifrequency vibrations with different amplitudes and phases. It is\na distinctive feature in vibrations driven by a coherent, multifrequency signal\n19\n)1-zH\n2N\n03-01(\nS\n1 0\n0.5\n0\n55 56 57\nχ )\n(gra\nπ -\n2\nπ -\nχ\n).mron(\n|\n|\nc 104\n102\nf\n100 d\nd\nf m\nf (MHz)\nFigure R11: Figs. 1c-d (c) Averaged power spectral density of driving force S(f) as a\nfunction of spectral frequency calculated with a raised cosine filter, for a drive power at the\ngate P = −31 dBm, and for R = 106 Hz. S(f) is superimposed on the background of\nd sym\nthermal forces. (d) Magnitude and argument of the linear mechanical response χ calculated\nfor Q = 100.\nm\nthat is unseen in vibrations actuated by single tone drives or by a force noise. It\nis a result of the resonator acting as a filter in the case where the spectrum of the drive is\nbroader than the response, W ≳ ∆f. If the drive were narrowband, W ≪ ∆f, then the symbol components (I ,Q )\nin in\nwould simply be transformed into (I ,Q ) by a rotation through the phase angle of the\nout out\nresponse. By contrast, every frequency component in the broadband drive gets transduced\ninto a vibrational frequency component with a different amplitude and a different phase.\nWhere f is far away from f , and provided that receiver noise is low, (I ,Q )\nd m out out\nstill result from a simple rotation through a single angle (≃ 0 or −π), since the magnitude\nand the phase of the response depend on frequency only weakly. In this case, I ∝ ±I\nout in\nand Q ∝ ±Q . Where f ≃ f , W ≃ ∆f, and where the response is linear, we find\nout in d m\nI ∝ Q and Q ∝ −I (Supplementary Note 6), so the symbols in the output stream are\nout in out in\nsimply exchanged (with no incidence on the video demodulation, as shown in Supplementary\nNote 7). However, the dephasing effect of the response on I and Q is strongest where\nout out\n|f − f | ≃ W/2 and W ≳ ∆f. This case is depicted in Figs. 1c, d [Figs. R11c, d], where the upper frequency edge of the drive spectrum\nd m\nis nearly resonant while the lower frequency edge is away from resonance. There, we find\n˜ ˜ ˜\nthat I (t) contains dephased frequency components both from I (t) and from Q (t), and\nout in in\n˜ ˜ ˜\nso does Q (t) [Supplementary Note 6]. As a result, the decimation of I (t) and Q (t)\nout out out\nproduces broad distributions of I and Q going through zero, which decreases ε. Exper-\nout out\nimentally, |f −f | ≃ W/2 is indeed where we find minima in ε(f ) for R = 0.5×106 Hz\nd m d sym\nand 106 Hz in the linear regime (low P ).\nd\nFinally, we explain how we build the driving force waveform in our new Supplementary\nNote 1 entitled “Driving voltage and driving force waveforms”. It reads:\nTo build the driving voltage and the driving force waveforms, we start from two streams\nof pulses p = ±1 and q = ±1 with pulse (symbol) rate R . The streams are filtered\nk k sym\neither with a raised cosine (RC) filter or with a root raised cosine (RRC) filter, yielding\nwaveforms p˜(t) and q˜(t) that vary continuously in time t. Where noise is added to the signal\nafter the transmitter and before the receiver, a RRC filter is used to filter the pulses on\nthe transmitter side and another is used to filter the baseband signal on the receiver side.\nA cascade of two RRC filters minimizes intersymbol interference, as does a single RC filter\nwithout noise. p˜(t) and q˜(t) can be expressed as:\n∞ ∞\n(cid:88) (cid:88)\np˜(t) = p g(t−kT ), q˜(t) = q g(t−kT ), (5)\nk sym k sym\nk=0 k=0\nwhere g is the impulse response of the filter and T is the reciprocal of the symbol rate,\nsym\nT = 1/R . For a RC filter, g reads:\nsym sym\n(cid:18) (cid:19)\nπ 1 T\nsym\ng(t) = sinc if t = ± (6)\n4T 2β 2β\nsym\n(cid:16) (cid:17)\n(cid:18) (cid:19) cos πβt\ng(t) =\n1\nsinc\nt Tsym\notherwise, (7)\nT T (cid:16) (cid:17)2\nsym sym 1− 2βt\nTsym\nwhere β is the roll-off factor of the filter. The filtered pulses are used to modulate the\namplitude of the in-phase component and that of the quadrature component of the waveform\n20\nx(t) defined as:\nx(t) = p˜(t)cos(ω t)+q˜(t)sin(ω t), (8)\nd d\nwhere ω is the angular frequency of the drive. Given a peak voltage amplitude V ,\nthe\nd in\ndriving voltage waveform s(t) is defined as:\n˜ ˜\ns(t) = V x(t) = I (t)cos(ω t)+Q (t)sin(ω t), (9)\nin in d in d\n˜ ˜\nwhere I (t) and Q (t) are the baseband in-phase and quadrature components of the drive,\nin in\nrespectively. They are defined as:\n˜ ˜\nI (t) = V p˜(t), Q (t) = V q˜(t). (10)\nin in in in\nThe power of s(t) that would be dissipated across a 50 resistor in units of dBm is:\n(cid:18) ⟨s2(t)⟩ (cid:19) (cid:18) V2⟨x2(t)⟩ (cid:19)\nP = 10 log = 10 log in , (11)\nin 10 50×10−3 10 50×10−3\nwhere ⟨·⟩ averages over time. Hence, the peak voltage V reads:\nin\n(cid:18) 10Pin/10 ×50×10−3(cid:19)1/2\nV = . (12)\nin ⟨x2(t)⟩\nThe total voltage between the gate and the resonator is the total voltage of the standing\nwave at the gate:\nV = (1+Γ)V , (13)\nd in\nwhere Γ ≃ 1 is the reflection coefficient of s(t) at the gate. The equivalent driving power\nthat would be dissipated across a 50 Ohm resistor in units of dBm is then:\n(cid:20) (1+Γ)2⟨s2(t)⟩ (cid:21)\nP = 10 log = 20 log (1+Γ)+P ≃ 6+P . (14)\nd 10 50×10−3 10 in in\nThe driving force waveform reads:\n(cid:12)dC (cid:12)\nF (t) = |Vdc|(1+Γ)(cid:12) g(cid:12)s(t), (15)\nd g (cid:12) dz (cid:12)\nwhere Vdc is the dc voltage applied between the resonator and the gate, and dC /dz is the\ng g\nderivative with respect to the dispacement of the resonator of the capacitance between the\nresonator and the gate.\nFor comparison, a driving force with the same peak amplitude but whose in-phase and\nquadrature components are not amplitude-modulated reads:\nFunmod(t) = Funmod[cos(ω t)+sin(ω t)] = 21/2Funmodcos(ω t−π/4), (16)\nd d d d d d\nwith\n(cid:12)dC (cid:12)\nFunmod = |Vdc|(1+Γ)V (cid:12) g(cid:12). (17)\nd g in(cid:12) dz (cid:12)\n21\n7\n6\n5\n4\n3\n2\n1\n0\n54 55 56 57 58\n/2 (MHz)\nd\n)mn(\nL\n106\n104\n102\n100\n54 55 56 57 58\n/2 (MHz)\n)zH/2m\n92-01(\nS\nzz\na\nb\nFigure R12: Supplementary Fig. 1. Linear response to the multifrequency driving force.\n(a) I−Q vector length L as a function of angular drive frequency ω for symbol rate R =\nd sym\n0.5×106 Hz(bluetrace)andR = 106 Hz(redtrace). Thegreytraceshowstheresponseto\nsym\nadrivingforcewhosein-phaseandquadraturecomponentsarenotamplitude-modulated, see\nEq. (16). Driving power P = −34 dBm. (b) Averaged single-sided power spectral density of\nd\namplitude fluctuations S as a function of angular spectral frequency ω in the linear regime\nzz\nfor P = −31 dBm. Blue trace: R = 0.5 × 106 Hz, ω /2π = 56 × 106 Hz. Red trace:\nd sym d\nR = 106 Hz, ω /2π = 56×106 Hz. Orange trace: R = 106 Hz, ω /2π = 55.5×106 Hz.\nsym d sym d\nIn both panels, driving force waveforms are computed using a raised cosine filter. Quality\nfactor Q = 100, effective mass 2×10−17 kg, resonant frequency 56 MHz.\nm\nFigure R12a shows our calculations of the linear response L = ⟨I +iQ ⟩ to F (t) for\nout out d\nP = −34 dBm, quality factor Q = 100, effective mass m ≃ 2×10−17 kg, and resonant\nd m eff\nfrequency ω /2π = 56 MHz. The symbol rates R = 0.5×106 Hz (blue trace) and R =\nm sym sym\nd\ndrive (grey trace). The linewidth broadening of L(ω ) as R increases underlies the fact\nd sym\nthat, even where ω is far away from resonance, vibrations can still be driven by the edge\nd\nof the spectrum of the drive near ω ±πW, where 2πW is the full bandwidth of the drive.\nd\nFor completeness, Fig. R12b displays the (averaged) single-sided power spectral density of\namplitude fluctuations S as a function of angular spectral frequency ω in the linear regime\nzz\nfor P = −31 dBm. S (ω) reads:\nd zz\nS (ω) = |χ(ω)|2S (ω), χ(ω) =\n(cid:2)\nm (ω2 −ω2 −iωω /Q )\n(cid:3)−1\n,\n(18)\nzz FF eff m m m\n22\nwhere S (ω) is the single-sided power spectral density of F (t) defined as:\nFF d\n2 ∞\nS (ω) = |F ˆ (ω)|2, F ˆ (ω) = dteiωtF (t), (19)\nFF t d d (cid:2) d\nmeas −∞\nwith t the measurement time of F (t). Figure R12b shows S (ω) calculated for R =\nmeas d zz sym\n0.5 × 106 Hz and ω = ω (blue trace); R = 106 Hz and ω = ω (red trace); and\nd m sym d m\nR = 106 Hz and ω = ω −πR (orange trace).\nsym d m sym\nThe Reviewer writes:\n3. The nonlinear regime features a critical amplitude and frequency at which\ntwo stable oscillation states emerge, as discussed extensively in Dykman’s “Meso-\nscopic Physics” review. The authors should determine this amplitude and\nfrequency and discuss with regards to the features in Figures 3, 4, S5 and S7.\nWe thank the Reviewer for addressing the important role of nonlinearities in our mea-\nsurements. We have created two new figures in the main text that illustrate the changes we\nobserve in the I − Q vector length L and in the reciprocal of the error vector magnitude\nε, as a function of drive frequency f , as the resonator crosses over from the linear to the\nd\nnonlinear regime. These new figures, Figs. 2 and 3 [Figs. R13, R14], appear below.\n20\n15\n10\n5\n18\n14\n10\n52 54 56 58 52 54 56 58 52 54 56 58 52 54 56 58\nf (MHz) f (MHz) f (MHz) f (MHz)\nd d d d\n)Vm(\nL\na b c d\n-31 dBm -28 dBm -25 dBm -22 dBm\ne f g h\nε\n)Bd(\n0\nsignal at R = 0.5×106 Hz. (a-d) Vector length L of I and Q as a function of drive\nsym out out\nfrequency f . (e-h) Signal-to-noise ratio ε(f ). Drive power at the gate P = −31 dBm (a,\nd d d\ne), −28 dBm (b, f), −25 dBm (c, g), and −22 dBm (d, h). Dots are measured data.\nSolid\nlines are calculated traces. Vdc = 13 V in all panels.\ng\n23\n10\n5\n54 56 58 54 56 58 54 56 58 54 56 58\nf (MHz) f (MHz) f (MHz) f (MHz)\nd d d d\n)Vm(\nL\na b c d\n-31 dBm -28 dBm -25 dBm -22 dBm\ne f g h\nε\n)Bd(\n0\n18\n14\n10\nsignal at R = 106 Hz. (a-d) Vector length L of I and Q as a function of drive\nsym out out\nfrequency f . (e-h) Signal-to-noise ratio ε(f ). Drive power at the gate P = −31 dBm\nd d d\n(a, e), −28 dBm (b, f), −25 dBm (c, g), and −22 dBm (d, h). Dots are measured data\n[measurement noise in (f) is a little higher than in (e, g, h)]. Blue (orange) solid lines are\ncalculated traces with noise on (off). Vdc = 13 V in all panels.\ng\nFurthermore, in the section “Discussion”, we now write:\nIn the nonlinear regime (high P ), our model also reproduces the deepening of the local\nd\nminimum (dip) in ε(f ) near the largest slope in L(f ) and the frequency shift of the dip\nd d\n∆f as P is increased. We show in Supplementary Note 8 that ∆f is nearly propor-\ndip d dip\ntional to the square of the change in vector length ∆L , ∆f ∝ ∆L2 . This behavior\ndip dip dip\nis consistent with the quartic relation between the frequency of a vibrational mode and the\nvibrational amplitude caused by weak conservative nonlinearities in the case where the mode\nis driven by a single tone drive [65, 66, 67]. It is also consistent with the lineshape broadening\nof the mechanical response induced by the interplay between conservative nonlinearities and\na force noise [68, 13, 14], as described in Supplementary Note 8.\n24\n60\n50\n40\n30\n20\n10\n0\n)mn(\nL\n25\n20\n15\n10\n5\n)Bd(\n0.4\n0.3\n0.2\n0.1\n0\n52 53 54 55 56 57 58 59\nf (MHz)\nd\nREB\na\nb\nc\n0\n-1\n-2\n-3\n-4\n0 1000 2000\nL2 (nm2)\ndip\n)zHM(\nf\npid\nFigure R15: Supplementary Fig. 12. Dip in ε and peak in BER in the nonlinear regime.\n(a) L(f ). Circles indicate the value of L at f where ε dips. (b) ε(f ). (c) BER(f ). In\nd d d d\n(a-c), P ranges from −31 dBm (lowest trace in blue) to −4 dBm (leftmost trace in red).\nd\nInset to (c): frequency shift ∆f of the dip in ε(f ) as a function of the square of the change\ndip d\nin vector length ∆L2 , where frequency shifts and vector length changes are with reference\ndip\nto calculations at P = −34 dBm. P ranges from −34 dBm (∆L = 0) to 0 dBm. The\nd d dip\neffective mass and the coefficients of the conservative and dissipative forces are taken from\nSupplementary Note 4. No addititive white Gaussian noise is used. R = 106 Hz in all\nsym\npanels.\nThe new Supplementary Note 8 is entitled “Dip in ε and peak in BER in the nonlinear\n25\nregime”. It reads:\nThe dip in ε(f ) and the peak in BER(f ) are striking features in our measurements.\nd d\nWe address them theoretically in the case of the linear regime in Supplementary Note 6.\nTo understand how they relate to L(f ) in the nonlinear regime, in Fig. R15 we show our\nd\ncalculations for L [Fig. R15a], ε [Fig. R15b], and BER [Fig. R15c] as a function of f at\nd\nR = 106 Hz, for P ranging from −31 dBm to −4 dBm, and without addititive white\nsym d\nGaussian noise. The dip in ε(f ) and the peak in BER(f ) appear concurrently.\nd d\nThe inset to Fig. R15c shows additional calculations with a finer f step where we resolve\nd\nthe dip in ε(f ) and the corresponding I −Q vector length at the dip, L , for P ranging\nd dip d\nfrom −34 dBm to 0 dBm. There, the frequency shift ∆f of the dip is plotted as a function\ndip\nof the square of the change in vector length ∆L , where frequency shifts and vector length\ndip\nchanges are with reference to calculations at P = −34 dBm. Our model shows that ∆f\nd dip\nis proportional to ∆L2 . This behavior is consistent with the quartic relation between the\ndip\nfrequency of a vibrational mode and the vibrational amplitude caused by weak conservative\nnonlinearities in the case where the mode is driven by a single tone drive [65, 66, 67]. It\nis also consistent with the lineshape broadening of the mechanical response induced by the\ninterplay between conservative nonlinearities and a force noise [68]. There, vibrational am-\nplitude fluctuations are converted into resonant frequency fluctuations. If the characteristic\namplitude of frequency fluctuations is larger than the linear decay rate of the resonator, the\nmeasured response is a superposition of instantaneous responses at various resonant frequen-\ncies. As a result, the frequency linewidth of the measured response is proportional to the\nvariance of the amplitude fluctuations [13, 14].\nThe Reviewer writes:\n4. The discussion of the two conditions for the BER is confusing, as the authors\nstate that both conditions result in zero error. The authors should clarify how\nthese two “zero-error” conditions result in the observed BER.\nWe thank the Reviewer for this comment. Conditions for low BER are better presented\nas conditions for high ε. In the section “Discussion”, we write:\nInterestingly, the lineshape of ε(f ) near f is not influenced by the noise at the receiver.\nd m\n(...) It is a result of the resonator acting as a filter in the case where the spectrum of the\ndrive is broader than the response, W ≳ ∆f. If the drive were narrowband, W ≪ ∆f,\nthen the symbol components (I ,Q ) would simply be transformed into (I ,Q ) by a\nin in out out\nrotation through the phase angle of the response. By contrast, every frequency component\nin the broadband drive gets transduced into a vibrational frequency component with a dif-\nferent amplitude and a different phase. Where f is far away from f , and provided that\nd m\nreceiver noise is low, (I ,Q ) still result from a simple rotation through a single angle\nout out\n(≃ 0 or −π), since the magnitude and the phase of the response depend on frequency only\nweakly. In this case, I ∝ ±I and Q ∝ ±Q . Where f ≃ f , W ≃ ∆f, and where\nout in out in d m\nthe response is linear, we find I ∝ Q and Q ∝ −I (Supplementary Note 6), so\nout in out in\nthe symbols in the output stream are simply exchanged (with no incidence on the video\n26\ndemodulation, as shown in Supplementary Note 7). However, the dephasing effect of the\nresponse on I and Q is strongest where |f − f | ≃ W/2 and W ≳ ∆f. This case is\nout out d m\ndepicted in Figs. 1c, d [Figs. R11c, d], where the upper frequency edge of the drive spectrum\nis nearly resonant while the lower frequency edge is away from resonance. There, we find\n˜ ˜ ˜\nthat I (t) contains dephased frequency components both from I (t) and from Q (t), and\nout in in\n˜ ˜ ˜\nso does Q (t) [Supplementary Note 6]. As a result, the decimation of I (t) and Q (t)\nout out out\nproduces broad distributions of I and Q going through zero, which decreases ε. Exper-\nout out\nimentally, |f −f | ≃ W/2 is indeed where we find minima in ε(f ) for R = 0.5×106 Hz\nd m d sym\nand 106 Hz in the linear regime (low P ).\nd\nThe Reviewer writes:\n5. The quality factor of the resonator should be mentioned in the main text or\ncaptions when discussing the graphene parameters, instead of at the end.\nWe thank the Reviewer for this remark. We now mention the quality factor of the res-\nonator Q early in the main text.\nm\nThe Reviewer writes:\n6. The BER will likely be highly dependent on the ratio of the relaxation time\nof the graphene to the correlation time of the data signal. So far, the authors\nhave shown the BER at signal correlation times below, and roughly equal to,\nthe graphene relaxation time (Rp = 1 MHz and 0.5 MHz, respectively). A plot\nof the minimum BER versus pulse rate and discussion of the oscillator response\nwith regards to this ratio would be very helpful.\nWe thank the Reviewer for suggesting we investigate the minimum BER and its depen-\ndence on the ratio of the relaxation time of the graphene to the correlation time of the data\nsignal. This suggestion was an eye-opener for us. In the section “Discussion”, we now write:\nFigures 4b-f [Figs. R16b-f] show BER as a function of f measured at R = 0.5×106 Hz\nd p\n(blue dots) and at R = 106 Hz (orange dots) from the linear regime to the nonlinear one\np\n[P increases from (b) to (f)]. At each f , BER is computed from a single measurement\nd d\nof an output bit stream acquired for 15 s. We find that BER at R = 0.5 × 106 Hz\nsym\nis always lower than at R = 106 Hz, indicating that BER , the minimum of BER,\nsym min\ndepends on R . At the largest P and at R = 0.5 × 106 Hz [Fig. 4f (Fig. R16f)], we\nsym d sym\nmeasure BER ≃ 5×10−6, corresponding to 5 misread bits for every million bits sent to\nmin\nthe resonator. It is limited by noise in our setup (Supplementary Notes 3 and 5).\n27\n1\n0\n0 20 40 60 80 100 120 140 160\nt (µs)\n10-6\n54 56 58 54 56 58 54 56 58 54 56 58 54 56 58\n20\n20\n10\n15\n0 10\n5\n-10\n0\n-20 10-6\n-20 -10 0 10 20 0.5 1 1.5\nREB\n100\n10-2\n10-4\nf (MHz) f (MHz) f (MHz) f (MHz) f (MHz)\nd d d d d\neulav\ntiB\nI (mV)\nout\n)Vm(\nQ\ntuo\na\nb c d e f\ng h i 100\n10-2\n10-4\n104 105 106 107\nR (Hz)\nsym\n)2-01(\nREB\nnim REB\nnim\nR (106 Hz)\nsym\nFigure R16: Fig. 4. Bit error ratio (BER) of the nanomechanical transduction. (a)\nBitstreamatthetransmitter(hollowsquares)andbitstreamatthereceiver(dots)measured\nat R = 0.5×106 Hz. Two bits make one symbol in 4-QAM, so the bit rate is twice the\nsym\nsymbol rate. (b-f) BER as a function of drive frequency f at R = 0.5 × 106 Hz (blue\nd sym\ndots) and at R = 106 Hz (orange dots) for drive power P = −34 dBm (b), −31 dBm\nsym d\n(c), −28 dBm (d), −25 dBm (e), and −22 dBm (f). (g) Constellation diagram for I\nout\nand Q at P = −25 dBm with f set to the BER peak frequency in (e). Light (dark)\nout d d\nblue dots are measurements (calculations). (h) Minimum of BER in the linear regime as\na function of R . Grey traces are calculated without (dark grey) and with added noise\nsym\nat the receiver (light grey). Up triangle: ∆f = 104 Hz. Down triangle: ∆f = 105 Hz.\nSquare: ∆f = 106 Hz. Red dots show measurements at P = −28 dBm. (i) Measured\nd\nBER as a function of R . From top (red) to bottom (blue), P = −28 dBm, −25 dBm,\nmin sym d\n−22 dBm, −19 dBm. Measurements in (h, i) were performed in a noisier environment than\nmeasurements shown in (b-g). Vdc = 13 V in all panels.\ng\nWith regard to applications, we now consider the dependence of BER on R .\nmin sym\nWe\nwish to find out how R∗ , the largest R at which BER is small, changes with the me-\nsym sym min\nchanical linewidth ∆f. We identify R∗ as the filtering bandwidth of the resonator for the\nsym\nvideo signal. We focus on BER at resonance in the linear regime, where BER(f ) = BER\nd min\n(Fig. 4b) [Fig. R16b]. In Fig. 4h [Fig. R16h], we plot BER as a function of R calculated\nmin sym\nusing Eq. (1) for ∆f = 104 Hz, 105 Hz, and 106 Hz (grey traces from left to right). Dark\ntraces are calculated with no noise added, while light traces are calculated with additive\nGaussian noise of variance σ2 used in Figs. 2 and 3 [Figs. R13, R14]. Every trace without\n28\nnoise shows a kink near R ≃ 3.5∆f, which marks an upper bound for R∗ . Additive\nsym sym\nnoise restricts R∗ to lower values, especially for larger ∆f (lower Q ), as BER starts to\nsym m min\nincrease from a lower R . Red dots in Fig. 4h [Fig. R16h] represent BER (R ) mea-\nsym min sym\nsured at P = −28 dBm, at the onset of the nonlinear regime, and in the presence of noise.\nd\nWe extract the experimental filtering bandwidth R∗ ≃ 106 Hz. BER data measured\nsym min\nover a range of P are displayed in Fig. 4i [Fig. R16i] (see also Supplementary Note 10).\nd\nThey show, somewhat expectedly, that nanomechanical information encoding incurs less in-\nformation loss at higher drive power.\nOnce again, we thank the Reviewer for carefully reading our manuscript and for offering\ninsightful comments.\nResponse to Reviewer #3\nWe thank the Reviewer for taking the time to consider our manuscript and for offering help-\nful comments.\nThe Reviewer writes:\nThe paper presents a graphene transducer that converts electrical signals into an\noptical signal. The authors use the graphene system to transmit a digital video\nsignal, a digitally modulated driving force, and create a ”video receiver”. On the\nonehand,themotivationtobuildavideoreceiverfromahighlycomplexgraphene\nresonator experiment is highly questionable. Electronic video receivers are a\ncommoditythatischeapand ubiquitous. Itis notclearhowagrapheneresonator\ncould contribute to any improvement. The chance, that such a graphene-based\nvideo receiver finds its way into a consumer product is quasi-zero.\nWe thank the Reviewer for this comment. Our manuscript was originally motivated by\nthe wish to share our amazement at demonstrating the encoding of a video in nanomechan-\nical vibrations. It was the result of much digital signal processing work that, we believed,\nwould be worth sharing with the community. It not only shows an improvement over ex-\nisting nanomechanical radio receivers for audio-modulated radio frequency signals, but it\nalso demonstrates an approach to freely customize nanomechanical measurements with a\nsoftware-defined radio, unconstrained by the limited measurement functions of standard\nmeasuring equipment. With the benefit of hindsight, and in light of the Reviewer’s com-\nments, we have realized that our earlier manuscript gave the false impression that we were\npresenting a prototype destined for commercialization. That impression may be the result of\nour enthusiasm for our nanomechanical video encoding. We have rewritten the paper in its\nentirety to clearly present our motivations and to place our work in its proper context, which\nis the physics of mesoscopic vibrations driven by a multifrequency coherent driving force. In\nthe concluding part of the paper, we propose that our work may inspire applications in radio\ncommunications. Finally, we have replaced the title of our manuscript with one that better\nsummarizes our work. The new title reads “A few-layer graphene nanomechanical resonator\ndriven by digitally modulated video signals.”\nchanics. In our new introduction, we write:\nNanomechanical resonators enable the study of mesoscopic vibrations within individual\nsystems that can be tuned and controlled [1, 2, 3, 4]. Owing to their small masses, they\ntransduce weak driving forces near resonance into vibrations of sizeable amplitude. Their\nfrequency response resembles that of RLC tuning circuits in analogue radio receivers, with\nthe force playing the role of the driving voltage and the mechanical displacement playing the\nrole of the circuit charge. Where the resonant frequency of the electrical response can be\ntuned with a variable capacitor or a variable inductor, the resonant frequencies of the me-\nchanical response, which are the frequencies of the vibrational modes, can be adjusted in-situ\nby harnessing capacitive [5, 6] and dielectric forces [7], and by straining the resonator either\nmechanically [8] or through its thermal expansion [9, 10]. As with the electrical response,\nthe frequency bandwidth of the mechanical response is determined by dissipation and by\nresonant frequency fluctuations [3]. A narrower electrical response enhances frequency selec-\ntivity, while a narrower mechanical response facilitates the sensing of smaller driving forces.\nA broader response is also advantageous. For example, a large-bandwidth tuning circuit\nis necessary to isolate a modulated radio signal and capture the entirety of its spectrum,\nwhile filtering away signals in adjacent bands. Similarly, nanomechanical resonators with a\nlarge bandwidth ∆f can respond to nearly resonant forces whose spectra are as broad as\nthe mechanical response [11, 12, 13, 14]. They can also be used to study resonant frequency\nfluctuations induced much below resonance by fluctuators with correlation times as short\nas 1/∆f [15, 16, 17, 18, 19, 20]. In essence, nanomechanical resonators behave as tunable\nbandpass filters for nearly resonant forces and as detectors for nonresonant fluctuators.\nWhile the nanomechanical response to broadband force noise and low frequency fluctua-\ntors has been thoroughly studied, the response to multifrequency coherent drives has thus far\nreceived less attention. Where multifrequency signals are applied, oftentimes only a single\nfrequency component is used to measure the response while the other components are used\nfor different tasks. For example, amplitude modulated [6, 21, 22] and frequency modulated\nsignals [23] have been employed to drive the resonator at the single frequency of the carrier,\nwhile the modulation sidebands have been used to detect the vibrations. Two-tone signals,\nwith one tone near resonance and the second tone outside of the response, have been used to\nstudy a variety of vibrational phenomena. Examples abound and include pump-probe spec-\ntroscopy, where intermode coupling is measured [24]; parametrically-induced mode coupling,\nused to heat or cool vibrations [25, 26, 27, 28] and to engineer nonlinear friction [29]; and\nsymmetry-breaking parametric resonators used for binary information encoding [30, 31, 32].\nBy contrast, the scope of applications for multifrequency signals whose entire spectra drive\nvibrations has been more limited. Two-tone drives have been used to demonstrate signal\nmixing with nonlinear nanomechanical resonators [33, 34]. The interplay of a two-tone drive\nnear resonance and parametric pumps has enabled the realization of mechanical logic gates\n[35]. The combination of a strong and a weak driving tone near resonance has been shown\nto squeeze thermomechanical noise [36]. Resonators driven nonlinearly by two large tones\nnear resonance have displayed unconventional dynamics, with the beating between the tones\n30\nacting as a slow modulation of the bifurcation points [37, 38]. Overall, studying nanome-\nchanical resonators using signals composed of multifrequency components sheds light on the\nphysics of mesoscopic vibrations and enables useful applications.\n(...) Here, we demonstrate that nanomechanical resonators based on few-layer graphene\n(FLG) can transduce a multifrequency, digital video signal into modulated vibrations that\nhost an accurate copy of the video. We quantify the transduction using signal quality met-\nrics for digital communications. We have chosen FLG resonators for their large resonant\nfrequency tunabilities within the high frequency (HF) and the very high frequency (VHF)\nbands of the radio spectrum, and for the compromise they offer between broad bandwidths\nand moderate force sensitivities. Of interest in our study is the fact that the signal band-\nwidth is larger than or comparable to that of the mechanical response. This is in contrast\nto nanomechanical radio receivers and transmitters mentioned above, which process audio\nbaseband signals whose bandwidth is necessarily much smaller than the mechanical band-\nwidth. A wide-bandwidth coherent drive acts as a useful probe to study the physics of the\nmechanical response. Because the resonator behaves as a filter, it shapes the multifrequency\ndriving signal and transduces it into multifrequency vibrations with different phases and\namplitudes that superimpose and interfere. We find that these interferences have a profound\nimpact on the signal-to-noise ratio and on the bit error ratio (BER) of the transduction.\nWe model the response to our multifrequency drive theoretically and are able to reproduce\nour data. We establish a relation between the mechanical bandwidth and the largest signal\nbandwidth that can be transduced without loss of information in the baseband components\nof the vibrations. Our work contributes to the recent research effort on the physics of res-\nonators driven nonlinearly by multiple tones. Yet, where only two tones were used earlier\n[37, 38], here we study an unexplored regime where the response is coherently driven by a\nnear continuum of frequency components within the whole mechanical bandwidth.\nWe then address our proposed applications in the field of radio communications. The\nend of our new introduction reads:\nWe also envision experimental applications at a time of renewed interest in communica-\ntions within the HF and the VHF bands [48]. Even though their data rates are low, HF\nand VHF communications still provide the most robust and reliable means of long-distance\ncommunications in situations where access to other channels is compromised. Future HF\ncommunications will require the ability to occasionally transmit and receive weak signals\nencoded with compressed images and videos [48], while narrowband digital television has al-\nready been experimented with in the VHF band [49]. Our results show that nanomechanical\nresonators may act as passive receivers for such signals, as they do for radiowaves modulated\nat audio frequencies.\nIn the concluding part of our manuscript, we write:\nIn the spirit of previous nanomechanical devices employed as signal processors [39], we\nenvision that our FLG resonators, and other resonators based on similar principles, may be\nemployed as small footprint, passive radio receivers in the HF and the VHF bands of the\n31\nradio spectrum. With a resonant frequency ranging from ≃ 40×106 Hz to ≃ 60×106 Hz\n(Fig. 5a), our resonator operates in the 6-meter subband that is internationally allocated to\nthe use of amateur radio. In this subband, digital amateur television (DATV) broadcasting\nhas been conducted using signals with a 300 kHz bandwidth [49]. In the United Kingdom,\nDATV with a broader bandwidth of up to 500 kHz is allowed in the 2-meter amateur radio\nband (144–148 MHz) [49]. Smaller diameter FLG resonators, or higher order vibrational\nmodes, can operate in this band. The filtering bandwidth of our resonator, and the maxi-\nmum 4-QAM data rate of its transduction 2R∗ ≃ 2×106 bps, are then fully compatible\nsym\nwith DATV. Arguably, our simple experimental setup may still not be practical enough for\nreceiving amateur radio messages, but it can be simplified further [51]. The resonator would\nserve the dual purpose of a front-end, tunable bandpass filter and of a passive optocoupler.\nIt would be placed at the ouput of the preamplifier that is usually connected to the feedpoint\nof a receiving antenna [69]. As a filter, it would reject jamming and other interfering signals\nthe way surface acoustic wave filters do, but with the additional benefit of a tunable cen-\nter frequency, and without the fabrication complexity of more advanced tunable front-end\nfilters [70]. As an optocoupler, it would passively convert the radio signal into modulations\nof reflected light intensity through the nanomechanical transduction, effectively eliminat-\ning the problem of radio frequency interferences caused by common-mode currents in the\ncoaxial cable that connects the antenna to the demodulator [69]. Admittedly, the idea of\nemploying nanomechanical resonators as devices for radio communcation mostly possesses\nan academic appeal, given the limitations posed by device reproducibility and thermal drifts\n[40]. However, these resonators may ultimately enable alternate architectures for shortwave\nradioreceivers, facilitatingthedecodingofimagesandvideoscarriedbyweaksignalsthrough\na noisy environment [61].\nThe Reviewer writes:\nOn the other hand, the implementation of the video receiver application with a\nnanoelectromechanicalsystemhasresultedininterestingnewinsightsintodriving\na NEMS resonator with a digitally modulated signal in the linear and nonlinear\nregime. Furthermore, the authors show the possibility of operating at different\nbase-band frequencies by electrostatic tuning of the resonators’ spring constant.\nWe thank the Reviewer for this comment and for expressing an interest in our results. In\nline with the comment, our present manuscript is concerned with the physics of mesoscopic\nvibrations driven by a multifrequency, coherent signal. We present what we believe is an in-\ndepth study of the nanomechanical response to such an unusual driving force. Our study is\nsupported by additional measurements of the I−Q vector length L and the reciprocal error\nvector magnitude ε of the response, along with measurements of the bit error ratio (BER)\nof the transduction. These signal quality metrics, usually reserved for digital communica-\ntions, have turned out to be surprisingly useful to analyze the vibrational dynamics of our\nresonator. For a quick overview of the physics at work in our study, we would like to invite\nthe Reviewer to play the animation we have embedded in Supplementary Note 9 (please use\nAdobe Reader). There, we show how our calculated L, ε, BER and constellation diagrams\ngracefully evolve as the drive frequency is stepped through resonance. This animation allows\n32\none to easily appreciate how these quantities relate to one another.\nThe Reviewer writes:\nApart from lacking a clear motivation and potential applications for the NEMS-\nbased video receiver, the authors erroneously compare it to the Nipkow wheel.\nThe only similarity between the two systems is that they are both mechanical\ndevices; no further analogies hold. While the Nipkow wheel was a pioneering\ntechnology eventually supplanted by electronic television, the graphene video\nreceiver does not represent a groundbreaking innovation and is already outdated\nupon introduction.\nWe thank the Reviewer for this comment. In the present version of our manuscript, we\nhave removed the mention of the Nipkow disk.\nOnce again, we thank the Reviewer for reading our manuscript and for offering helpful\ncomments.\nResponse to Reviewer #4\nThe Reviewer writes:\nI co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. 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Accessed April 4, 2024.\n[58] HackRF software-defined radio transceiver. https://greatscottgadgets.\ncom/hackrf/. NESDR radio receiver. https://www.nooelec.com/store/sdr/\nsdr-receivers/nesdr.html. Accessed April 4, 2024.\n[59] Franks, L. E. Carrier and Bit Synchronization in Data Communication - A Tutorial\nReview. IEEE Trans. Commun. 28, no. 8, 1107–1121 (1980).\n[60] Harris, F. J. & Rice, M. Multirate digital filters for symbol timing synchronization in\nsoftware defined radios. IEEE J. Sel. Areas Commun. 19, no. 12, 2346–2357 (2001).\n[61] Crockett, L. H., Northcote, D. & Stewart, R. W. (Editors). Software Defined Radio\nwith Zynq UltraScale+ RFSoC (First Edition). Strathclyde Academic Media (2023).\nhttps://www.rfsocbook.com/\n[62] FFmpeg. https://ffmpeg.org. Accessed April 4, 2024.\n[63] Dykman, M. I. & Krivoglaz, M. A. Spectral distribution of nonlinear oscillators with\nnonlinear friction due to a medium. Phys. Status Solidi B 68, no. 1, 111–123 (1975).\n[64] Zaitsev, S., Shtempluck, O., Buks, E. & Gottlieb, O. Nonlinear damping in a microme-\nchanical oscillator. Nonlinear Dyn. 67, no. 1, 859–883 (2012).\n[65] Carr, D. W., Evoy, S., Sekaric, L., Craighead, H. G. & Parpia, J. M. Measurement of\nmechanical resonance and losses in nanometer scale silicon wires. Appl. Phys. Lett. 75,\nno. 7, 920–922 (1999). https://doi.org/10.1063/1.124554\n[66] Husain, A., Hone, J., Postma, H. W. Ch., Huang, X. M. H., Drake, T., Barbic, M.,\nScherer, A. & Roukes, M. L. Nanowire-based very-high-frequency electromechanical\nresonator. Appl. Phys. Lett. 83, no. 6, 1240–1242 (2003). https://doi.org/10.1063/\n1.1601311\n[67] Kozinsky, I., Postma, H. W. Ch., Bargatin, I. & Roukes, M. L. Tuning nonlinearity,\ndynamic range, and frequency of nanomechanical resonators. Appl. Phys. Lett. 88, no.\n25, Art. no. 253101 (2006). https://doi.org/10.1063/1.2209211\n[68] Dykman, M. I. & Krivoglaz, M. A. Classical theory of nonlinear oscillators interacting\nwith a medium. Physica Status Solidi (b) 48, no. 2, 497–512 (1971). https://doi.org/\n10.1002/pssb.2220480206.\n[69] ARRL—The National Association for Amateur Radio. The ARRL Antenna Book for\nRadio Communications (24th Ed). The American Radio Relay League (2019).\n[70] Hashimoto, K.-Y., Kimura, T., Matsumura, T., Hirano, H., Kadota, M., Esashi, M.\n& Tanaka, S. Moving Tunable Filters Forward. IEEE Microwave Magazine 16, no. 7,\n89–97 (2015). https://doi.org/10.1109/MMM.2015.2431237\n(Manuscript #NCOMMS-24-26392A)\nWe are grateful for the time and the care that the Editor and the Reviewers have taken\nin assessing our manuscript. Please find below our point-by-point response to the comments\nwe received.\nResponse to Reviewer #5\nThe Reviewer writes:\nI think that the paper could be published in its present form.\nWe thank the Reviewer for taking the time to consider our manuscript. We are heartened\nby the Reviewer’s appreciation.\nResponse to Reviewer #6\nThe Reviewer writes:\nI have thoroughly reviewed the manuscript and carefully considered the feed-\nback from the previous round of reviews. The manuscript introduces a NEMS\ngraphenedrumcapableoftransmittingvideodatabyconvertingdigitalelectronic\nsignals into vibrations. As a demonstration, the authors claim that a video was\ntransmitted via the graphene. Compared to previous methods, the tunability of\nthe graphene could facilitate frequency-selective filtering of RF signals.\nWe thank the Reviewer for taking the time to consider our manuscript. We agree with\nthe Reviewer that the tunability of graphene nanomechanical resonators can be effectively\nleveraged to select specific frequency bands within the radio spectrum and suppress adjacent\nbands. Signal filtering plays a critical role in radio-frequency telecommunication systems.\nThe Reviewer continues:\nWhiletheconceptpresentedinthispaperisintriguing,Isharetheconcernsraised\nby the reviewers regarding several scientific and technical issues that require\nfurther clarification.\ning points in order to make our discussion as precise as possible and to ensure our work is\naccessible to a broad audience.\nThe Reviewer writes:\n1. Although the manuscript primarily focuses on modulation/demodulation and\nsignal processing, I strongly recommend a more detailed and quantitative eval-\nuation of the fundamental performance of the proposed NMES graphene drum\nstructure.\nWe thank the Reviewer for this comment. Our manuscript focuses on the physics of\nnanomechanical vibrations in our resonators. Elements of signal processing, such as the\nmodulation of the driving signal and the demodulation of the measured signal, are essential\ntools used to quantify the nanomechanical response to a multifrequency drive, a topic that\nhas not been addressed thus far. We believe that our study of this response, as presented, is\nboth detailed and quantitative. We provide a step-by-step description of our experimental\nsetup and of our data analysis protocol, including all necessary parameters, to enable readers\nto reproduce our measurements and analyze the data. We also present a theoretical model\nto explain our results. Our model only requires two fit parameters (a transduction factor\nand a voltage noise variance), whose values are independently confirmed through separate\nexperiments (please see Supplementary Notes 5 and 6). Finally, our analysis of the res-\nonator’s response to a multifrequency signal is based on both numerical calculations (main\ntext) and on closed-form solutions (Supplementary Note 7).\nThe Reviewer continues:\nFor example, molecular dynamics simulations could reveal the resonant charac-\nteristics of the drum and provide a visualization of atomic distributions.\nWe thank the Reviewer for this comment. The computational cost and time required for\nmolecular dynamics simulations of our devices –graphene drums with diameters of ≃ 3 µm–\nwould be astronomical. Previous molecular dynamics simulations of graphene membranes\nhave been limited to nanoscale systems [e.g., 0.7 nm×14.1 nm, or 424 atoms, in the following\nreference: Kang, J.-W., Kim, H.-W., Kim, K.-S. & Lee, J. H. Molecular dynamics model-\ning and simulation of a graphene-based nanoelectromechanical resonator. Current Applied\nPhysics, 13, 789–794 (2013)].\nThe Reviewer continues:\nAdditionally, woulditbepossibletomodelthisNEMSresonatorusingcontinuum\ntheory and solve the models numerically?\nWe thank the Reviewer for this question. Continuum elastic theory has been used to\nmodel the mechanical response of graphene resonators [see, e.g., Atalaya, J., Isacsson, A.\n& Kinaret, J. M. Continuum Elastic Modeling of Graphene Resonators. Nano Lett. 8,\n4196–4200 (2008)]. This formalism describes bending and stretching based on Timoshenko’s\n2\nplate and shell theory. Taking into account the symmetry of the resonator and assuming\nuniformly distributed strain, the dynamic response described by continuum elastic theory\ncan be simplified into a form resembling the wave equation for the transverse displacement\nof the membrane (the so-called Duffing equation), with an effective mass representing the\nresonator’s mass normalized by its mode shape [see, e.g., Lifshitz, R. & Cross, M. C. Nonlin-\near dynamics of nanomechanical and micromechanical resonators. In Reviews of Nonlinear\nDynamics and Complexity, Vol. 1. (ed. H.G. Schuster), 1–52 (Wiley-VCH, 2008)]. This is\nthe approach most commonly used in the 2D materials NEMS community and is also the\none we employ in our work [see Eq. (1) in our main text].\nThe Reviewer continues:\nA more in-depth mechanism analysis would help readers –especially from a basic\nphysics standpoint– understand the superposition and interference effects in the\ngraphene, as well as uncover the characteristics of nonlinearity, which is claimed\nto be a significant factor influencing the bit error rate.\nWe thank the Reviewer for this comment. In response, we have amended the main text\nto include the following:\nThe fact that the in-phase component of the vibrations depends on both the in-phase and\nquadrature components of the drive, and that the quadrature response likewise depends on\nboth drive components, can already be seen by solving Eq. (1) in the linear regime, yielding:\nκ ∞ (cid:104) (cid:105)\nI ˜ (t) ≃ dt′exp(−Γ t′) −I ˜ (t−t′)sin∆ωt′ −Q ˜ (t−t′)cos∆ωt′ ,\nout 4m ω (cid:2) m in in\neff d 0 (1)\nκ ∞ (cid:104) (cid:105)\nQ ˜ (t) ≃ dt′exp(−Γ t′) I ˜ (t−t′)cos∆ωt′ −Q ˜ (t−t′)sin∆ωt′ ,\nout 4m ω (cid:2) m in in\neff d 0\nwhere κ = |ξVdc(1+Γ)C′|, ω = 2πf , Γ = πW , and ∆ω = 2π(f −f ).\ng g d d m m d m\n˜ ˜\nIn the above expressions, I (t) and Q (t) are the in-phase and quadrature components\nout out\n˜ ˜\nof the voltage signal at the output of the photodetector, I (t) and Q (t) are the in-phase\nin in\nand quadrature components of the driving force, ∆ω = ω −ω with ω the angular carrier\nd m d\nfrequency of the driving force and ω the angular resonant frequency of the fundamental\nm\nvibration mode, m is the effective mass of the mode, Γ is the linear damping rate, and κ\neff m\nis a constant that depends on the transduction factor in V m−1 and on the amplitude of the\nforce.\nIn Supplementary Note 7, In-phase and quadrature baseband components of\nthe response, we present our theory describing the superposition and interference effects\nthat are the hallmark of vibrations driven by a multifrequency signal. We write:\nBoth systems of equations (48) and (50) show that, away from resonance (∆ω ̸= 0), the\nin-phaseresponsedependsonboththein-phaseandquadraturecomponentsofthedrive, and\nlikewise, the quadrature response depends on both drive components. These equations show\n3\nthat the linear response is a superposition of waves with frequency-dependent amplitudes\nand phases:\nκ ∞ −∆ωI ˆ (ω)−(Γ −iω)Q ˆ (ω)\n˜ in m in\nI (t) ≃ Re dωexp(−iωt) ,\nout 4πm ω (cid:2) (Γ −iω)2 +(∆ω)2\neff d 0 m (2)\nκ ∞ (Γ −iω)I ˆ (ω)−∆ωQ ˆ (ω)\n˜ m in in\nQ (t) ≃ Re dωexp(−iωt) ,\nout 4πm ω (cid:2) (Γ −iω)2 +(∆ω)2\neff d 0 m\nwhere we used the property f ˆ (−ω) = f ˆ (ω)∗ if f(t) is real.\nˆ ˆ\nIn the above expressions, I (ω) and Q (ω) are the Fourier transforms in the frequency\nin in\ndomain of the in-phase and quadrature components of the driving force.\nRegarding nonlinearities in the response, the negative value of the Duffing elastic con-\nstant suggests an electrical origin for the conservative nonlinearity. In the main text, we\nbegin the Discussion section as follows:\nTounderstandtheseobservations,wenumericallysolvetheresonator’sequationofmotion\nin response to the multifrequency driving force F (t). The equation reads [1, 2]:\nd\n(cid:20) (cid:21) (cid:20) (cid:21)\nω η α F (t)\nz¨+ m + z2 z˙ + ω2 + z2 z = d , (3)\nQ m m m m\nm eff eff eff\nwhere z is the resonator displacement, ω = 2πf is the angular resonant frequency, η is the\nm m\nnonlinear damping coefficient, and α is the Duffing elastic constant. The values Q ≃ 100,\nm\nη ≃ 1.2×105 kg m−2 s−1 and α ≃ −1014 kg m−2 s−2 (arising from the electrical potential\nacting on the resonator, see Ref. [3]) are estimated from the response to a single-tone drive\n(Supplementary Note 5).\nThe Reviewer writes:\n2. The discussion on experimental implementation should be expanded. For\ninstance, couldyouprovidemoredetailsonthefabricationprocessofthesample?\nThiswouldhelpreadersbetterunderstandtheresults. [...] Additionally,Isuggest\nincluding photographs of the sample along with zoomed-in details to enhance\nclarity.\nWe thank the Reviewer for suggesting that we provide more details about the fabrica-\ntion process of our devices. The new Supplementary Note 1, Device fabrication, reads:\nThe structure of our devices and their fabrication process are similar to those used in\nRef. [4]. The substrate is a thermally oxidized wafer of high-resistivity silicon. Fabrication\nbegins with the dry etching of a mesa in the silicon oxide layer of the substrate (white-\nshaded structure in Figs. R1a, b). At the center of the mesa is a cylindrical cavity. Two\nmetal electrodes are then patterned on the etched susbtrate. The electrode patterned at the\nbottom of the cavity serves as a local gate electrode, while the one patterned around the\n4\nmesa serves as a source. The vertical distance between the top surface of the mesa and the\ntop surface of the gate is ≃ 230 nm. A few-layer graphene (FLG) flake, estimated to be\nabout 10 layers thick based on optical reflectometry [5], is transferred onto the substrate,\ncontacting the source electrode and covering the mesa. The part of the flake covering the\ncavity is suspended and forms the resonator. Figure R1c shows an optical image of the\ndevice used in the measurements presented in both the main text and the supplementary\ninformation.\na b c\nFLG\nS\nSiO on Si\nFigure R1: Structure of the few-layer graphene nanomechanical resonator used in this work.\n(a) Atomic force microscopy image (AFM) of the patterned substrate. The white structure\nis a mesa with a central cavity etched in silicon oxide. At the bottom of the cavity is a local\ngate electrode. Surrounding the mesa is a source electrode. The electrodes are patterned\nusing electron-beam lithography, followed by the deposition of 5 nm of titanium and 50 nm\nof gold. The vertical distance between the top surface of the mesa and the top surface of the\nelectrodes is ≃ 230 nm. Scale bar: 5 µm. (b) Tilted AFM image highlighting the local gate\nelectrode at the bottom of the cavity and the narrow venting trench connected to the cavity.\n(c) Optical image of the final device, obtained by transferring a ≃ 10-layer-thick few-layer\ngraphene (FLG) flake onto the patterned substrate. The blue-shaded structure is the mesa.\nG: local gate. S: source. Scale bar: 5 µm.\nThe Reviewer continues:\nThe manuscript claims that the device is capable of transmitting and receiving\nweak signals. Will the device function in environments with strong background\nnoise and low signal-to-noise ratio?\nWe thank the Reviewer for this question. Supplementary Note 4, I − Q vector\nlength and reciprocal of error vector magnitude at low drive power, presents our\nmeasurements of graphene vibrations driven by the weakest signal capable of enabling suc-\ncessful video decoding. Furthermore, in the last paragraph of our main text, we now write:\nThe detection of weaker radio signals may be improved by using more advanced SDR\nsystems with lower receiver noise and by engineering the mirror underneath the resonator\nto enhance the device’s responsivity [6].\nReference [6] is one of our recent works, in which we demonstrate that the optical re-\nsponsivity of the resonator can be enhanced by one order of magnitude simply by replacing\n5\nthe planar mirror with a photonic crystal. This enhancement would enable the detection of\nsignals nearly 20 dB weaker than those detectable with our current low-cost software-defined\nradio receiver.\nThe Reviewer writes:\n3. Beyond the points mentioned above, one of my main concerns is the real\ntechnological significance and potential of the proposed device.\nWethanktheReviewerforthiscomment. AsmentionedinourresponsetotheReviewer’s\ncomment #1, the primary aim of our work is rooted in basic science. Specifically, we inves-\ntigate the response of graphene nanomechanical resonators to a broadband, multifrequency\ndrive. Any potential applications discussed are meant solely for illustrative purposes. In our\nabstract, we write:\nOur work reveals distinctive features in vibrations driven by a coherent, multifrequency\ndrive that are not observed in vibrations actuated by single-tone drives or by a force noise.\nFurthermore, in the introduction, we write:\nOverall, studying nanomechanical resonators driven by multifrequency signals sheds light\non the physics of mesoscopic vibrations and leads to proof-of-concept demonstrations for po-\ntential applications. [...] Our work contributes to the recent study of resonators driven\nnonlinearly by multiple tones. Whereas earlier studies explored the case of two-tone driving\n[7, 8], we investigate a previously unexplored regime in which the resonator is coherently\ndriven by a near continuum of frequency components across its full mechanical bandwidth.\nAs Reviewer #5 writes, “Only rev 3 raises serious doubts about the advantages of the\nproposed encoding technique with respect to other options, but I do not think this is the\npoint of an academic publication.” Nevertheless, we find the prospect of future applications\nappealing. In response to the Reviewer’s comment, we write in the second-to-last paragraph\nof our main text:\nIn the spirit of previous nanomechanical devices used as signal processors [9], we envision\nthat our FLG resonators, as well as other resonators based on similar principles, may serve\nas passive radio receivers operating in the HF and VHF bands of the radio spectrum. With\na resonant frequency ranging from ≃ 40×106 Hz to ≃ 60×106 Hz (Fig. 5a), our resonator\noperates within the 6-meter subband, which is internationally allocated for amateur radio\nuse. In this subband, digital amateur television (DATV) broadcasts have been conducted\nusing signals with a 300 kHz bandwidth [10]. In the United Kingdom, DATV with a broader\nbandwidth of up to 500 kHz is permitted in the 2-meter amateur radio band (144–148 MHz)\n[10]. Smaller-diameter FLG resonators, or higher-order vibrational modes, can operate in\nthis frequency range. The filtering bandwidth of our resonator, along with the maximum\n4-QAM data rate of its transduction, 2R∗ ≃ 2×106 bps, is fully compatible with DATV.\nsym\nThe Reviewer continues:\nthan those of its competitors, such as SAW filters, which are widely used in\nvarious consumer electronics. The authors should clarify whether graphene offers\nsignificant advantages that would justify its further development in this context.\nWe thank the Reviewer for this comment. In the last paragraph of our main text, we\naddress the comparison between our devices and SAW filters, with an important caveat.\nIt reads:\nWhile our experimental setup may not yet be practical for receiving amateur radio broad-\ncasts, it can be further simplified [5]. The resonator would serve a dual purpose: as a\nfront-end, tunable bandpass filter, and as a passive optocoupler. It could be placed at the\noutput of the preamplifier typically connected to the feedpoint of a receiving antenna [11].\nAs a filter, it would reject jamming and other interfering signals in a manner similar to\nsurface acoustic wave filters, but with the added benefit of a tunable center frequency, and\nwithout the fabrication complexity of more advanced tunable front-end filters [12]. As an\noptocoupler, it would passively convert the radio signal into modulations of reflected light\nintensity through the nanomechanical transduction, effectively eliminating the issue of radio\nfrequency interference caused by common-mode currents in the coaxial cable that connects\nthe antenna to the demodulator [11]. The detection of weaker radio signals may be improved\nby using more advanced SDR systems with lower receiver noise and by engineering the mir-\nror underneath the resonator to enhance the device’s responsivity [6]. Admittedly, the idea\nof using nanomechanical resonators for radio communication remains largely academic due\nto challenges related to device reproducibility and thermal drift [13]. Nevertheless, these\nresonators may eventually enable alternative architectures for shortwave radio receivers, fa-\ncilitating the decoding of images and videos carried by weak signals in noisy environments\n[14].\nOnce again, we thank the Reviewer for carefully reading our manuscript and providing\nthoughtful comments.\nReferences\n[1] Dykman, M. I. & Krivoglaz, M. A. Spectral distribution of nonlinear oscillators with\nnonlinear friction due to a medium. Phys. Status Solidi B 68, no. 1, 111–123 (1975).\n[2] Zaitsev, S., Shtempluck, O., Buks, E. & Gottlieb, O. Nonlinear damping in a microme-\nchanical oscillator. Nonlinear Dyn. 67, no. 1, 859–883 (2012).\n[3] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechan-\nical resonators. In Reviews of Nonlinear Dynamics and Complexity, Vol. 1. (ed. H.G.\nSchuster), 1–52 (Wiley-VCH, 2008). https://doi.org/10.1002/9783527629374.ch8\n[4] Zhang, C., Zhang, Y., Yang, C., Lu, H., Chen, F., Yan, Y. & Moser, J. Graphene\nnanomechanical vibrations measured with a phase-coherent software-defined radio.\nComms. Eng. 3, Art. no. 45 (2024). https://doi.org/10.1038/s44172-024-00186-4\n7\n[5] Lu, H., Yang, C., Tian, Y., Lu, J., Xu, F., Zhang, C., Chen, F., Yan, Y., Schaedler, K.\nG., Wang, C., Koppens, F. H. L., Reserbat-Plantey, A. & Moser, J. Imaging vibrations\nof electromechanical few layer graphene resonators with a moving vacuum enclosure.\nPrecis. Eng. 72, 769–776 (2021). https://doi.org/10.1016/j.precisioneng.2021.\n06.012\n[6] Lu, H., Li, T., Hu, H., Chen, F., Sun, T., Yan, Y., Wang, C. & Moser, J. Force\nsensing with a graphene nanomechanical resonator coupled to photonic crystal guided\nresonances. arXiv:2507.06755. https://arxiv.org/abs/2507.06755\n[7] Houri, S., Ohta, R., Asano, M., Blanter, Y.M.&Yamaguchi, H.Pulse-widthmodulated\noscillationsinanonlinearresonatorundertwo-tonedrivingasameansforMEMSsensor\nreadout. Jpn. J. Appl. Phys. 58, no. SB, Art. no. SBBI05 (2019). https://doi.org/\n10.7567/1347-4065/aaffb9\n[8] Catalini, L., del Pino, J., Kumar, S. S., Dumont, V., Margiani, G., Zilberberg, O.\n& Eichler, A. Adiabatic and diabatic responses in a Duffing resonator driven by two\ndetuned tones. arXiv:2408.15794v1 (2024). https://doi.org/10.48550/arXiv.2408.\n15794\n[9] Nguyen, C. T.-C. Radios with micromachined resonators. IEEE Spectrum (30 Nov.\n2009).https://spectrum.ieee.org/radios-with-micromachined-resonators.Ac-\ncessed: October 24, 2024.\n[10] Radio Society of Great Britain. Amateur television. https://rsgb.org/main/\ntechnical/amateur-television/\n[11] ARRL—The National Association for Amateur Radio. The ARRL Antenna Book for\nRadio Communications (24th Ed). The American Radio Relay League (2019).\n[12] Hashimoto, K.-Y., Kimura, T., Matsumura, T., Hirano, H., Kadota, M., Esashi, M.\n& Tanaka, S. Moving Tunable Filters Forward. IEEE Microwave Magazine 16, no. 7,\n89–97 (2015). https://doi.org/10.1109/MMM.2015.2431237\n[13] Nguyen, C. T.-C. MEMS-based RF channel selection for true software-defined cognitive\nradio and low-power sensor communications. IEEE Communications Magazine 51, no.\n4, pp. 110–119 (2013). https://doi.org/10.1109/MCOM.2013.6495769\n[14] Crockett, L. H., Northcote, D. & Stewart, R. W. (Editors). Software Defined Radio\nwith Zynq UltraScale+ RFSoC (First Edition). Strathclyde Academic Media (2023).\nhttps://www.rfsocbook.com/\n8"}, {"round": 2, "reviewer_comments": {"reviewer_6": "I have thoroughly reviewed the manuscript and carefully considered the feedback from the previous round of reviews.\nThe manuscript introduces a NEMS graphene drum capable of transmitting video data by converting digital electronic signals\ninto vibrations. As a demonstration, the authors claim that a video was transmitted via the graphene. Compared to previous\nmethods, the tunability of the graphene could facilitate frequency-selective filtering of RF signals. While the concept\npresented in this paper is intriguing, I share the concerns raised by the reviewers regarding several scientific and technical\nissues that require further clarification.\n1. Although the manuscript primarily focuses on modulation/demodulation and signal processing, I strongly recommend a\nmore detailed and quantitative evaluation of the fundamental performance of the proposed NMES graphene drum structure.\nFor example, molecular dynamics simulations could reveal the resonant characteristics of the drum and provide a\nvisualization of atomic distributions. Additionally, would it be possible to model this NEMS resonator using continuum\ntheory and solve the models numerically? A more in-depth mechanism analysis would help readers—especially from a basic\nphysics standpoint—understand the superposition and interference effects in the graphene, as well as uncover the\ncharacteristics of nonlinearity, which is claimed to be a significant factor influencing the bit error rate.\n2. The discussion on experimental implementation should be expanded. For instance, could you provide more details on the\nfabrication process of the sample? This would help readers better understand the results. The manuscript claims that the\ndevice is capable of transmitting and receiving weak signals. Will the device function in environments with strong background\nnoise and low signal-to-noise ratio? Additionally, I suggest including photographs of the sample along with\nzoomed-in details to enhance clarity.\n3. Beyond the points mentioned above, one of my main concerns is the real technological significance and potential of the\nproposed device. It appears that the operational conditions for this device are much more stringent than those of its\ncompetitors, such as SAW filters, which are widely used in various consumer electronics. The authors should clarify whether\ngraphene offers significant advantages that would justify its further development in this context."}, "author_response": "Response to the Comments of the Reviewers\n(Manuscript #NCOMMS-24-26392A)\nWe are grateful for the time and the care that the Editor and the Reviewers have taken\nin assessing our manuscript. Please find below our point-by-point response to the comments\nwe received.\nResponse to Reviewer #5\nThe Reviewer writes:\nI think that the paper could be published in its present form.\nWe thank the Reviewer for taking the time to consider our manuscript. We are heartened\nby the Reviewer’s appreciation.\nResponse to Reviewer #6\nThe Reviewer writes:\nI have thoroughly reviewed the manuscript and carefully considered the feed-\nback from the previous round of reviews. The manuscript introduces a NEMS\ngraphenedrumcapableoftransmittingvideodatabyconvertingdigitalelectronic\nsignals into vibrations. As a demonstration, the authors claim that a video was\ntransmitted via the graphene. Compared to previous methods, the tunability of\nthe graphene could facilitate frequency-selective filtering of RF signals.\nWe thank the Reviewer for taking the time to consider our manuscript. We agree with\nthe Reviewer that the tunability of graphene nanomechanical resonators can be effectively\nleveraged to select specific frequency bands within the radio spectrum and suppress adjacent\nbands. Signal filtering plays a critical role in radio-frequency telecommunication systems.\nThe Reviewer continues:\nWhiletheconceptpresentedinthispaperisintriguing,Isharetheconcernsraised\nby the reviewers regarding several scientific and technical issues that require\nfurther clarification.\ning points in order to make our discussion as precise as possible and to ensure our work is\naccessible to a broad audience.\nThe Reviewer writes:\n1. Although the manuscript primarily focuses on modulation/demodulation and\nsignal processing, I strongly recommend a more detailed and quantitative eval-\nuation of the fundamental performance of the proposed NMES graphene drum\nstructure.\nWe thank the Reviewer for this comment. Our manuscript focuses on the physics of\nnanomechanical vibrations in our resonators. Elements of signal processing, such as the\nmodulation of the driving signal and the demodulation of the measured signal, are essential\ntools used to quantify the nanomechanical response to a multifrequency drive, a topic that\nhas not been addressed thus far. We believe that our study of this response, as presented, is\nboth detailed and quantitative. We provide a step-by-step description of our experimental\nsetup and of our data analysis protocol, including all necessary parameters, to enable readers\nto reproduce our measurements and analyze the data. We also present a theoretical model\nto explain our results. Our model only requires two fit parameters (a transduction factor\nand a voltage noise variance), whose values are independently confirmed through separate\nexperiments (please see Supplementary Notes 5 and 6). Finally, our analysis of the res-\nonator’s response to a multifrequency signal is based on both numerical calculations (main\ntext) and on closed-form solutions (Supplementary Note 7).\nThe Reviewer continues:\nFor example, molecular dynamics simulations could reveal the resonant charac-\nteristics of the drum and provide a visualization of atomic distributions.\nWe thank the Reviewer for this comment. The computational cost and time required for\nmolecular dynamics simulations of our devices –graphene drums with diameters of ≃ 3 µm–\nwould be astronomical. Previous molecular dynamics simulations of graphene membranes\nhave been limited to nanoscale systems [e.g., 0.7 nm×14.1 nm, or 424 atoms, in the following\nreference: Kang, J.-W., Kim, H.-W., Kim, K.-S. & Lee, J. H. Molecular dynamics model-\ning and simulation of a graphene-based nanoelectromechanical resonator. Current Applied\nPhysics, 13, 789–794 (2013)].\nThe Reviewer continues:\nAdditionally, woulditbepossibletomodelthisNEMSresonatorusingcontinuum\ntheory and solve the models numerically?\nWe thank the Reviewer for this question. Continuum elastic theory has been used to\nmodel the mechanical response of graphene resonators [see, e.g., Atalaya, J., Isacsson, A.\n& Kinaret, J. M. Continuum Elastic Modeling of Graphene Resonators. Nano Lett. 8,\n4196–4200 (2008)]. This formalism describes bending and stretching based on Timoshenko’s\n2\nplate and shell theory. Taking into account the symmetry of the resonator and assuming\nuniformly distributed strain, the dynamic response described by continuum elastic theory\ncan be simplified into a form resembling the wave equation for the transverse displacement\nof the membrane (the so-called Duffing equation), with an effective mass representing the\nresonator’s mass normalized by its mode shape [see, e.g., Lifshitz, R. & Cross, M. C. Nonlin-\near dynamics of nanomechanical and micromechanical resonators. In Reviews of Nonlinear\nDynamics and Complexity, Vol. 1. (ed. H.G. Schuster), 1–52 (Wiley-VCH, 2008)]. This is\nthe approach most commonly used in the 2D materials NEMS community and is also the\none we employ in our work [see Eq. (1) in our main text].\nThe Reviewer continues:\nA more in-depth mechanism analysis would help readers –especially from a basic\nphysics standpoint– understand the superposition and interference effects in the\ngraphene, as well as uncover the characteristics of nonlinearity, which is claimed\nto be a significant factor influencing the bit error rate.\nWe thank the Reviewer for this comment. In response, we have amended the main text\nto include the following:\nThe fact that the in-phase component of the vibrations depends on both the in-phase and\nquadrature components of the drive, and that the quadrature response likewise depends on\nboth drive components, can already be seen by solving Eq. (1) in the linear regime, yielding:\nκ ∞ (cid:104) (cid:105)\nI ˜ (t) ≃ dt′exp(−Γ t′) −I ˜ (t−t′)sin∆ωt′ −Q ˜ (t−t′)cos∆ωt′ ,\nout 4m ω (cid:2) m in in\neff d 0 (1)\nκ ∞ (cid:104) (cid:105)\nQ ˜ (t) ≃ dt′exp(−Γ t′) I ˜ (t−t′)cos∆ωt′ −Q ˜ (t−t′)sin∆ωt′ ,\nout 4m ω (cid:2) m in in\neff d 0\nwhere κ = |ξVdc(1+Γ)C′|, ω = 2πf , Γ = πW , and ∆ω = 2π(f −f ).\ng g d d m m d m\n˜ ˜\nIn the above expressions, I (t) and Q (t) are the in-phase and quadrature components\nout out\n˜ ˜\nof the voltage signal at the output of the photodetector, I (t) and Q (t) are the in-phase\nin in\nand quadrature components of the driving force, ∆ω = ω −ω with ω the angular carrier\nd m d\nfrequency of the driving force and ω the angular resonant frequency of the fundamental\nm\nvibration mode, m is the effective mass of the mode, Γ is the linear damping rate, and κ\neff m\nis a constant that depends on the transduction factor in V m−1 and on the amplitude of the\nforce.\nIn Supplementary Note 7, In-phase and quadrature baseband components of\nthe response, we present our theory describing the superposition and interference effects\nthat are the hallmark of vibrations driven by a multifrequency signal. We write:\nBoth systems of equations (48) and (50) show that, away from resonance (∆ω ̸= 0), the\nin-phaseresponsedependsonboththein-phaseandquadraturecomponentsofthedrive, and\nlikewise, the quadrature response depends on both drive components. These equations show\n3\nthat the linear response is a superposition of waves with frequency-dependent amplitudes\nand phases:\nκ ∞ −∆ωI ˆ (ω)−(Γ −iω)Q ˆ (ω)\n˜ in m in\nI (t) ≃ Re dωexp(−iωt) ,\nout 4πm ω (cid:2) (Γ −iω)2 +(∆ω)2\neff d 0 m (2)\nκ ∞ (Γ −iω)I ˆ (ω)−∆ωQ ˆ (ω)\n˜ m in in\nQ (t) ≃ Re dωexp(−iωt) ,\nout 4πm ω (cid:2) (Γ −iω)2 +(∆ω)2\neff d 0 m\nwhere we used the property f ˆ (−ω) = f ˆ (ω)∗ if f(t) is real.\nˆ ˆ\nIn the above expressions, I (ω) and Q (ω) are the Fourier transforms in the frequency\nin in\ndomain of the in-phase and quadrature components of the driving force.\nRegarding nonlinearities in the response, the negative value of the Duffing elastic con-\nstant suggests an electrical origin for the conservative nonlinearity. In the main text, we\nbegin the Discussion section as follows:\nTounderstandtheseobservations,wenumericallysolvetheresonator’sequationofmotion\nin response to the multifrequency driving force F (t). The equation reads [1, 2]:\nd\n(cid:20) (cid:21) (cid:20) (cid:21)\nω η α F (t)\nz¨+ m + z2 z˙ + ω2 + z2 z = d , (3)\nQ m m m m\nm eff eff eff\nwhere z is the resonator displacement, ω = 2πf is the angular resonant frequency, η is the\nm m\nnonlinear damping coefficient, and α is the Duffing elastic constant. The values Q ≃ 100,\nm\nη ≃ 1.2×105 kg m−2 s−1 and α ≃ −1014 kg m−2 s−2 (arising from the electrical potential\nacting on the resonator, see Ref. [3]) are estimated from the response to a single-tone drive\n(Supplementary Note 5).\nThe Reviewer writes:\n2. The discussion on experimental implementation should be expanded. For\ninstance, couldyouprovidemoredetailsonthefabricationprocessofthesample?\nThiswouldhelpreadersbetterunderstandtheresults. [...] Additionally,Isuggest\nincluding photographs of the sample along with zoomed-in details to enhance\nclarity.\nWe thank the Reviewer for suggesting that we provide more details about the fabrica-\ntion process of our devices. The new Supplementary Note 1, Device fabrication, reads:\nThe structure of our devices and their fabrication process are similar to those used in\nRef. [4]. The substrate is a thermally oxidized wafer of high-resistivity silicon. Fabrication\nbegins with the dry etching of a mesa in the silicon oxide layer of the substrate (white-\nshaded structure in Figs. R1a, b). At the center of the mesa is a cylindrical cavity. Two\nmetal electrodes are then patterned on the etched susbtrate. The electrode patterned at the\nbottom of the cavity serves as a local gate electrode, while the one patterned around the\n4\nmesa serves as a source. The vertical distance between the top surface of the mesa and the\ntop surface of the gate is ≃ 230 nm. A few-layer graphene (FLG) flake, estimated to be\nabout 10 layers thick based on optical reflectometry [5], is transferred onto the substrate,\ncontacting the source electrode and covering the mesa. The part of the flake covering the\ncavity is suspended and forms the resonator. Figure R1c shows an optical image of the\ndevice used in the measurements presented in both the main text and the supplementary\ninformation.\na b c\nFLG\nS\nSiO on Si\nFigure R1: Structure of the few-layer graphene nanomechanical resonator used in this work.\n(a) Atomic force microscopy image (AFM) of the patterned substrate. The white structure\nis a mesa with a central cavity etched in silicon oxide. At the bottom of the cavity is a local\ngate electrode. Surrounding the mesa is a source electrode. The electrodes are patterned\nusing electron-beam lithography, followed by the deposition of 5 nm of titanium and 50 nm\nof gold. The vertical distance between the top surface of the mesa and the top surface of the\nelectrodes is ≃ 230 nm. Scale bar: 5 µm. (b) Tilted AFM image highlighting the local gate\nelectrode at the bottom of the cavity and the narrow venting trench connected to the cavity.\n(c) Optical image of the final device, obtained by transferring a ≃ 10-layer-thick few-layer\ngraphene (FLG) flake onto the patterned substrate. The blue-shaded structure is the mesa.\nG: local gate. S: source. Scale bar: 5 µm.\nThe Reviewer continues:\nThe manuscript claims that the device is capable of transmitting and receiving\nweak signals. Will the device function in environments with strong background\nnoise and low signal-to-noise ratio?\nWe thank the Reviewer for this question. Supplementary Note 4, I − Q vector\nlength and reciprocal of error vector magnitude at low drive power, presents our\nmeasurements of graphene vibrations driven by the weakest signal capable of enabling suc-\ncessful video decoding. Furthermore, in the last paragraph of our main text, we now write:\nThe detection of weaker radio signals may be improved by using more advanced SDR\nsystems with lower receiver noise and by engineering the mirror underneath the resonator\nto enhance the device’s responsivity [6].\nReference [6] is one of our recent works, in which we demonstrate that the optical re-\nsponsivity of the resonator can be enhanced by one order of magnitude simply by replacing\n5\nthe planar mirror with a photonic crystal. This enhancement would enable the detection of\nsignals nearly 20 dB weaker than those detectable with our current low-cost software-defined\nradio receiver.\nThe Reviewer writes:\n3. Beyond the points mentioned above, one of my main concerns is the real\ntechnological significance and potential of the proposed device.\nWethanktheReviewerforthiscomment. AsmentionedinourresponsetotheReviewer’s\ncomment #1, the primary aim of our work is rooted in basic science. Specifically, we inves-\ntigate the response of graphene nanomechanical resonators to a broadband, multifrequency\ndrive. Any potential applications discussed are meant solely for illustrative purposes. In our\nabstract, we write:\nOur work reveals distinctive features in vibrations driven by a coherent, multifrequency\ndrive that are not observed in vibrations actuated by single-tone drives or by a force noise.\nFurthermore, in the introduction, we write:\nOverall, studying nanomechanical resonators driven by multifrequency signals sheds light\non the physics of mesoscopic vibrations and leads to proof-of-concept demonstrations for po-\ntential applications. [...] Our work contributes to the recent study of resonators driven\nnonlinearly by multiple tones. Whereas earlier studies explored the case of two-tone driving\n[7, 8], we investigate a previously unexplored regime in which the resonator is coherently\ndriven by a near continuum of frequency components across its full mechanical bandwidth.\nAs Reviewer #5 writes, “Only rev 3 raises serious doubts about the advantages of the\nproposed encoding technique with respect to other options, but I do not think this is the\npoint of an academic publication.” Nevertheless, we find the prospect of future applications\nappealing. In response to the Reviewer’s comment, we write in the second-to-last paragraph\nof our main text:\nIn the spirit of previous nanomechanical devices used as signal processors [9], we envision\nthat our FLG resonators, as well as other resonators based on similar principles, may serve\nas passive radio receivers operating in the HF and VHF bands of the radio spectrum. With\na resonant frequency ranging from ≃ 40×106 Hz to ≃ 60×106 Hz (Fig. 5a), our resonator\noperates within the 6-meter subband, which is internationally allocated for amateur radio\nuse. In this subband, digital amateur television (DATV) broadcasts have been conducted\nusing signals with a 300 kHz bandwidth [10]. In the United Kingdom, DATV with a broader\nbandwidth of up to 500 kHz is permitted in the 2-meter amateur radio band (144–148 MHz)\n[10]. Smaller-diameter FLG resonators, or higher-order vibrational modes, can operate in\nthis frequency range. The filtering bandwidth of our resonator, along with the maximum\n4-QAM data rate of its transduction, 2R∗ ≃ 2×106 bps, is fully compatible with DATV.\nsym\nThe Reviewer continues:\nthan those of its competitors, such as SAW filters, which are widely used in\nvarious consumer electronics. The authors should clarify whether graphene offers\nsignificant advantages that would justify its further development in this context.\nWe thank the Reviewer for this comment. In the last paragraph of our main text, we\naddress the comparison between our devices and SAW filters, with an important caveat.\nIt reads:\nWhile our experimental setup may not yet be practical for receiving amateur radio broad-\ncasts, it can be further simplified [5]. The resonator would serve a dual purpose: as a\nfront-end, tunable bandpass filter, and as a passive optocoupler. It could be placed at the\noutput of the preamplifier typically connected to the feedpoint of a receiving antenna [11].\nAs a filter, it would reject jamming and other interfering signals in a manner similar to\nsurface acoustic wave filters, but with the added benefit of a tunable center frequency, and\nwithout the fabrication complexity of more advanced tunable front-end filters [12]. As an\noptocoupler, it would passively convert the radio signal into modulations of reflected light\nintensity through the nanomechanical transduction, effectively eliminating the issue of radio\nfrequency interference caused by common-mode currents in the coaxial cable that connects\nthe antenna to the demodulator [11]. The detection of weaker radio signals may be improved\nby using more advanced SDR systems with lower receiver noise and by engineering the mir-\nror underneath the resonator to enhance the device’s responsivity [6]. Admittedly, the idea\nof using nanomechanical resonators for radio communication remains largely academic due\nto challenges related to device reproducibility and thermal drift [13]. Nevertheless, these\nresonators may eventually enable alternative architectures for shortwave radio receivers, fa-\ncilitating the decoding of images and videos carried by weak signals in noisy environments\n[14].\nOnce again, we thank the Reviewer for carefully reading our manuscript and providing\nthoughtful comments.\nReferences\n[1] Dykman, M. I. & Krivoglaz, M. A. Spectral distribution of nonlinear oscillators with\nnonlinear friction due to a medium. Phys. Status Solidi B 68, no. 1, 111–123 (1975).\n[2] Zaitsev, S., Shtempluck, O., Buks, E. & Gottlieb, O. Nonlinear damping in a microme-\nchanical oscillator. Nonlinear Dyn. 67, no. 1, 859–883 (2012).\n[3] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechan-\nical resonators. In Reviews of Nonlinear Dynamics and Complexity, Vol. 1. (ed. H.G.\nSchuster), 1–52 (Wiley-VCH, 2008). https://doi.org/10.1002/9783527629374.ch8\n[4] Zhang, C., Zhang, Y., Yang, C., Lu, H., Chen, F., Yan, Y. & Moser, J. Graphene\nnanomechanical vibrations measured with a phase-coherent software-defined radio.\nComms. Eng. 3, Art. no. 45 (2024). https://doi.org/10.1038/s44172-024-00186-4\n7\n[5] Lu, H., Yang, C., Tian, Y., Lu, J., Xu, F., Zhang, C., Chen, F., Yan, Y., Schaedler, K.\nG., Wang, C., Koppens, F. H. L., Reserbat-Plantey, A. & Moser, J. Imaging vibrations\nof electromechanical few layer graphene resonators with a moving vacuum enclosure.\nPrecis. Eng. 72, 769–776 (2021). https://doi.org/10.1016/j.precisioneng.2021.\n06.012\n[6] Lu, H., Li, T., Hu, H., Chen, F., Sun, T., Yan, Y., Wang, C. & Moser, J. Force\nsensing with a graphene nanomechanical resonator coupled to photonic crystal guided\nresonances. arXiv:2507.06755. https://arxiv.org/abs/2507.06755\n[7] Houri, S., Ohta, R., Asano, M., Blanter, Y.M.&Yamaguchi, H.Pulse-widthmodulated\noscillationsinanonlinearresonatorundertwo-tonedrivingasameansforMEMSsensor\nreadout. Jpn. J. Appl. Phys. 58, no. SB, Art. no. SBBI05 (2019). https://doi.org/\n10.7567/1347-4065/aaffb9\n[8] Catalini, L., del Pino, J., Kumar, S. S., Dumont, V., Margiani, G., Zilberberg, O.\n& Eichler, A. Adiabatic and diabatic responses in a Duffing resonator driven by two\ndetuned tones. arXiv:2408.15794v1 (2024). https://doi.org/10.48550/arXiv.2408.\n15794\n[9] Nguyen, C. T.-C. Radios with micromachined resonators. IEEE Spectrum (30 Nov.\n2009).https://spectrum.ieee.org/radios-with-micromachined-resonators.Ac-\ncessed: October 24, 2024.\n[10] Radio Society of Great Britain. Amateur television. https://rsgb.org/main/\ntechnical/amateur-television/\n[11] ARRL—The National Association for Amateur Radio. The ARRL Antenna Book for\nRadio Communications (24th Ed). The American Radio Relay League (2019).\n[12] Hashimoto, K.-Y., Kimura, T., Matsumura, T., Hirano, H., Kadota, M., Esashi, M.\n& Tanaka, S. Moving Tunable Filters Forward. IEEE Microwave Magazine 16, no. 7,\n89–97 (2015). https://doi.org/10.1109/MMM.2015.2431237\n[13] Nguyen, C. T.-C. MEMS-based RF channel selection for true software-defined cognitive\nradio and low-power sensor communications. IEEE Communications Magazine 51, no.\n4, pp. 110–119 (2013). https://doi.org/10.1109/MCOM.2013.6495769\n[14] Crockett, L. H., Northcote, D. & Stewart, R. W. (Editors). Software Defined Radio\nwith Zynq UltraScale+ RFSoC (First Edition). Strathclyde Academic Media (2023).\nhttps://www.rfsocbook.com/\n8"}, {"round": 3, "reviewer_comments": {"reviewer_6": "I am pleased to note that the authors have addressed most of my concerns in the revised manuscript.\nThe newly\nincorporated discussions have improved the paper's clarity. Consequently, I recommend the manuscript for acceptance in its\ncurrent form.\n(Remarks on code availability)"}, "author_response": ""}]

Decision letter