Abstract
Driven nonlinear resonators can display sharp resonances or even multistable behaviours amenable to induce strong enhancements of weak signals. Such enhancements can make use of the phenomenon of vibrational resonance, whereby a weak lowfrequency signal applied to a bistable resonator can be amplified by driving the nonlinear oscillator with another appropriatelyadjusted nonresonant highfrequency field. Here we demonstrate experimentally and theoretically a significant resonant enhancement of a weak signal by use of a vibrational force, yet in a monostable system consisting of a driven nanoelectromechanical nonlinear resonator. The oscillator is subjected to a strong quasiresonant drive and to two additional tones: a weak signal at lower frequency and a nonresonant driving at an intermediate frequency. We analyse this phenomenon in terms of coherent nonlinear resonance manipulation. Our results illustrate a general mechanism which might have applications in the fields of microwave signal amplification or sensing for instance.
Introduction
In bistable systems, weak periodic signals can be amplified by use of external driving. Such external driving can be some noise of appropriate strength in the case of stochastic resonance^{1}, or a highfrequency harmonic signal of appropriate amplitude in the case of vibrational resonance^{2}. Both physical phenomena share qualitative features including a resonantlike behaviour, though the underlying mechanisms differ. Time matching criterion dependent on the applied noise amplitude required for stochastic resonance is replaced, in the case of vibrational resonance, by an amplitude criterion equivalently to a parametric amplification near the critical point. Both phenomena have been reported in many different areas including electronics^{3,4}, optics^{5,6,7,8} or neurobiology^{9,10}. In nanomechanics, the bistable system is usually a simple nonlinear resonator and bistability arises thanks to a quasiresonant forcing.
This phenomenon can be fully understood thanks to the ubiquitous Duffing model which, beyond nanomechanics, can be used for superconducting Josephson amplifier^{11}, ionisation waves in plasma^{12,13} to describe complex spatiotemporal behaviours such as chimera states^{14}. In the frame of the wellknown Duffing model, the oscillator features two equilibrium states of different amplitudes and phases for the same values of parameters. In this regime, substantial resonant enhancement of a weak and slowly modulated signal through stochastic resonance can be achieved either by use of amplitude^{15,16,17} or phase^{18} noise. When the external driving is no more stochastic but rather a harmonic signal of high frequency, a little bit of care has to be taken. The system is then subjected to forces occurring on three different timescales: the one of the signal, the one of the external drive and the one of the forcing. In the standard picture of vibrational resonance, the signal must have a much smaller frequency than the one of the external drive. We here show an enhancement by a factor up to 20 of a weak modulated signal thanks to vibrational resonance. Moreover, the occurence of vibrational resonance in a forced system requires the external driving frequency not only to be higher than the signal frequency but also to be lower than the forcing frequency. Most importantly, we show in that case that the highfrequency driving amplitude renormalises the forced nonlinear resonator response through the manipulation of the nonlinear resonance. We argue that this effect could be used besides the one we are presenting here, as a general mechanism for nonlinear resonance manipulation. Beyond these fundamental aspects, potential applications encompass various fields such as telecommunications, where the data are encoded on a lowfrequency modulation applied to a highfrequency carrier, or, more prospectively, microwave signals processing and sensing such as accelerometers featuring enhanced sensitivities via optomechanically driven signal amplification^{19} or for torque magnetometry^{20}. Bichromatic signals are in addition pervasive in many other fields including braininspired architecture mimicking neural networks^{21} where bursting neurons may exhibit two widely different time scales. Finally, enhancement of weak signal by vibrational resonance might be valuable for binary logic gate based on phase transition for reprogrammable logic operation^{22,23} as well as for memory operation^{24}, physical simulators^{25,26} and more prospective application as quantum computing^{27}.
Results
Nonlinear driving of the nanoelectromechanical resonator
In our experiments, the resonator consists of a nonlinear nanoelectromechanical oscillator formed by a thin micronscale InP suspended membrane (see Fig. 1a). The membrane’s outofplane motion is induced by applying an AC voltage V(t) on integrated metallic interdigitated electrodes placed underneath the membrane at a submicron distance (see Fig. 1a; see Chowdhury et al.^{28} and Methods for more details). It is placed in a lowpressure chamber (10^{−4} mbar) in order to reduce mechanical damping. The outofplane motion of the membrane is probed optically (see Fig. 1b) thanks to a Michelson interferometer whose one end mirror is formed by the oscillating membrane (see Fig. 1a). The membrane mechanical fundamental mode of oscillation lies at 2.82 MHz in the linear regime with a mechanical quality factor of Q_{M} ~ 10^{3}.
The AC forcing voltage lies in the MHz regime and writes:
Here V_{0} is the amplitude of the applied voltage while ν_{f} denotes the frequency of the quasiresonant forcing. When sweeping up and down the frequency ν_{f} in the vicinity of the fundamental mode frequency, asymmetry in the mechanical response spectrum appears for a sufficiently high driving amplitude V_{0} > 5.5V (see Fig. 1b). Hysteresis behaviour becomes prominent and two stable points, represented by the low and the high amplitude values of the mechanical motion, coexist. The evolution of the bistable region width as a function of V_{0} is shown in Fig. 1b Right. The closing of this bistable region for increasing V_{0} cannot be observed due to limited voltage handled at the electrodes terminals. In the following, V_{0} will be set to 9V in order to be deeply in the bistable regime, and the forcing frequency is set at ν_{f} = 2.825 MHz, close to the middle of the hysteresis region, in order to get symmetrical potentials^{18}.
Externally induced dynamics in the time domain
Jumps between the two stable states of oscillation can be induced by slowly modulating the forcing amplitude. This scenario can be implemented by applying to the electrodes a voltage in the form of:
where γ and ν_{m} denote respectively the modulation index and the frequency of the amplitude modulated signal. Yet, a sufficiently high modulation amplitude is needed to drive the system in order to overcome the barrier height and to induce interwell motion following the applied modulation. In the case of a weak amplitude modulated signal (as in our experiments with γ set at 0.1), the system is solely subjected to intrawell modulated motion as can be seen in Fig. 2 top for the resonator being initially prepared in its upper state. Amplification of the weak modulated signal following jumps of the system between the two states can however still be induced in that case by adding an external driving with a frequency that is much higher than the frequency of the weak modulation, but still lower than the forcing frequency.
In this scenario reproduced in our experiment, the external drive takes the form of an additive amplitude modulation voltage of amplitude V_{HF} ≡ δ ∗ V_{0} and frequency ν_{HF}. The criteria for enabling the onset of vibrational resonance as predicted by theory (see section theoretical analysis), requires strong frequency inequalities: ν_{m} ≪ ν_{HF} ≪ ν_{f}. The total applied voltage then writes:
Eq. (3) describes the total applied signal required to achieve amplification of the weak signal at ν_{m}. The signals at ν_{f} and ν_{HF} can be externally controlled and triggered in order to respectively probe and enhance amplification of the weak signal which needs to be detected. Figure 2 shows time series of the mechanical motion amplitude for ν_{m} = 30 Hz and increasing amplitudes of the amplitude modulation at high frequency ν_{HF} = 200 kHz > 6500 ⋅ ν_{m}. The system starts in its upper state (high amplitude state) where the small signal modulation is visible as a small intrawell motion. As the amplitude of the external driving increases, switching events between the two stable states become more prevalent. At first, occasional interwell transitions occur, weakly locked to the modulation signal. For V_{HF} = 6.4V, the system response gets completely synchronised with the applied weak and lowfrequency modulation. Further increase of the additional external drive amplitude worsens the synchronisation and the system drops into its lower amplitude state, where a small intrawell modulation is visible. There is thus an optimal amplitude of the external drive which maximises the response amplitude. When the system is modulated close to the hysteresis turning points, it is more sensitive to noise induced fluctuations which are inherent in the experimental system, and this results in the observed aperiodic switchings in the weakly locked regions (cf. Fig. 2).
Gain factor
The gain or amplification factor can be inferred by quantifying the achieved spectral power amplification. For every time traces recorded on a time scale of 600 s, Discrete Fourier Transform (DFT) are performed. The resulting DFT spectra are presented in Fig. 2. They feature peaks, the most prominent being at the modulation frequency ν_{m}. The achieved gain M is then given by the ratio between the strength of the peak in the DFT spectrum at ν_{m} for a given amplitude of the external driving and its strength without external driving (V_{HF} = 0V). The induced gain factor is presented in Fig. 3. The gain factor features a resonantlike behaviour: the gain factor first rapidly rises with the strength of the external driving, reaches a maximum for V_{HF} = 6.4 V and then drops. The maximum achieved gain factor is M = 20. Experimental noise modifies either the amplitude or the bistability region of the response. The probe at the frequency of the quasiresonant forcing ν_{f} being fixed, this noise lead to fluctuations visible in the gain.
Vibrational resonance is governed by an amplitude condition. It occurs close to the transition from bistability to monostability, during which the effective potential of the slow variable evolves from a rapidly oscillating double well to a single well with a parametric dependence on the highfrequency signal amplitude and frequency^{29}. As such, this phenomenon has some features in common with parametric amplification near the critical point.
Theoretical analysis
To figure out the origin of this resonant response, we introduce a simplified theoretical model and compare its results to our experimental findings. The original treatment of vibrational resonance in refs. ^{2,30} considers the motion of a nonlinear oscillator in a bistable potential, subject to a lowfrequency signal and a highfrequency drive. Theoretical studies so far have mostly concentrated on studying the impact of the potential shape on the resonance^{29,31,32}, or the response to multifrequency signals^{33}. Interestingly, it was also noted in Gitterman^{30} that one particularly important aspect of vibrational resonance was the ability to change the stability of some equilibria, or to have control over the shift of the resonance frequency. Our system only becomes a nonlinear oscillator if it is resonantly driven. Conversely, it cannot show a bistable response per se, whatever the sign of the stiffness parameter α. However, with a quasiresonant harmonic forcing, the nonlinear oscillator can become bistable. It is then interesting to examine in more details if an additional “high” frequency forcing can induce a resonance on a small amplitude signal.
The nanoelectromechanical system can be described in a good approximation as a forced nonlinear (cubic) Duffing oscillator^{18}. Its dynamics can be modelled, in the limit of the small injection and the dissipation of energy by
where x(t) accounts for the outofplane displacement of the membrane, η is the effective damping, ω_{0}/2π is the natural oscillation frequency of the membrane, α is the nonlinear stiffness coefficient, F is the amplitude of the modulated forcing with frequency ω_{f}/2π ≡ (ω_{0} + Δ)/2π, introducing the small detuning from resonance Δ. The highfrequency amplitude modulation has an amplitude Fδ and a frequency Ω/2π = ν_{HF}. The oscillation amplitude of the oscillator is the result of the beating of two frequencies: one fast at Ω and one slow at ω_{m}. The parameters γ and δ characterise the amplitude of the beating. By considering the following separation of timescales for the forcing frequencies ω_{m} ≪ Ω ≪ ω_{0}, an amplitude equation for the timeaveraged dynamics can be derived (see Methods). We start by deriving the equation for the amplitude of the forced nonlinear oscillator close to resonance (ω_{f} ~ ω_{0}) by looking for a solution in the form \(x(t)=C(t){e}^{i({\omega }_{0}+\Delta )t}+{\mathrm{cc}}\) (where cc accounts for the complex conjugate term):
The strong timescale separation of the modulation frequencies motivates the introduction of a timeaveraged variable A over the short period 2π/Ω^{34} such that
The amplitude equation for the averaged response writes
where we have introduced rescaled quantities: \(\frac{F}{{\omega }_{0}}\to F\), \(\frac{\delta }{\Omega }\to \delta\) and \(\frac{\alpha }{{\omega }_{0}}\to \alpha\).
The averaged equation satisfies an amplitude equation with a renormalised detuning Δ − 3αF^{2}δ^{2}/16 which depends on the highfrequency driving amplitude. The most important aspect to note here is that the nonresonant and “high” frequency driving can modify the resonance behaviour of a nonlinear system. To study how the intermediate frequency Ω modifies the resonance region, we consider the polar representation A = Re^{iϕ}/2 with γ = 0, and solve for the steady state \(\dot{R}=\dot{\phi }=0\). We get the characteristic equation
Note that in the limit of zero highfrequency amplitude modulation (δ → 0) we recover the deterministic forced Duffing resonator model. At this point, we highlight that the timescale separation hypothesis, ω_{m} ≪ Ω ≪ ω_{0}, is central to obtain this result. Indeed, if we suppose ω_{m} ≪ ω_{0} ≪ Ω, i.e. a very highfrequency driving and average Eq. (4) before deriving the amplitude equation, then we cannot show evidence for vibrational resonance.
We numerically investigate the amplitude equation given by Eq. (7) for the parameters σ = 0.0016, η = 0.001, α = 0.4 and ω_{m} = 2π/200,000. These parameters values are chosen to match the experimental ones. By fitting the nonlinear resonance curve (see Fig. 1b) we get the mechanical quality factor Q_{M} = η^{−1} and the nonlinear spring term α^{18}. The signal modulation frequency is chosen to be much larger than mechanical quality factor to ensure almost adiabatic evolution. Figure 4 shows the steadystate response curves (inferred from Eq. (7)) versus the driving amplitude for different highfrequency amplitude modulation. Without any highfrequency drive (δ = 0), the system displays a large hysteretic response (see Fig. 4a). A slow modulation of amplitude less than the hysteresis width would not produce any jump between the branches, hence would not produce any strong amplification of the signal at ω_{m}. The addition of the highfrequency drive introduces an extra detuning which deforms the nonlinear response: in Fig. 4a, we observe that the centre of the hysteresis loop is shifted towards lower driving forces F, and that the width of the hysteresis shrinks as well. Since the signal modulation amplitude scales as Fγ, this means that a smaller slow modulation amplitude γ will be necessary to overcome the hysteresis width and produce large jumps between the lower and upper branch. This is the essence of the vibrational resonance phenomenon. In order to check this, we integrated numerically Eq. (7) with a slow amplitude signal at ω_{m}. The results are shown on the time traces plotted in Fig. 4a, c. In the absence of highfrequency amplitude modulation (δ = 0) the response is quasi linear since the system cannot jump between the lower and higher branches for the chosen modulation amplitude. The amplification factor is close to one if the system resides on the lower branch (same as the linear regime), or can be even smaller if it resides in the upper branch (deamplification). When the driving δ is increased, the system is tuned into resonance and undergoes synchronous jumps with the signal driving frequency between the lower and upper branches (Fig. 4c). This corresponds to a large signal amplification, provided the signal amplitude is large enough, i.e. larger than the hysteresis width for the chosen parameters.
The amplification is shown in Fig. 5, rescaled to the response with zero highfrequency drive in the quasi linear case, i.e., on the lower branch. We plot both the amplitude ratio M_{a} (ratio of the response amplitude with and without highfrequency drive) and the powerspectral density (PSD) ratio M. M_{a} displays a sharp transition corresponding to the tuning of the system into the bistable region. When the highfrequency drive is not large enough, the system stays in the lower branch and the response is quasilinear, leading to an amplification factor close to 1. When the bistable regime is reached, a large amplitude amplification is obtained. And for still higher δ the system stays in the upper branch where the response is sublinear, thus leading to deamplification as expected. The same thing occurs in the PSD, except that the transition is less marked because the response of the system is highly nonlinear, hence the spectral energy is spread among the different harmonics. Note that we reach here, with the chosen parameters, a PSD amplification of the same order of magnitude as in the experiment. However, much larger amplification factors can be reached for other slightly different parameters, as illustrated in Fig. 5 where M ~ 140 is obtained for still smaller linear driving signal not accessible in experiments. This important point is illustrated in Fig. 4b, d. If the signal strength is too small to overcome the hysteresis width, it is possible to increase the highfrequency drive to tune it into the resonance. As shown in Fig. 4b for δ = 1000, a higher highfrequency modulation shifts the hysteresis curve further to the left, i.e., to lower overall forcing, but most importantly reduces the width of the hysteresis while not changing the hysteresis height too much. This makes it possible to amplify a much weaker signal by the vibrational resonance phenomenon. Note also that the amplification factor is even much larger in that case because of the already discussed different effect on the width and on the height of the hysteresis. This shows that it is necessary to tune both the highfrequency drive δ and the modulation strength F to amplify optimally a signal of a given amplitude.
Discussion
The previous analyses clearly indicate the primordial role of the highfrequency amplitude modulation and of the proper timescale separation in such vibrational resonance phenomenon. The former allows to control the nonlinear resonance in order to amplify weak signals. The latter, while being compulsory for technical reasons in the theoretical analysis, could potentially be relaxed in experiments. The exact value of the external drive frequency Ω is not critical at all, as long as it satisfies the timescale separation condition with ω_{m} ≪ ω_{f}, Ω and if it remains nonresonant. Signal amplification results from the tuning into resonance of the nonlinear response of the system. Amplification could occur also in the case of a non multivalued response, as long as the slope of the tuned response is large enough to ensure appropriate amplification. In principle, for any given signal amplitude, it is always possible to adjust both the forcing strength F and the highfrequency modulation strength δ for vibrational resonance to occur. However, the maximum gain achievable will be a complicated function of all the system parameters. It will occur at the nascent bistability, i.e. when the hysteresis curve has an infinite tangent. Indeed, in this case, any small nonzero amplitude signal will be maximally amplified until saturation on the lower or upper branches. Ultimately, maximum gain achievable is limited by signal noise or noise in the system. Concerning the signal amplification frequency, its maximum is limited by the frequency response of the oscillator which is given by the damping rate (~1 kHz here). In order to push it further it is necessary to either decrease the mechanical quality factor of the nanomembrane or increase the resonant frequency with a similar quality factor.
By comparing the amplitude magnification curves in Figs. 3 and 5 we note a slight softening effect on the experimental gain curve whereas the theoretical one shows a sharp transition to high gain when the signal modulation is larger than the hysteresis curve turning points. This difference can be attributed to residual noise in the experiment which can modify the behaviour of the system close to the turning points of the nonlinear response.
At last, as observed in previous optical implementations of vibrational resonance^{8,35} in nonparametrically forced bistable systems, the gain obtained seems higher than to the one observed in stochastic resonance for the same system. Even though vibrational resonance and stochastic resonance are based on different physical principles, this observation is verified in our system from a raw quantitative comparison with the stochastic resonance amplification^{18}.
In conclusion, we established and analysed the conditions for using vibrational resonance in order to enhance weak signals in a forced nonlinear oscillator, even if the system is initially monostable. The physical phenomenon is based on the resonance manipulation, thanks to a nonresonant, highfrequency amplitudemodulation drive obeying a timescale separation condition. We derived a model to describe vibrational resonance in a monostable, forced nonlinear oscillator which shows good agreement with our experimental results obtained on a forced nanoelectromechanical membrane. This deterministic amplification method gives rise to high amplification factors, especially when compared to stochastic resonance^{36}. As such, these results pave the way towards the design of novel architectures based on nonlinear dynamic resonances for weak signal amplification, as currently done by quantumlimited Josephson parametric amplifiers^{37} or, in the optical domain, by phasesensitive amplifiers in the optical domain^{38} to name a few. In a more general framework, it may open new avenues for the manipulation of nonlinear resonances with the addition of a nonresonant driving field.
Methods
Fabrication of the InP resonator membrane
The fabrication of the whole platform is based on a 3D heterogeneous integration process involving mainly four steps. First, a 400nmthick SiO_{2} layer is deposited on the 260nmthick InP membrane, which is grown along with a 500nmthick InGaAs etchstop layer on top of an InP (100) substrate by metal organic vapour phase epitaxy. Simultaneously, interdigitated electrodes (IDTs) arrays, displaying a finger period of 2 μm, finger length of 10 μm and electrode width of 500 nm, are deposited on a Si substrate. The patterning process involves an electronic lithography, deposition of a 200nmthick gold layer and standard liftoff. The Si chip is then spincoated with a 200nmthick DiVinylSiloxaneBenzoCycloButene (DVSBCB) layer, thereby planarising the Si substrate. In the second step, the InP wafer is bonded on the Si substrate at hightemperature (300 °C) by positioning the SiO_{2} layer atop the DVSBCB layer and by using a vacuum wafer bonding technique^{39}. The InP substrate and InGaAs etchstop layer are then removed by chemical etching, leaving the residual 260nmthick InP membrane on the DVSBCBSiO_{2} layer. In the third step, the InP membrane is patterned by standard ebeam lithography and dryetching, to form a twodimensional squarelattice photonic crystal of periodicity 532 nm, hole radius 181 nm and whole surface of 10 × 20 μm^{2}. It is clamped by four tethers of 2 μm length and 1 μm width, in order to reduce clamping losses. The alignment of the photonic crystal mirror with respect to the IDT’s arrays, is performed with an accuracy better than 20 nm, by making use of alignment marks deposited beforehand on the Si substrate. Last, the photonic crystal membranes are released by underetching the underlying 400nmthick SiO_{2} layer, followed by a critical point drying step. The lateral InP suspension pads act as protective structures for the SiO_{2} layer beneath them, leaving them anchored to the substrate.
Measurement of the outofplane motion
A He–Ne laser with wavelength of 633 nm is sent to the membrane. The reflectivity of the membrane is enhanced up to 50% by piercing a square lattice photonic crystal in it^{40}. The laser is focused on the membrane via an objective with a NA of 0.4. The light reflected by the membrane is brought to interference with a strong local oscillator. A balanced homodyne detector locked on the drive frequency at the interferometer output is then used to decipher the amplitude and phase of the mechanical motion.
Actuation of the mechanical oscillator
The membrane is driven via the electrostatic force induced by the electrodes placed underneath. These electrodes are connected to an external signal generator which can go up to 50 MHz and is synchronised with a lockin amplifier (HF2LI) which demodulates the detected signal at the actuation frequency. For vibrational resonance, the weak (ν_{m}) modulated signal is generated by an another signal generator (Model Agilent 33522A) and combined with the additive (ν_{HF}) signal in the HF2LI. This signal is then modulated at the frequency of the quasiresonant forcing (ν_{f}) and sent to the electrodes. The electrical signal obtained from the photodiodes (Thorlabs APD120A2) is timerecorded with the oscilloscope function of the HF2LI. The sampling frequency is 900 Hz, a much higher frequency than the one of the amplified signal.
Derivation of the theoretical model
Let us consider the timescale separation ω_{m} ≪ Ω ≪ ω_{0} and the resonance condition ω_{f} ~ ω_{0}. We first look for a solution to Eq. (4) using the ansatz \(x(t)=C(t){e}^{i({\omega }_{0}+\Delta )t}+cc\) (where cc accounts for the complex conjugate term). After straightforward algebra, one gets an amplitude equation for the slow envelope C(t), assuming that \({\partial }_{tt}C\ll {\omega }_{0}^{2}C\) and ∂_{t}C ≪ ω_{0}C:
The amplitude equation (Eq. (8)) corresponds to the one of a forced oscillator with temporally modulated amplitude. Since we have a strong timescale separation of the modulation frequencies, ω_{m} ≪ Ω, one can consider the averaged variable on the short period 2π/Ω
and homogenise the scales^{41} by writing
Considering that the envelope A(τ) is a slow variable (∂_{τ}A ≪ ω_{m}A), using the ansatz (9) in Eq. (8) and averaging over the period 2π/Ω, we get the amplitude equation for the averaged response
Introducing the notation \(F^{\prime} =\frac{F}{{\omega }_{0}}\), \(\delta ^{\prime} =\frac{\delta }{\Omega }\) and \(\alpha ^{\prime} =\frac{\alpha }{{\omega }_{0}}\) and omitting \(^{\prime}\), the equation reads
We further introduce a Madelung transform A = Re^{iϕ}/2 and γ = 0, and get
At steady state, \(\dot{R}=\dot{\phi }=0\) and we finally get the characteristic equation
Highfrequency modulated forcing (Ω ≫ ω _{0})
Let us consider the nanoelectromechanical system described by model Eq. (4) in the limit Ω ≫ ω_{0} ~ ω_{f}. Due to the separation of temporal scales, one can analogously separate the temporal scales by means of the following change of variable
where the average over the rapid temporal variable reads
Introducing the ansatz (15) in Eq. (4) and taking the average over the rapid temporal variable, one gets after straightforward calculations,
Therefore, within this limit, the resonator natural frequency is modified to
In addition, the system has an extra parametric forcing term. Considering the forcing frequency close to the new resonant frequency \({\omega }_{{\rm{f}}}={\tilde{\omega }}_{\mathrm{o}}+\tilde{\Delta }\) and looking for a solution of the form \(z={\mathrm{c}}{{\mathrm{e}}}^{i{\omega }_{{\rm{f}}}t}+{\mathrm{c}}.{\mathrm{c}}.\) in Eq. (17), we obtain the amplitude equation for C
By comparing Eq. (19) with Eq. (6), we notice that the terms are similar except for the complex conjugate term which is a signature of the additional parametric forcing. Therefore, we can conclude that the resonance of the system can be manipulated as in the previous case with the highfrequency forcing amplitude. However, the presence of a simultaneous parametric resonance does not allow to draw a simple conclusion on the amplification of the weak signal in that case, and we expect the parametric resonance to modify substantially the general physical picture.
Manipulation of the resonance
The characteristic equation Eq. (14) can be rearranged to yield a third order polynomial equation in z = R^{2} such that
We look for extrema of the characteristic curve p(z) = 0 in the plane (z, F) to compute the hysteresis width which yields two roots z_{±}
The hysteresis width can now be obtained by solving for F_{±} in p(z±) = 0. One obtains the implicit formula
with \({\Delta }_{\pm }=\sqrt{{\left(3\alpha {\delta }^{2}{F}_{\pm }^{2}16\Delta \right)}^{2}192{\eta }^{2}}\). An analytic expression for the hysteresis width ΔF = F_{+} − F_{−} or the hysteresis centre \({F}_{{\mathrm{c}}}=\frac{1}{2}\left({F}_{+}+{F}_{}\right)\) cannot be expressed easily. However, for the set of parameters used in Fig. 4 we can compute both ΔF and F_{c}. With these parameters, the hysteresis width can be almost closed for a large highfrequency amplitude modulation δ, whereas the hysteresis central frequency can be tuned in a large range, the same order of size as the original width.
Data availability
All data and figures that support the findings of this study are available in Zenodo with the identifier https://doi.org/10.5281/zenodo.3595858.
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Acknowledgements
This work is supported by the French RENATECH network, the Marie Curie Innovative Training Networks (ITN) cQOM and the European Union’s Horizon 2020 research and innovation programme under grant agreement No 732894 (FET Proactive HOT), and the Agence Nationale de Recherche projet ADOR (grant agreement no. ANR19CE24001101). M.G.C. thanks the Millennium Institute for Research in Optics (MIRO) and FONDECYT projects Grants No. 1180903 for financial support.
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All authors planned the experiment and discussed the data. The sample was fabricated by A.C. and R.B., the measurement was carried out by A.C. in a setup build by A.C. and R.B. and A.C. analysed the data. M.G.C. and S.B. developped the theoretical model and I.R., M.G.C., S.B. and R.B. wrote the manuscript.
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Chowdhury, A., Clerc, M.G., Barbay, S. et al. Weak signal enhancement by nonlinear resonance control in a forced nanoelectromechanical resonator. Nat Commun 11, 2400 (2020). https://doi.org/10.1038/s41467020158273
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