Nano-mechanical resonators have gained an increasing importance in nanotechnology owing to their contributions to both fundamental and applied science. Yet, their small dimensions and mass raises some challenges as their dynamics gets dominated by nonlinearities that degrade their performance, for instance in sensing applications. Here, we report on the precise control of the nonlinear and stochastic bistable dynamics of a levitated nanoparticle in high vacuum. We demonstrate how it can lead to efficient signal amplification schemes, including stochastic resonance. This work contributes to showing the use of levitated nanoparticles as a model system for stochastic bistable dynamics, with applications to a wide variety of fields.
Nano-mechanical resonators need to meet criteria of light mass and high-Q factor in order to maximize their performances, when operated as linear force transducers. However, these features lead to intrinsically nonlinear behaviour1, with consequent vanishing dynamical range. To overcome this limitation, modern nanotechnology requires new sensing schemes that take nonlinearities into account and even benefit from them2. Many of the proposed solutions operate inside an instability region3,4 or close to a bifurcation point5,6, where the system ideally becomes infinitely sensitive. Others exploit fluctuations of noisy environments to trigger stochastic resonances7 that amplify weak harmonic signals8,9,10. For all these sensing applications, a single nanopartticle optimally decoupled from the environment represents a particularly interesting system in the family of high-Q resonators. In fact, since its very first realization11, optical levitation of a nano-scale object in vacuum has enabled several ground-breaking experiments, including demonstration of zeptonewton force sensitivity1,12, tests of fluctuation theorems and stochastic thermodynamics13,14, as well as the observation of photon recoil heating15.
In this work, we demonstrate full control on the linear, non-linear and bistable dynamics of a levitated nanoparticle in high vacuum and under the effect of external noise. The potential of our platform is validated by the implementation of two nonlinear amplification schemes, including stochastic resonance7. Remarkably, we demonstrate, in excellent agreement with theory, up to ∼50 dB amplification of non-resonant harmonic force at the atto-newton scale. In addition, the unprecedented level of control achieved will enable the use of levitated nanoparticles as a model system for stochastic bistable dynamics with applications to a wide range of fields including biophysics16,17, chemistry18,19 and nanotechnology20.
Our experiment is sketched in Fig. 1a. A single silica nanoparticle (d∼177 nm in diameter) is optically trapped in vacuum by a tightly focused laser beam, and its 3D trajectory monitored with a balanced split detection scheme. The detector signal controls an electro-optic modulator (EOM) that feedback-cools all three translational degrees of freedom by modulating the laser intensity. In high-vacuum conditions (P∼10−6 mbar) the mechanical Q-factor, given by the gas damping γ0, is measured to be Q=1.2 × 108. Under feedback, the nanoparticle oscillates at non-degenerate angular frequencies Ω0(x,y,z) with very small amplitudes, corresponding to sub-Kelvin effective temperatures11. By optimizing the feedback settings (Supplementary Note 2) and carefully screening important sources of noise such as mechanical vibrations and air turbulences, we obtain a highly stable resonator with frequency fluctuations improved by one to two orders of magnitude compared to previous works1,21 (Supplementary Note 3). When one of the spatial modes is parametrically driven at resonance21, the particle explores the anharmonic part of the optical potential, which can be modelled as a Duffing nonlinearity1. As a result, the equation of motion for the driven coordinate, which we choose to be x, reads
where η is the non-linear feedback-induced damping coefficient, ξ is the Duffing term prefactor, and Ωm are, respectively, the modulation depth and modulation frequency of the driving signal. Finally, the fluctuating force has two contributions. The first one, , represents the stochastic force arising from random collisions with residual air molecules in the chamber, while the second, , represents artificial parametric noise that we add through the EOM. See Supplementary Note 1 for further experimental set-up details.
The nonlinear response of the resonator is fully characterized by measuring the oscillation amplitude in the driving parameters phase space , where δm=Ωm/Ω0−2 is the normalized detuning21. When the modulation frequency is decreased (down-sweep) across the resonant condition δm=0, the amplitude response spans a triangular region corresponding to the first instability tongue22. This is shown in the top 2D false colour map of Fig. 1d. In the opposite case of a frequency up-sweep, however, the system displays hysteresis, and the tongue results in a narrower width. The red and blue cross cuts in Fig. 1d clearly show this behaviour, identifying a region where two stable oscillation states coexist (Fig. 1b,c). The bistability region, shown at the bottom of Fig. 1d, is obtained by subtracting up- from down-sweeps maps. By fitting our experimental data with the analytical solution (See Supplementary equation 1; ref. 21) we retrieve the nonlinear coefficients η=(18.1±0.4) μm−2 and ξ=(−9.68±0.15) μm−2. Such precision measurements are made possible by the highly stable oscillation frequencies that lead to minimal drifts of the detuning parameter (Δδm≈2ΔΩ/Ω0⪅10−3, over the entire measurement time), therefore allowing to predict with excellent agreement the boundaries of the instability tongues (white lines in Fig. 1d). Further details on spectral features of the nonlinear response are provided in Supplementary Note 4.
Inside the bistable region, natural thermal noise can activate spontaneous transitions between low- and high-amplitude states. However, in high vacuum these events are extremely rare, which makes our system a promising candidate for nonlinear sensing. To observe stochastic activation within a reasonable time we therefore add to the EOM a Gaussian noise ζ(t) of amplitude NV, that in turns produces a position dependent parametric stochastic force of the form . Noise calibration can be performed by looking at the particle’s oscillation amplitude A in the absence of parametric driving23. For low noise levels, the particle’s dynamics remains unaffected, being still dominated by the thermomechanical noise of the environment. However, for sufficiently high noise, the particle’s amplitude starts to increase. Assuming an effective temperature , the noise amplitudes NV are calibrated into meaningful temperature units TN (Supplementary Note 5). Interestingly, the measured trend differs from what reported in other systems8, where a quadratic dependence between TN and NV was found. This is to be ascribed to the parametric (that is, multiplicative) nature of the noise injected in the present case, and prompts the interest of future studies.
The stochastic switching dynamics of the levitated nanoparticle is investigated via the injection of noise. Figure 2a illustrates position and amplitude time traces corresponding to few switching events and emphasizes the remarkable ability of our tracking scheme to resolve considerably different time scales. The amplitude probability distribution ρ(A), obtained by analysing the particle trajectory over ∼1 min, is shown in Fig. 2b. Even though the particle is in a non-thermal state due to the applied parametric control13, we introduce a simplified model that describes the dynamics of its amplitude as the motion of a thermal fictitious particle in a double well potential. This justifies the use of the Boltzmann–Gibbs distribution to extract the amplitude effective potential that models the bistable dynamics. Note that this approximation does not induce any inconsistency in the subsequent analysis, which relies in the experimental determination of the symmetric condition for . The high-frequency stability of the system allows us to finely modify the probability distribution ρ(A), and hence to engineer almost at will the shape of following any path inside the bistable regime. An interesting case consists of the so called iso-amplitude lines: paths along which the amplitude , and consequently the amplitude gap , is kept constant. These particular subsets of the phase space are nonlinear features generally hidden by the frequency noise that blurs the measured instability tongue10,21. Thanks to the reduced frequency fluctuations, they are clearly visible in our system. Few examples of iso-amplitude lines are shown in the lower panel of Fig. 1d (solid lines), and satisfy the following relation (See Supplementary Note 4 for derivation):
being . By following an iso-amplitude line with a fixed , we obtain a very smooth evolution of the effective potential (Fig. 2c). Starting from an asymmetrically tilted configuration, it progressively undergoes inversion of its shape, passing through a quasi-symmetric condition where the potential barrier determines equal depth of the two wells. It is important to stress that, upon this ad-hoc dynamical sweep of and δm, the two minima of the potential corresponding to the amplitudes and , maintain their position fixed. Clearly, this would not be the case when following any other path, for example moving along =const lines (see cross-cuts in Fig. 1d). The populations of the two states Σhigh and Σlow, defined as the normalized integrals of the amplitude distribution respectively above and below a threshold , are shown in Fig. 2d, and display an inversion consistent with the potential dynamics observed in Fig. 2c. The switching rate Γ also depends significantly on the detuning (Fig. 2e), with a maximum corresponding to the symmetric potential configuration. Under this symmetry condition the switching rate, as a function of noise temperature TN, is expected to follow Kramers’ law . However, the injection of noise and its interplay with nonlinearities leads to a drift of the oscillation frequency, equivalent to a change in the detuning δm, thereby distorting the symmetric potential (see Supplementary Note 6). If this change were not taken into account, a decay of the switching rate would be observed8. Conversely, if an adjusted detuning is applied (Supplementary Fig. 8), the symmetric potential is preserved and the prediction of Kramers’ law is perfectly met, as we show in Fig. 2f.
The achieved control over the bistable dynamics makes our system a very flexible platform for studying two different signal amplification schemes. The first relies on a common phenomenon in bistable systems called stochastic resonance (SR)7. A weak periodic perturbation induces a modulation of the potential barrier separating the two states, and leads to an overall synchronized (that is, quasi-coherent) switching dynamics when the interwell transition rate matches twice the frequency of the perturbation. Given that this rate depends monotonically on the amount of noise in the system, SR leads to a noise-induced rise (and then fall) of the signal-to-noise ratio (SNR), and can therefore be exploited to amplify a narrow-band signal in a nonlinear system under appropriate conditions.
To this aim, we prepare the bistable system driving the particle with suitably chosen parameters . Once more, the noise-dependent detuning avoids the escape from the bistable region as a consequence of hysteresis quenching8,24, which in turn would prevent the observation of the full SR curve when noise is increased. The periodic () perturbation is then introduced modulating the depth of the parametric driving signal, namely , where the modulation strength χAM corresponds to optical forces of ten to a hundred atto-Newtons for the parameters used in our experiment. Amplitudes are measured with long acquisition times (∼103 s) and repeated for increasing noise temperatures.
At TN∼10 K (point marked A) switching events are only partially correlated with the modulation signal (Fig. 3b). Instead, when noise is increased to TN∼16 K (point B) the system clearly exhibits an overall synchronization. A spectral analysis of the corresponding amplitude trace shows that SA features an extremely sharp peak precisely at (see inset in Fig. 3a), for which spectral amplification G and noise floor N can be defined. When noise is further increased, the switching dynamics loses correlation with the modulation signal. Interestingly, a similar and counterintuitive behaviour is observed in the noise floor N, which presents a similar non-monotonic trend caused by a redistribution of noise intensity towards higher frequencies. Linear perturbation theory of stochastic resonance7 (Supplementary Note 7) predicts a SNR=G/N of the form:
where χ is the modulation of the potential, proportional to χAM.
Figure 3c shows the experimental SNR curves for different values of χAM, together with a fitting of equation (3) that displays very good agreement. Similarly, the ratios of the fitted modulation depths (2: 5.4: 11) are also in good agreement with the expected ratios (2: 5: 10).
The second SR experiment was carried out for frequency modulation of the driving signal, , which according to our study (Fig. 2c), induces a potential modulation consistent with SR requirements. The corresponding results are shown in Fig. 3d. We emphasize the fact that previous experiments have only explored amplitude modulation, and that this is the first experimental demonstration of SR with frequency modulated signals. Again the fitted modulation strengths give ratios (1.9: 2.6: 4.9), in good agreement with the expected ones (2: 2.5: 5). Interestingly, the resonance appears at a noise temperature equivalent to the one observed in the amplitude modulation case. This is consistent with the fixed value of along the two experiments that requires an equal ΓK in order to fulfil the time-scale matching of the resonant condition.
Our platform enables us to directly compare the SR experiment with another amplification scheme that is implemented by changing the driving parameters to zero detuning. For δm=0, the particle is driven outside of the bistable regime, but still inside the instability tongue (see inset in Fig. 4). In this configuration, the effective potential is monostable, and the AM and FM modulations applied (having same χAM/FM as in the SR case) result in a modulation of the oscillation amplitude for which a spectral analysis still leads to the observation of a peak at . The corresponding SNR as a function of noise temperature TN are shown in Fig. 4, together with fitted exponential decays that properly follow the experimental data for TN<30 K, and the fits of Fig. 3c,d in order to ease a direct comparison of the two amplification schemes. Independently of the noise level, direct amplification always features a higher SNR than the corresponding SR, and the two methods give similar outcomes at high TN. This result is an experimental verification of the central dogma of signal detection theory25, namely that stochastic resonance can decrease the SNR degradation of a noisy signal but it does not provide a mechanism by which the undetectable becomes detectable26,27. The couterinuitive nature of SR, in fact, mainly relies on suboptimal parameter ranges28, here exemplified by the fact that the same modulation becomes remarkably detectable when the detuning is set to zero, without the need of adding noise. Nonetheless, the fact of SR being observed in a wide variety of settings poses the question of why this phenomenon is omnipresent and favoured by nature. We surmise that suboptimal balanced configurations are generally preferable when dealing with very complex systems, for which optimal conditions would be difficult to achieve. Thus, exploiting ambient noise and stochastic resonance appears to be a successful strategy to ensure a robust amplification method that, at least above certain thresholds, is less sensitive to noise changes than other detection schemes (Fig. 4).
In conclusion, we presented an extensive study of the stochastic bistable dynamics of a levitated nanoparticle in high vacuum. The high stability achieved—in particular close to the bifurcation point—enabled the application of our system as a test platform for different amplification schemes, leading (to our knowledge) to the first qualitative and quantitative agreement between the stochastic resonance of a nanomechanical resonator and the corresponding analytical models. The present state of the experiment allowed up to ∼50 dB amplification of a parametric force at the attonewton scale that was mimicked by modulating the trap stiffness. A similar perturbation could arise from a time varying nonlinear potential. However, to evaluate the performance of our system to detect a conventional linear force, a direct actuation is required. This could be implemented by electrically coupling a charged particle29 to an external field12 or via scattering force from a weakly focused beam and is the subject of future work. The unprecedented performances demonstrated could enable the realization of novel ultra-sensitive threshold sensors, capable of detecting tiny perturbations via a state change in the system, or other detection schemes based on nonlinear nanomechanical resonators3,4. Likewise, a high-Q parametrically driven Duffing resonator could boost the state-of-the-art in nanomechanical memory elements30,31 introducing additional bits by simultaneous manipulation of the orthogonal oscillation modes32. In the parametrically driven regime our system could also enable fundamental search for classical to quantum transitions33, and prompt the realization of quantum-enhanced sensing techniques. Finally, we foresee in optically levitated nanoparticles a suitable platform for mimicking very complex stochastic nonlinear dynamics, able to shine a light on natural phenomena such as bio-molecule folding16,17, hearing34,35 and neural signalling36.
The amplitude is tracked by sampling position x at fs=625 kHz and integrating over n successive positions measurements. The value of n depends on the temporal resolution needed in the particular measurement to be carried out. We set n=8,192 for the data shown in Fig. 1, n=128 for the amplitude time traces (blue data) of Fig. 2a and to n=4,096 for the SR experiment of Fig. 3. In this latter data sets, the acquisition time of a single amplitude trace at constant noise temperature (Fig. 3b) was 103 s. However, in order to display a meaningful dynamics, we show only 5 s time traces.
The data that support the findings of this study are available from the corresponding author on reasonable request.
How to cite this article: Ricci, F. et al. Optically levitated nanoparticle as a model system for stochastic bistable dynamics. Nat. Commun. 8, 15141 doi: 10.1038/ncomms15141 (2017).
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We acknowledge financial support from the ERC- QnanoMECA (Grant No. 64790), the Spanish Ministry of Economy and Competitiveness, under grant FIS2016-80293-R and through the ‘Severo Ochoa’ Programme for Centres of Excellence in R&D (SEV-2015-0522), Fundació Privada CELLEX and from the CERCA Programme/Generalitat de Catalunya. J.G. has been supported by H2020-MSCA-IF-2014 under REA grant Agreement No. 655369. L.R. acknowledges support from an ETH Marie Curie Cofund Fellowship.
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Nature Nanotechnology (2017)