We propose a novel methodology to estimate parameters characterizing a weakly nonlinear Duffing oscillator represented by an optically levitating nanoparticle. The method is based on averaging recorded trajectories with defined initial positions in the phase space of nanoparticle position and momentum and allows us to study the transient dynamics of the nonlinear system. This technique provides us with the parameters of a levitated nanoparticle such as eigenfrequency, damping, coefficient of nonlinearity and effective temperature directly from the recorded transient particle motion without any need for external driving or modification of an experimental system. Comparison of this innovative approach with a commonly used method based on fitting the power spectrum density profile shows that the proposed complementary method is applicable even at lower pressures where the nonlinearity starts to play a significant role and thus the power spectrum density method predicts steady state parameters. The technique is applicable also at low temperatures and extendable to recent quantum experiments. The proposed method is applied on experimental data and its validity for one-dimensional and three-dimensional motion of a levitated nanoparticle is verified by extensive numerical simulations.

Department of Optics Palacký University 17 listopadu 1192 12 771 46 Olomouc Czech Republic

Institute of Scientific Instruments of the Czech Academy of Sciences Královopolská 147 612 64 Brno Czech Republic

Zobrazit více v PubMed

Chowdhury A, Barbay S, Clerc MG, Robert-Philip I, Braive R. Phase stochastic resonance in a forced nanoelectromechanical membrane. Phys. Rev. Lett. 2017;119:234101. PubMed

Chowdhury A, Barbay S, Robert-Philip I, Braive R. Weak signal enhancement by nonlinear resonance control in a forced nano-electromechanical resonator. Nat. Commun. 2020;11:2400. PubMed PMC

Güttinger J, et al. Energy-dependent path of dissipation in nanomechanical resonators. Nat. Nanotechnol. 2017;12:631. PubMed

Ganesan A, Do C, Seshia A. Phononic frequency comb via intrinsic three-wave mixing. Phys. Rev. Lett. 2017;118:033903. PubMed

Chen C, Zanette DH, Czaplewski DA, Shaw S, López D. Direct observation of coherent energy transfer in nonlinear micromechanical oscillators. Nat. Commun. 2017;8:15523. PubMed PMC

Huang L, et al. Frequency stabilization and noise-induced spectral narrowing in resonators with zero dispersion. Nat. Commun. 2019;10:3930. PubMed PMC

Sun F, Dong X, Zou J, Dykman MI, Chan HB. Correlated anomalous phase diffusion of coupled phononic modes in a sideband-driven resonator. Nat. Commun. 2016;7:12694. PubMed PMC

Leuch A, et al. Parametric symmetry breaking in a nonlinear resonator. Phys. Rev. Lett. 2016;117:214101. PubMed

Meucci R, et al. Optimal phase-control strategy for damped-driven Duffing oscillators. Phys. Rev. Lett. 2016;116:044101. PubMed

Amarouchene Y, et al. Nonequilibrium dynamics induced by scattering forces for optically trapped nanoparticles in strongly inertial regimes. Phys. Rev. Lett. 2019;122:183901. PubMed

Wen Y, et al. A coherent nanomechanical oscillator driven by single-electron tunnelling. Nat. Phys. 2020;16:75–82. PubMed PMC

Huang P, et al. Generating giant and tunable nonlinearity in a macroscopic mechanical resonator from a single chemical bond. Nat. Commun. 2016;7:11517. PubMed PMC

Abdi M, Degenfeld-Schonburg P, Sameti M, Navarrete-Benlloch C, Hartmann MJ. Dissipative optomechanical preparation of macroscopic quantum superposition states. Phys. Rev. Lett. 2016;116:233604. PubMed

Ricci F, et al. Optically levitated nanoparticle as a model system for stochastic bistable dynamics. Nat. Commun. 2017;8:15141. PubMed PMC

Papariello L, Zilberberg O, Eichler A, Chitra R. Ultrasensitive hysteretic force sensing with parametric nonlinear oscillators. Phys. Rev. E. 2016;94:022201–22207. PubMed

Ranjit G, Cunningham M, Casey K, Geraci AA. Zeptonewton force sensing with nanospheres in an optical lattice. Phys. Rev. A. 2016;93:053801.

Aldana S, Bruder C, Nunnenkamp A. Detection of weak forces based on noise-activated switching in bistable optomechanical systems. Phys. Rev. A. 2014;90:063810–63818.

Gieseler J, Novotny L, Quidant R. Thermal nonlinearities in a nanomechanical oscillator. Nat. Phys. 2013;9:806–810.

Kuhn S, et al. Full rotational control of levitated silicon nanorods. Optica. 2017;4:356–360.

Kuhn S, et al. Optically driven ultra-stable nanomechanical rotor. Nat. Commun. 2017;8:1670–1675. PubMed PMC

Rajasekar SP, Pitchaimani M, Zhu Q. Dynamic threshold probe of stochastic SIR model with saturated incidence rate and saturated treatment function. Physica A. 2019;535:122300.

Rajasekar SP, Pitchaimani M. Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence. Appl. Math. Comput. 2020;377:125143.

Rifhat R, Wang L, Teng Z. Dynamics for a class of stochastic SIS epidemic models with nonlinear incidence and periodic coefficients. Physica A. 2017;481:176–190.

Nørrelykke SF, Flyvbjerg H. Harmonic oscillator in heat bath: Exact simulation of time-lapse-recorded data and exact analytical benchmark statistics. Phys. Rev. E. 2011;83:041103. PubMed

Berg-Sørensen K, Flyvbjerg H. Power spectrum analysis for optical tweezers. Rev. Sci. Instrum. 2004;75:594–612.

Kiesel N, et al. Cavity cooling of an optically levitated submicron particle. Proc. Natl. Acad. Sci. USA. 2013;110:14180–14185. PubMed PMC

Fonseca PZG, Aranas EB, Millen J, Monteiro TS, Barker PF. Nonlinear dynamics and strong cavity cooling of levitated nanoparticles. Phys. Rev. Lett. 2016;117:173602. PubMed

Gieseler J, Deutsch B, Quidant R, Novotny L. Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 2012;109:103603. PubMed

Hebestreit E, et al. Calibration and energy measurement of optically levitated nanoparticle sensors. Rev. Sci. Instrum. 2018;89:033111. PubMed

Romero-Isart O, et al. Optically levitating dielectrics in the quantum regime: theory and protocols. Phys. Rev. A. 2011;83:013803.

Tebbenjohanns F, Frimmer M, Militaru A, Jain V, Novotny L. Cold damping of an optically levitated nanoparticle to microkelvin temperatures. Phys. Rev. Lett. 2019;122:223601–223606. PubMed

Ralph JF, et al. Dynamical model selection near the quantum-clasical boundary. Phys. Rev. A. 2018;98:010102–10107.

Jain V, et al. Direct measurement of photon recoil from a levitated nanoparticle. Phys. Rev. Lett. 2016;116:243601. PubMed

Millen J, Fonseca PZG, Mavrogordatos T, Monteiro TS, Barker PF. Cavity cooling a single charged levitated nanosphere. Phys. Rev. Lett. 2015;114:123602. PubMed

Delić U, et al. Cavity cooling of a levitated nanosphere by coherent scattering. Phys. Rev. Lett. 2019;122:123602. PubMed

Windey D, et al. Cavity-based 3D cooling of a levitated nanoparticle via coherent scattering. Phys. Rev. Lett. 2019;122:123601–123605. PubMed

Delić U, et al. Cooling of a levitated nanoparticle to the motional quantum ground state. Science. 2020;30:eaba3993. PubMed

MacPherson WN, Jones DC, Mangan BJ, Knight JC, Russell PSJ. Two-core photonic crystal fibre for Doppler difference velocimetry. Opt. Commun. 2003;223:375–380.

Kheifets S, Simha A, Melin K, Li T, Raizen MG. Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss. Science. 2014;343:1493–1496. PubMed

Setter A, Vovrosh J, Ulbricht H. Characterization of non-linearities through mechanical squeezing in levitated optomechanics. Appl. Phys. Lett. 2019;115:153106.

Harada Y, Asakura T. Radiation forces on a dielectric sphere in the Rayleigh scattering regime. Opt. Commun. 1996;124:529–541.

Jones P, Maragò O, Volpe G. Optical Tweezers: Principles and Applications. Cambridge: Cambridge University Press; 2015.

Siegman AE. Lasers. Sausalito, CA: Univ. Sci. Books; 1986.

Yoneda M, Aikawa K. Thermal broadening of the power spectra of laser-trapped particles in vacuum. J. Phys. B At. Mol. Opt. Phys. 2017;50:245501–245509.

Strogatz S. Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering. Boulder: Westview Press; 2015.

Gieseler J, Spasenovic M, Novotny L, Quidant R. Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle. Phys. Rev. Lett. 2014;112:103603. PubMed

Li T, Kheifets S, Raizen MG. Millikelvin cooling of an optically trapped microsphere in vacuum. Nat. Phys. 2011;7:527–530.

Mangeat M, Amarouchene Y, Louyer Y, Guérin T, Dean DS. Role of nonconservative scattering forces and damping on Brownian particles in optical traps. Phys. Rev. E. 2019;99:052107. PubMed

Miles RN. An approximate solution for the spectral response of Duffing’s oscillator with random input. J. Sound Vib. 1989;132:43–49.

Ge W, Bhattacharya M. Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence. New J. Phys. 2016;18:103002–103016.

Rakhubovsky AA, Moore DW, Filip R. Nonclassical states of levitated macroscopic objects beyond the ground state. Quantum Sci. Technol. 2019;4:024006–24011.

Rakhubovsky, A. A. & Filip, R. Stroboscopic high-order nonlinearity in quantum optomechanics. arXiv: 1904.00773 [quant-ph] (2019).

Černotík O, Filip R. Strong mechanical squeezing for a levitated particle by coherent scattering. Phys. Rev. Res. 2020;2:013052.

Moore DW, Rakhubovsky AA, Filip R. Estimation of squeezing in a nonlinear quadrature of a mechanical oscillator. New J. Phys. 2019;21:113050.

Litshitz R, Cross MC. Nonlinear dynamics of nanomechanical and micromechanical resonators. In: Schuster HG, editor. Review of Nonlinear Dynamics and Complexity. New York: Wiley; 2008. pp. 1–52.

Kovacic I, Brennan MJ. The Duffing Equation Nonlinear Oscillators and Their Behaviour. New York: Wiley; 2011.

Hebestreit E, Reimann R, Frimmer M, Novotny L. Measuring the internal temperature of a levitated nanoparticle in high vacuum. Phys. Rev. A. 2018;97:043803.

Grønbech-Jensen N, Hayre NR, Farago O. Application of the G-JF discrete-time thermostat for fast and accurate molecular simulations. Comput. Phys. Commun. 2014;185:524–527.

Pérez García L, Donlucas Pérez J, Volpe G, Arzola AV, Volpe G. High-performance reconstruction of microscopic force fields from Brownian trajectories. Nat. Commun. 2018;9:5166. PubMed PMC

Kuhn S, et al. Cavity-assisted manipulation of freely rotating silicon nanorods in high vacuum. Nano. Lett. 2015;15:5604–5608. PubMed PMC

Stroboscopic thermally-driven mechanical motion

Uncertainty-induced instantaneous speed and acceleration of a levitated particle