[Paper Review] Resonant nonlinear response of a nanomechanical system with broken symmetry

J. S. Ochs, Gianluca Rastelli+3 more|arXiv (Cornell University)|Jan 1, 2021

Mechanical and Optical Resonators30 references11 citations

TL;DR

This paper investigates the resonant nonlinear response of a nanomechanical string resonator with broken inversion symmetry due to asymmetric dielectric electrodes. By going beyond the standard rotating-wave approximation, the authors show that cubic anharmonicity induces significant deviations from the Duffing model at moderate drive amplitudes, including measurable second-harmonic generation, and quantitatively extract the nonlinear potential parameters from experimental amplitude and phase data of the fundamental and second-harmonic modes.

ABSTRACT

We study the response of a weakly damped vibrational mode of a nanostring resonator to a moderately strong resonant driving force. Because of the geometry of the experiment, the studied flexural vibrations lack inversion symmetry. As we show, this leads to a nontrivial dependence of the vibration amplitude on the force parameters. For a comparatively weak force, the response has the familiar Duffing form, but for a somewhat stronger force, it becomes significantly different. Concurrently there emerge vibrations at twice the drive frequency, a signature of the broken symmetry. Their amplitude and phase allow us to establish the cubic nonlinearity of the potential of the mode as the mechanism responsible for both observations. The developed theory goes beyond the standard rotating-wave approximation. It quantitatively describes the experiment and allows us to determine the nonlinearity parameters.

Motivation & Objective

• To understand the nonlinear dynamics of a nanomechanical resonator with broken inversion symmetry.
• To identify the origin of deviations from the standard Duffing model under resonant driving.
• To measure and quantify the cubic nonlinearity in the potential energy of the vibrational mode.
• To demonstrate the breakdown of the rotating-wave approximation in asymmetric systems under moderate driving.
• To experimentally extract nonlinear parameters via simultaneous detection of fundamental and second-harmonic response.

Proposed method

• Experimental system: a doubly clamped silicon nitride string resonator with asymmetric gold electrodes inducing a broken-symmetry potential.
• Use of microwave cavity-enhanced heterodyne detection to measure both fundamental and second-harmonic vibration amplitudes and phases.
• Application of a theoretical model extending Hamiltonian nonlinear dynamics to include weak damping and cubic anharmonicity beyond the rotating-wave approximation.
• Derivation of a modified Duffing-like equation with cubic and quartic terms, where the cubic term arises from broken symmetry.
• Comparison of theoretical predictions with experimental data to extract nonlinear parameters such as the effective Duffing coefficient and cubic nonlinearity.
• Analysis of the amplitude and phase of the second-harmonic response to confirm its origin in the cubic nonlinearity of the potential.

Experimental results

Research questions

• RQ1 How does broken inversion symmetry in a nanomechanical resonator alter its nonlinear response under resonant driving?
• RQ2 To what extent does the standard rotating-wave approximation fail in describing the nonlinear dynamics of asymmetric nanomechanical systems?
• RQ3 What is the role of cubic anharmonicity in generating observable second-harmonic vibrations in such systems?
• RQ4 How do the amplitude and phase of the second-harmonic response depend on the drive parameters and system parameters like DC control voltage?
• RQ5 Can the nonlinear potential parameters be quantitatively extracted from experimental measurements of fundamental and second-harmonic modes?

Key findings

• The resonant response deviates significantly from the standard Duffing model at drive amplitudes where the frequency shift is less than 10−4ω0, indicating early breakdown of the rotating-wave approximation.
• Second-harmonic vibrations at twice the drive frequency emerge with measurable amplitude and well-defined phase, confirming the presence of cubic anharmonicity.
• The amplitude and phase of the second harmonic are experimentally measured and used to quantitatively determine the cubic nonlinearity parameter in the potential.
• Theoretical modeling beyond the rotating-wave approximation successfully describes the experimental data, including the non-monotonic dependence of the response on drive frequency.
• The effective Duffing parameter is renormalized by the cubic term, but its influence becomes subdominant at higher amplitudes, where the quartic term dominates.
• The system exhibits a transition in nonlinear response behavior at moderate drive levels, where the cubic term's effect on frequency renormalization is overtaken by the hardening effect of the quartic term.