# Non-linear Frequency Transduction of Nano-mechanical Brownian Motion

## TL;DR

This paper investigates how non-linear interactions in nano-mechanical resonators convert Brownian motion into frequency noise, revealing phase diffusion effects and fundamental limits on frequency resolution.

## Contribution

It experimentally demonstrates the non-linear transduction of Brownian motion into frequency noise and validates a theoretical model without free parameters.

## Key findings

- Observation of phase diffusion due to non-linear frequency transduction
- Identification of two limit behaviors analogous to NMR phenomena
- Matching experimental results with theoretical predictions over a wide range

## Abstract

We report on experiments addressing the non-linear interaction between a nano-mechanical mode and position fluctuations. The Duffing non-linearity transduces the Brownian motion of the mode, and of other non-linearly coupled ones, into frequency noise. This mechanism, ubiquitous to all weakly-nonlinear resonators thermalized to a bath, results in a phase diffusion process altering the motion: two limit behaviors appear, analogous to motional narrowing and inhomogeneous broadening in NMR. Their crossover is found to depend non-trivially on the ratio of the frequency noise correlation time to its magnitude. Our measurements obtained over an unprecedented range covering the two limits match the theory of Y. Zhang and M. I. Dykman, Phys. Rev. B 92, 165419 (2015), with no free parameters. We finally discuss the fundamental bound on frequency resolution set by this mechanism, which is not marginal for bottom-up nanostructures.