  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.
