Fig. 1: Vibration amplitude xnl for which nonlinearities emerge divided by the zero-point motion xzp as a function of the mass of the mechanical eigenmode for a large range of different vibrational systems.
From: Nonlinear nanomechanical resonators approaching the quantum ground state
Different colours correspond to different types of vibrational system. The stars correspond to systems that have been experimentally cooled to the quantum ground state. Supplementary Fig. 11 indicates the reference for each system. When both displacement and frequency fluctuations are negligible, the effect of Duffing nonlinearity is sizable when \({x}_{{{{\rm{nl}}}}}/{x}_{{{{\rm{zp}}}}} > {(\beta {m}^{2}{\omega }_{{{{\rm{m}}}}}^{2}{{{\varGamma }}}_{{{{\rm{m}}}}}/\hslash \gamma )}^{1/2}\), where β ≃ 3.1 is a constant; ωm, the resonance frequency; Γm, the mechanical linewidth; and γ, the Duffing constant12.
