Fig. 2: Power spectral densities of laser field and mechanical oscillation.

a Numerically simulated spectrum \({S}_{{{{{{\rm{photon}}}}}}}(\omega )\) of the intracavity light field for different cavity \(Q\) factors and pump rates \(R\). The atom–cavity detuning is set at \(\Delta =-\kappa\) with the cavity loss rate \(\kappa\), and \({R}_{0}\) denotes the minimum pump threshold. At \(\Delta =-\kappa\), the optomechanical laser has stable steady-state solutions for different \(Q\) and \(R\). The laser frequency \({\omega }_{{{\rm{L}}}}\) is shifted from the atomic transition frequency \({\omega }_{{{\rm{A}}}}\) and the laser linewidth (i.e. full width half maximum) is \(\Delta {\omega }_{{{\rm{L}}}}\). The mechanical oscillation frequency \(\Omega\) is chosen as the frequency unit. b Dependence of laser frequency shift \(({\omega }_{{{\rm{L}}}}-{\omega }_{{{\rm{A}}}})\) and linewidth \(\Delta {\omega }_{{{\rm{L}}}}\) on the pump rate \(R\) for different \(Q\) factors with \(\Delta =-\kappa\). Symbols corresponds to the numerical results. The curves derived from the cavity pulling effect and Schawlow–Townes formula are also plotted for comparison. c Numerically simulated spectrum \({S}_{\delta }(\omega )\) of the mechanical-displacement-induced detuning \(\delta (t)\) for different \(Q\) and \(R\) with \(\Delta =-\kappa\).