Fig. 1: Active optomechanics in the macroscopic limit.

a Schematic of a generic cavity optomechanical system that operates in an active fashion. An ensemble of two-level (upper \(|{{{{{\rm{e}}}}}}\rangle\) and lower \(|{{{{{\rm{g}}}}}}\rangle\)) atoms is coupled to an optical cavity with a single-photon coupling strength \(\mu\). A pump light excites the atoms onto \(|{{{{{\rm{e}}}}}}\rangle\) at a rate \(R\). The total decay rate of \(|{{{{{\rm{e}}}}}}\rangle\) is \({\gamma }_{{{\rm{e}}}}^{{\prime} }\), wherein the decay rate of the atom from \(|{{{{{\rm{g}}}}}}\rangle\) to \(|{{{{{\rm{e}}}}}}\rangle\) is \({\gamma}_{{{\rm{e}}}}\). The fast decay rate \({\gamma }_{{{\rm{g}}}}\) of \(|{{{{{\rm{g}}}}}}\rangle\) ensures the population inversion between \(|{{{{{\rm{e}}}}}}\rangle\) and \(|{{{{{\rm{g}}}}}}\rangle\). The active atoms emit photons into the optical cavity. The position of one cavity mirror is fixed while the other mirror is movable. The motion of the movable mirror is modelled as a mechanical oscillator with a displacement \(x(t)\), oscillation frequency \(\Omega\), and damping rate \(\Gamma\). The radiation pressure of the intracavity field drives the mechanical oscillator. b Steady-state photon number \({N}_{{{{{{\rm{photon}}}}}},{{{{{\rm{ss}}}}}}}\) (in units of the saturation photon number \({N}_{{{{{{\rm{photon}}}}}}}^{({{{{{\rm{sat}}}}}})}\)) as a function of the detuning \(\Delta ={\omega }_{{{{{{\rm{C}}}}}}}-{\omega }_{{{\rm{A}}}}\) between the cavity mode frequency \({\omega }_{{{{{{\rm{C}}}}}}}\) and the atomic transition frequency \({\omega }_{{{\rm{A}}}}\). The pump rate is set at \(R={10}^{3}{R}_{0}\) with the minimum pump threshold \({R}_{0}\) and the cavity quality factor is \(Q={10}^{7}\). The cavity loss rate \(\kappa ={\omega }_{{{{{{\rm{C}}}}}}}/Q\) is chosen as the frequency unit in the plot. The solid curves denote the stable steady-state solutions while the dashed lines correspond to the unstable steady-state solutions. c Time evolutions of the mechanical-displacement-induced detuning \(\delta (t)\) and the intracavity photon number \({N}_{{{{{{\rm{photon}}}}}}}(t)\) for the optomechanical system operating at an unstable steady state. The dashed line corresponds to the average value of \(\delta (t)\). d, e Dependences of the steady-state \({N}_{{{{{{\rm{photon}}}}}},{{{{{\rm{ss}}}}}}}\) on the detuning \(\Delta\) and the pump rate \(R\) for \(Q={10}^{5}\) and \(Q={10}^{7}\), respectively. The boundaries of unstable steady-state regions have been plotted. The red shading denotes the stable steady states while the blue shading corresponds to the unstable steady states. f, g Photon differences \(\Delta {N}_{{{{{{\rm{photon}}}}}},{{{{{\rm{ss}}}}}}}\) between the active optomechanical system and the corresponding conventional laser for \(Q={10}^{5}\) and \(Q={10}^{7}\), respectively.