We theoretically calculate normal-mode splitting (NMS) in a linear and quadratic optomechanical system (OMS) with an ensemble of two-level atoms, where the interaction between the mechanical membrane and the optical cavity includes linear optomechanical coupling and quadratic optomechanical coupling (QOC). In the presence of atomic ensemble, a negative QOC strength is instrumental for generating NMS, while the positive QOC restricts NMS, and eventually it disappears. Further, for the hybrid OMS assisted with the atomic ensemble, the displacement spectrum of the mechanical resonator displays three peaks, where the middle peak results from the effective coupling strength between the cavity field and the atomic ensemble. Here the negative QOC strength and the effective ensemble-field coupling can provide an efficient control of the amplitude and position of the three peaks.

中文翻译：

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线性和二次光机电系统中具有两级原子集合的正模分裂

我们从理论上计算线性和二次光机电系统（OMS）中具有两级原子集合的正模分裂（NMS），其中机械膜和光学腔之间的相互作用包括线性光机耦合和二次光机耦合（QOC） ）。在存在原子团的情况下，负QOC强度有助于生成NMS，而正QOC会限制NMS，并最终消失。此外，对于辅助有原子团的混合OMS，机械谐振器的位移谱显示三个峰，中间峰来自腔场与原子团之间的有效耦合强度。