Effective description of a suspended mirror coupled to cavity light: Limitations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> enhancement due to normal-mode splitting by an optical spring

Pendulums have long been used as force sensors due to their ultimately low dissipation (high-quality factor) characteristic. They are widely in the measurement of gravitational constant, detection waves, and determination ultralight dark matter. Furthermore, it is expected that quantum nature gravity will be demonstrated by performing control for macroscopic pendulums. Recently, we entanglement between two pendulums can generated using an optical spring [D. Miki, N. Matsumoto, A. Matsumura, T. Shichijo, Y. Sugiyama, K. Yamamoto, arXiv:2210.13169 (2022)]; however, ignored reduce quality factor (Q-factor) applying normal-mode splitting pendulum rotational modes possessing relatively high dissipation. Herein, analyze a system composed cylinder suspended beam (a mirror, i.e., pendulum) consider splitting. The reduction Q-factor determined only parameters: ratio radius mirror length beam, frequency mode absence cavity photons. In our analysis, find $4.38$ reproduced, which consistent with experimental result Matsumoto \textit{et al.} [N. S. B. Catan$\tilde{\text{o}}$-Lopez, M. Sugawara, Suzuki, Abe, Komori, Michimura, Aso, Edamatsu, Phys. Rev. Lett. 122, 071101 (2019)]. Our analysis shows (high quality) reached realistic considering degree freedom.