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We propose a scheme for cooling a mechanical resonator to its ground state in a quadratic optomechanical system, assisted by an atomic ensemble in the unresolved sideband regime. The system features an auxiliary cavity directly coupled to an optical cavity, with a portion of the optical cavity's output field being fed back through an asymmetric beam splitter. Utilizing quantum Langevin and master equations, we derive the optical fluctuation spectrum, the cooling rate, and the mean phonon number of the mechanical resonator. Our results demonstrate that the feedback mechanism substantially enhances the cooling rate. Furthermore, under optimal cooling conditions, the mechanical resonator achieves ground state cooling even with weaker optomechanical coupling strengths and higher auxiliary cavity dissipation rates, thereby mitigating the experimental constraints associated with these parameters. Additionally, we provide the feasible ranges for optomechanical coupling strength and atomic decay rates. Our findings suggest promising avenues for quantum manipulation in nonlinear systems and its applications in macroscopic optical devices.
Chang et al. (Mon,) studied this question.