We propose a coherent-control scheme for engineering quantum correlations in a cavity optomechanical (COM) system consisting of a driven optical cavity with an embedded nonlinear medium and a membrane, assisted by a coherent feedback loop. The nonlinear medium and the membrane are pumped to implement optical and mechanical parametric amplifications with controllable modulation frequencies and pump amplitudes. Within the stable regime identified by Floquet analysis, the feedback-induced enhancement of optomechanical entanglement persists when the period-averaged mean-field intracavity photon number is matched, showing that the enhancement cannot be attributed simply to an increase in this quantity. A systematic comparison of the four configurations further reveals a nonadditive interplay between dual parametric amplification and coherent feedback. Varying the amplitude reflectivity of the beam splitter and the feedback-loop phase enables effective control of optomechanical entanglement and directional Einstein-Podolsky-Rosen steering, including a transition from one-way to two-way steering. The combined scheme also improves the robustness of both correlations against thermal noise, providing a potential route toward robust quantum-state engineering in COM systems.

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• Quantum correlations in quantum information