Free-Particle State Realized via Optimal Filtering in Optomechanics: Implications for Gravity-Induced Entanglement

Abstract: We analyze the conditional quantum state of a mechanical mirror in an optomechanical system under continuous measurement, feedback control, and quantum filtering. Our study identifies a regime in which the mirror's momentum is squeezed beyond the standard quantum limit achievable through appropriate tuning of the homodyne detection angle. We show that in this regime, optimal filtering effectively realizes a free-particle-like state. Notably, when applied to a setup involving two optomechanical systems, this phenomenon significantly boosts the signal of gravity-induced entanglement (GIE) because the momentum squeezing not only accentuates the difference between the common mode and the differential mode but, when the purity is high, also increases the position uncertainty due to the uncertainty principle, thereby enlarging the spatial extent of the superposition. Our results provide new insights into strategies for verifying the quantum nature of gravity using optomechanical platforms.