Distinguishing quantum and classical gravity via nonstationary test mass dynamics

Classical gravity theory predicts a state-dependent gravitational potential for a quantum test mass, leading to nonlinear Schr"odinger-Newton (SN) state evolution that contrasts with quantum gravity. Testing the effect of SN evolution can provide evidence for distinguishing quantum gravity and classical gravity, which is challenging to realize in the stationary optomechanical systems as analyzed in previous works [Yubao Liu, Haixing Miao, Yanbei Chen, and Yiqiu Ma, Semiclassical gravity phenomenology under the causal-conditional quantum measurement prescription, Phys. Rev. D 107, 024004 (2023).; Yubao Liu, Wenjie Zhong, Yanbei Chen, and Yiqiu Ma, Semiclassical gravity phenomenology under the causal-conditional quantum measurement prescription. II. Heisenberg picture and apparent optical entanglement, Phys. Rev. D 111, 062004 (2025).]. This Letter is devoted to analyzing the possibility of capturing the signature of SN theory during the nonstationary evolution of the test mass under the optomechanical measurement, where the second-order moments of a test mass can exhibit a distinctive oscillatory behavior. We show that this feature manifests in the nonstationary noise spectrum of outgoing light as additional peak structures, although resolving these structures in practical experiments requires a larger number of repetitive trials with our sampling parameters, which is cost prohibitive. To address this issue, we further employ statistical inference methods to extract more comprehensive information, thereby reducing the required number of experimental repetitions. Through mock data simulations, we demonstrate that only ten experimental trials of 40 sec each are sufficient to reduce the false alarm rate for distinguishing between the two models to below 1%.