Quantum Effects in Gravity Beyond the Newton Potential from a Delocalized Quantum Source

Recent progress in tabletop experiments offers the opportunity to show, for the first time, that gravity is not compatible with a classical description. In all current experimental proposals, such as the generation of gravitationally induced entanglement between two quantum sources of gravity, gravitational effects can be explained solely with the Newton potential, namely, in a regime that is consistent with the weak-field limit of general relativity and does not probe the field nature of gravity. Hence, the Newtonian origin of the effects is a limitation to the conclusions on the nature of gravity that can be drawn from these experiments. Here, we identify two effects that overcome this limitation: They cannot be reproduced using the Newton potential, and they are independent of graviton emission. First, we show that the interaction between two generic quantum sources of gravity, e.g., in wide Gaussian states, cannot be reproduced with the Newton potential nor with a known classical theory of gravity. Second, we show that the quantum commutator between the gravitational field and its canonically conjugate momentum appears as an additional term in the relative phase of a generic quantum source interacting with a test particle. Observing these two effects would give further quantitative information and stronger evidence, compared to experiments only involving the Newton potential, to show that gravity is nonclassical. More broadly, identifying stronger quantum aspects of gravity than those reproducible with the Newton potential is crucial to prove the nonclassicality of gravity and to plan a new generation of experiments testing quantum aspects of gravity in a broader sense than what has been proposed so far.