Amplification of gravitational motion via Quantum weak measurement

We investigate a new experimental possibility of measuring the Newtonian gravitational constant $G$ by using the weak measurement. Amplification via weak measurement is one of the interesting phenomena of quantum mechanics. In this letter, we consider it in a system consisting of many cold atoms which are gravitationally interacting with an external macroscopic source and show that it is possible to obtain ${\mathcal{O}}(10^{3})$ amplification of their relative motion compared with the classical motion when the number of atoms are ${\mathcal{O}}(10^{15})$ and the observing time is $\sim 0.5$s. This result suggests that it might be possible to use this system as a new experimental set up for determining $G$. Besides, our study indicates that the gravitational force can behave as a repulsive force because of the weak measurement.