Quantum precision measurement of two-dimensional forces with 10^-28-Newton stability

High-precision sensing of vectorial forces has broad impact on both fundamental research and technological applications such as the examination of vacuum fluctuations and the detection of surface roughness of nanostructures. Recent years have witnessed much progress on sensing alternating electromagnetic forces for the rapidly advancing quantum technology-orders of magnitude improvement has been accomplished on the detection sensitivity with atomic sensors, whereas such high-precision measurements for static electromagnetic forces have rarely been demonstrated. Here, based on quantum atomic matter waves confined by a two-dimensional optical lattice, we perform precision measurement of static electromagnetic forces by imaging coherent wave mechanics in the reciprocal space. The lattice confinement causes a decoupling between real-space and reciprocal dynamics, and provides a rigid coordinate frame for calibrating the wavevector accumulation of the matter wave. With that we achieve a state-of-the-art sensitivity of 2.30(8)×10^-26 N/Hz. Long-term stabilities on the order of 10^-28 N are observed in the two spatial components of a force, which allows probing atomic Van der Waals forces at one millimeter distance. As a further illustrative application, we use our atomic sensor to calibrate the control precision of an alternating electromagnetic force applied in the experiment. Future developments of this method hold promise for delivering unprecedented atom-based quantum force sensing technologies.