Precise estimation of the coupling strength between two nanomechanical modes from four Ramsey fringes

We experimentally determine the coupling strength between two strongly coupled nanomechanical modes using a Ramsey-inspired technique optimized for signals as short as four fringes. The method is applied to precisely probe the change of the coupling rate induced by a modification of the microwave-cavity readout field. It opens a pathway towards sensing electrostatic field fluctuations approaching single-charge resolution.