Measurement of a microwave field amplitude beyond the standard quantum limit

We report a quantum measurement beyond the standard quantum limit (SQL) for the amplitude of a small displacement acting on a cavity field. This measurement uses as resource an entangled mesoscopic state, prepared by the resonant interaction of a circular Rydberg atom with a field stored in a superconducting cavity. We analyse the measurement process in terms of Fisher information and prove that it is, in principle, optimal. The achieved experimental precision, 2.4 dB below the SQL, is well understood in terms of experimental imperfections.This method could be transposed to other systems, particularly to circuit QED, for the precise measurements of weak forces acting on oscillators.