  We consider the phase sensing via weak optical coherent state at quantum
limit precision. A new detection scheme for the phase estimation is proposed
which is inspired by the suboptimal quantum measurement in coherent optical
communication. We theoretically analyze a performance of our detection scheme,
which we call the displaced-photon counting, for phase sensing in terms of the
Fisher information and show that the displaced-photon counting outperforms the
static homodyne and heterodyne detections in wide range of the target phase.
The proof-of-principle experiment is performed with linear optics and a
superconducting nanowire single photon detector. The result shows that our
scheme overcomes the limit of the ideal homodyne measurement even under
practical imperfections.
