Deterministic super-resolved estimation towards angular displacements based upon a Sagnac interferometer and parity measurement

Super-resolved angular displacement estimation is of crucial significances for quantum information process and optical lithography. Here we report on and experimentally demonstrate a protocol for angular displacement estimation based on a coherent state containing orbital angular momentum. In the lossless scenario, with using parity measurement, this protocol can theoretically achieve 4$\ell$-fold super-resolution with quantum number $\ell$, and shot-noise-limited sensitivity saturating the quantum Cram\'er-Rao bound. Several realistic factors and their effects are considered, including nonideal state preparation, photon loss, and imperfect detector. Finally, given mean photon number $\bar N=2.297$ and $\ell=1$, we show an angular displacement super-resolution effect with a factor of 7.88, and the sensitivity approaching shot-noise limit is reachable.