Experimental demonstration of phase measurement precision beating standard quantum limit by projection measurement

We propose and demonstrate experimentally a projection scheme to measure the quantum phase with a precision beating the standard quantum limit. The initial input state is a twin Fock state $\mathrm{\mid }N,N>$ proposed by Holland and Burnett [Phys. Rev. Lett. {\bf 71}, 1355 (1993)] but the phase information is extracted by a quantum state projection measurement. The phase precision is about $1.4\mathrm{/}N$ for large photon number $N$, which approaches the Heisenberg limit of 1/N. Experimentally, we employ a four-photon state from type-II parametric down-conversion and achieve a phase uncertainty of $0.291\pm 0.001$ beating the standard quantum limit of $1\mathrm{/}\sqrt{N}=1\mathrm{/}2$ for four photons.