Optimal phase estimation with photon-number difference measurement using twin-Fock states of light

Quantum phase measurement with multiphoton twin-Fock states has been shown to be optimal for detecting equal numbers of photons at the output ports of a Mach-Zehnder interferometer (i.e., the so-called single-fringe detection), since the phase sensitivity can saturate the quantum Cramér-Rao lower bound at certain values of phase shift. Here we report a further step to achieve a global phase estimation at the Heisenberg limit by detecting the particle-number difference (i.e., the Ĵ_z measurement). We show the role of experimental imperfections on the ultimate estimation precision with the six-photon twin-Fock state of light. Our results show that both the precision and the sensing region of the Ĵ_z measurement are better than those of the single-fringe detection, due to combined contributions of the measurement outcomes. We numerically simulate the phase estimation protocol using an asymptotically unbiased maximum likelihood estimator.