Optimal phase estimation with photon-number difference measurement using twin-Fock states of light**Project supported by the National Natural Science Foundation of China (Grant Nos. 91636108, 11775190, and 11774024), Science Foundation of Zhejiang Sci-Tech University, China (Grant No. 18062145-Y), Open Foundation of Key Laboratory of Optical Field Manipulation of Zhejiang Province, China (Grant No. ZJOFM-2019-002), and Science Challenge Project, China (Grant No.

Abstract

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 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 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.