Optimal phase estimation with photon-number difference measurement using twin-Fock states of light<xref rid="cpb_28_12_120303fn1" ref-type="fn">*</xref><fn id="cpb_28_12_120303fn1"><label>*</label><p>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. TZ2018003).</p></fn>

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. TZ2018003).

Xu J H¹, Wang J Z¹, Chen A X¹, Li Y², Jin G R^{1, †}

Maximum likelihood estimator. For a given true value of phase shift θ₀ = π/4, the phase distribution P(θ|{Nμ}) as a function of θ (the red dashed line), which approaches to a Gaussian (solid blue line) and hence can be used to determine the MLE θ_est and its variance σ (i.e, 68.3% confidence interval). The inset: four peaks of P(θ|{Nμ}) over the entire phase interval θ ∈ (−π, π). The occurrence number of each outcome Nμ=Nμ(θ0) is obtained from N=500 random numbers.