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, †}

The conditional probabilities for detecting particle-number difference: p_μ(θ) for (a) μ = 0, (b) μ = ± 1, (c) μ = ± 2, and (d) μ = ± 3. The red dashed lines show the ideal results; the blue solid lines give the occurrence probabilities with the experimental imperfections, obtained from Eq. (6). Numerical simulations give the statistical average of the occurrence frequency 〈Nμ(i)〉s/N (the solid circles) and its standard derivation (the bars) of each outcome, simulated by M = 20 replicas of N=500 random numbers at each θ ∈ (−π, π). Hereinafter, 〈A(i)〉s=∑i=1MA(i)/M denotes the statistical average.