Estimation of P1
This protocol is extracted from research article:
High-efficiency single-photon generation via large-scale active time multiplexing
Sci Adv, Oct 4, 2019; DOI: 10.1126/sciadv.aaw8586

We estimated a single-photon probability P1, assuming that the probability of more than two photons is negligibly small; a multiplexed k-photon probability Pk for k ≥ 3 is expected <1.5% of P1 for μ = 0.18 in our experiment. We modeled our SPDs as bucket detectors that discriminate between zero and one or more photons. The NPBS’s transmission/reflection ratio is also assumed to be 1. With those assumptions, single and coincidence count rates are given bySi=P1Rη2+P2R{η2+η(2η)4}(4)C=P2Rη22(5)where P2 is the probability of a two-photon state after multiplexing. The second and third terms in Eq. 4 are the respective detector click probabilities that one and two of two photons arrive at detector i. Equation 2 is obtained by substituting Eq. 5 into Eq. 4.

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