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Image reconstruction process
This protocol is extracted from research article:
Quantum image distillation
Sci Adv, Oct 18, 2019; DOI: 10.1126/sciadv.aax0307

The camera was an EMCCD Andor iXon Ultra 897 and was operated at −60°C with a horizontal pixel readout rate of 17 MHz, a vertical pixel shift every 0.3 μs, and a vertical clock amplitude voltage of +4 V above the factory setting. In each acquisition, N frames are collected with an exposure time of τ = 6 ms. No threshold was applied, and all calculations were performed directly using gray values returned by the camera (26). For r2r1, Γ(r1, r2) was calculated using the formulaΓ(r1,r2)=I(r1)I(r2)I(r1)I(r2)(2)

The first term is the average intensity productI(r1)I(r2)=limN+1Nl=1NIl(r1)Il(r2)(3)where Il(r1) [Il(r2)] corresponds to the intensity value measured at pixel r1 [r2] in the jth frame. Experimentally, this term is estimated by multiplying intensity values in each frame and averaging over a large number of frames (typically N on the order of 106 to 107). Intensity correlations in this term originate from detections of both real coincidence (two photons from the same entangled pair) and accidental coincidence (two photons from different entangled pairs). The second term in Eq. 2 is defined asI(r1)I(r2)=limN+1N2l=1Nl=1NIl(r1)Il(r2)(4)

Experimentally, this term is estimated by multiplying intensity values between successive frames and averaging over a large number of framesI(r1)I(r2)1Nl=1NIl(r1)Il+1(r2)(5)

Since there is zero probability for two photons from the same entangled pair to be detected in two different images, intensity correlations in this term originate only from photons from different entangled pairs (accidental coincidence). A subtraction between these two terms (Eq. 2) leaves only genuine coincidences, which is proportional to the joint probability distribution of photon pairs. Moreover, the use of intensity products between successive frames, rather than the products of the averaged intensities, allows the reduction of artifacts such as spatial distortions in the retrieved Γ that are due to fluctuations of the camera amplification gain during the time of an acquisition (26).

Since Eq. 2 is only valid for r2r1, diagonal values Γ(r, r) are approximated to intensity correlation values between neighboring pixels Γ(r, r) ≈ Γ(r, r + δr), where δr = −δ ex with δ = 16 μm and ex is a unit vector. This approximation is justified because the Andor Ultra 897 has a fill factor near 100%, and the correlation width on the camera is estimated to be σr ≈ 10 μm (27). More details about the image reconstruction process are provided in sections S1 and S2.

A convenient method to visualize Γ is to use conditional projections. The conditional projection relative to an arbitrarily chosen position A, denoted Γ(rA), is an image of intensity correlations between any position r and the position A. For example, two positions A and B are selected in the direct-intensity image in Fig. 5A, and their corresponding conditional projections are shown in Fig. 5 (B and C). Γ(rA) shows an intense peak demonstrating that photon pairs from the SPDC source are transmitted together through the object around position A. On the contrary, the flat and null pattern of Γ(rB) shows that both photons are absorbed by the object around position B.

Direct-intensity image (A) measured under simultaneous illumination of classical and quantum light. Conditional image Γ(rA) (B) shows an intense peak centered around position A. Conditional images Γ(rB) (C) are null and flat.

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