Fig. 3: Frequency-binned energy-time entanglement verification via Franson interference.

a The measured Franson interference fringes, after accidentals were subtracted, for symmetric frequency-bin pairs. All the fringes were obtained at zero relative delay between arm 1 and arm 2 for optimum Franson interference. Maximum frequency-binned Franson interference is observed for S₀&I₀ pair with visibility up to 99.66 ± 1.67%. The averaged frequency-binned Franson visibility for 5 pairs is 98.03 ± 1.10%. The coincidence window for all the measurements was 2 ns. The error bars represent one standard deviation of the mean assuming Poissonian statistics. b The Franson interference visibility map for frequency-bin pairs within the SPDC bandwidth. Frequency bins were selected using a pair of tunable BPFs with 300 pm bandwidths, that were manually tuned to scan from the −2 to +2 frequency bins from frequency degeneracy. Only the symmetric frequency-bin pairs in anti-diagonal terms show high Franson interference visibility. c Recorded signal-idler cross-correlation function for constructive (top) and destructive (bottom) Franson interference. The inset is the zoom-in of the central correlation peak, showing the single-sided decay temporal signature of a singly-filtered BFC. d Extracted Schmidt eigenvalues (color bars represent different corresponding frequency-bin pairs in the frequency-correlation matrix) and Franson visibilities (red dots) for 5 symmetric frequency-bin pairs. Frequency-binned Franson interference visibilities with (without) background subtracted are 97.24 (82.88) ± 1.96% for S₂&I_-2, 98.42 (90.90) ± 1.81% for S₁&I_-1, 99.66 (92.06) ± 0.60% for S₀&I₀, 98.38 (88.86) ± 1.62% for S_-1&I₁ and 96.45 (81.53) ± 2.07% for S_-2&I₂, respectively. The error bars represent one standard deviation of the mean. We extract the lower bound of the Schmidt number to be 4.17 across 5 symmetric frequency-bin pairs (detailed calculation is presented in Supplementary Note 4).