This corresponds to $\sim210000$ bins of $\left(1.317\pm0.002\right)\,\mathrm{ns}$ width, surrounding the signal 2 detection events. It also implies that, due to the repetition time of our pump laser system of $100\,\mathrm{ns}$ and the sixteen-fold merging of the time-bins, we have access to $41$ neighboring pump pulses within the time window for the estimation of higher-order photons and other accidental contributions to the three-fold coincidence rate.

Indeed, we find accidental three-fold coincidences at multiple integers of the pump repetition time for both temporal directions. We illustrate this in Fig. \[16-bin\] (right), where only a fraction of the analyzed time window is shown for clarity. The graph shows that higher-order photons contribute to the three-fold coincidences along the $(\tau_1-\tau_2)$-axis, indicating an increase on the primary idler detection probability. The appearance of three-fold coincidences in neighboring pulses along the $(\tau_3-\tau_2)$-axis, where only secondary photons should reside, indicates other parasitic influences: either higher-order idler 1 photons survive the demultiplexing on-chip and the subsequent CWDM-filtering, or primary PDC processes involving higher-order mode combinations produce idler 1 photons at secondary PDC photon wavelengths. It is also possible that nonlinear Cherenkov-type PDC [@Suhara2003] is generated in the primary PDC stage with a broad idler distribution in the secondary 