mean photon pair number per pulse of only $\langle m\rangle=0.215\pm0.02$ behind the first PDC stage. Thus, we expect that predominantly $\left(\sim88\,\%\right)$ genuine photon-triplet states $\ket{\psi_\mathrm{triplet}}=\ket{1,1,1}$ are generated. Likewise, we deduce that a high conversion efficiency in our primary PDC stage limits the cleanliness of the generated photon-triplets by generating higher-order photon pairs [@Krapick2015].

We chose the signal 2 photon detection as the reference events for our data analysis, because these occur at the lowest detection rates due to the InGaAs detector efficiency. A three-fold coincidence is given, if a signal 2 detection event announces the detection of its idler 2 twin photon, and if the corresponding idler 1 photon is also registered. Hence, the relative arrival times for idler 1 events are labeled $\tau_1-\tau_2$, whereas $\tau_3-\tau_2$ denotes the relative arrival time of idler 2 photons with respect to the signal 2 photons. This pseudo-heralding method significantly reduces the computational effort for the post-selection. Additionally, we merge the bins of our time-tagging module sixteen-fold in order to include the joint timing jitter of our apparatus. Thus, the time-bins for the data analysis have widths of $\left(1.317\pm0.002\right)\,\mathrm{ns}$ in both temporal directions.

![Long-term stability of our device: we recorded the idler 1 count 