and at $1625\,\mathrm{nm}$ (idler 1) in the periodically poled area I. The integrated directional coupler splits up the generated photon pairs spatio-spectrally. While idler 1 photons couple to the adjacent waveguide, the corresponding signal photons remain in the original waveguide and decay to secondary photon pairs at $\left(1551\pm25\right)\,\mathrm{nm}$ (signal 2) and $\left(1611\pm25\right)\,\mathrm{nm}$ (idler 2) in the periodically poled area II. Anti-reflective dielectric coatings on the waveguide end-faces provide the reduction of Fresnel-losses at telecom wavelengths, while the green pump is reflected in order to reduce additional filtering efforts.[]{data-label="triplet-chip"}](fig01.eps){width="0.95\linewidth"}

The first PDC stage in the cascade is pumped with a classical field. It is known that parametric down-conversion processes, which are not pumped by single-photons, produce not only single, but also higher-order photon pairs [@Takesue2010]. Thus, we must expect to generate a statistical mixture of genuine photon triplets, $\ket{\psi_\mathrm{triplet}}=\ket{1\,1\,1}$, and states including higher-order photon contributions in our process. This means that we also generate states of the form $\ket{\psi_\mathrm{m-plet}}=\ket{m\,1\,1}$ with certain probabilities. The mean photon number per optical pulse behind the primary PDC is denoted by $\langle m\rangle$. We write for the vector containing the photon-number-occupation probabilities of the first PDC process: $$\label{eq:01}
\boldsymbol{\rho}=
 \begin{pmatrix}
 \rho_0\\
 \vdots\\
 \rho_n
 \end{pmatrix},$$ where the 