at $\lambda_\mathrm{s1}=\lambda_\mathrm{p2}=790.3\,\mathrm{nm}$ and idler 1 photons (i1) at $\lambda_\mathrm{i1}=1625\,\mathrm{nm}$. The WDM separates the generated primary photons pairs with high probability in a spatio-spectral manner. The signal 1 photons remain in the original arm, whereas the idler 1 photons are transferred to the adjacent waveguide. By deploying the signal 1 photons as the pump (p2) in the second type-0 PDC stage we generate secondary signal photons (s2) at $\lambda_\mathrm{s2}=\left(1551\pm25\right)\,\mathrm{nm}$ and idler photons (i2) at $\lambda_\mathrm{i2}=\left(1611\pm25\right)\,\mathrm{nm}$. In summa, we are able to create three time-correlated photons in the telecom wavelength regime with the cascaded PDC processes, while the energy conservation, $\hbar\omega_\mathrm{p}=\hbar\omega_\mathrm{i1}+\hbar\omega_\mathrm{s2}+\hbar\omega_\mathrm{i2}$, must be fulfilled. Likewise, the wavelength-dependent conservations of momenta in both PDC processes (commonly called phase-matchings) have to be made mutually compatible as described in Appendix A. Note that our compact monolithic approach provides intrinsic spatial mode matching and conveniently tunable spectral mode matching of the intermediate signal 1/pump 2 photons along the coupled-waveguide structure. Additionally, our titanium-diffused waveguides exhibit very low propagation loss of $\sim0.08\,\mathrm{dB/cm}$ on average at telecom wavelengths.

![Device layout and on-chip functionalities: picosecond pulses at $532\,\mathrm{nm}$ are injected to the coupled waveguide structure and pump the cascaded PDC process. They decay to photons at $790.3\,\mathrm{nm}$ (signal 1) 