# Generation of Pure State Photon Triplets in the C-Band
## 2 Tripartite State and Joint Spectrum
### Hamiltonian and Probability Amplitude Function  
For the convenience of calculation, our model adopts a one-dimensional collinear phase-matching structure. Since the pump field is strong, the field is treated as an electric classical field \(E_{p}\) (\(\mathbf{r}\), \(t\)) = \(\widetilde{\alpha}(t)\) exp [\(ik_{p}(\omega_{p})z\)], rather than using the annihilation operator of the pump photon. A Gauss envelope is chosen as the pump function \(\widetilde{\alpha}_{p}(t)\) = \(\widetilde{\alpha}_{p}(0)\) exp(\(-t^{2}/2\tau_{p}{}^{2}\)). The expression corresponding to the frequency domain is  
\[\alpha\Big{(}\Omega_{p}\Big{)}=\frac{\tau_{p}}{\sqrt{2\pi}}\exp(-\frac{\tau_{ p}^{2}\Omega_{p}^{2}}{2}) \tag{1}\]  
where \(\tau_{p}\) is the pump duration and \(\Omega_{p}=\omega_{p}-\overline{\omega}_{p}\) is the frequency difference. After calculating the integral of Hamiltonian [31] and simplifying the statements, the final expression of the two-photon state is  
\[\big{|}\psi_{2}\big{>}=\int_{I_{0}}^{t}dt^{\prime}\hat{H}_{I}(t^{\prime})= A\int d\omega_{s}\int d\omega_{i}d_{s}^{\dagger}(\omega_{s})d_{i}^{ \dagger}(\omega_{i})\alpha(\omega_{s},\omega_{i})\varphi(\omega_{s},\omega_{ i})|0\rangle+c.c. \tag{2}\]  
where \(\alpha(\omega_{s}\), \(\omega_{i})\) and \(\varphi(\omega_{s}\), \(\omega_{i})\) are pump envelope function and phase-matching function, respectively. Their product is the two-photon amplitude function  
\[F(\omega_{s},\omega_{i})=\alpha(\omega_{s}+\omega_{i})\varphi(\omega_{s}, \omega_{i}) \tag{3}\]  
The phase matching function in upper equation is  
\[\varphi(\omega_{s},\omega_{i})=\sin\varsigma\left[\frac{\Big{(}k_{s}(\omega_{ s})+k_{i}(\omega_{i})-k_{p}(\omega_{s}+\omega_{i})+k_{g}\Big{)}L}{2}\right] \tag{4}\]  
For simpler operation, a coefficient \(\gamma\) = 0.193 is introduced to approximate the sinc function to a Gauss function to ensure that they have the same full width at half maximum (FWHM). This approximation only removes the small peak of sinc function and has no effect on the calculation of biphoton joint spectrum. Assuming a perfect phase-matching condition, we carry out the Taylor expansion of the wave vector and preserve the first order term. That is \(k_{m}(\omega_{m})=k_{m0}+k^{\prime}{}_{m}(\omega_{m}-\overline{\omega}_{m})+\cdots\), \(k^{\prime}{}_{m}=\partial k_{m}(\omega)/\partial\omega\big{|}_{\omega= \overline{\omega}_{m}}(m=p,\,s,\,i)\). The influence of group velocity dispersion and higher order terms are not considered. The second derivative of wave vector does not change obviously with the  
Figure 1: Theoretical model of cascaded second-order SPDC. Photons \(\omega_{0}\) and \(\omega_{1}\) are generated in the first crystal and then photons \(\omega_{2}\) and \(\omega_{3}\) are generated from the second crystal. The two parts of this model are periodically poled lithium niobite (PPLN) with lengths \(L_{1}\) and \(L_{2}\) respectively. Two Gauss filters are used to manipulate the joint spectrum of photonic pairs. wavelength. In addition, in the actual system, the error caused by dispersion can be overcome by compensation. The phase-matching function is described by  
\[\varphi(\omega_{s},\omega_{i})\approx\exp\left\{-\gamma\left(\frac{\Omega_{s} \left(k^{\prime}{}_{s}-k^{\prime}{}_{p}\right)+\Omega_{i}\left(k^{\prime}{}_{i }-k^{\prime}{}_{p}\right)L}{2}\right)\right\} \tag{5}\]  
where \(\Omega_{s}\) and \(\Omega_{i}\) are the frequency difference. In addition to the phase-matching condition, we also consider the matching condition of group velocity [32]. But in the first SPDC, the derivative of the pump wave vector is always larger than that of the two down-converted photons. We use two Gaussian filters to remove the correlation of the two photons [33]. The two photons' amplitude function becomes  
\[F(\omega_{s},\omega_{i})=T(\omega_{0})T(\omega_{1})\alpha(\omega_{s}+\omega_{ i})\varphi(\omega_{s},\omega_{i}) \tag{6}\]  
where \(T(\omega_{i})=\exp(-\Omega_{i}^{2}/z_{i}^{2})\) is the corresponding filter. \(\varsigma_{i}\) is the FWHM.