# Generation of Pure State Photon Triplets in the C-Band
## 3 Numerical Simulation Results
### Realization of Pure-State Photon Triplets in PPLN  
The pump wavelength is 520 nm. Relevant data in the first down-conversion is shown in Figure 2. The z axis in Figure 2a describes the variation of the spectral purity with the parameters. The x-axis represents the range of the selected crystal lengths from 0 to 1 cm while the y-axis is the variation of pump duration in the range of 0-1 ps. The wavelengths of the pair of entangled photons are \(\lambda_{1}=1560\) nm and \(\lambda_{0}=780\) nm. The periodicity of the first PPLN is 38.47 \(\upmu\)m. Due to the Gaussian filter with a bandwidth of 0.8 THz, the spectrum purity between photons \(\omega_{0}\) and \(\omega_{1}\) is almost 1 in the region where the crystal length and pump duration are smaller. Considering the realizability, we selected a pump duration of 100 fs and a crystal length of 0.2 cm.  
Figure 2b describes the joint spectral intensity of photons \(\omega_{1}\) and \(\omega_{0}\). It is intuitive to see that there is no frequency correlation between the two photons. Figure 2c,d are the bandwidth of photons \(\omega_{1}\) and \(\omega_{0}\), respectively. Because the transmission of the filter is related to the bandwidth, the down-conversion photons of the two channels have the identical frequency distribution.  
In the second SPDC process, we select the generation wavelengths of \(\lambda_{2}\) = 1570 nm and \(\lambda_{3}\) = 1550 nm in consideration of the matching condition of the group velocity. The polarization period of the second PPLN is 88.76 \(\upmu\)m. Relevant data are shown in Figure 3. Figure 3 describes the calculation of the spectral purity of photons \(\omega_{2}\) and \(\omega_{3}\) with the crystal length \(L_{2}\) as the independent variable. It can be seen that with the increase of crystal length, the purity increases to the maximum value of 1. We chose the best crystal length \(L_{2}\) as 9.16 cm. Figure 3b is the joint spectral intensity of photon pair \(\omega_{2}\) and \(\omega_{3}\). It can be seen that the photon pairs are still frequency uncorrelated in the second SPDC process. As shown in Figure 3c,d, the bandwidth of photons \(\omega_{2}\) and \(\omega_{3}\) is different because the perfect group velocity match is not achieved, but this does not affect the correlation between them.  
Figure 3: **(a)** Spectral purity data in the second SPDC. **(b)** The joint spectrum of photon 2 and 3. **(c,d)** The bandwidth of the photon pairs.  
Figure 2: **(a)** The spectral purity in the first SPDC using PPLN. **(b)** The joint spectrum of photon 1 and 0. **(c,d)** The bandwidth of the photon pairs.  
Since photons \(\omega_{1}\) and \(\omega_{0}\) are not correlated in frequency, both photon (1,2) and photon (1,3) should be irrelevant theoretically. Figure 4 shows the joint spectrum of photon triplets. From the relationship between each two-photon, as shown in three projection planes, there is no correlation between photons \(\omega_{1}\) and \(\omega_{2}\), \(\omega_{1}\) and \(\omega_{3}\). So far, we have obtained photon triplets which are not related in the frequency dimension. At the same time, all three of them are in the C-band.