**3. Numerical Simulation Results**

In this section, we discuss the generation of photon pairs from different materials, and finally obtain the pure-state photon triplets. Among the numerous nonlinear crystals, lithium niobite has a relatively higher nonlinear coefficient [34,35], which leads to a greater conversion efficiency. There is a wide range of transparency, from 420 nm to 5200 nm. In addition, lithium niobite doped with MgO has higher damage threshold, thus the periodically poled lithium niobate doped with 5% MgO (PPMgLN) will also be used as a reference for comparison. We get a set of crystal lengths which are optimum for each SPDC process through theoretical arithmetic.

In the first SPDC, the crystal and pump parameters are taken as the variables and the calculation of the Schmidt number is done. We select the appropriate pumping duration and crystal length *L*<sup>1</sup> by analyzing the obtained data. After calculation, the frequency distribution of photonic ω<sup>0</sup> is obtained. That is to say, the envelope information of the pump in the second SPDC is determined, which is exp[−(ω<sup>0</sup> <sup>−</sup> <sup>ω</sup>0)/σ<sup>2</sup> <sup>0</sup>], where σ<sup>0</sup> is the bandwidth of the new source ω0. Then, the amplitude function of photon ω<sup>2</sup> and ω<sup>3</sup> is described as

$$F(\Omega\_2, \Omega\_3) = \exp\left[-\frac{\left(\Omega\_2 + \Omega\_3\right)^2}{\sigma\_0^2}\right] \exp\left(-i\frac{\Delta k\_2 L\_2}{2}\right) \sin c \left[\frac{(k\_0(\omega\_0) - k\_2(\omega\_2) - k\_3(\omega\_3) - k\_{\emptyset})L\_2}{2}\right] \tag{12}$$

In the second SPDC, the Schmidt numbers of photon state between ω<sup>2</sup> and ω<sup>3</sup> are calculated by using the bandwidth information of the generated photons ω<sup>0</sup> and taking the crystal length *L*<sup>2</sup> as the variable. Then we select the appropriate crystal length *L*2. Each time the most appropriate parameters are determined, the two-photon joint spectrum and the final three-photon joint spectrum are given to verify the theoretical calculation.

## *3.1. 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 λ<sup>1</sup> = 1560 nm and λ<sup>0</sup> = 780 nm. The periodicity of the first PPLN is 38.47 μm. Due to the Gaussian filter with a bandwidth of 0.8 THz, the spectrum purity between photons ω<sup>0</sup> and ω<sup>1</sup> 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 ω<sup>1</sup> and ω0. It is intuitive to see that there is no frequency correlation between the two photons. Figure 2c,d are the bandwidth of photons ω<sup>1</sup> and ω0, 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.

**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.

In the second SPDC process, we select the generation wavelengths of λ<sup>2</sup> = 1570 nm and λ<sup>3</sup> = 1550 nm in consideration of the matching condition of the group velocity. The polarization period of the second PPLN is 88.76 μm. Relevant data are shown in Figure 3. Figure 3a describes the calculation of the spectral purity of photons ω<sup>2</sup> and ω<sup>3</sup> with the crystal length *L*<sup>2</sup> 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*<sup>2</sup> as 9.16 cm. Figure 3b is the joint spectral intensity of photon pair ω<sup>2</sup> and ω3. 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 ω<sup>2</sup> and ω<sup>3</sup> 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.

Since photons ω<sup>1</sup> and ω<sup>0</sup> 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 ω<sup>1</sup> and ω2, ω<sup>1</sup> and ω3. 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.

**Figure 4.** Joint spectrum of the three photons generated from the cascaded PPLN.
