*3.2. Realization of Pure-State Photon Triplets in PPMgLN*

We also chose 520 nm as the pump wavelength for comparison. The group velocity matching condition of the second SPDC is not satisfied. The wavelength of photon ω<sup>1</sup> and ω<sup>0</sup> generated in the first down-conversion are 1520 nm and 790.4 nm, respectively.

The results of correlated data are given in Figure 5. The photon wavelengths generated by the second down-conversion are λ<sup>2</sup> = 1590 nm and λ<sup>3</sup> = 1571.7 nm, pumped by photon ω0. Similar data are shown in Figure 6. The pump duration is 0.27 ps while the crystal lengths are *L*<sup>1</sup> = 0.2 cm and *L*<sup>2</sup> = 10.74 cm (corresponding Λ<sup>1</sup> = 34.85 μm and Λ<sup>2</sup> = 83.51 μm).

We also produce three photons with a purity of 100%. Due to the material differences, the center wavelengths of the photons ω<sup>2</sup> and ω<sup>3</sup> are longer than in PPLN. It takes a slightly longer crystal than PPLN to achieve the phase-matching condition. The bandwidth of the photons ω<sup>1</sup> and ω<sup>2</sup> generated in the PPMgLN is relatively wider. Figure 7 shows the joint spectral intensity of photon triplets generated by cascaded PPMgLN. The three projection planes reflect the correlation between two of the three photons.

PPLN is more suitable for weak light due to the better phase-matching conditions. According to our theoretical results, the wavelength distribution of the three-photon generated in PPLN is closer. For the same pump (λ*<sup>p</sup>* = 520 nm), three photons with wavelengths of 1550 nm, 1560 nm and 1570 nm can be realized in PPLN. PPMgLN is more suitable for the pump with higher intensity, because doping MgO can increase the damage threshold of the material and obtain higher brightness photon triplets. But we can only obtain photons with wavelengths of 1520 nm, 1590 nm and 1571.7 nm. In the preparation of the light source, spectral purity is one of the core indicators. The purpose of our work is to prepare photon triplets of spectral pure-state (frequency uncorrelated), which provides a reliable scheme for the preparation of high quality sources in the field of quantum technology. There is no prior research on pure state photon triplets in the C-band before our work. Our method can also provide a heralding pure-state biphoton source with higher interference visibility. Compared with the unpredicted conditions, the heralding two-photons have superior advantages, such as avoiding the detection of noise photons, which greatly reduces the bit error rate (BER). It guarantees the realization of many of these tasks relying on qubits that are encoded in the polarization states of single photons.

**Figure 5.** (**a**) The spectral purity in the first SPDC using MgO-doped PPLN. (**b**) The joint spectrum of photon 1 and 0. (**c**,**d**) The bandwidth of the photon pairs.

**Figure 6.** (**a**) The spectral purity data in the second SPDC. (**b**) The joint spectrum of photon 2 and 3. (**c**,**d**) The bandwidth of the photon pairs.
