**1. Introduction**

The scheme of generating photon pairs using cascaded second-order spontaneous parametric down-conversion (SPDC) [1,2] is an indispensable ingredient of modern quantum technology and has great potential in many applications, such as quantum cryptography [3], quantum teleportation [4] and quantum entanglement swapping [5]. Recently, a wide variety of methods have been proposed to produce photon triplets. Common methods include direct generation of photon triplets [6–8], the process of four wave mixing (FWM) [9–12] and generation of three entangled photons by cascaded second-order SPDC [13–16]. Some studies propose implementing third-order SPDC in optical fibers and bulk crystals. There are always low count rates for schemes based on the χ(3) process. The FWM techniques consists of stimulated SPDC and cascaded FWM. The latter can be divided into three categories according to the different ways of cascading. The cascaded second-order SPDC is considered because of the simple model, which consists of two second order SPDC processes. The mature theory and substantial experiments make it a reliable scheme.

The research about quantum correlation among individual photons lies at the core of quantum technologies. Under different conditions, the two-photon generated by SPDC will present a state of frequency positive correlation, inverse correlation or uncorrelation. The last method is used to provide a heralded source [17,18]. Previous experiments have failed to give a specific theoretical numerical analysis to judge the spectral purity of the generated photon pairs. The full use of filters [19,20] will greatly reduce the coincidence counting rate. Recently, Zhang et al. decomposed the factor mathematically to manipulate the tripartite frequency correlation [21]. But the spectrum of photons after the first SPDC and the effect on the second order down-conversion were not taken into account. They produce photons with wavelength of ~3000 nm, which is almost unavailable. So far, there is little theoretical work about pure states photon triplets in the C-band.

Quantum interference is vital for quantum information science. It is not only the basis of quantum manipulation technology, but also an important tool to implement quantum computing and quantum communication. The realization of quantum computation [22] depends on the measurement and reading of quantum states, and quantum interference is one of the most simple and feasible methods for quantum measurement. Quantum communication [23,24] is more dependent on the transmission and acquisition of information by means of interference. Three-photon interference is critical for the exploitation of quantum information in higher dimensions [25]. The GHZ interference is observed in the experiment, which lays the foundation for the subsequent quantum secret sharing [26]. In general, the photon triplets generated by the cascaded SPDC will have correction in frequency. This allows the photon pair to be resolved in the frequency dimension, thereby reducing the visibility of the interference [27]. For instance, the interference of indistinguishable photons makes the entanglement swapping and teleportation possible, which in turn opens up prospects for distributing of entanglement between distant matter qubits. The goal of our work is to prepare three photons with hyperspectral purity, which are critical for research into quantum information processes.

In this work, the suitable pump duration and crystal length are selected to eliminate the frequency correlation between the photon pairs in each SPDC process. In terms of the theoretical analysis, spectral purity of photon pairs is mainly measured by means of Schmidt number [28,29]. The conclusion of our theoretical calculation is supported by the two photons' and three photons' joint spectrum. Relevant theories will be discussed in Section 2. The common pump source used to acquire polarization-entangled photon pairs from SPDC is narrow-band or continuous wave (CW) laser, but the subsequent photon pairs have a strong correlation in frequency [30]. A broadband pumping source is adopted in our work, and the optimal pumping duration is chosen by numerical investigation in Section 3. Finally, we obtain pure-state photon triplets with two kinds of periodically poled crystals under different parameters.

#### **2. Tripartite State and Joint Spectrum**

#### *2.1. Model*

Quasi-phase matching is adopted because of the simpler and more flexible matching condition. The theoretical model consists of two parts, which are two nondegenerate SPDC processes [13]. A pair of photons called idler photons ω<sup>0</sup> and signal photons ω<sup>1</sup> are generated from the first SPDC process. The idler photons continue to be the pump source of the second SPDC process, producing photons ω<sup>2</sup> and ω3.

The phase-matching conditions of the two processes are type e → o + o and type e → e + o, respectively. As shown in Figure 1, the lengths of the two crystals are *L*<sup>1</sup> and *L*<sup>2</sup> while the periodicities are Λ<sup>1</sup> and Λ2, respectively. When the pump light with center frequency of ω<sup>0</sup> is incident into the first crystal, the generated photon pairs will be correlated in time and frequency due to the conservation of energy and momentum. The relation between the wave vectors and the frequency of the three photons are *kp* = *k*<sup>1</sup> + *k*<sup>0</sup> + *kg*<sup>1</sup> and *h*¯ω*<sup>p</sup>* = *h*¯ω<sup>1</sup> + *h*¯ω0, where *kg*<sup>1</sup> = 2π*m*/Λ<sup>1</sup> is the compensated wave vector. In the second down-conversion process, photon ω<sup>0</sup> splits into ω<sup>2</sup> and ω<sup>3</sup> while the conservation conditions are also satisfied, which are *k*<sup>0</sup> = *k*<sup>2</sup> + *k*<sup>3</sup> + *kg*<sup>2</sup> and *h*¯ω<sup>0</sup> = *h*¯ω<sup>2</sup> + *h*¯ω3, where *k*g2 = 2π*m*/Λ2. Therefore, in the whole frequency conversion process, the energy conservation and momentum conservation are also satisfied between the initial pump photon and the resulting three photons.

**Figure 1.** Theoretical model of cascaded second-order SPDC. Photons ω<sup>0</sup> and ω<sup>1</sup> are generated in the first crystal and then photons ω<sup>2</sup> and ω<sup>3</sup> are generated from the second crystal. The two parts of this model are periodically poled lithium niobite (PPLN) with lengths *L*<sup>1</sup> and *L*<sup>2</sup> respectively. Two Gauss filters are used to manipulate the joint spectrum of photonic pairs.

## *2.2. 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 *Ep* (**r**, *t*) = <sup>∼</sup> α(*t*) exp [*ikp*(ω*p*)*z*], rather than using the annihilation operator of the pump photon. A Gauss envelope is chosen as the pump function <sup>∼</sup> α*p*(*t*) = <sup>∼</sup> α*p*(0) exp(−*t* <sup>2</sup>/2τ*<sup>p</sup>* 2). The expression corresponding to the frequency domain is

$$\ln \left( \Omega\_p \right) = \frac{\tau\_p}{\sqrt{2\pi}} \exp(-\frac{\tau\_p^2 \Omega\_p^2}{2}) \tag{1}$$

where τ*<sup>p</sup>* is the pump duration and Ω*<sup>p</sup>* = ω*<sup>p</sup>* − ω*<sup>p</sup>* is the frequency difference.

After calculating the integral of Hamiltonian [31] and simplifying the statements, the final expression of the two-photon state is

$$\left|\psi\_{2}\right\rangle = \int\_{t\_{0}}^{t} dt' \hat{H}\_{l}(t') = A \int d\omega\_{s} \int d\omega\_{l} \hat{a}\_{s}^{\dagger}(\omega\_{s}) \hat{a}\_{l}^{\dagger}(\omega\_{l}) a(\omega\_{s}, \omega\_{l}) \varphi(\omega\_{s}, \omega\_{l}) |0\rangle + \text{c.c.} \tag{2}$$

where α(ω*s*, ω*i*) and ϕ(ω*s*, ω*i*) are pump envelope function and phase-matching function, respectively. Their product is the two-photon amplitude function

$$F(\omega\_{\mathfrak{s}}, \omega\_{\mathfrak{i}}) = \mathfrak{a}(\omega\_{\mathfrak{s}} + \omega\_{\mathfrak{i}})q(\omega\_{\mathfrak{s}}, \omega\_{\mathfrak{i}}) \tag{3}$$

The phase matching function in upper equation is

$$\varphi(\omega\_{\mathfrak{s}}, \omega\_{i}) = \sin \mathbf{c} \left[ \frac{(k\_{\mathfrak{s}}(\omega\_{\mathfrak{s}}) + k\_{i}(\omega\_{i}) - k\_{p}(\omega\_{\mathfrak{s}} + \omega\_{i}) + k\_{\mathfrak{g}})L}{2} \right] \tag{4}$$

For simpler operation, a coefficient γ = 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 *km*(ω*m*) = *km*<sup>0</sup> + *k <sup>m</sup>*(ω*<sup>m</sup>* − ω*m*) + ··· , *k <sup>m</sup>* = ∂*km*(ω)/∂ω <sup>ω</sup>=ω*<sup>m</sup>* (*<sup>m</sup>* <sup>=</sup> *<sup>p</sup>*, *<sup>s</sup>*, *<sup>i</sup>*). 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

wavelength. In addition, in the actual system, the error caused by dispersion can be overcome by compensation. The phase-matching function is described by

$$\varphi(\boldsymbol{\omega}\_{s}, \boldsymbol{\omega}\_{i}) \approx \exp\left\{-\gamma \left(\frac{\Omega\_{s}(\boldsymbol{k}'\_{s} - \boldsymbol{k}'\_{p}) + \Omega\_{i}(\boldsymbol{k}'\_{i} - \boldsymbol{k}'\_{p})L}{2}\right)\right\} \tag{5}$$

where Ω*<sup>s</sup>* and Ω*<sup>i</sup>* 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

$$F(a\nu\_{\mathfrak{s}}, a\mathfrak{e}) = T(a\nu\_{\mathfrak{t}})T(a\nu\_{\mathfrak{t}})a(a\nu\_{\mathfrak{s}} + a\nu\_{\mathfrak{t}})q(a\nu\_{\mathfrak{s}}, a\mathfrak{e})\tag{6}$$

where *T*(ω*i*) = exp(−Ω<sup>2</sup> *i*/ς<sup>2</sup> *<sup>i</sup>*) is the corresponding filter. ς*<sup>i</sup>* is the FWHM.

In the total cascaded process, the holistic Hamiltonian is the product of the Hamiltonian of two parts, that is *Hˆ* = *Hˆ* <sup>1</sup>*Hˆ* 2. The expression of the last three photon states is

$$\langle \psi\_3 \rangle = \int dt\_1 dt\_2 \hat{H}\_1(t\_1) \hat{H}\_2(t\_2) = B \int d\omega\_1 \int d\omega\_2 \int d\omega\_3 \hat{\mathfrak{f}}\_1^\dagger(\omega\_1) \hat{\mathfrak{f}}\_2^\dagger(\omega\_2) \hat{\mathfrak{f}}\_3^\dagger(\omega\_3) F(\omega\_1, \omega\_2, \omega\_3) |0\rangle + \text{c.c.} \tag{7}$$

When we determine the frequency distribution of the down-conversion of three photons, and there is no correlation between them, then the three photons amplitude can be equivalent to

$$F(\omega\_1, \omega\_2, \omega\_3) = \exp(-\Omega\_1^2/\sigma\_1^2) \exp(-\Omega\_2^2/\sigma\_2^2) \exp(-\Omega\_3^2/\sigma\_3^2) \tag{8}$$

In practice, there are two sensitive parameters of the system that need to be strictly controlled: (1) the polarization stability of the light source and the optical path, and; (2) the temperature of the non-linear material. Both of them directly affect the refractive index of materials, thus affecting the phase-matching conditions.
