*2.3. Joint Spectrum and Purity*

The simplest method to judge the frequency dependence of two photons produced by second order SPDC is to analyze their joint spectrum which is determined by

$$JSI(\omega\_{\sf s}, \omega\_{\sf i}) = \left| F(\omega\_{\sf s}, \omega\_{\sf i}) \right|^2 \tag{9}$$

The two photons are frequency uncorrelated if their joint spectrum is a circle or an ellipse parallel to the axis, which means the distribution of photons in frequency is independent of each other. It is impossible to obtain an optimal value simply by judging the shape or the angle with the coordinate axis because of the lack of a specific parameter to quantify the two-photon frequency correlation. Calculating the Schmidt number is the effective scheme to measure spectral correlation because it reflects the purity of correlation over frequency. It is defined as follows

$$\mathbf{K} = \frac{1}{\text{Tr}\{\rho\_1^2\}} = \frac{1}{P} \tag{10}$$

In this formula, ρ<sup>1</sup> is the density operator of photon ω<sup>1</sup> and *P* represents the spectral purity. There is no frequency correlation between photon pairs when the Schmidt number K reaches the minimum value of 1. After calculating the Schmidt numbers with the parameters of crystal length and pump duration in each SPDC, the results are verified and analyzed by the joint spectrum of two photons under the optimum parameters. The joint spectral intensity of the photon triplets can be written as

$$JSI(\omega\_1, \omega\_2, \omega\_3) = \left| F(\omega\_1, \omega\_2, \omega\_3) \right|^2 \tag{11}$$

We use the symbol quantity to carry on the maximum precision calculation. The result is converted to double type with 16 bits precision. The precision is enough that an ideal numerical simulation result can be obtained. Therefore, the error caused by the accuracy of software calculation can also be ignored.
