# Second Harmonic Revisited: An Analytic Quantum Approach
## 2 Results
### Quantum Regime
#### 2.2.1 Photon-Number Distribution  
[PERSON] and [PERSON] in [7] provided an expansion of the first moment of the output statistics \(\langle\hat{a}_{\text{SH}}^{\dagger}\hat{a}_{\text{SH}}\rangle\) up to the fourth order as a function of the Glauber autocorrelation function of the input field. The Glauber autocorrelation function for the modes \(\hat{a}_{\text{F}}\) and \(\hat{a}_{\text{F}}^{\dagger}\) is defined as follows [12; 16]:  
\[g_{\text{F}}^{(n)}\equiv\frac{\langle(\hat{a}_{\text{F}}^{\dagger})^{n}\hat{a }_{\text{F}}^{n}\rangle}{\langle\hat{a}_{\text{F}}^{\dagger}\hat{a}_{\text{F} }\rangle^{n}}. \tag{32}\]  
As shown in Appendix A, it is possible to demonstrate that every order of the perturbative expansion of the output mean photon-number can be expressed as a linear combination of the Glauber autocorrelation functions in Equation (32). The theorem allows us to express the SH moments as convergent series over the order of the autocorrelation function of the fundamental field. In particular, we found out that up to the sixth order in \(\gamma\), the output mean value reads:  
\[\langle\hat{n}_{\text{SH}}\rangle=g_{\text{F}}^{(2)}N^{2}\tilde{\gamma}^{2}- \frac{2}{3}(g_{\text{F}}^{(2)}N^{2}+2g_{\text{F}}^{(3)}N^{3})\tilde{\gamma}^{ 4}+\frac{4}{45}(2g_{\text{F}}^{(2)}N^{2}+16g_{\text{F}}^{(3)}N^{3}+17g_{\text{ F}}^{(4)}N^{4})\tilde{\gamma}^{6}+o(\tilde{\gamma}^{8}), \tag{33}\]  
where \(N\equiv\langle\hat{a}_{\text{F}}^{\dagger}\hat{a}_{\text{F}}\rangle\). For the sake of simplicity, we assumed \(\gamma\in\mathbb{R}\) and replaced \(\gamma t\) with \(\tilde{\gamma}\). Moreover, for the Fock state, the theorem provides a general expression for the asymptotic value of \(\langle\hat{n}_{\rm SH}\rangle\) as a function of \(N\) (see Appendix B), namely:  
\[\langle\hat{n}_{\rm SH}\rangle\sim N\sum_{j}\tilde{\gamma}^{2j}N^{j}. \tag{34}\]  
Note that, upon defining the conversion efficiency as:  
\[\eta\equiv\frac{\langle\hat{n}_{\rm SH}\rangle}{N}, \tag{35}\]  
we find from Equation (34) that:  
\[\eta\sim\sum_{j}\tilde{\gamma}^{2j}N^{j} \tag{36}\]  
meaning that the conversion efficiency related to every single interaction is proportional to the input intensity. This fact will be relevant in the analysis of the evolution of the input light statistics. Furthermore, note that, if \(\tilde{\gamma}^{2}N<1\), Equation (34) is a geometric series, and \(\langle\hat{n}_{\rm SH}\rangle\sim\tilde{\gamma}^{2}N^{2}/(1-\tilde{\gamma}^{ 2}N)\). The physical meaning of the coefficients in the perturbative terms of \(\langle\hat{n}_{\rm SH}\rangle\) can be pointed out by writing the autocorrelations in Equation (33) for the Fock state as a function of \(N\):  
\[\langle\hat{n}_{\rm SH}\rangle=\sum_{k=1}^{K}\tilde{\gamma}^{2k}\left[\frac{N! }{k!(N-2k)!}k+\alpha_{k}(N)+\beta_{k}(N)\right]+o(\tilde{\gamma}^{2(K+1)}) \tag{37}\]  
for any given \(K\). Up to the sixth order in \(\tilde{\gamma}\), we have:  
\[\alpha_{1} = 0\qquad\beta_{1}=0\qquad\alpha_{2}=0\qquad\beta_{2}=-\frac{1}{3} \left[\left(\frac{N!}{(N-2)!}\right)^{2}+2\frac{N!}{(N-4)!}\right]\] \[\alpha_{3} = \frac{1}{36}\left[\left(\frac{N!}{(N-2)!}\right)^{3}+4\frac{N!(N- 2)!}{[(N-4)!]^{2}}\right]\] \[\beta_{3} = \frac{1}{60}\left(\frac{N!}{(N-2)!}\right)^{3}+\frac{1}{90}\frac{ (N!)^{2}}{(N-2)!(N-4)!}-\frac{4}{15}\frac{N!(N-2)!}{[(N-4)!]^{2}}+\frac{1}{10 }\frac{N!}{(N-6)!}. \tag{38}\]  
Note that the three terms in Equation (37) represent the three different processes concurring in SHG. The first one  
\[\sum_{k}\tilde{\gamma}^{2k}N!/(N-2k)!\]  
is related to the creation of \(k\) SH photons from \(N\). This contribution comes from all and only the terms that in the expansion of \(U\) in Equation (29) are of the kind \((\hat{a}_{\rm SH}^{\rm t})^{k}\). They correspond to pure up-conversion events, i.e., the creation of \(2k\) SH photons from \(N\) at each order \(k\). Then, for \(k\geq 3\), we have the \(\alpha_{k}\) terms, corresponding to the alternating creation and annihilation of SH photons.