# Second Harmonic Revisited: An Analytic Quantum Approach
## 2 Results
### Quantum Regime
#### 2.2.1 Photon-Number Distribution
For the sake of simplicity, we name these as annihilation processes since, at variance with the former term, they consist of back-conversion events. For instance, for \(k=3\), we have three different processes leading to a non-null value of \(\langle\hat{n}_{\rm SH}\rangle\), which are the following: the creation process \((\hat{a}_{\rm SH}^{\rm t})^{3}\) (creation of three SH photons from \(N\) after three interactions) and the annihilation processes \(\hat{a}_{\rm SH}^{\rm t}\hat{a}_{\rm SH}^{\rm t}\hat{a}_{\rm SH}^{\rm t}\) and \(\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\rm t}\hat{a}_{\rm SH}^{\rm t}\) (creation of one SH photon from \(N\) after three interactions). The last two provide the first and the second term in \(\alpha_{3}\) (Equation (38)), respectively, and differ for the ordering of the operators only. In particular, both these processes lead to the same output state (\(|N-2,1\rangle\)), but, in the first one, we have that an SH photon is created, annihilated, and then created again (each of these events happening in \(N!/(N-2)!\) possible ways), while in the second one, two SH photons are created (which is likely to happen in \(N!/(N-4)!\) equivalent configurations), and then, one of the two is down-converted in two photons of the fundamental field (\((N-2)!/(N-4)!\) possible configurations). Finally, if \(k\geq 2\), the annihilation processes can interfere either with one another or with creation events, giving rise to the third kind of contribution (the \(\beta_{k}\) terms in Equation (37)). This superposition happens whenever different processes lead to the same output state. For instance, for \(k=2\), we find, together with the two-photon-creation process \((\hat{a}_{\rm SH}^{\dagger})^{2}\) (creation of two SH photons from \(N\) after two interactions), the superposition \(\beta_{2}\) of the one-photon-creation process \(\hat{a}_{\rm SH}^{\dagger}\) creation of one SH photon from \(N\) after one interaction) with the annihilation processes \(\hat{a}_{\rm SH}^{\dagger}\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\dagger}\) and \(\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\dagger}\hat{a}_{\rm SH}^{\dagger}\) mentioned above. Note that all of these three terms provide the output state \(|N-2,1\rangle\). More explicitly, the superposition of \(\hat{a}_{\rm SH}^{\dagger}\) with \(\hat{a}_{\rm SH}^{\dagger}\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\dagger}\) corresponds to a term \((\tilde{\gamma}\hat{a}_{\rm SH})(\tilde{\gamma}^{3}\hat{a}_{\rm SH}^{\dagger} \hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\dagger})\) in the Taylor expansion of \(U\). In particular, from that expansion, it is straightforward that the factor \(\tilde{\gamma}\hat{a}_{\rm SH}\) contributes with an amplitude \(\sqrt{N!/(N-2)! }\), whereas \(\tilde{\gamma}^{3}\hat{a}_{\rm SH}^{\dagger}\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{ \dagger}\) contributes with \([N!/(N-2)!]^{3/2}\). Similarly, the superposition of \(\hat{a}_{\rm SH}^{\dagger}\) with \(\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\dagger}\hat{a}_{\rm SH}^{\dagger}\) is given by a term \((\tilde{\gamma}\hat{a}_{\rm SH})(\tilde{\gamma}^{3}\hat{a}_{\rm SH}\hat{a}_{ \rm SH}^{\dagger}\hat{a}_{\rm SH}^{\dagger})\), where \(\tilde{\gamma}^{3}\hat{a}_{\rm SH}\hat{a}_{\rm SH}^{\dagger}\hat{a}_{\rm SH}^{ \dagger}\) contributes with an amplitude \(\sqrt{N!/(N-2)!(N-2)!/(N-4)!}\). The role of these different contributions will be outlined in the following, in connection with the conversion probability and the evolution of the input light statistics.