arXiv2026
This article is a theoretical and quantitative exploration of the limit regarding the pump intensity between the two regimes of spontaneous-parametric down-conversion (SPDC) as well as of four-wave-mixing (FWM) in the framework of a semi-classical model and analytical calculations. A dimensionless parameter has been defined at this limit, corresponding to the photon-pairs flux per frequency unit: it has been found equal to 0.369. The ratio between the electric field of the generated photons and the quantum fluctuations of vacuum calculated at the limit is equal to 1.718. These quantitative results confirm that below the limit, the pump photon splitting leading to photon-pairs can be considered as spontaneous, i.e. mainly seeded by the quantum fluctuations of vacuum, while it is stimulated by the generated signal and idler photons above the limit, which corresponds to an optical parametric amplification regime. Our calculations also show that this limit can be easily reached in the case of SPDC according to the typical values of non-linearities and available crystal dimensions. In the case of FWM, it would be only possible in kilometric optical fibers. This corpus is a useful tool box for designing further quantum experiments performed from either side of the limit, as well as at the limit it-self where the influence of the quantum fluctuations of vacuum and of the generated photons should have the same weight. Furthermore, the quantum significance of the numerical values of the two criteria defined here remains to be established, which should motivate future theoretical quantum studies.