The work is devoted to the antibunching and sub-Poissonian photon statistics in the degenerate frequency up-conversion (FUC) process using a short-time approximation in the Heisenberg picture. We analyse how pump photon numbers, interaction times, and coupling strengths affect nonclassicality, distinguishing between first- and second-order Hamiltonian interactions. Results show that first-order interactions lead to stronger nonclassical effects, though only second-order interactions enable antibunching in the harmonic mode due to higher-order pump contributions. Photon antibunching intensity increases with pump strength and shorter interaction times. In the first-order coupling interactions, the harmonic mode does not exhibit antibunching because a coherent or vacuum pump induces photon clustering. However, second-order interactions, where higher powers of the pump field contribute significantly, facilitate antibunching in the harmonic mode. Third-order antibunching exhibits the strongest nonclassical behaviour within the observed effects, followed by second- and first-order antibunching. It is more apparent that increased pump intensity and reduced interaction time both strengthen antibunching and sub-Poissonian photon statistics. While the pump mode exhibits clear antibunching, the harmonic mode displays milder nonclassicality. These results reveal the higher-order transitions of antibunching as an inherent quantum feature of light, valuable for quantum communication and single-photon sources with a probabilistic destination.
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