Deterministic multiphoton bundle emission via interference-interaction control

The controlled generation of nonclassical light beyond single photons remains a central challenge in quantum optics, due to the difficulty of enhancing multiphoton processes while suppressing lower-order excitations. Here we propose an interference-interaction-engineered scheme for programmable few-photon emission in a cavity-QED system of three atoms coupled to orthogonal cavity modes. By adiabatically eliminating an auxiliary Fabry-P\'erot cavity, we generate a tunable cavity-mediated spin-exchange interaction $\chi$, which, combined with a controllable geometric phase $\phi$, reshapes the many-body dressed-state spectrum. This interplay enables selective addressing of excitation manifolds ($N=1,2,3$), establishing a direct mapping between excitation structure and photon-emission channels. For $\phi=0$, constructive interference enhances the spectral anharmonicity of low-excitation manifolds, yielding tunable single- and two-photon emission associated with the $N=1$ and $N=2$ manifolds. In contrast, for $\phi=2\pi/3$, destructive interference suppresses lower-order excitation pathways and activates a resonant three-photon channel originating from the $N=3$ manifold. Importantly, the cavity-mediated interaction $\chi$ further enhances spectral separation between manifolds, enabling a substantial improvement in multiphoton purity while maintaining a sizable photon population. We demonstrate a three-order-of-magnitude enhancement in two-photon purity and more than two orders of magnitude improvement in three-photon emission. Our results establish a unified interference-interaction framework in which effective optical nonlinearities can be programmably engineered through phase and interaction, providing a scalable route toward high-purity multiphoton sources and programmable quantum photonic devices.