Triphoton generation near atomic resonance via SSWM: Harmonic expansion for accurate optical response

Quantum correlations of time-frequency-entangled photon pairs generated via parametric processes are critically influenced by both the linear and nonlinear optical responses of the medium. This sensitivity is especially significant in schemes utilizing atomic ensembles with well-defined energy level structures near resonance. However, conventional theoretical approaches often fall short in accurately calculating the optical responses--particularly when a single atomic transition is simultaneously driven by multiple light fields with (significantly) different intensities. To address this limitation, we generalize the harmonic expansion method originally introduced by Wen for biphoton generation near atomic resonance. As a case study, we apply this generalized approach to the reliable direct generation of time-energy-entangled W-state triphotons via spontaneous six-wave mixing in a five-level asymmetric-M atomic system. Our results demonstrate the method's superior accuracy and self-consistency, offering clear advantages over traditional calculation techniques.