Entanglement as Complex Astigmatism: Geometry of the Multimode Gaussian Ray

Recent semiclassical approaches suggest that many optical phenomena traditionally attributed to quantization can be described by wave mechanics interacting with quantized matter. However, multipartite entanglement—exemplified by the Hong–Ou–Mandel (HOM) effect—remains a significant hurdle for such geometric theories. In this work, we propose a resolution by extending the "Complex Ray" (Thawed Gaussian) formalism to N-mode systems. We demonstrate that the dynamics of a multimode quantum system are governed exactly by a complex matrix trajectory Z(t) obeying generalized Newtonian mechanics. Within this framework, we identify quantum entanglement not as non-local influence, but as complex astigmatism: the emergence of off-diagonal imaginary correlations in the matrix Z that violate established Gaussian separability bounds (e.g., Simon's criterion). We rigorously derive the HOM dip by projecting the complex trajectory onto the antisymmetric normal mode. We show that symplectic area conservation forces this mode into the vacuum state at perfect temporal overlap, forbidding coincidence detection. This extends the "Conic Trinity" of Gaussian dynamics to the many-body regime, interpreting entanglement as a rigid geometric coupling in the complex phase space.