A two-level emitter chirally coupled to a waveguide imprints a conditional phase of π on a two-photon pulse, a key ingredient in quantum photonic neural network architectures. Until now it was unknown whether that phase survives the pure dephasing of a solid-state emitter, since no two-photon scattering treatment included it. A new study derives the dephased two-photon amplitude in closed form and validates it against an independent Fock-state master equation.

The central result is that the conditional phase remains pinned at exactly π for every dephasing rate examined: dephasing destroys the amplitude carrying the nonlinear phase rather than rotating it. However, the one- and two-photon channels are not attenuated equally, so dephasing does not simply scale the node down but reweights it between photon-number sectors. It is an error that no phase setting can train away.

The study further shows that the standard factorization of two-photon correlators, exact for arbitrary loss, fails at first order in γ_φ. The approximation discards between 16 % and 81 % of photon pairs for dephasing ratios γ_φ/γ between 0.1 and 1, a range covering typical solid-state emitter conditions.

The implications bear on the design of photonic nonlinear gates for quantum neural networks: the π phase is preserved, but the probability that the process occurs degrades asymmetrically across photon numbers. The calculations provide a closed-form way to predict that loss and indicate that simulations assuming correlator factorization overestimate gate performance. The work paves the way for explicitly including pure dephasing in the design of integrated quantum photonics with solid-state emitters.