We analytically treat the scattering of two counter-propagating photons on a two-level emitter embedded in an optical waveguide. We find that the non-linearity of the emitter can give rise to significant pulse-dependent directional correlations in th
e scattered photonic state, which could be quantified via a reduction in coincident clicks in a Hong-Ou-Mandel measurement setup, analogous to a linear beam splitter. Changes to the spectra and phase of the scattered photons, however, would lead to reduced interference with other photons when implemented in a larger optical circuit. We introduce suitable fidelity measures which account for these changes, and find that high values can still be achieved even when accounting for all properties of the scattered photonic state.
We develop a wavefunction approach to describe the scattering of two photons on a quantum emitter embedded in a one-dimensional waveguide. Our method allows us to calculate the exact dynamics of the complete system at all times, as well as the transm
ission properties of the emitter. We show that the non-linearity of the emitter with respect to incoming photons depends strongly on the emitter excitation and the spectral shape of the incoming pulses, resulting in transmission of the photons which depends crucially on their separation and width. In addition, for counter-propagating pulses, we analyze the induced level of quantum correlations in the scattered state, and we show that the emitter behaves as a non-linear beam-splitter when the spectral width of the photon pulses is similar to the emitter decay rate.
