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We develop a wavefunction approach to describe the scattering of two photonson a quantum emitter embedded in a one-dimensional waveguide. Our method allowsus to calculate the exact dynamics of the complete system at all times, as wellas the transmission properties of the emitter. We show that the non-linearityof the emitter with respect to incoming photons depends strongly on the emitterexcitation and the spectral shape of the incoming pulses, resulting intransmission of the photons which depends crucially on their separation andwidth. In addition, for counter-propagating pulses, we analyze the inducedlevel of quantum correlations in the scattered state, and we show that theemitter behaves as a non-linear beam-splitter when the spectral width of thephoton pulses is similar to the emitter decay rate.

Scattering of two photons on a quantum emitter in a one-dimensional waveguide: Exact dynamics and induced correlations