  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 transmission 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.
