  We investigate the origin of imperfections in the fidelity of a two-photon
controlled-phase gate based on two-level-emitter non-linearities. We focus on a
passive system that operates without external modulations to enhance its
performance. We demonstrate that the fidelity of the gate is limited by
opposing requirements on the input pulse width for one- and two-photon
scattering events. For one-photon scattering, the spectral pulse width must be
narrow compared to the emitter linewidth, while two-photon scattering processes
require the pulse width and emitter linewidth to be comparable. We find that
these opposing requirements limit the maximum fidelity of the two-photon
controlled-phase gate for Gaussian photon pulses to 84%.
