Efficiencies for the single mode operation of a quantum optical nonlinear shift gate

We investigate the single mode operation of a quantum optical nonlinear \pi phase shift gate implemented by a single two-level atom in one-dimensional free space. Since the single mode property of the input photons at the atom is not preserved in the interaction at the atom, we analyze the effeciency of single mode operation that can still be achieved. We show how the input pulse shape can be optimized to obtain high efficiencies for the nonlinear single mode operation. With this analysis, we obtain an optimal single mode transmittance per photon of 78% for the successful nonliner \pi phase shift operation