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 efficiency 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 nonlinear pi phase-shift operation.