  Photon-photon interactions are an essential requirement of quantum photonic
information processing. One way to generate these interactions is to utilize an
atom strongly coupled to an optical cavity. This system exhibits the photon
blockade effect which enables single photon switching and creation of
non-classical light. But the nonlinear effects enabled by this system suffer
from a fundamental time-bandwidth constraint. For the the simple case of
splitting an input pulse of two indistinguishable photons, this constraint
imposes a limit on the efficiency of routing photons to different output ports.
We show that this limit can be exceeded by combining the strongly-coupled atom
with linear optics. By optimizing the unitary of the linear optical
transformation, we achieve improved splitting efficiency for both un-entangled
and entangled photons. Our results suggest that it may be possible to improve
the efficiency of nonlinear optical processes at the single photon level by
making suitable use of linear optics. These results could have implications for
quantum information processing with photons.
