  Directing indistinguishable photons from one input port into separate output
ports is a fundamental operation in quantum information processing. The
simplest scheme for achieving routing beyond random chance uses the photon
blockade effect of a two-level emitter. But this approach is limited by a
time-energy uncertainty relation. We show that a linear optical unitary
transformation applied after the atom enables splitting efficiencies that
exceed this time-energy limit. We show that the linear optical unitary improves
the splitting efficiency from 67\% to 82\% for unentangled photon inputs, and
from 77\% to 90\% for entangled photon inputs. We then optimize the temporal
mode profile of the entangled photon wavefunction to attain the optimal
splitting efficiency of 92\%, a significant improvement over previous limits
derived using a two-level atom alone. These results provide a path towards
optimizing single photon nonlinearities and engineering programmable and robust
photon-photon interactions for practical, high-fidelity quantum operations.
