  We investigate the two-photon scattering properties of a Jaynes-Cummings (JC)
nonlinearity consisting of a two-level system (qubit) interacting with a single
mode cavity, which is coupled to two waveguides, each containing a single
incident photon wave packet initially. In this scattering setup, we study the
interplay between the Hong-Ou-Mandel effect arising due to quantum interference
and effective photon-photon interactions induced by the presence of the qubit.
We calculate the two-photon scattering matrix of this system analytically and
identify signatures of interference and interaction in the second order auto-
and cross-correlation functions of the scattered photons. In the dispersive
regime, when qubit and cavity are far detuned from each other, we find that the
JC nonlinearity can be used as an almost linear, in-situ tunable beam splitter
giving rise to ideal Hong-Ou-Mandel interference, generating a highly
path-entangled two-photon NOON state of the scattered photons. The latter
manifests itself in strongly suppressed waveguide cross-correlations and
Poissonian photon number statistics in each waveguide. If the two-level system
and the cavity are on resonance, the JC nonlinearity strongly modifies the
ideal HOM conditions leading to a smaller degree of path entanglement and
sub-poissonian photon number statistics. In the latter regime, we find that
photon blockade is associated with bunched auto-correlations in both
waveguides, while a two-polariton resonance can lead to bunched as well as
anti-bunched correlations.
