  We report on criteria to detect entanglement between the light modes of two
crossed optical cavities by analyzing the transverse deflection patterns of an
atomic beam. The photon exchange between the modes and the atoms occurs around
the overlapping nodes of associated standing waves, which generates the
two-dimensional (2D) version of the Optical Stern-Gerlach (OSG) effect. In this
optical cross-cavity setup, we show that the discrete signatures of the fields
states, left in the momentum distribution of the deflected atoms, may reveal
entanglement for a certain class of two-mode states. For a single photon, we
present the possibility of quantifying entanglement by the rotation of the
momentum distribution. For a larger number of photons, we demonstrate that
quantum interference precludes the population of specific momentum states
revealing maximum entanglement between the light modes.
