  Using two crystals for spontaneous parametric down-conversion in a parallel
setup, we observe two-photon interference with high visibility. The high
visibility is consistent with complementarity and the absence of which-path
information. The observations are explained as effects of entanglement or
equivalently in terms of interfering probability amplitudes, and also by the
calculation of a second-order field correlation function in the Heisenberg
picture. The latter approach brings out explicitly the role of the vacuum
fields in the down-conversion at the crystals and in the photon coincidence
counting. For comparison we show that the Hong-Ou-Mandel dip can be explained
by the same approach in which the role of the vacuum signal and idler fields,
as opposed to entanglement involving vacuum states, is emphasized.
