  We perform the experimental demonstration of the method proposed in [Phys.
Rev. A 91, 063615 (2015)] to extract the differential phase in dual atom
interferometers. From a single magneto-optical trap, we generate two atomic
sources, vertically separated and free-falling synchronously, with the help of
an accelerated lattice. We drive simultaneous Raman interferometers onto the
two sources, and use the correlation with the vibration signal measured by a
seismometer to extract the phase of each interferometer. We demonstrate an
optimal sensitivity of the extracted differential phase between the two
interferometers, free from vibration noise and limited by detection noise, when
the two interferometers are in phase.
