  We present here the performance of a simultaneous dual-species matter-wave
accelerometer for measuring the differential acceleration between two different
atomic species ($^{87}$Rb and $^{85}$Rb). We study the expression and the
extraction of the differential phase from the interferometer output. The
differential accelerometer reaches a short-term sensitivity of
$1.23\times10^{-7}g/\sqrt{Hz}$ limited by the detection noise and a resolution
of $2\times10^{-9}g$ after 11000 s, the highest reported thus far with a
dual-species atom interferometer to our knowledge. Thanks to the simultaneous
measurement, such resolution levels can still be achieved even with vibration
levels up to $3\times10^{-3}g$, corresponding to a common-mode vibration noise
rejection ratio of 94 dB (rejection factor of 50 000). These results prove the
ability of such atom sensors for realizing a quantum based test of the weak
equivalence principle (WEP) at a level of $\eta\sim10^{-9}$ even with high
vibration levels and a compact sensor.
