Already in the early years of quantum mechanics,Kennard showed that the wave function of a particle of mass m exposed for a time T to a constant linear potential with the force F accumulates a phase scaling as $F^2T^3/m$. Being position-independent, this phase does not contribute to the phase shift (proportional to $FT^2/m$) of a conventional matter-wave interferometer used to measure a constant acceleration. More recently, several theoretical and experimental works have discussed this cubic phase
in the context of an interferometer. We present here our own unique realization of a $T^3$ matter-wave interferometer based on a Bose-Einstein Condensate (BEC) near an atom chip and utilizing the Stern-Gerlach effect in order to create a state-dependent force. We show that this configuration allows the interferometer's phase to be amplified unproportionally to the space-time area encompassed by the interferometer's arms, unlike other atom interferometers. This may enable enhanced sensitivities.