  Particle-wave duality enables the construction of interferometers for matter
waves, which complement optical interferometers in precision measurement
devices. This requires the development of atom-optics analogs to beam
splitters, phase shifters, and recombiners. Integrating these elements into a
single device has been a long-standing goal. Here we demonstrate a full
Mach-Zehnder sequence with trapped Bose-Einstein condensates (BECs) confined on
an atom chip. Particle interactions in our BEC matter waves lead to a
non-linearity, absent in photon optics. We exploit it to generate a
non-classical state having reduced number fluctuations inside the
interferometer. Making use of spatially separated wave packets, a controlled
phase shift is applied and read out by a non-adiabatic matter-wave recombiner.
We demonstrate coherence times a factor of three beyond what is expected for
coherent states, highlighting the potential of entanglement as a resource for
metrology. Our results pave the way for integrated quantum-enhanced matter-wave
sensors.
