  Interference is fundamental to wave dynamics and quantum mechanics. The
quantum wave properties of particles are exploited in metrology using atom
interferometers, allowing for high-precision inertia measurements [1, 2].
Furthermore, the state-of-the-art time standard is based on an interferometric
technique known as Ramsey spectroscopy. However, the precision of an
interferometer is limited by classical statistics owing to the finite number of
atoms used to deduce the quantity of interest [3]. Here we show experimentally
that the classical precision limit can be surpassed using nonlinear atom
interferometry with a Bose-Einstein condensate. Controlled interactions between
the atoms lead to non-classical entangled states within the interferometer;
this represents an alternative approach to the use of non-classical input
states [4-8]. Extending quantum interferometry [9] to the regime of large atom
number, we find that phase sensitivity is enhanced by 15 per cent relative to
that in an ideal classical measurement. Our nonlinear atomic beam splitter
follows the "one-axis-twisting" scheme [10] and implements interaction control
using a narrow Feshbach resonance. We perform noise tomography of the quantum
state within the interferometer and detect coherent spin squeezing with a
squeezing factor of -8.2dB [11-15]. The results provide information on the
many-particle quantum state, and imply the entanglement of 170 atoms [16].
