  Entanglement is a fundamental resource that allows quantum sensors to surpass
the standard quantum limit set by the quantum collapse of independent atoms.
Collective cavity-QED systems have succeeded in generating large amounts of
directly observed entanglement involving the internal degrees of freedom of
laser-cooled atomic ensembles. Here we demonstrate cavity-QED entanglement of
external degrees of freedom to realize a matter-wave interferometer of 700
atoms in which each individual atom falls freely under gravity and
simultaneously traverses two paths through space while also entangled with the
other atoms. We demonstrate both quantum non-demolition measurements and
cavity-mediated spin interactions for generating squeezed momentum states with
directly observed metrological gain $3.4^{+1.1}_{-0.9}$ dB and
$2.5^{+0.6}_{-0.6}$ dB below the standard quantum limit respectively. An
entangled state is for the first time successfully injected into a Mach-Zehnder
light-pulse interferometer with $1.7^{+0.5}_{-0.5}$ dB of directly observed
metrological enhancement. Reducing the fundamental quantum source of
imprecision provides a new resource that can be exploited to directly enhance
measurement precision, bandwidth, and accuracy or operate at reduced size.
These results also open a new path for combining particle delocalization and
entanglement for inertial sensors, searches for new physics, particles, and
fields, future advanced gravitational wave detectors, and accessing beyond
mean-field quantum many-body physics.
