  Bragg interferometers, operating using pseudospin-1/2 systems composed of two
momentum states, have become a mature technology for precision measurements.
State-of-the-art Bragg interferometers are rapidly surpassing technical
limitations and are soon expected to operate near the projection noise limit
set by uncorrelated atoms. Despite the use of large numbers of atoms, their
operation is governed by single-atom physics. Motivated by recent proposals and
demonstrations of Raman gravimeters in cavities, we propose a scheme to squeeze
directly on momentum states for surpassing the projection noise limit in Bragg
interferometers. In our modeling, we consider the unique issues that arise when
a spin squeezing protocol is applied to momentum pseudospins. Specifically, we
study the effects of the momentum width of the atomic cloud and the coupling to
momentum states outside the pseudospin manifold, as these atoms interact via a
mode of the cavity. We show that appreciable levels of spin squeezing can be
demonstrated in suitable parameter regimes in spite of these complications.
Using this setting, we show how beyond mean-field techniques developed for spin
systems can be adapted to study the dynamics of momentum states of interacting
atoms. Our scheme promises to be feasible using current technology and is
experimentally attractive because it requires no additional setup beyond what
will be required to operate Bragg interferometers in cavities.
