Momentum based entanglement via phase modulation of a cavity

Matter-wave interferometry creates an important interface between quantum mechanics and relativity. Using a light-matter interactions, one can manipulate each atom into a superposition of two momentum states which can be subsequently treated as a pseudo-two level system, whereupon either momentum acquires a phase due to the differing kinetic energy. Through this phase, acceleration may be encoded and measured. Recently, an experiment demonstrated measurements of gravity below the standard quantum limit using quantum non-demolition measurements on hyperfine levels coupled to different momentum states. We introduce a method which similarly creates useful entanglement for matter-wave interferometry below the standard quantum limit, but with no need for coupling to an electronic degree of freedom. This is achieved via the atoms self-interacting through a momentum dependent phase modulation on a vertical cavity in the presence of gravity. The self-interaction leads to entanglement on momentum states which can created near cavity resonances without corresponding increases to spontaneous or collective decay, because the dynamics are decoupled from any electronic degrees of freedom.