Effective Models for Atom-interferometry in Quantized Electromagnetic Fields in Free Space and in Cavities

Quantum resources in form of entangled atoms promise to boost the sensitivity of atom interferometers beyond the regime achievable with classical sensor devices. Future dark matter or gravitational wave detectors, tests of the equivalence principle and even simple inertial sensors based on atom interferometry in combination with entanglement are already envisioned to apply these techniques to reach their projected potential. However, proper characterization and interpretation of these devices necessitates a full description including the entanglement dynamics between the light and matter subsystems during typical experimental sequences combined from free propgation and the light-matter interactions. Beginning with a few-mode model of the optical field, coupled to a few-level atom with quantized motional degrees of freedom we show: (i) how effective Jaynes-Cummings-Paul like multi-mode Rabi models with center-of-mass motion can be derived for multi-photon transitions, (ii) a master equation characterizing imperfect beamsplitters can be derived and (iii) our approach and the resulting models are not limited to atom interferometry but have possible applications in cavity optomechanics or ion traps. Lastly, we highlight how and under which appropriate classical limit the usual theory of atom interferometry arises.