Recoil-induced errors and their correction in photon-mediated entanglement between atom qubits

Photonically-interconnected matter qubit systems have wide-ranging applications across quantum science and technology, with entanglement between distant qubits serving as a universal resource. While state-of-the-art heralded entanglement generation performance thus far has been achieved in trapped atomic systems modelled as stationary emitters, the improvements to fidelities and generation rates demanded by large-scale applications require taking into account their motional degrees of freedom. Here, we derive the effects of atomic motion on spontaneous emission coupled into arbitrary optical modes, and study the implications for commonly-used atom-atom entanglement protocols. We arrive at a coherent physical picture in the form of "kick operators" associated with each instant in the photonic wavepackets, which also suggests a method to mitigate motional errors by disentangling qubit and motion post-herald. This proposed correction technique removes overheads associated with the thermal motion of atoms, and may greatly increase entanglement rates in long-distance quantum network links by allowing single-photon-based protocols to be used in the high-fidelity regime.