  Rydberg-aided atomic electrometry using alkali-metal atoms is gaining
increased research interest for detecting external electric fields. However,
the inability of Rydberg probes to detect phase is a serious impediment to
their realistic deployment. In this paper, we derive a novel phase retrieval
algorithm for use in a phased array or synthetic aperture applications where
only measurements of electric field intensity are possible at each spatial
sample. These array configurations exist if a Rydberg atom probe is used in
place of an antenna. We employ three-stage alternating projections to solve the
resulting optimization problem. Our numerical experiments demonstrate the
effectiveness of the proposed algorithm in terms of beamformed array output.
