Optical coherence transfer in atom arrays

A $J=0$ to $J=1$ to $J=0$ ladder scheme can be used as a prototype for probing optical coherence transfer in an atom array. Two incident ${\ensuremath{\sigma}}_{\ensuremath{-}}$ polarized fields drive transitions from the ground state to the intermediate state and the intermediate state to the uppermost state in the atoms. The uppermost state is taken to be a long-lived Rydberg level. In the absence of any atom-atom interactions, excitation to the Rydberg level is not allowed, owing to the selection rules. However, dipole-dipole interactions can provide a transfer of the state amplitude associated with the $m=\ensuremath{-}1$ sublevel in one atom to the $m=+1$ sublevel in another atom. The overall process of field excitation and optical coherence transfer can then lead to an excitation of the Rydberg level. One can then apply a readout pulse to obtain phase-matched emission from the array. For a linear array of atoms, the phase-matched signal can increase more rapidly than ${N}^{2}$, where $N$ is the number of atoms in the array. For arrays that are invariant under a rotation $\ensuremath{\delta}\ensuremath{\phi}<\ensuremath{\pi}$ about the $z$ axis, however, it turns out that the total field intensity vanishes identically in the phase-matched direction. For such arrays, the Rydberg state population can serve as a measure of the optical coherence transfer.