Theory of cascade correlated emission from atom arrays

We present a theory of cascade emission from an array of N fixed, ``three-level'' atoms. The total angular momentum of the ground state of each atom is $J=0$, that of the intermediate state is $J=1$, and that of the upper state is $J=0$. The atoms are prepared in a spatially phased superposition of their ground and upper states, collectively sharing a single excitation. We calculate the time-integrated, joint probability distribution for radiation to be emitted in a given direction with a given polarization on each transition and use this joint probability distribution to calculate the time-integrated intensity emitted on each transition. The dipole-dipole interaction between the atoms is taken into account. Analytic expressions are obtained for two atoms and the calculation is then extended formally to an ensemble of N atoms at arbitrary positions. As expected, the radiation emitted on the upper transition is unpolarized and isotropic. However, somewhat surprisingly, we find that for excitation by counterpropagating fields and for nearly equal ground to intermediate and intermediate to upper level transition frequencies, the radiation on the lower transition is also isotropic and unpolarized, provided the atoms are prepared in a fully symmetric state. We also establish conditions for which there can be a phase-matched enhancement of the joint probability density for specific directions of emissions on the upper and lower transitions.