Directed spontaneous emission from an extended ensemble of N atoms: timing is everything.

TLDR: It is shown that a single photon absorbed by the N atoms will be followed by spontaneous emission in the same direction, and phase matched emission is found when one photon is absorbed by N atoms followed by two-photon down-conversion.

Abstract: A collection of $N$ static atoms is fixed in a crystal at a low temperature and prepared by a pulse of incident radiation of wave vector ${\stackrel{\ensuremath{\rightarrow}}{k}}_{0}$. The $N$ atoms are well described by an entangled Dicke-like state, in which each atom carries a characteristic phase factor $\mathrm{exp}(i{\stackrel{\ensuremath{\rightarrow}}{k}}_{0}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{r}}_{j})$, where ${\stackrel{\ensuremath{\rightarrow}}{r}}_{j}$ is the atomic position in the crystal. It is shown that a single photon absorbed by the $N$ atoms will be followed by spontaneous emission in the same direction. Furthermore, phase matched emission is found when one photon is absorbed by $N$ atoms followed by two-photon down-conversion.