Collective spontaneous emission from a line of atoms

We study collective spontaneous emission from a linear array of N two-state atoms using quantum trajectory theory and without an a priori single-mode assumption. Assuming a fully excited initial state, we calculate the angular distribution of the $k\mathrm{th}$ emitted photon, $k=1,\dots ,N.\mathrm{}$ We investigate the evolution of the distribution from a dipole radiation pattern for the first photon emission to a distribution characteristic of directional superradiance. The formalism is developed around an unravelling of the master equation in terms of source-mode quantum jumps. Exact calculations for 11 and fewer atoms do not show directional superradiance, but are characterized by delayed (subradiant) photon emissions directed along the axis of the linear array. A modified boson approximation is made to treat the many-atom case, where it is found that strong directional superradiance occurs for a few hundred atoms; the decay of subradiant excitations is preserved in the tail of the superradiant pulse.