Theoretical study of early-time superradiance for atom clouds and arrays

We explore conditions for Dicke superradiance in a cloud of atoms by examining the Taylor series expansion of the photon emission rate at t = 0. By defining superradiance as an increasing photon emission rate for t similar to 0, we have calculated the conditions for superradiance for a variety of cases. We investigate superradiance as defined for photon emission into all angles as well as directional superradiance where the photon emission is only detected in a particular direction. Although all of the examples are for two-level atoms that are fully inverted at t = 0, we also give equations for partially inverted two-level atoms and for fully inverted multilevel atoms. We give an algorithm for efficiently evaluating these equations for atom arrays and determine superradiance conditions for large atom number.