  Interfaces of light and matter serve as a platform for exciting many-body
physics and photonic quantum technologies. Due to the recent experimental
realization of atomic arrays at sub-wavelength spacings, collective interaction
effects such as superradiance have regained substantial interest. Their
analytical and numerical treatment is however quite challenging. Here we
develop a semiclassical approach to this problem that allows to describe the
coherent and dissipative many-body dynamics of interacting spins while taking
into account lowest-order quantum fluctuations. For this purpose we extend the
discrete truncated Wigner approximation, originally developed for unitarily
coupled spins, to include collective, dissipative spin processes by means of
truncated correspondence rules. This maps the dynamics of the atomic ensemble
onto a set of semiclassical, numerically inexpensive stochastic differential
equations. We benchmark our method with exact results for the case of Dicke
decay, which shows excellent agreement. For small arrays we compare to exact
simulations and a second order cumulant expansion, again showing good agreement
at early times and at moderate to strong driving. We conclude by studying the
radiative properties of a spatially extended three-dimensional, coherently
driven gas and compare the coherence of the emitted light to experimental
results.
