  Ultra-cold atomic systems are among the most promising platforms that have
the potential to shed light on the complex behavior of many-body quantum
systems. One prominent example is the case of a dense ensemble illuminated by a
strong coherent drive while interacting via dipole-dipole interactions. Despite
being subjected to intense investigations, this system retains many open
questions. A recent experiment carried out in a pencil-shaped geometry reported
measurements that seemed consistent with the emergence of strong collective
effects in the form of a ``superradiant'' phase transition in free space, when
looking at the light emission properties in the forward direction. Motivated by
the experimental observations, we carry out a systematic theoretical analysis
of the system's steady-state properties as a function of the driving strength
and atom number, $N$. We observe signatures of collective effects in the weak
drive regime, which disappear with increasing drive strength as the system
evolves into a single-particle-like mixed state comprised of randomly aligned
dipoles. Although the steady-state features some similarities to the reported
superradiant to normal non-equilibrium transition, also known as cooperative
resonance fluorescence, we observe significant qualitative and quantitative
differences, including a different scaling of the critical drive parameter
(from $N$ to $\sqrt{N}$). We validate the applicability of a mean-field
treatment to capture the steady-state dynamics under currently accessible
conditions. Furthermore, we develop a simple theoretical model that explains
the scaling properties by accounting for interaction-induced inhomogeneous
effects and spontaneous emission, which are intrinsic features of interacting
disordered arrays in free space.
