  Photon-mediated interactions between atomic systems can arise via coupling to
a common electromagnetic mode or by quantum interference. Here, we probe the
role of coherence in cooperative emission arising from two distant but
indistinguishable solid-state emitters because of path erasure. The primary
signature of cooperative emission, the emergence of "bunching" at zero delay in
an intensity correlation experiment, is used to characterise the
indistinguishability of the emitters, their dephasing, and the degree of
correlation in the joint system that can be coherently controlled. In a stark
departure from a pair of uncorrelated emitters, in Hong-Ou-Mandel type
interference measurements we observe photon statistics from a pair of
indistinguishable emitters resembling that of a weak coherent state from an
attenuated laser. Our experiments establish techniques to control and
characterize cooperative behavior between matter qubits using the full quantum
optics toolbox, a key step toward realizing large-scale quantum photonic
networks.
