Quantifying the breakdown of the rotating-wave approximation in single-photon superradiance

We study quantitatively the breakdown of the rotating-wave approximation when calculating collective light emission by quantum emitters, in particular in the weak-excitation limit. Our starting point is a known multiple-scattering formalism where the full light-matter interaction leads to induced inter-emitter interactions described by the classical Green function of inhomogeneous dielectric media. When making the RWA in the light-matter interaction, however, these induced interactions differ from the classical Green function, and for free space we find a reduction of the interatomic interaction strength by up to a factor of two. By contrast, for the corresponding scalar model the relative RWA error for the inter-emitter interaction even diverges in the near field. For two identical emitters, the errors due to the RWA in collective light emission will show up in the emission spectrum, but not in the sub- and superradiant decay rates. In case of two non-identical emitters, also the collective emission rates will differ by making the RWA. For three or more identical emitters, the RWA errors in the interatomic interaction in general affect both the collective emission spectra and the collective decay rates. Ring configurations with discrete rotational symmetry are an interesting exception. Interestingly, the maximal errors in the collective decay rates due to making the RWA do not occur in the extreme near-field limit.