Collective magnetic splitting in single-photon superradiance

In an ensemble of identical atoms, cooperative effects like sub- or superradiance may alter the decay rates and the energy of specific transitions may be shifted from the single-atom value by the so-called collective Lamb shift. While such effects in ensembles of two-level systems are by now well understood, realistic multi-level systems are more difficult to handle. In this work we show that in a system of atoms or nuclei under the action of an external magnetic field, the collective contribution to the level shifts can amount to seizable deviations from the single-atom Zeeman or magnetic hyperfine splitting picture. We develop a formalism to describe single-photon superradiance in multi-level systems in the small sample limit and quantify the parameter regime for which the collective Lamb shift leads to measurable deviations in the magnetic-field-induced splitting. In particular, we show that this effect should be observable in the nuclear magnetic hyperfine splitting in M\"ossbauer nuclei embedded in thin-film x-ray cavities.