Scattering of light dark matter in atomic clocks

We present a detailed analysis of the effect of light Dark Matter (DM) on atomic clocks, for the case where DM mass and density are such that occupation numbers are low and DM must be considered as particles scattering off the atoms, rather than a classical field. We show that the resulting atomic clock frequency shifts are first order in the scattering amplitudes, and particularly suited to constrain DM models in the regime where the DM mass $m_\chi \ll$ GeV. We provide some rough order of magnitude estimates of sensitivity that can be confronted to any DM model that allows for non zero differential scattering amplitudes of the two atomic states involved in the clock.