Fluctuations of atomic energy levels due to axion dark matter

The amplitude of the pseudoscalar (axion) or scalar field fluctuates on a time scale of order of million field oscillation periods which is a typical coherence time in the virialized axion galactic dark matter halo model. This causes fluctuations of frequencies of atomic clocks on the same time scale. We show that this effect may be employed to search for the axion and scalar field dark matter with atomic and nuclear clocks. We re-purpose the results of the atomic clocks experiments comparing the variations of frequencies of hyperfine transitions in Rb and Cs atoms as well as in hydrogen atom vs cavity frequency fluctuations, and extract new limits on the axion coupling constant $f_a$ for masses in the range $2.4\times 10^{-17}\text{ eV}\lesssim m \lesssim 10^{-13}\text{ eV}$. We also show that similar energy shifts arise in the second-order perturbation theory with linear in the pseudoscalar field interaction. These shifts may be potentially measured with nuclear clocks based on the low-energy transition in $^{229}$Th nucleus. We propose a procedure which could, in principle, help determine the axion mass if the axion dark matter signal is present in experimental data sets.