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The amplitude of the pseudoscalar (axion) or scalar field fluctuates on atime scale of order of million field oscillation periods which is a typicalcoherence time in the virialized axion galactic dark matter halo model. Thiscauses fluctuations of frequencies of atomic clocks on the same time scale. Weshow that this effect may be employed to search for the axion and scalar fielddark matter with atomic and nuclear clocks. We re-purpose the results of theatomic clocks experiments comparing the variations of frequencies of hyperfinetransitions in Rb and Cs atoms as well as in hydrogen atom vs cavity frequencyfluctuations, and extract new limits on the axion coupling constant $f_a$ formasses in the range $2.4\times 10^{-17}\text{ eV}\lesssim m \lesssim10^{-13}\text{ eV}$. We also show that similar energy shifts arise in thesecond-order perturbation theory with linear in the pseudoscalar fieldinteraction. These shifts may be potentially measured with nuclear clocks basedon the low-energy transition in $^{229}$Th nucleus. We propose a procedurewhich could, in principle, help determine the axion mass if the axion darkmatter signal is present in experimental data sets.

Fluctuations of atomic energy levels due to axion dark matter