The equations of electrodynamics are altered in the presence of a classical coherent axion dark matter background field, changing the dispersion relation for electromagnetic waves. Careful measurements of the frequency stability in sensitive atomic clocks could in principle provide evidence for such a background for $f_a ge 10^7$ GeV. Turning on a background magnetic field might enhance these effects in a controllable way, and interferometric measurements might also be useful for probing the time-varying photon dispersion relation that results from a coherent cosmic axion background.
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