  The axion-gluon coupling is the defining feature of the QCD axion. This
feature induces additional and qualitatively different interactions of the
axion with standard model particles -- quadratic couplings. Previously,
hadronic quadratic couplings have been studied and experimental implications
have been explored especially in the context of atomic spectroscopy and
interferometry. We investigate additional quadratic couplings to the
electromagnetic field and electron mass. These electromagnetic quadratic
couplings are generated at the loop level from threshold corrections and are
expected to be present in the absence of fine-tuning. While they are generally
loop-suppressed compared to the hadronic ones, they open up new ways to search
for the QCD axion, for instance via optical atomic clocks. Moreover, due to the
velocity spread of the dark matter field, the quadratic nature of the coupling
leads to low-frequency fluctuations in any detector setup. These distinctive
low-frequency fluctuations offer a way to search for heavier axions. We provide
an analytic expression for the power spectral density of this low-frequency
background and briefly discuss experimental strategies for a low-frequency
background search.
