  Axion-like particles (ALPs) with couplings to electromagnetism have long been
postulated as extensions to the Standard Model. String theory predicts an
"axiverse" of many light axions, some of which may make up the dark matter in
the universe and/or solve the strong CP problem. We propose a new experiment
using superconducting radiofrequency (SRF) cavities which is sensitive to light
ALPs independent of their contribution to the cosmic dark matter density.
Off-shell ALPs will source cubic nonlinearities in Maxwell's equations, such
that if a SRF cavity is pumped at frequencies $\omega_1$ and $\omega_2$, in the
presence of ALPs there will be power in modes with frequencies $2\omega_1 \pm
\omega_2$. Our setup is similar in spirit to light-shining-through-walls (LSW)
experiments, but because the pump field itself effectively converts the ALP
back to photons inside a single cavity, our sensitivity scales differently with
the strength of the external fields, allowing for superior reach as compared to
experiments like OSQAR while utilizing current technology. Furthermore, a
well-defined program of increasing sensitivity has a guaranteed physics result:
the first observation of the Euler-Heisenberg term of low-energy QED at
energies below the electron mass. We discuss how the ALP contribution may be
separated from the QED contribution by a suitable choice of pump modes and
cavity geometry, and conclude by describing the ultimate sensitivity of our
proposed program of experiments to ALPs.
