  We revisit the superstrong coupling regime of multi-mode cavity quantum
electrodynamics (QED), defined to occur when the frequency of vacuum Rabi
oscillations between the qubit and the nearest cavity mode exceeds the cavity's
free spectral range. A novel prediction is made that the cavity's linear
spectrum, measured in the vanishing power limit, can acquire an intricate fine
structure associated with the qubit-induced cascades of coherent single-photon
down-conversion processes. This many-body effect is hard to capture by a
brute-force numerics and it is sensitive to the light-matter coupling
parameters both in the infra-red and the ultra-violet limits. We focused at the
example case of a superconducting fluxonium qubit coupled to a long
transmission line section. The conversion rate in such a circuit QED setup can
readily exceed a few MHz, which is plenty to overcome the usual decoherence
processes. Analytical calculations were made possible by an unconventional
gauge choice, in which the qubit circuit interacts with radiation via the
flux/charge variable in the low-/high-frequency limits, respectively. Our
prediction of the fine spectral structure lays the foundation for the "strong
down-conversion" regime in quantum optics, in which a single photon excited in
a non-linear medium spontaneously down-converts faster than it is absorbed.
