  Light and matter can now interact in a regime where their coupling is
stronger than their bare energies. This deep-strong coupling (DSC) regime of
quantum electrodynamics promises to challenge many conventional assumptions
about the physics of light and matter. Here, we show how light and matter
interactions in this regime give rise to electromagnetic nonlinearities
dramatically different from those of naturally existing materials. Excitations
in the DSC regime act as photons with a linear energy spectrum up to a critical
excitation number, after which, the system suddenly becomes strongly
anharmonic, thus acting as an effective intensity-dependent nonlinearity of an
extremely high order. We show that this behavior allows for N-photon blockade
(with $N \gg 1$), enabling qualitatively new kinds of quantum light sources.
For example, this nonlinearity forms the basis for a new type of gain medium,
which when integrated into a laser or maser, produces large Fock states (rather
than coherent states). Such Fock states could in principle have photon numbers
orders of magnitude larger than any realized previously, and would be protected
from dissipation by a new type of equilibrium between nonlinear gain and linear
loss. We discuss paths to experimental realization of the effects described
here.
