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Light and matter can now interact in a regime where their coupling isstronger than their bare energies. This deep-strong coupling (DSC) regime ofquantum electrodynamics promises to challenge many conventional assumptionsabout the physics of light and matter. Here, we show how light and matterinteractions in this regime give rise to electromagnetic nonlinearitiesdramatically different from those of naturally existing materials. Excitationsin the DSC regime act as photons with a linear energy spectrum up to a criticalexcitation number, after which, the system suddenly becomes stronglyanharmonic, thus acting as an effective intensity-dependent nonlinearity of anextremely 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 (ratherthan coherent states). Such Fock states could in principle have photon numbersorders of magnitude larger than any realized previously, and would be protectedfrom dissipation by a new type of equilibrium between nonlinear gain and linearloss. We discuss paths to experimental realization of the effects describedhere.

Nonperturbative electromagnetic nonlinearities, n-photon reflectors, and Fock-state lasers based on deep-strong coupling of light and matter