  We explore the nonlinear response of tailor-cut light-matter hybrid states in
a novel regime, where both the Rabi frequency induced by a coherent driving
field and the vacuum Rabi frequency set by a cavity field are comparable to the
carrier frequency of light. In this previously unexplored strong-field limit of
ultrastrong coupling, subcycle pump-probe and multi-wave mixing nonlinearities
between different polariton states violate the normal-mode approximation while
ultrastrong coupling remains intact, as confirmed by our mean-field model. We
expect such custom-cut nonlinearities of hybridized elementary excitations to
facilitate non-classical light sources, quantum phase transitions, or cavity
chemistry with virtual photons.
