  Nonlinear light-matter interactions in structured materials are the source of
exciting properties and enable vanguard applications in photonics. However, the
magnitude of nonlinear effects is generally small, thus requiring high optical
intensities for their manifestation at the nanoscale. Here, we reveal a large
nonlinear response of monolayer hexagonal boron nitride (hBN) in the
mid-infrared phonon-polariton region, triggered by the strongly anharmonic
potential associated with atomic vibrations in this material. We present robust
first-principles theory predicting a threshold light field $\sim40\,$MV/m to
produce order-unity effects in Kerr nonlinearities and harmonic generation,
which are made possible by a combination of the long lifetimes exhibited by
optical phonons and the strongly asymmetric landscape of the configuration
energy in hBN. We further foresee polariton blockade at the few-quanta level in
nanometer-sized structures. In addition, by mixing static and optical fields,
the strong nonlinear response of monolayer hBN gives rise to substantial
frequency shifts of optical phonon modes, exceeding their spectral width for
in-plane DC fields that are attainable using lateral gating technology. We
therefore predict a practical scheme for electrical tunability of the
vibrational modes with potential interest in mid-infrared optoelectronics. The
strong nonlinear response, low damping, and robustness of hBN polaritons set
the stage for the development of applications in light modulation, sensing, and
metrology, while triggering the search for intense vibrational nonlinear
response in other ionic materials.
