  Strong light-matter coupling within electromagnetic environments provides a
promising path to modify and control chemical and physical processes. The
origin of the enhancement of nonlinear optical processes such as
second-harmonic and third-harmonic generation (SHG and THG) due to strong
light-matter coupling is attributed to distinct physical effects which
questions the relevance of strong coupling in these processes. In this work, we
leverage a first-principles approach to investigate the origins of the
experimentally observed enhancement of resonant SHG and THG under strong
light-matter coupling. We find that the enhancement of the nonlinear conversion
efficiency has its origins in a modification of the associated nonlinear
optical susceptibilities as polaritonic resonances emerge in the nonlinear
spectrum. Further, we find that the nonlinear conversion efficiency can be
tuned by increasing the light-matter coupling strength. Finally, we provide a
general framework to compute the harmonic generation spectra from the
displacement field as opposed to the standard approach which computes the
harmonic spectrum from the matter-only induced polarization. Our results
address a key debate in the field, and pave the way for predicting and
understanding quantum nonlinear optical phenomena in strongly coupled
light-matter systems.
