  Nonlinear optical processes associated with even-order nonlinear
susceptibilities are critical for both classical and quantum technologies.
Inversion symmetry, however, prevents nonlinear optical responses mediated by
even-order susceptibilities in several material systems pertinent for
applications in nanophotonics. Here, we demonstrate induced nonlinear optical
processes, namely second- and fourth-harmonic generation that are naturally
forbidden in an inversion symmetric system, by strongly coupling to a photon
mode of a high-Q optical cavity. As an illustrative system with an inversion
symmetry, we consider a semiconductor quantum ring of GaAs that features a
single effective electron. For the coupled system, we control the inversion
symmetry breaking by changing the light-matter coupling strength which at the
same time allows to tune the nonlinear conversion efficiency. We find that the
harmonic generation yield can be significantly increased by increasing the
light-matter coupling strength in an experimentally feasible way. In the
few-photon limit where the incident pump field is a coherent state with just a
few photons, we find that the harmonic conversion efficiency is increased for
strong coupling as opposed to using intense pump fields. This new approach is
applicable to a wide variety of centrosymmetric systems as the symmetry
breaking rest on the properties of the photonic environment used to achieve
strong light-matter interaction. Our work constitutes a step forward in the
direction of realizing physically forbidden nonlinear optical processes in
centrosymmetric materials widely adopted for applications in integrated
photonics.
