  Optical microcavities and metallic nanostructures have been shown to
significantly modulate the dynamics and spectroscopic response of molecular
systems. We present a study of the nonlinear optics of a model consisting of
$N$ anharmonic multilevel systems (e.g., Morse oscillators) undergoing
collective strong coupling with a resonant infrared microcavity. We find that,
under experimentally accessible conditions, molecular systems in microcavities
may have nonlinear phenomena significantly intensified due to the high quality
of polariton resonances and the enhanced microcavity electromagnetic energy
density relative to free space. Particularly large enhancement of multiphoton
absorption happens when multipolariton states are resonant with bare molecule
multiphoton transitions. In particular, our model predicts two-photon
absorption cross section enhancements by several orders of magnitude relative
to free space when the Rabi splitting $\Omega_R$ is approximately equal to the
molecular anharmonic shift $2\Delta$. Our results provide rough upper bounds to
resonant nonlinear response enhancement factors as relaxation to dark states is
treated phenomenologically. Notably, ensembles of two-level systems undergoing
strong coupling with a cavity (described by the Tavis-Cummings model) show no
such optical nonlinearity enhancements, highlighting the rich phenomenology
afforded by multilevel anharmonic systems. Similar conclusions are expected to
hold for excitonic systems that share features with our model (e.g., molecular
dyes with accessible S_0 -> S_1 -> S_2 transitions) and strongly interact with
a UV-visible cavity.
