  The interaction between anharmonic quantum emitters (e.g., molecular
vibrations) and confined electromagnetic fields gives rise to quantum states
with optical and chemical properties that are different from those of their
precursors. The exploration of these properties has been typically constrained
to the first excitation manifold, the harmonic approximation, ensembles of
two-level systems [Tavis-Cummings (TC) model], or the anharmonic
single-molecule case. The present work studies, for the first time, a
collective ensemble of identical multi-level anharmonic emitters and their
dipolar interaction with a photonic cavity mode. The permutational properties
of the system allow identifying symmetry classified submanifolds in the energy
spectrum. Notably, in this approach, the number of particles, typically in the
order of several millions, becomes only a parameter from the operational
standpoint, and the size of the dimension of the matrices to diagonalize is
independent of it. The formalism capabilities are illustrated by showing the
energy spectrum structure, up to the third excitation manifold, and the
calculation of the photon contents as a permutationally invariant quantity.
Emphasis is placed on (a) the collective (superradiant) scalings of
light-matter couplings and the various submanifolds of dark (subradiant) states
with no counterpart in the single-molecule case, as well as (b) the delocalized
modes containing more than one excitation per molecule with no equivalent in
the TC model. We expect these findings to be applicable in the study of
non-linear spectroscopy and chemistry of polaritons.
