  Strong coupling between light and matter is possible with a variety of
organic materials. In contrast to the simpler inorganic case, organic materials
often have a complicated spectrum, with vibrationally dressed electronic
transitions. Strong coupling to light competes with this vibrational dressing,
and if strong enough, can suppress the entanglement between electronic and
vibrational degrees of freedom. By exploiting symmetries, we can perform exact
numerical diagonalization to find the polaritonic states for intermediate
numbers of molecules, and use these to define and validate accurate expressions
for the lower polariton states and strong-coupling spectrum in the
thermodynamic limit. Using this approach, we find that vibrational decoupling
occurs as a sharp transition above a critical matter-light coupling strength.
We also demonstrate how the polariton spectrum evolves with the number of
molecules, recovering classical linear optics results only at large $N$.
