  We present an experimental study of cavity assisted Rydberg atom
electromagnetically induced transparency (EIT) using a high-finesse optical
cavity ($F \sim 28000$). Rydberg atoms are excited via a two-photon transition
in a ladder-type EIT configuration. A three-peak structure of the cavity
transmission spectrum is observed when Rydberg EIT is generated inside the
cavity. The two symmetrically spaced side peaks are caused by bright-state
polaritons, while the central peak corresponds to a dark-state polariton.
Anti-crossing phenomenon and the effects of mirror adsorbate electric fields
are studied under different experimental conditions. We determine a lower bound
on the coherence time for the system of $7.26 \pm 0.06 \,\mu$s, most likely
limited by laser dephasing. The cavity-Rydberg EIT system can be useful for
single photon generation using the Rydberg blockade effect, studying many-body
physics, and generating novel quantum states amongst many other applications.
