  Achieving strong coupling between light and matter is usually a challenge in
Cavity Quantum Electrodynamics (cQED), especially in solid state systems. For
this reason is useful taking advantage of alternative approaches to reach this
regime, and then, generate reliable quantum polaritons. In this work we study a
system composed of a quantized single mode of a mechanical resonator
interacting linearly with both a single mode nanocavity and a two-level single
quantum dot. In particular, we focus on the behavior of the indirect
light-matter interaction when the phonon mode interfaces both parts. By
diagonalization of the Hamiltonian and computing the density matrix in a master
equation approach, we evidence several features of strong coupling between
photons in optical cavities and excitons in a quantum dot. For large energy
detuning between the cavity and the mechanical resonator it is obtained a
phonon-dispersive effective Hamiltonian which is able to retrieve much of the
physics of the conventional Jaynes-Cummings model (JCM). In order to
characterize this mediated coupling, we make a quantitative comparison between
both models and analyze light-matter entanglement and purity of the system
leading to similar results in cQED.
