  The strong coupling between elementary excitations of the electromagnetic
field (photons) and quantized mechanical vibrations (phonons) produces hybrid
quasi-particle states, known as phonon-polaritons. Their typical signature is
the avoided crossing between the eigenfrequencies of the coupled system, as
paradigmatically illustrated by the Jaynes-Cummings Hamiltonian, and observed
in quantum electrodynamics experiments where cavity photons are coupled to
atoms, ions, excitons, spin ensambles and superconducting qubits. In this work,
we demonstrate the generation of phonon-polaritons in the quantum motion of an
optically-levitated nanosphere. The particle is trapped in high vacuum by an
optical tweezer and strongly coupled to a single cavity mode by coherent
scattering of the tweezer photons. The two-dimensional motion splits into two
nearly-degenerate components that, together with the optical cavity mode,
define an optomechanical system with three degrees-of-freedom. As such, when
entering the strong coupling regime, we observe hybrid light-mechanical states
with a dispersion law typical of tripartite quantum systems. Remarkably, the
independent components of motion here identify a physical vibration direction
on a plane that, similarly to the polarization of light, confers a vectorial
nature to the polariton field. Our results pave the way to novel protocols for
quantum information transfer between photonic and phononic components and
represent a key-step towards the demonstration of optomechanical entangled
states at room temperature.
