  Achieving cavity-optomechanical strong coupling with high-frequency phonons
provides a rich avenue for quantum technology development including quantum
state-transfer, memory, and transduction, as well as enabling several
fundamental studies of macroscopic phononic degrees-of-freedom. Reaching such
coupling with GHz mechanical modes however has proved challenging, with a
prominent hindrance being material- and surface-induced-optical absorption in
many materials. Here, we circumvent these challenges and report the observation
of optomechanical strong coupling to a high frequency (11 GHz) mechanical mode
of a fused-silica whispering-gallery microresonator via the electrostrictive
Brillouin interaction. Using an optical heterodyne detection scheme, the
anti-Stokes light backscattered from the resonator is measured and normal-mode
splitting and an avoided crossing are observed in the recorded spectra,
providing unambiguous signatures of strong coupling. The optomechanical
coupling rate reaches values as high as $G/2\pi = 39 \ \text{MHz}$ through the
use of an auxiliary pump resonance, where the coupling dominates both the
optical ($\kappa/2\pi = 3 \ \text{MHz}$) and the mechanical
($\gamma_\text{m}/2\pi = 21 \ \text{MHz}$) amplitude decay rates. Our findings
provide a promising new approach for optical quantum control using light and
sound.
