  Strong vibrational coupling has been realized in a variety of mechanical
systems from cavity optomechanics to electromechanics.$^{1, 2, 3, 4, 5}$ It is
an essential requirement for enabling quantum control over the vibrational
states.$^{6, 7, 8, 9, 10, 11}$ The majority of the mechanical systems that have
been studied to date are vibrational resonances of dielectric or semiconductor
nanomaterials coupled to optical modes.$^{12, 13, 14, 15}$ While there are
fewer studies of coupling between two mechanical modes,$^{3, 9}$ particularly,
there have been no experimental observation of strong coupling of the
ultra-high frequency acoustic modes of plasmonic nanostructures, due to the
rapid energy dissipation in these systems. Here we realized strong vibrational
coupling in ultra-high frequency plasmonic nanoresonators by increasing the
vibrational quality factors by an order of magnitude. This is achieved through
blocking an energy dissipation pathway in the form of out-going acoustic waves.
We achieved the highest frequency quality factor products of
$\mathbf{f}\times\mathbf{Q}=1.0\times10^{13}$ Hz for the fundamental mechanical
modes in room temperature plasmonic nanoresonators reported to date, which
exceeds the value of $0.1\times10^{13}$ Hz required for ground state cooling.
Avoided crossing were observed between the vibrational modes of two plasmonic
nanoresonators with a coupling rate of $\mathbf{g}=7.5\pm 1.2$ GHz, an order of
magnitude larger than the dissipation rates. The intermodal strong coupling was
consistent with theoretical calculations using a coupled oscillator model. Our
results expanded the strong coupling systems for mechanical resonators and
enabled a platform for future observation and control of the quantum behavior
of phonon modes in metallic nanoparticles.
