  Chip-scale multimode optomechanical systems have unique benefits for sensing,
metrology and quantum technologies relative to their single-mode counterparts.
Slot-mode optomechanical crystals enable sideband resolution and large
optomechanical couplings of a single optical cavity to two microwave-frequency
mechanical modes. Still, previous implementations have been limited to nanobeam
geometries, whose effective quantum cooperativity at ultralow temperatures is
limited by their low thermal conductance. In this work, we design and
experimentally demonstrate a two-dimensional mechanical-optical-mechanical
(MOM) platform that dispersively couples a slow-light slot-guided
photonic-crystal waveguide mode and two slow-sound $\sim 7$ GHz phononic wire
modes localized in physically distinct regions. We first demonstrate
optomechanical interactions in long waveguide sections, unveiling acoustic
group velocities below 800 m/s, and then move on to mode-gap adiabatic
heterostructure cavities with a tailored mechanical frequency difference.
Through optomechanical spectroscopy, we demonstrate optical quality factors $Q
\sim 10^5$, vacuum optomechanical coupling rates, $g_o/2\pi$, of 1.5 MHz and
dynamical backaction effects beyond the single-mode picture. At larger power
and adequate laser-cavity detuning, we demonstrate regenerative optomechanical
oscillations involving a single mechanical mode, extending to both mechanical
modes through modulation of the input laser drive at their frequency
difference. This work constitutes an important advance towards engineering MOM
systems with nearly degenerate mechanical modes as part of hybrid multipartite
quantum systems.
