  We present the design, fabrication, and characterization of a planar silicon
photonic crystal cavity in which large position-squared optomechanical coupling
is realized. The device consists of a double-slotted photonic crystal structure
in which motion of a central beam mode couples to two high-Q optical modes
localized around each slot. Electrostatic tuning of the structure is used to
controllably hybridize the optical modes into supermodes which couple in a
quadratic fashion to the motion of the beam. From independent measurements of
the anti-crossing of the optical modes and of the optical spring effect, the
position-squared vacuum coupling rate is measured to be as large as 245 Hz to
the fundamental in-plane mechanical resonance of the structure at 8.7MHz, which
in displacement units corresponds to a coupling coefficient of 1 THz/nm$^2$.
This level of position-squared coupling is approximately five orders of
magnitude larger than in conventional Fabry-Perot cavity systems.
