  Optomechanical devices in which a flexible SiN membrane is placed inside an
optical cavity allow for very high finesse and mechanical quality factor in a
single device. They also provide fundamentally new functionality: the cavity
detuning can be a quadratic function of membrane position. This enables a
measurement of "position squared" ($x^2$) and in principle a QND phonon number
readout of the membrane. However, the readout achieved using a single
transverse cavity mode is not sensitive enough to observe quantum jumps between
phonon Fock states.
  Here we demonstrate an $x^2$-sensitivity that is orders of magnitude stronger
using two transverse cavity modes that are nearly degenerate. We derive a
first-order perturbation theory to describe the interactions between
nearly-degenerate cavity modes and achieve good agreement with our measurements
using realistic parameters. We also demonstrate theoretically that the
$x^2$-coupling should be easily tunable over a wide range.
