  We consider a nanomechanical analogue of a nonlinear interferometer,
consisting of two parallel, flexural nanomechanical resonators, each with an
intrinsic Duffing nonlinearity and with a switchable beamsplitter-like coupling
between them. We calculate the precision with which the strength of the
nonlinearity can be estimated and show that it scales as $1/n^{3/2}$, where $n$
is the mean phonon number of the initial state. This result holds even in the
presence of dissipation, but assumes the ability to make measurements of the
quadrature components of the nanoresonators.
