  We propose a design for a superconducting nonlinear interferometer operating
at microwave frequencies which allows the measurement of the optical
nonlinearity \eta, with a precision which scales better than the
Heisenberg-like limit as \delta \eta similar to R^{-3/2}, with R the
quantification of resources. By designing the nonlinear optical element to
possess physically moving parts we are able to use the superconducting
nonlinear interferometer to measure the physical displacement r, of the moving
parts to a spatial precision of \delta (rt) on the order of 10^{-21}m/Hz
