We propose a technique to couple the position operator of a nano mechanical
resonator to a SQUID device by modulating its magnetic flux bias. By tuning the
magnetic field properly, either linear or quadratic couplings can be realized,
with a discretely adjustable coupling strength. This provides a way to realize
coherent nonlinear effects in a nano mechanical resonator by coupling it to a
Josephson quantum circuit. As an example, we show how squeezing of the nano
mechanical resonator state can be realized with this technique. We also propose
a simple method to measure the uncertainty in the position of the nano
mechanical resonator without quantum state tomography
