Nonlinear coupling of nano mechanical resonators to Josephson quantum circuits

We study 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 allowes us to realize nonlinear dynamics on the nano mechanical resonator by coupling it to a Josephson quantum circuits. As an example we show how squeezing of the nano mechanical resonator 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.