Nonlinear coupling of nanomechanical resonators to josephson quantum circuits.

We propose a technique to couple the position operator of a nanomechanical 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 nanomechanical resonator by coupling it to a Josephson quantum circuit. As an example, we show how squeezing of the nanomechanical resonator state can be realized with this technique. We also propose a simple method to measure the uncertainty in the position of the nanomechanical resonator without quantum state tomography.