Two-mode squeezed states and entangled states of two mechanical resonators

We study a device consisting of a dc superconducting quantum interference device (SQUID) with two sections of its loop acting as two mechanical resonators. An analog of the parametric down-conversion process in quantum optics can be realized with this device. We show that a two-mode squeezed state can be generated for two overdamped mechanical resonators, where the damping constants of the two mechanical resonators are larger than the coupling strengths between the dc-SQUID and the two mechanical resonators. Thus we show that entangled states of these two mechanical resonators can be generated.