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.

Figure 1

(Color online) Schematic diagram of the top view of the device considered here. It consists of a rectangular-shaped dc-SQUID and two mechanical resonators shown in blue on the left and right sides. These two opposite segments of the dc-SQUID are freely suspended and are treated as two nanomechanical resonators (NAMRs). The dotted lines indicate the equilibrium positions of the left and right NAMRs. Each “×” in the top segment of the loop represents a Josephson junction. $X_L$ $\left(X_R\right)$ is the displacement of the center of the left (right) resonator. The two NAMRs could be located sufficiently “far apart” by using a SQUID with an appropriate aspect ratio. This could provide EPR-type correlations on the two NAMRs located sufficiently “far apart” for wide-enough SQUIDs.

Figure 2

(Color online) The potential energy $U(\phi ,{\Phi }_X)$ (scaled by $E_J$) of the dc-SQUID as a function of the phase variable $\phi$ and the magnetic flux ${\Phi }_X$ originating from the displacements of the two NAMRs. The red color represents higher potential energy $U$, and the blue color represents lower potential energy. Both $\phi$ and ${\Phi }_X$ are shown in units of $\pi ∕{\Phi }_0$. In (a)–(c), the bias magnetic flux ${\Phi }_b$ threading the loop of the dc-SQUID is set at $2n{\Phi }_0$, $(2n+\frac{1}{4}){\Phi }_0$, and $(2n+\frac{1}{2}){\Phi }_0$, respectively; and the bias currents are all set at $I_b=0.1I_c$. In (d), ${\Phi }_b=2n{\Phi }_0$ and $I_b=0.5I_c$.

Figure 3

(Color online) The squared variance of the collective coordinates $X_T$ of the two NAMRs as the function of damping rates ${\kappa }_L$ and ${\kappa }_R$ of the left and right NAMRs, both ${\kappa }_L$ and ${\kappa }_R$ are normalized by the effective coupling constant $\xi$. Here, ${\delta }_X$ is the zero fluctuation of the collective coordinates of the two NAMRs.