Entanglement of nanoelectromechanical oscillators by Cooper-pair tunneling

We demonstrate that entanglement of two macroscopic nanoelectromechanical resonators—coupled to each other via a common detector, a tunnel junction—can be generated by running a current through the device. We introduce a setup that overcomes generic limitations of proposals suggesting to entangle systems via a shared bath. At the heart of the proposal is an Andreev entangler setup, representing an experimentally feasible way of entangling two nanomechanical oscillators. Instead of relying on the coherence of a (fermionic) bath, in the Andreev entangler setup, a split Cooper pair that coherently tunnels to each oscillator mediates their coupling and thereby induces entanglement between them. Since entanglement is in each instance generated by Markovian and non-Markovian noisy open system dynamics in an out-of-equilibrium situation, we argue that the present scheme also opens up perspectives to observe dissipation-driven entanglement in a condensed-matter system.

Figure 1

(a) Schematic setup of a bipartite continuous variable quantum system, realized as two nanoelectromechanical oscillators (green) in a tunnel junction setup which share common fermionic reservoirs (blue). Yellow lines indicate tunnel junctions. (b) Two nanomechanical oscillators are effectively coupled in an Andreev entangler setup due to a Cooper pair that is split. Each of the two electrons of the Cooper pair is assumed to tunnel from the superconductor (orange) onto a different oscillator.

Figure 2

Logarithmic negativity for $L_c=0.3\Omega$ (red) and $L_c=0.6\Omega$ (blue). Here, the temperature is $T=0$, $V=0.1\Omega$, and ${\gamma }_x=0.1\sqrt{m\Omega }$. As can be seen, the qualitative dependence on the parameter $L_c$ is rather small.

Figure 3

Logarithmic negativity for $L_c=0.3$. Here, the electronic temperature is $T=0$, $V=0.1\Omega$, ${\gamma }_0=0.3$, and ${\gamma }_x=0.05\sqrt{m\Omega }$. The inset shows the influence of a finite bosonic heat bath with inverse temperature ${\beta }_{\mathrm{Bos}}$ as described in the text.