Quantum entanglement of nanocantilevers

We propose a scheme to entangle two mechanical nanocantilevers through indirect interactions mediated by a gas of ultracold atoms. We envisage a system of nanocantilevers magnetically coupled to a Bose-Einstein condensate of atoms and focus on studying the dark states of the system. These dark states are entangled states of the two nanocantilevers, with no coupling to the atomic condensate. In the absence of dissipation, the degree of entanglement is found to oscillate with time, while if dissipation is included, the system is found to relax to a statistical mixture of dark states which remains time independent until the inevitable thermal dephasing destroys the nanocantilever coherence. This opens up the possibility of achieving long-lived entangled nanocantilever states.