Cooling and entanglement of multimode graphene resonators via vacuum fluctuations

Sympathetic laser cooling of a single mode graphene membrane coupled to an atomic cloud interacting via Casimir-Polder forces has been recently proposed. Here, we extend this study to the effect of secondary graphene membrane whose frequency may be far or close to resonance. We show that if the two mechanical modes are close together, it is possible to simultaneously cool both modes. Conversely, if the two frequencies are set far apart, the secondary mode does not affect the cooling of the first one. We also study the entanglement properties of the steady-state using the logarithmic negativity. We show how stationary mechanical entanglement between two graphene sheets can be generated by means of vacuum fluctuations. Moreover, we find that, within feasible experimental parameters, large steady-state acoustomechanical entanglement, i.e. entanglement between the phononic and mechanical mode, $E_N \approx 5$, can be generated.