  Realizing quantum mechanical behavior in micro- and nanomechanical resonators
has attracted continuous research effort. One of the ways for observing quantum
nature of mechanical objects is via the mechanism of phonon blockade. Here, we
show that phonon blockade could be achieved in a system of two weakly nonlinear
mechanical resonators coupled by a Coulomb interaction. The optimal blockade
arises as a result of the destructive quantum interference between paths
leading to two-phonon excitation. It is observed that, in comparison to a
single drive applied on one mechanical resonator, driving both the resonators
can be beneficial in many aspects; such as, in terms of the temperature
sensitivity of phonon blockade and also with regard to the tunability, by
controlling the amplitude and the phase of the second drive externally. We also
show that via a radiation pressure induced coupling in an optomechanical
cavity, phonon correlations can be measured indirectly in terms of photon
correlations of the cavity mode.
