  The observation of quantized nanomechanical oscillations by detecting
femtometer-scale displacements is a significant challenge for experimentalists.
We propose that phonon blockade can serve as a signature of quantum behavior in
nanomechanical resonators. In analogy to photon blockade and Coulomb blockade
for electrons, the main idea for phonon blockade is that the second phonon
cannot be excited when there is one phonon in the nonlinear oscillator. To
realize phonon blockade, a superconducting quantum two-level system is coupled
to the nanomechanical resonator and is used to induce the phonon
self-interaction. Using Monte Carlo simulations, the dynamics of the induced
nonlinear oscillator is studied via the Cahill-Glauber $s$-parametrized
quasiprobability distributions. We show how the oscillation of the resonator
can occur in the quantum regime and demonstrate how the phonon blockade can be
observed with currently accessible experimental parameters.
