  Phonon blockade is a purely quantum phenomenon, analogous to Coulomb and
photon blockades, in which a single phonon in an anharmonic mechanical
resonator can impede the excitation of a second phonon. We propose an
experimental method to realize phonon blockade in a driven harmonic
nanomechanical resonator coupled to a qubit, where the coupling is proportional
to the second-order nonlinear susceptibility $\chi^{(2)}$. This is in contrast
to the standard realizations of phonon and photon blockade effects in Kerr-type
$\chi^{(3)}$ nonlinear systems. The nonlinear coupling strength can be adjusted
conveniently by changing the coherent drive field.As an example, we apply this
model to predict and describe phonon blockade in a nanomechanical resonator
coupled to a Cooper-pair box (i.e., a charge qubit) with a linear longitudinal
coupling. By obtaining the solutions of the steady state for this composite
system, we give the conditions forobserving strong antibunching and
sub-Poissonian phonon-number statistics in this induced second-order nonlinear
system. Besides using the qubit to produce phonon blockade states, the qubit
itself can also be employed to detect blockade effects by measuring its states.
Numerical simulations indicate that the robustness of the phonon blockade, and
the sensitivity of detecting it, will benefit from this strong induced
nonlinear coupling.
