  A single phonon in a nonlinear nanomechanical resonator (NAMR) can block the
excitation of a second phonon [Phys. Rev. A 82, 032101 (2010)]. This
intrinsically quantum effect is called phonon blockade, and is an analog of
Coulomb blockade and photon blockade. Here we predict tunable multiphonon
blockade in coupled nonlinear NAMRs, where nonlinearity is induced by two-level
systems (TLSs) assuming dispersive (far off-resonance) interactions.
Specifically, we derive an effective Kerr-type interaction in a hybrid system
consisting of two nonlinearly-interacting NAMRs coupled to two TLSs and driven
by classical fields. The interaction between a given NAMR and a TLS is
described by a Jaynes-Cummings-like model. We show that by properly tuning the
frequency of the driving fields one can induce various types of phonon
blockade, corresponding to the entangled phonon states of either two qubits,
qutrit and quartit, or two qudits. Thus, a $k$-phonon Fock state (with
$k=1,2,3$) can impede the excitation of more phonons in a given NAMR, which we
interpret as a $k$-phonon blockade (or, equivalently, phonon tunneling). Our
results can be explained in terms of resonant transitions in the Fock space and
via phase-space interference using the $s$-parametrized Cahill-Glauber
quasiprobability distributions including the Wigner function. We study the
nonclassicality, entanglement, and dimensionality of the blockaded phonon
states during both dynamics and in the stationary limits.
