  We study coherent phonon oscillations and tunneling between two coupled
nonlinear nanomechanical resonators. We show that the coupling between two
nanomechanical resonators creates an effective phonon Josephson junction which
exhibits two different dynamical behaviors: Josephson oscillation (phonon-Rabi
oscillation) and macroscopic self-trapping (phonon blockade). Self-trapping
originates from mechanical nonlinearities, meaning that when the nonlinearity
exceeds its critical value, the energy exchange between the two resonators is
suppressed, and phonon-Josephson oscillations between them are completely
blocked. An effective classical Hamiltonian for the phonon Josephson junction
is derived and its mean-field dynamics is studied in phase space. Finally, we
study the phonon-phonon coherence quantified by the mean fringe visibility, and
show that the interaction between the two resonators may lead to the loss of
coherence in the phononic junction.
