  We propose a concrete experiment to probe the non-equilibrium local dynamics
of the one-dimensional Bose-Hubbard model using a trapped ion system consisting
of a linear chain of few Ba^+ ions prepared in a state of transverse motional
mode which corresponds to a fixed number of phonons per ion. These phonons are
well-known to be described by an effective Bose-Hubbard model. We propose a
protocol which leads to a sudden local sign reversal of the on-site interaction
strength of this Hubbard model at one of the sites and demonstrate that the
subsequent non-equilibrium dynamics of the model can be experimentally probed
by measuring the time-dependent phonon number in a specific motional state of
the Ba+ ions. We back our experimental proposal with exact numerical
calculation of the dynamics of a Bose-Hubbard model subsequent to a local
quench.
