  Multi-party entangled states have important applications in quantum metrology
and quantum computation. Experimental preparation of large entangled state, in
particular, the NOON states, however, remains challenging as the particle
number $N$ increases. Here we develop a deterministic method to generate
arbitrarily high NOON states for phonons and experimentally create the states
up to $N=9$ phonons in two radial modes of a single trapped \Yb ion. We
demonstrate that the fidelity of the NOON states are significantly above the
classical limit by measuring the interference contrast and the population
through the projective phonon measurement of two motional modes. We also
measure the quantum Fisher information of the generated NOON state and observe
the Heisenberg scaling in the lower bounds of the phase sensitivity as the $N$
increases. Our scheme is generic and applicable to other photonic or phononic
systems.
