NOON States of Nine Quantized Vibrations in Two Radial Modes of a Trapped Ion.

We develop a deterministic method to generate and verify arbitrarily high NOON states of quantized vibrations (phonons), through the coupling to the internal state. We experimentally create the entangled states up to N=9 phonons in two vibrational modes of a single trapped ^{171}Yb^{+} ion. We observe an increasing phase sensitivity of the generated NOON state as the number of phonons N increases and obtain the fidelity from the contrast of the phase interference and the population of the phonon states through the two-mode projective measurement, which are significantly above the classical bound. We also measure the quantum Fisher information of the generated state and observe Heisenberg scaling in the lower bounds of phase sensitivity as N increases. Our scheme is generic and applicable to other photonic or phononic systems such as circuit QED systems or nanomechanical oscillators, which have Jaynes-Cummings-type of interactions.