  Precise quantum control and measurement of several harmonic oscillators, such
as the modes of the electromagnetic field in a cavity or of mechanical motion,
are key for their use as quantum platforms. The motional modes of trapped ions
can be individually controlled and have good coherence properties. However,
achieving high-fidelity two-mode operations and nondestructive measurements of
the motional state has been challenging. Here we demonstrate the coherent
exchange of single motional quanta between spectrally separated harmonic
motional modes of a trapped-ion crystal. The timing, strength, and phase of the
coupling are controlled through an oscillating electric potential with suitable
spatial variation. Coupling rates that are much larger than decoherence rates
enable demonstrations of high fidelity quantum state transfer and beamsplitter
operations, entanglement of motional modes, and Hong-Ou-Mandel-type
interference. Additionally, we use the motional coupling to enable repeated
non-destructive projective measurement of a trapped-ion motional state. Our
work enhances the suitability of trapped-ion motion for continuous-variable
quantum computing and error correction and may provide opportunities to improve
the performance of motional cooling and motion-mediated entangling
interactions.
