  In recent years, remarkable progress has been made towards encoding and
processing quantum information in the large Hilbert space of bosonic modes.
Mechanical resonators are of great interest for this purpose, since they
confine many high quality factor modes into a small volume and can be easily
integrated with many different quantum systems. An important yet challenging
task is to create direct interactions between different mechanical modes. Here
we demonstrate an in-situ tunable beam-splitter-type interaction between
several mechanical modes of a high-overtone bulk acoustic wave resonator. The
engineered interaction is mediated by a parametrically driven superconducting
transmon qubit, and we show that it can be tailored to couple pairs or triplets
of phononic modes. Furthermore, we use this interaction to demonstrate the
Hong-Ou-Mandel effect between phonons. Our results lay the foundations for
using phononic systems as quantum memories and platforms for quantum
simulations.
