  We study the collective dynamics of two distant magnets coherently coupled by
acoustic phonons that are transmitted through an intercalated crystal. By
tuning the ferromagnetic resonances of the two magnets to an acoustic resonance
of the crystal, we control a coherent three levels system. We show that the
parity of the phonon mode governs the nature of the indirect coupling between
the magnets: the resonances with odd / even phonon modes correspond to
out-of-phase / in-phase lattice displacements at the magnets, leading to bright
/ dark states in response to uniform microwave magnetic fields, respectively.
The sample is a tri-layer garnet consisting of two thin magnetic layers
epitaxially grown on both sides of a half-millimeter thick non-magnetic single
crystal. In spite of the relatively weak magneto-elastic interaction, the long
lifetimes of the magnon and phonon modes in the sample are the key to unveil
this long range strong coupling. This demonstrates that garnets are a great
platform to study multi-partite hybridization process between magnon and
phonons at microwave frequencies.
