  Cavity magnonics, which studies the interaction of light with magnetic
systems in a cavity, is a promising platform for quantum transducers and
quantum memories. At microwave frequencies, the coupling between a cavity
photon and a magnon, the quasi-particle of a spin wave excitation, is a
consequence of the Zeeman interaction between the cavity's magnetic field and
the magnet's macroscopic spin. For each photon/magnon interaction, a coupling
phase factor exists, but is often neglected in simple systems. However, in
"loop-coupled" systems, where there are at least as many couplings as modes,
the coupling phases become relevant for the physics and lead to synthetic gauge
fields. We present experimental evidence of the existence of such coupling
phases by considering two spheres made of Yttrium-Iron-Garnet and two different
re-entrant cavities. We predict numerically the values of the coupling phases,
and we find good agreement between theory and the experimental data. These
results show that in cavity magnonics, one can engineer synthetic gauge fields,
which can be useful for cavity-mediated coupling and engineering dark mode
physics.
