  Synthetic gauge fields have recently emerged, arising in the context of
quantum simulations, topological matter, and the protected transportation of
excitations against defects. For example, an ultracold atom experiences a
light-induced effective magnetic field when tunnelling in an optical lattice,
and offering a platform to simulate the quantum Hall effect and topological
insulators. Similarly, the magnetic field associated with photon transport
between sites has been demonstrated in a coupled resonator array. Here, we
report the first experimental demonstration of a synthetic gauge field in the
virtual dimension of bosonic modes in a single optomechanical resonator. By
employing degenerate clockwise (CW) and counter-clockwise (CCW) optical modes
and a mechanical mode, a controllable synthetic gauge field is realized by
tuning the phase of the driving lasers. The non-reciprocal conversion between
the three modes is realized for different synthetic magnetic fluxes. As a
proof-of-principle demonstration, we also show the dynamics of the system under
a fast-varying synthetic gauge field. Our demonstration not only provides a
versatile and controllable platform for studying synthetic gauge fields in high
dimensions but also enables an exploration of ultra-fast gauge field tuning
with a large dynamic range, which is restricted for a magnetic field.
