  Optomechanics offers a natural way to implement synthetic dynamical gauge
fields, leading to synthetic electric fields for phonons and, as a consequence,
to unidirectional light transport. Here we investigate the quantum dynamics of
synthetic gauge fields in the minimal setup of two optical modes coupled by
phonon-assisted tunneling where the phonon mode is undergoing
self-oscillations. We use the quantum van-der-Pol oscillator as the simplest
dynamical model for a mechanical self-oscillator that allows us to perform
quantum master equation simulations. We identify a single parameter, which
controls the strength of quantum fluctuations, enabling us to investigate the
classical-to-quantum crossover. We show that the generation of synthetic
electric fields is robust against noise and that it leads to unidirectional
transport of photons also in the quantum regime, albeit with a reduced
isolation ratio. Our study opens the path for studying dynamical gauge fields
in the quantum regime based on optomechanical arrays.
