  We solve the dynamics of an ensemble of interacting rotors coupled to two
leads at different chemical potential letting a current flow through the system
and driving it out of equilibrium. We show that at low temperature the
coarsening phase persists under the voltage drop up to a critical value of the
applied potential that depends on the characteristics of the electron
reservoirs. We discuss the properties of the critical surface in the
temperature, voltage, strength of quantum fluctuations and coupling to the bath
phase diagram. We analyze the coarsening regime finding, in particular, which
features are essentially quantum mechanical and which are basically classical
in nature. We demonstrate that the system evolves via the growth of a coherence
length with the same time-dependence as in the classical limit, $R(t) \simeq
t^{1/2}$ -- the scalar curvature driven universality class. We obtain the
scaling function of the correlation function at late epochs in the coarsening
regime and we prove that it coincides with the classical one once a prefactor
that encodes the dependence on all the parameters is factorized. We derive a
generic formula for the current flowing through the system and we show that,
for this model, it rapidly approaches a constant that we compute.
