  We study the effects of dissipation and time-independent nonequilibrium drive
on an open superconducting graphene. In particular, we investigate how
dissipation and nonequilibrium effects modify the semi-metal-BCS quantum phase
transition that occurs at half-filling in equilibrium graphene with attractive
interactions. Our system consists of a graphene sheet sandwiched by two
semi-infinite three-dimensional Fermi liquid reservoirs, which act both as a
particle pump/sink and a source of decoherence. A steady-state charge current
is established in the system by equilibrating the two reservoirs at different,
but constant, chemical potentials. The nonequilibrium BCS superconductivity in
graphene is formulated using the Keldysh path integral formalism, and we obtain
generalized gap and number density equations valid for both zero and finite
voltages. The behaviour of the gap is discussed as a function of both
attractive interaction strength and electron densities for various
graphene-reservoir couplings and voltages. We discuss how tracing out the
dissipative environment (with or without voltage) leads to decoherence of
Cooper pairs in the graphene sheet, hence to a general suppression of the gap
order parameter at all densities. For weak enough attractive interactions we
show that the gap vanishes even for electron densities away from half-filling,
and illustrate the possibility of a dissipation-induced metal-superconductor
quantum phase transition. We find that the application of small voltages does
not alter the essential features of the gap as compared to the case when the
system is subject to dissipation alone (i.e. zero voltage).
