  Stochastic processes with absorbing states feature remarkable examples of
non-equilibrium universal phenomena. While a broad understanding has been
progressively established in the classical regime, relatively little is known
about the behavior of these non-equilibrium systems in the presence of quantum
fluctuations. Here we theoretically address such a scenario in an open quantum
spin model which in its classical limit undergoes a directed percolation phase
transition. By mapping the problem to a non-equilibrium field theory, we show
that the introduction of quantum fluctuations stemming from coherent, rather
than statistical, spin-flips alters the nature of the transition such that it
becomes first-order. In the intermediate regime, where classical and quantum
dynamics compete on equal terms, we highlight the presence of a bicritical
point with universal features different from the directed percolation class in
low dimension. We finally propose how this physics could be explored within
gases of interacting atoms excited to Rydberg states.
