Phase transitions to absorbing states are among the simplest examples of critical phenomena out of equilibrium. The characteristic feature of these models is the presence of a fluctuationless configuration which the dynamics cannot leave, which has proved a rather stringent requirement in experiments. Recently, a proposal to seek such transitions in highly tuneable systems of cold atomic gases offers to probe this physics and, at the same time, to investigate the robustness of these transitions to quantum coherent effects. Here we specifically focus on the interplay between classical and quantum fluctuations in a simple driven open quantum model which, in the classical limit, reproduces a contact process, which is known to undergo a continuous transition in the "directed percolation" universality class. We derive an effective long-wavelength field theory for the present class of open spin systems and show that, due to quantum fluctuations, the nature of the transition changes from second to first order, passing through a bicritical point which appears to belong instead to the "tricritical directed percolation" class.
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