  An open quantum system, whose time evolution is governed by a master
equation, can be driven into a given pure quantum state by an appropriate
design of the system-reservoir coupling. This points out a route towards
preparing many body states and non-equilibrium quantum phases by quantum
reservoir engineering. Here we discuss in detail the example of a \emph{driven
dissipative Bose Einstein Condensate} of bosons and of paired fermions, where
atoms in an optical lattice are coupled to a bath of Bogoliubov excitations via
the atomic current representing \emph{local dissipation}. In the absence of
interactions the lattice gas is driven into a pure state with long range order.
Weak interactions lead to a weakly mixed state, which in 3D can be understood
as a depletion of the condensate, and in 1D and 2D exhibits properties
reminiscent of a Luttinger liquid or a Kosterlitz-Thouless critical phase at
finite temperature, with the role of the ``finite temperature'' played by the
interactions.
