The paper discusses the preparation of many-body quantum states and non-equilibrium phases in driven open quantum systems using quantum reservoir engineering. Specifically, it focuses on a driven dissipative Bose Einstein Condensate (BEC) of bosons and paired fermions, where atoms in an optical lattice are coupled to a bath of Bogoliubov excitations via local dissipation. The authors show that for non-interacting atoms, the system can be driven into a pure state with long-range order. Weak interactions lead to a weakly mixed state, which in 3D results in a depletion of the condensate, while in 1D and 2D exhibits properties similar to a Luttinger liquid or a Kosterlitz-Thouless critical phase at finite temperature. The paper also discusses the preparation of an $\eta$-condensate, an excited eigenstate of a Hubbard Hamiltonian, which exhibits superfluidity and non-decaying off-diagonal long-range order in any spatial dimension. The authors provide a detailed theoretical analysis and numerical simulations to support their findings, demonstrating the potential of this approach for preparing complex quantum states in cold atomic systems.The paper discusses the preparation of many-body quantum states and non-equilibrium phases in driven open quantum systems using quantum reservoir engineering. Specifically, it focuses on a driven dissipative Bose Einstein Condensate (BEC) of bosons and paired fermions, where atoms in an optical lattice are coupled to a bath of Bogoliubov excitations via local dissipation. The authors show that for non-interacting atoms, the system can be driven into a pure state with long-range order. Weak interactions lead to a weakly mixed state, which in 3D results in a depletion of the condensate, while in 1D and 2D exhibits properties similar to a Luttinger liquid or a Kosterlitz-Thouless critical phase at finite temperature. The paper also discusses the preparation of an $\eta$-condensate, an excited eigenstate of a Hubbard Hamiltonian, which exhibits superfluidity and non-decaying off-diagonal long-range order in any spatial dimension. The authors provide a detailed theoretical analysis and numerical simulations to support their findings, demonstrating the potential of this approach for preparing complex quantum states in cold atomic systems.