This paper explores the preparation of many-body quantum states and non-equilibrium phases in driven open quantum systems using cold atoms. The authors demonstrate how a driven dissipative Bose-Einstein condensate (BEC) of bosons and paired fermions can be stabilized by coupling atoms in an optical lattice to a bath of Bogoliubov excitations. The system is governed by a master equation, which describes the time evolution of the density matrix under the influence of both the Hamiltonian and dissipative terms. The key idea is to design the system-reservoir coupling such that the system evolves into a desired many-body quantum state, such as a pure state with long-range order or a state resembling a Luttinger liquid or Kosterlitz-Thouless critical phase at finite temperature. The paper discusses the dynamics of a lattice of N bosonic atoms in d dimensions, where the atoms are coupled to a bath of Bogoliubov excitations. The system is driven into a pure state with long-range order by quasi-local dissipation. The authors show that weak interactions lead to a weakly mixed state, which in 3D can be understood as a depletion of the condensate, while in 1D and 2D exhibits properties reminiscent of a Luttinger liquid or a Kosterlitz-Thouless critical phase at finite temperature. The results are illustrated using cold bosonic atoms in an optical lattice immersed in a superfluid bath. The paper also discusses the preparation of an η-state, an exact excited eigenstate of the d-dimensional two-species fermionic Hubbard Hamiltonian. This state is created by the η-operator and exhibits superfluidity with non-decaying off-diagonal long-range order. The authors show that the dissipative coupling to the bath can be used to prepare this state as a dark state, which is an eigenstate of the system's Hamiltonian and immune to the dissipative terms. The paper concludes by emphasizing the potential of quantum reservoir engineering to prepare interesting entangled states of qubits for quantum information. The results highlight the importance of understanding the competition between Hamiltonian and Liouvillian dynamics in driven open quantum systems, and the role of effective temperature in determining the behavior of the system. The study provides a framework for understanding and controlling the dynamics of many-body quantum systems in driven open quantum systems.This paper explores the preparation of many-body quantum states and non-equilibrium phases in driven open quantum systems using cold atoms. The authors demonstrate how a driven dissipative Bose-Einstein condensate (BEC) of bosons and paired fermions can be stabilized by coupling atoms in an optical lattice to a bath of Bogoliubov excitations. The system is governed by a master equation, which describes the time evolution of the density matrix under the influence of both the Hamiltonian and dissipative terms. The key idea is to design the system-reservoir coupling such that the system evolves into a desired many-body quantum state, such as a pure state with long-range order or a state resembling a Luttinger liquid or Kosterlitz-Thouless critical phase at finite temperature. The paper discusses the dynamics of a lattice of N bosonic atoms in d dimensions, where the atoms are coupled to a bath of Bogoliubov excitations. The system is driven into a pure state with long-range order by quasi-local dissipation. The authors show that weak interactions lead to a weakly mixed state, which in 3D can be understood as a depletion of the condensate, while in 1D and 2D exhibits properties reminiscent of a Luttinger liquid or a Kosterlitz-Thouless critical phase at finite temperature. The results are illustrated using cold bosonic atoms in an optical lattice immersed in a superfluid bath. The paper also discusses the preparation of an η-state, an exact excited eigenstate of the d-dimensional two-species fermionic Hubbard Hamiltonian. This state is created by the η-operator and exhibits superfluidity with non-decaying off-diagonal long-range order. The authors show that the dissipative coupling to the bath can be used to prepare this state as a dark state, which is an eigenstate of the system's Hamiltonian and immune to the dissipative terms. The paper concludes by emphasizing the potential of quantum reservoir engineering to prepare interesting entangled states of qubits for quantum information. The results highlight the importance of understanding the competition between Hamiltonian and Liouvillian dynamics in driven open quantum systems, and the role of effective temperature in determining the behavior of the system. The study provides a framework for understanding and controlling the dynamics of many-body quantum systems in driven open quantum systems.