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 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 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 KosterlitzThouless critical phase at finite temperature, with the role of the "finite temperature" played by the interactions.

We investigate the possibility of using a dissipative process to prepare a quantum system in a desired state. We derive for any multipartite pure state a dissipative process for which this state is the unique stationary state and solve the corresponding master equation analytically. For certain states, like the Cluster states, we use this process to show that the jump operators can be chosen quasi-locally, i.e. they act non-trivially only on a few, neighboring qubits. Furthermore, the relaxation time of this dissipative process is independent of the number of subsystems. We demonstrate the general formalism by considering arbitrary MPS-PEPS states. In particular, we show that the ground state of the AKLT-model can be prepared employing a quasi-local dissipative process.

Throughout physics, stable composite objects are usually formed via attractive forces, which allow the constituents to lower their energy by binding together. Repulsive forces separate particles in free space. However, in a structured environment such as a periodic potential and in the absence of dissipation, stable composite objects can exist even for repulsive interactions. Here we report on the first observation of such an exotic bound state, comprised of a pair of ultracold atoms in an optical lattice. Consistent with our theoretical analysis, these repulsively bound pairs exhibit long lifetimes, even under collisions with one another. Signatures of the pairs are also recognised in the characteristic momentum distribution and through spectroscopic measurements. There is no analogue in traditional condensed matter systems of such repulsively bound pairs, due to the presence of strong decay channels. These results exemplify on a new level the strong correspondence between the optical lattice physics of ultracold bosonic atoms and the Bose-Hubbard model [1,2], a correspondence which is vital for future applications of these systems to the study of strongly correlated condensed matter systems and to quantum information.Cold atoms loaded into a 3D optical lattice provide a realisation of a quantum lattice gas [1,2]. An optical lattice can be generated by pairs of counterpropagating laser beams, where the resulting standing wave intensity pattern forms a periodic array of microtraps for the cold atoms, with period a given by half the wavelength of the light, λ /2. The periodicity of the potential gives rise to a bandstructure for the atom dynamics with Bloch bands separated by band gaps, which can be controlled via the laser parameters and beam configuration. The dynamics of ultracold atoms loaded into the lowest band of a sufficiently deep optical lattice is well described by the BoseHubbard model with Hamiltonian[1, 3]are destruction (creation) operators for the bosonic atoms at site i. J/h denotes the nearest neighbour tunnelling rate, U the on-site collisional energy shift, and ε i the background potential. The high degree of control available over the parameters in this system, e.g., changing the relative values of U and J by varying the lattice depth, V 0 , has led to seminal experiments on strongly correlated gases in optical lattices, e.g., the study of the superfluidMott insulator transition[4], the realisation of 1D quantum liquids with atomic gases [5,6] (see also [7, 8]), and the investigation of disordered systems [9]. 3D optical lattices have also opened new avenues in cold collision physics and chemistry [10,11,12,13].A striking prediction of the Bose-Hubbard Hamiltonian (1) is the existence of stable repulsively bound atom pairs. These are most intuitively understood for strong repulsive interaction |U| ≫ J, U > 0, where an example of such a pair is a state of two atoms occupying a single site,This state has a potential energy offset U with respect to states where the atoms are separated (see Fig. ...