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.

There is growing interest to investigate states of matter with topological order, which support excitations in the form of anyons, and which underly topological quantum computing. Examples of such systems include lattice spin models in two dimensions. Here we show that relevant Hamiltonians can be systematically engineered with polar molecules stored in optical lattices, where the spin is represented by a single electron outside a closed shell of a heteronuclear molecule in its rotational ground state. Combining microwave excitation with the dipole-dipole interactions and spin-rotation couplings allows us to build a complete toolbox for effective twospin interactions with designable range and spatial anisotropy, and with coupling strengths significantly larger than relevant decoherence rates. As an illustration we discuss two models: a 2D square lattice with an energy gap providing for protected quantum memory, and another on stacked triangular lattices leading to topological quantum computing.

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.

We discuss techniques to tune and shape the long-range part of the interaction potentials in quantum gases of bosonic polar molecules by dressing rotational excitations with static and microwave fields. This provides a novel tool towards engineering strongly correlated quantum phases in combination with low-dimensional trapping geometries. As an illustration, we discuss the 2D superfluid-crystal quantum phase transition for polar molecules interacting via an electric-field-induced dipole-dipole potential.

We discuss techniques to generate long-range interactions in a gas of ground state alkali atoms, by weakly admixing excited Rydberg states with laser light. This provides a tool to engineer strongly correlated phases with reduced decoherence from inelastic collisions and spontaneous emission. As an illustration, we discuss the quantum phases of dressed atoms with dipole-dipole interactions confined in a harmonic potential, as relevant to experiments. We show that residual spontaneous emission from the Rydberg state acts as a heating mechanism, leading to a quantum-classical crossover.

We discuss an open driven-dissipative many-body system, in which the competition of unitary Hamiltonian and dissipative Liouvillian dynamics leads to a nonequilibrium phase transition. It shares features of a quantum phase transition in that it is interaction driven, and of a classical phase transition, in that the ordered phase is continuously connected to a thermal state. We characterize the phase diagram and the critical behavior at the phase transition approached as a function of time. We find a novel fluctuation induced dynamical instability, which occurs at long wavelength as a consequence of a subtle dissipative renormalization effect on the speed of sound.

We show that polar molecules driven by microwave fields give naturally rise to strong three-body interactions, while the two-particle interaction can be independently controlled and even switched off. The derivation of these effective interaction potentials is based on a microscopic understanding of the underlying molecular physics, and follows from a well controlled and systematic expansion into many-body interaction terms. For molecules trapped in an optical lattice, we show that these interaction potentials give rise to Hubbard models with strong nearest-neighbor two-body and three-body interaction. As an illustration, we study the one-dimensional Bose-Hubbard model with dominant three-body interaction and derive its phase diagram.Fundamental interactions between particles, such as the Coulomb law, involves pairs of particles, and our understanding of the plethora of phenomena in condensed matter physics rests on models involving effective twobody interactions. On the other hand, exotic quantum phases, such as topological phases or spin liquids, are often identified as ground states of Hamiltonians with three or more body terms. While the study of these phases and properties of their excitations is presently one of the most exciting developments in theoretical condensed matter physics, it is difficult to identify real physical systems exhibiting such properties -a noticeable exception being the Fractional Quantum Hall effect. Here we show that polar molecules in optical lattices driven by microwave fields give naturally rise to Hubbard models with strong nearest-neighbor three-body interactions, while the twobody terms can be tuned (even switched off) with external fields.The many-body Hamiltonians underlying condensed matter physics are derived within an effective low energy theory, obtained by integrating out the high energy excitations. In general, this gives rise to interaction termswhere V (r) describes the two-particle interaction depending only on the separation between the particles. The second term W (r i , r j , r k ) is the three-body interaction, which depends on the distance and orientation of three particles, and vanishes if one particle is far apart from the other two. The ellipsis denotes possible higher many-body term terms. While for Helium atoms in the context of superfluidity the two-particle interaction dominates and determines the ground state properties with the three-body interactions providing small corrections, 1 model Hamiltonians with strong three-body interactions have attracted a lot of interest in the search for microscopic Hamiltonians exhibiting exotic ground state properties. Well known examples are the fractional quantum Hall states described by the Pfaffian wave functions which appear as ground states of a Hamiltonian with three-body interaction. 2,3,4 These topological phases admit anyonic excitations with non-abelian braiding statistic. Of special interest are also spin systems and bosonic Hamiltonians with complex many-body interactions, such as ring exchange model, whic...

We investigate schemes to dynamically create many particle entangled states of a two component Bose-Einstein condensate in a very short time proportional to 1/N where N is the number of condensate particles. For small N we compare exact numerical calculations with analytical semiclassical estimates and find very good agreement for N ≥ 50. We also estimate the effect of decoherence on our scheme, study possible scenarios for measuring the entangled states, and investigate experimental imperfections.