Following Feynman and as elaborated on by Lloyd, a universal quantum simulator (QS) is a controlled quantum device which reproduces the dynamics of any other many particle quantum system with short range interactions. This dynamics can refer to both coherent Hamiltonian and dissipative open system evolution. Here we show that laser excited Rydberg atoms in large spacing optical or magnetic lattices provide an efficient implementation of a universal QS for spin models involving (high order) n-body interactions. This includes the simulation of Hamiltonians of exotic spin models involving n-particle constraints such as the Kitaev toric code, color code, and lattice gauge theories with spin liquid phases. In addition, it provides the ingredients for dissipative preparation of entangled states based on engineering n-particle reservoir couplings. The key basic building blocks of our architecture are efficient and high-fidelity n-qubit entangling gates via auxiliary Rydberg atoms, including a possible dissipative time step via optical pumping. This allows to mimic the time evolution of the system by a sequence of fast, parallel and high-fidelity nparticle coherent and dissipative Rydberg gates.Laser excited Rydberg atoms [1-7] stored in large spacing optical lattices [8] or magnetic trap arrays [9] offer unique possibilities for implementing scalable quantum information processors. In such a setup single atoms can be loaded and kept effectively frozen at each lattice site, with long-lived atomic ground states representing qubits or effective spin degrees of freedom. Lattice spacings of the order of a few µm allow single site addressing with laser light, and thus individual manipulation and readout of atomic spins. Exciting atoms with lasers to high-lying Rydberg states and exploiting the strong and long-range dipole-dipole or Van der Waals interactions between Rydberg states provides fast and addressable 2-qubit entangling operations or effective spin-spin interactions; recent theoretical proposals have extended Rydberg-based protocols towards a single step, high-fidelity entanglement of a mesoscopic number of atoms [10,11]. Remarkably, the basic building blocks behind such a setup have been demonstrated recently in the laboratory by several groups [12,13].Motivated by and building on these new experimental possibilities, we discuss below a Rydberg QS for many body spin models. As a key ingredient of our setup (see Fig. 1) we introduce additional auxiliary qubit atoms in the lattice, which play a two-fold role: First, they control and mediate effective n-body spin interactions among a subset of n system spins residing in their neighborhood in the lattice. In our scheme this is achieved efficiently making use of single-site addressability and a parallelized multi-qubit gate, which is based on a combination of strong and long-range Rydberg interactions and electromagnetically induced transparency (EIT), as suggested recently in Ref. [11]. Second, the auxiliary atoms can be optically pumped, thereby providing a dissipative...

We review experimental and theoretical tools to excite, study and understand strongly interacting Rydberg gases. The focus lies on the excitation of dense ultracold atomic samples close to, or within quantum degeneracy, to high lying Rydberg states. The major part is dedicated to highly excited S-states of Rubidium, which feature an isotropic van-der-Waals potential. Nevertheless are the setup and the methods presented also applicable to other atomic species used in the field of laser cooling and atom trapping.

We demonstrate theoretically a parallelized C-NOT gate which allows us to entangle a mesoscopic ensemble of atoms with a single control atom in a single step, with high fidelity and on a microsecond time scale. Our scheme relies on the strong and long-ranged interaction between Rydberg atoms triggering electromagnetically induced transparency. By this we can robustly implement a conditional transfer of all ensemble atoms between two logical states, depending on the state of the control atom. We outline a many-body interferometer which allows a comparison of two many-body quantum states by performing a measurement of the control atom.

We study the appearance of correlated many-body phenomena in an ensemble of atoms driven resonantly into a strongly interacting Rydberg state. The ground state of the Hamiltonian describing the driven system exhibits a second order quantum phase transition. We derive the critical theory for the quantum phase transition and show that it describes the properties of the driven Rydberg system in the saturated regime. We find that the suppression of Rydberg excitations known as blockade phenomena exhibits an algebraic scaling law with a universal exponent.