**Magnetic-Atom Quantum Simulator**

We propose to realize a novel quantum simulator made of magnetic atoms in periodic potentials, which will enable the investigation of quantum-many body problems associated with long-range dipole-dipole interactions.

Our proposal is based on a series of key new developments. We will develop new tools to increase the strength of dipole-dipole interactions (shorter-period UV lattices, magneto-association of magnetic atoms into molecules with a stronger magnetic moment), and to control and measure their interaction at the nano-scale (using super-resolution techniques and narrow spectroscopic lines). We will develop new probes to certify the presence of quantum correlations, which are expected to be particularly strong in these many-body long-range interacting systems. We will either probe correlations in real space (microscope, double-well lattices), in momentum space (Doppler spectroscopy), or in the spin sector. These probes will be developed in a joint theory-experiment endeavor, to find the best ways to define and quantify entanglement.

The breakthrough realization of quantum simulators based on lattice-trapped magnetic atoms will allow us to explore for the first time two families of problems. First, we will probe low energy quantum phases stabilized by dipolar interactions; and second, out-of-equilibrium dynamics and quantum thermalization dominated by long-range interactions. A number of exotic phases will be within experimental reach, such as the checkerboard or stripe phases, or peculiar phases of spin systems with long-range interactions. We will aim at protocols to certify the nature of the quantum correlations within these systems. Such correlations can be explored in four different complementary setups: 1) an Er lattice gas within a Dy bath (Innsbruck); strongly dipolar lattice gases made of either 2) Dy atoms in UV lattices (Stuttgart) or 3) Dy2 molecules in standard lattices (Pisa/Florence), and 4) Cr atoms realizing lattice spin models (Paris).

**CONSORTIUM**

**Coordinator: Bruno Laburthe-Tolra (CNRS, FR)**- Tommaso Roscilde (ENS of Lyon, FR)
- Francesca Ferlaino (Institut für Quantenoptik und Quanten-information, AT)
- Tilman Pfau (Universität Stuttgart, DE)
- Giovanni Modugno (Istituto Nazionale di Ottica, IT)
- Maciej Lewenstein (Institute of Photonic Sciences, ES)
- Mariusz Gajda (Instytut Fizyki Polskiej Akademii Nauk, PL)