One of the major challenges of many-body physics is to understand the physical behavior of unconventional states of matter which cannot be pertubatively mapped onto non-interacting particle systems. We propose to create topological quantum states of minimal size with ultracold atoms in an optical lattice of four-site plaquettes. The experimental setup is based on a two-dimensional superlattice structure which will provide a full dynamical control of the plaquette characteristics. Our first objective is to create in isolated plaquettes the probable building blocks of high-Tc superconductivity, the so-called resonating valence bond states, and to observe their d-wave symmetry. Minimal instances of Laughlin states, which play a central role in the description of the fractional quantum Hall effect, will also be created and we aim at observing the fractional statistics of their low-lying excitations. By inhibiting super-exchange interactions, we also plan to observe coherent dynamical evolutions of four-particle states driven by ring-exchange interactions. This would provide a first step towards the realization of lattice gauge models with ultracold atoms. We will then address open questions of many-body physics that can be efficiently simulated with coupled plaquettes. We will couple an ensemble of resonating valence bond states, underdoped with d-wave hole pairs, and observe whether the system develops long-range phase coherence and superfluidity. This original bottom up approach could constitute an important step forward in the domain of quantum magnetism. Our experimental setup will also allow us to create the very lattice structure of Copper oxides CuO2. By preparing atoms in the deep Cu wells into p orbitals, we will investigate new many-body quantum phases in higher orbitals, such as a p-band superfluid phase. Each orbital flavor px or py playing the role of an effective spin, it will also be possible to realize magnetic phases and exotic bond algebraic liquids.
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