Minimal instances for ℤ₂ lattice gauge theories and spin pumps

The implementation and characterization of minimal instances of complex many-body systems have fundamental importance for the development of quantum simulators. This thesis reports on the implementation of two such minimal instances: a minimal instance for ℤ₂ lattice gauge theories coupled to matter and a minimal instance for spin pumping. ℤ₂ lattice gauge theories are characterized by a ℤ₂ gauge degree-of-freedom on every lattice link. Matter particles moving across these links pick up a phase depending on the traversed link's gauge field. To conserve the local gauge symmetry during this motion, also the gauge field itself changes. A corresponding minimal instance consists of a single matter particle on two lattice sites, which are connected by a link. The presented implementation of this minimal instance is based on two-component ultracold rubidium atoms in a periodically-driven optical double-well potential with a state-dependent energy offset between neighboring sites. In this implementation, one component represents the matter, while the other component represents the gauge field. The resulting effective Floquet Hamiltonian exhibits ℤ₂ symmetry for specific modulation parameters, if the periodic driving is resonant with the on-site interaction energy. The dynamics was investigated for different initial states and compared to a numerical analysis. This numerical analysis is in agreement with the measurements and reveals nontrivial dynamics as expected from a ℤ₂ lattice gauge theory coupled to matter. Furthermore, symmetry-breaking terms are identified and routes to overcome these limitations are discussed. Finally, a way to couple minimal instances to a one-dimensional extended system is proposed. The results constitute a first step towards quantum simulation of gauge theories and provide important insights for future studies using Floquet-based techniques. A quantum spin pump generates a spin transport in response to a cyclic and adiabatic variation of a one-dimensional Hamiltonian. The spins are transported in opposite directions and thereby exhibit a vanishing charge transport. In a spin-conserving system with homogeneously filled bands, the spin transport per pump cycle is quantized and deeply connected to the pump cycle's topology. Therefore, such a spin pump can be interpreted as a dynamical version of a two-dimensional time-reversal symmetric quantum spin Hall system. In the experiment, a minimal instance of such a spin pump was implemented with ultracold rubidium atoms in an optical double-well potential. In the limit of piecewise isolated instances throughout the pump cycle, where each instance shifts by one lattice site per half pump cycle, a one-dimensional spin pump is realized. To characterize the response of the pump, the spin separation and the occurring spin currents are measured. For the detection of the spin separation, the center-of-mass positions of both spin components are analyzed in in-situ absorption images. For the detection of spin currents, a novel detection method was developed, which connects the superexchange oscillations after a projection onto static double wells to the instantaneous spin current. This newly developed method can be applied to a general class of systems and in combination with single-site detection, it also allows for a local detection of instantaneous spin currents. The results demonstrate a corner stone for the implementation of a spin pump, which can be extended, e.g. by adding time-reversal-invariant spin--orbit interactions or by breaking time-reversal symmetry. This extension then results either in a system described by a nontrivial ℤ₂ invariant or a quantum spin Hall system described by a spin-Chern number.