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The study of fractional Chern insulators and their exotic anyonic excitationsposes a major challenge in current experimental and theoretical research.Quantum simulators, in particular ultracold atoms in optical lattices, providea promising platform to realize, manipulate, and understand such systems with ahigh degree of controllability. Recently, an atomic $\nu=1/2$ Laughlin statehas been realized experimentally for a small system of two particles on 4 by 4sites. The next challenge concerns the preparation of Laughlin states inextended systems, ultimately giving access to anyonic braiding statistics orgapless chiral edge-states in systems with open boundaries. Here, we proposeand analyze an experimentally feasible scheme to grow larger Laughlin states byconnecting multiple copies of the already existing 4-by-4-system. First, wepresent a minimal setting obtained by coupling two of such patches, producingan extended 8-by-4-system with four particles. Then, we analyze differentpreparation schemes, setting the focus on two shapes for the extended system,and discuss their respective advantages: While growing strip-like latticescould give experimental access to the central charge, square-like geometriesare advantageous for creating quasi-hole excitations in view of braidingprotocols. We highlight the robust quantization of the fractional quasi-holecharge upon using our preparation protocol. We benchmark the performance of ourpatchwork preparation scheme by comparing it to a protocol based on couplingone-dimensional chains. We find that the patchwork approach consistently giveshigher target-state fidelities, especially for elongated systems. The resultspresented here pave the way towards near-term implementations of extendedLaughlin states in quantum gas microscopes and the subsequent exploration ofexotic properties of topologically ordered systems in experiments.

Growing Extended Laughlin States in a Quantum Gas Microscope: A Patchwork Construction