A precisely controlled quantum system may reveal a fundamental understanding of another, less accessible system of interest. A universal quantum computer is currently out of reach, but an analogue quantum simulator that makes relevant observables, interactions and states of a quantum model accessible could permit insight into complex dynamics. Several platforms have been suggested and proof-of-principle experiments have been conducted. Here, we operate two-dimensional arrays of three trapped ions in individually controlled harmonic wells forming equilateral triangles with side lengths 40 and 80 μm. In our approach, which is scalable to arbitrary two-dimensional lattices, we demonstrate individual control of the electronic and motional degrees of freedom, preparation of a fiducial initial state with ion motion close to the ground state, as well as a tuning of couplings between ions within experimental sequences. Our work paves the way towards a quantum simulator of two-dimensional systems designed at will.

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We demonstrate Floquet engineering in a basic yet scalable 2D architecture of individually trapped and controlled ions. Local parametric modulations of detuned trapping potentials steer the strength of long-range inter-ion couplings and the related Peierls phase of the motional state. In our proof-ofprinciple, we initialize large coherent states and tune modulation parameters to control trajectories, directions and interferences of the phonon flow. Our findings open a new pathway for future Floquetbased trapped-ion quantum simulators targeting correlated topological phenomena and dynamical gauge fields.

Quantum mechanics dominates various effects in modern research from miniaturizing electronics, up to potentially ruling solid-state physics, quantum chemistry and biology 1, 2 . To study these effects experimental quantum systems may provide the only effective access 3, 4 . Seminal progress has been achieved in a variety of physical platforms, 2 highlighted by recent applications 5-8 . Atomic ions are known for their unique controllability and are identical by nature, as evidenced, e.g., by performing among the most precise atomic clocks 9 and providing the basis for one-dimensional simulators 10 . However, controllable, scalable systems of more than one dimension are required to address problems of interest and to reach beyond classical numerics with its powerful approximative methods 1, 4 . Here we show, tunable, coherent couplings and interference in a two-dimensional ion microtrap array, completing the toolbox for a reconfigurable quantum simulator. Previously, couplings 11, 12 and entangling interactions 13 between sites in one-dimensional traps have been realized, while coupling remained elusive in microtrap approaches [14][15][16] . Our architecture is based on well isolatable ions as identical quantum entities hovering above scalable CMOS chips. 15 In contrast to other multi-dimensional approaches 17 , it allows individual control in arbitrary, even non-periodic, lattice structures 18 . Embedded control structures can exploit the long-range Coulomb interaction to configure synthetic, fully connected many-body systems to address multi-dimensional problems 19 .Our approach of a synthetic dense lattice (Fig. 1a) is designed for controlling interactions between tens to hundred sites 15 . To engineer the desired interactions at short and long range within microtrap arrays, we can 1 arXiv:1812.08552v1 [quant-ph]

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