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]

We present two methods for characterization of motional-mode configurations that are generally applicable to the weak and strong-binding limit of single or multiple trapped atomic ions. Our methods are essential to realize control of the individual as well as the common motional degrees of freedom. In particular, when implementing scalable radio-frequency trap architectures with decreasing ion-electrode distances, local curvatures of electric potentials need to be measured and adjusted precisely, e.g., to tune phonon tunneling and control effective spin-spin interaction. We demonstrate both methods using single 25 Mg + ions that are individually confined 40 µm above a surface-electrode trap array and prepared close to the ground state of motion in three dimensions. [14], yielding increasing interaction strengths by decreasing system length scales. Correspondingly, higher-order terms need to be considered, in order to enable precise control of interaction potentials. For example, in microfabricated surface-electrode ion trap arrays [15-17] local potentials, dominated by applied electric trapping potentials, define motional modes. For envisioned quantum simulations, motional degrees of freedom can be exploited either within individual sites or between different sites. This, in turn, requires adjustment of motional-mode configurations, i.e., individual orientation of the normal-mode vectors and related motional frequencies, to enable individual, tunable inter-site interactions [17]. In this letter, we introduce and experimentally demonstrate two distinct methods for the analysis of motional-mode configurations that are generally applicable to the weak and strong-binding limit. For the latter, we cool single ions close to the ground state of motion in three dimensions.To introduce our system, we consider a single ion with charge Q and mass m, harmonically bound in three di-