Traversal of a symmetry-breaking phase transition at finite rates can lead to causally separated regions with incompatible symmetries and the formation of defects at their boundaries, which has a crucial role in quantum and statistical mechanics, cosmology and condensed matter physics. This mechanism is conjectured to follow universal scaling laws prescribed by the Kibble-Zurek mechanism. Here we determine the scaling law for defect formation in a crystal of 16 laser-cooled trapped ions, which are conducive to the precise control of structural phases and the detection of defects. The experiment reveals an exponential scaling of defect formation g b , where g is the rate of traversal of the critical point and b ¼ 2.68 ± 0.06. This supports the prediction of b ¼ 8/3E2.67 for finite inhomogeneous systems. Our result demonstrates that the scaling laws also apply in the mesoscopic regime and emphasizes the potential for further tests of non-equilibrium thermodynamics with ion crystals.

The accurate characterization of eigenmodes and eigenfrequencies of two-dimensional ion crystals provides the foundation for the use of such structures for quantum simulation purposes. We present a combined experimental and theoretical study of two-dimensional ion crystals. We demonstrate that standard pseudopotential theory accurately predicts the positions of the ions and the location of structural transitions between different crystal configurations. However, pseudopotential theory is insufficient to determine eigenfrequencies of the two-dimensional ion crystals accurately but shows significant deviations from the experimental data obtained from resolved sideband spectroscopy. Agreement at the level of 2.5×10 −3 is found with the full time-dependent Coulomb theory using the Floquet-Lyapunov approach and the effect is understood from the dynamics of two-dimensional ion crystals in the Paul trap. The results represent initial steps towards an exploitation of these structures for quantum simulation schemes. Accurate control of ion crystals is of major importance for spectroscopy, quantum simulation, or quantum computing with such experimental platform. Since the invention of dynamical trapping by Paul [1], this versatile instrument has been adapted and optimized for specific purposes. Charged particles, more specifically singly charged ions, are confined in a radio frequency (rf) potential, which is formed by tailored electrode structures. In the case of the linear Paul trap, one aims for a quadrupole field along one z axis, such that a harmonic pseudopotential in x and y direction is formed. This radial potential strongly confines the ions, while an additional weaker axial potential in z direction is generated with static (dc) voltages applied to end cap electrodes. Trapped ions are cooled by laser radiation [2] in the potential described by three trap frequencies ω x,y,z eventually forming a crystalized structure.The conditions of operation are characterized by two anisotropy parameters where the radial confinement ω (x,y) typically exceeds the axial dc confinement ω z . For sufficiently small values of α (x,y) ≡ ω 2 z /ω 2 (x,y) , the ion crystal is linear and aligned along the weakest axis, the z trap axis; all ions are placed in the node of the rf potential. Spectacular highlights using linear crystals of cold ions are the demonstration of quantum logic operations [3,4], the generation of entangled states [5,6], sympathetic cooling of ions of different species [7,8], or the quantum-logic clock [9]. To reach the level of quantum control, as required in the experiments listed above, the first precondition was a complete understanding of eigenmodes and eigenfrequencies for such stored linear ion crystals [10][11][12].For larger numbers of ions, or for larger values of α, the linear crystal undergoes a transition to a zigzag structure and eventually to a fully crystalline two-or threedimensional structure [13,14]. Especially interesting are planar ion crystals, where usually one of the confining radial potentia...

We realize a single particle microscope by using deterministically extracted laser-cooled ^{40}Ca^{+} ions from a Paul trap as probe particles for transmission imaging. We demonstrate focusing of the ions to a spot size of 5.8±1.0  nm and a minimum two-sample deviation of the beam position of 1.5 nm in the focal plane. The deterministic source, even when used in combination with an imperfect detector, gives rise to a fivefold increase in the signal-to-noise ratio as compared with conventional Poissonian sources. Gating of the detector signal by the extraction event suppresses dark counts by 6 orders of magnitude. We implement a Bayes experimental design approach to microscopy in order to maximize the gain in spatial information. We demonstrate this method by determining the position of a 1  μm circular hole structure to a precision of 2.7 nm using only 579 probe particles.

The quest for experimental platforms that allow for the exploration, and even control, of the interplay of low dimensionality and frustration is a fundamental challenge in several fields of quantum many-body physics, such as quantum magnetism. Here, we propose the use of cold crystals of trapped ions to study a variety of frustrated quantum spin ladders. By optimizing the trap geometry, we show how to tailor the low dimensionality of the models by changing the number of legs of the ladders. Combined with a method for selectively hiding ions provided by laser addressing, it becomes possible to synthesize stripes of both triangular and Kagome lattices. Besides, the degree of frustration of the phonon-mediated spin interactions can be controlled by shaping the trap frequencies. We support our theoretical considerations by initial experiments with planar ion crystals, where a high and tunable anisotropy of the radial trap frequencies is demonstrated. We take into account an extensive list of possible error sources under typical experimental conditions, and describe explicit regimes that guarantee the validity of our scheme.

We report on transport operations with linear crystals of 40 Ca + ions by applying complex electric time-dependent potentials. For their control we use the information obtained from the ions' fluorescence. We demonstrate that by means of this feedback technique, we can transport a predefined number of ions and also split and unify ion crystals. The feedback control allows for a robust scheme, compensating for experimental errors as it does not rely on a precisely known electrical modeling of the electric potentials in the ion trap beforehand. Our method allows us to generate a self-learning voltage ramp for the required process. With an experimental demonstration of a transport with more than 99.8 % success probability, this technique may facilitate the operation of a future ion based quantum processor.

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