Ferdinand Schmidt-Kaler

Can a high-quality quantum gate be certified when uncharacterized state-preparation and measurement errors are dominant? Can this be achieved with low experimental overhead? Here, we introduce a sound black-box certification protocol for a single-qubit gate based on a small set of fixed, deterministic sequences. From the data, the protocol derives finite-sample bounds on the gate's rotation eigenvalue, a gauge-invariant property. Its phase reveals the accuracy of the rotation angle, while its modulus quantifies the loss of coherence under repeated gate applications. We implement the protocol on a $^{40}\mathrm{Ca}^{+}$ trapped-ion processor and certify the $\sqrt{\mathrm{X}}$-gate rotation eigenvalue using $22\,000$ circuit executions, and demonstrate the robustness of certification to state-preparation and measurement errors by deliberately degrading the readout. Finally, we prove that these spectral constraints imply, up to a physically meaningful unitary change of basis, a rigorous average gate-fidelity lower bound for every time-independent qubit model compatible with the data. In both readout settings, the spectral bounds yield the same fidelity certificate of $99.94(3)\%$ with $99\%$ confidence. Our results establish a new standard for quantum-gate certification by combining soundness and experimental efficiency without requiring trusted reference operations, randomized circuits, or model fitting.

Trapped-ion quantum computers rely on shuttling compilers, which cast an input algorithm into a sequence of ion-qubit movements within a given architecture. We present the first study in which a single frontier large language model (LLM), Claude Opus 4.7, generates and iteratively refines the full Python code of shuttling compilers from written specifications. We start with a compiler for (i) a linear segmented trap, extend it to (ii) a trap with junctions, and finally achieve efficient compilation for (iii) a broad class of connected trap graphs. The compilers for the more general cases are seeded with code from the previous ones. We benchmark the LLM-generated compilers against state-of-the-art hand-crafted ones using a common suite of quantum circuits. The number of shuttling timesteps is reduced by up to 76% for (i) and up to 39% for (ii). For the broad case (iii) of freely connected architectures, we find large variations in the required number of shuttling timesteps, depending on the connectivity. A densely connected, junction-rich architecture yields an order-of-magnitude reduction in shuttling timesteps compared to a corridor-like one. Repeating the complete generation and evaluation with a second frontier LLM, Claude Fable 5, reproduces these findings, with the Fable 5 compilers surpassing the hand-crafted ones more often on the largest circuits. Our results show that an unmodified frontier LLM can produce working, correct, and competitive shuttling compilers without additional manual algorithmic engineering, thus reducing the development time for new architectures from several months to a few days.

Scalable trapped-ion quantum computing requires fast and reliable transport of ions through complex, segmented radiofrequency trap architectures without inducing excessive motional excitation. We present a numerical toolchain for the systematic generation of time-dependent electrode voltages enabling fast, low-excitation ion shuttling in segmented radiofrequency traps. Based on a model of the trap electrode geometry, the framework combines an electrostatic field solver, efficient unconstrained optimization, waveform postprocessing, and dynamical simulations of ion motion to compute voltage waveforms that realize prescribed transport trajectories while respecting experimental constraints such as voltage limits and bandwidth. The toolchain supports arbitrary trap geometries, including junctions and multi-zone layouts, and allows for the flexible incorporation of optimization objectives. We provide a detailed assessment of the accuracy of the framework by investigating its numerical stability and by comparing measured and predicted secular frequencies. The framework is optimized for numerical performance, enabling rapid numerical prototyping of trap architectures of increasing complexity. As application examples, we apply the framework to the transport of a potential well along a linear, uniformly segmented trap, and we compute a solution for shuttling a potential well around the corner of an X-type trap junction. The presented approach provides an extensible and highly efficient numerical foundation for designing and validating transport protocols in current and next-generation trapped-ion processors.

Quantum simulation of interacting many-body spin systems is routinely performed with cold trapped ions, and systems with hundreds of spins have been studied in one and two dimensions. In the most common realizations of these platforms, spin degrees of freedom are encoded in low-lying electronic levels, and interactions among the spins are mediated through crystal vibrations. Here we propose a new approach which enables the quantum simulation of two-dimensional spin systems with interaction strengths that are increased by orders of magnitude. This, together with the unprecedented longevity of trapped ions, opens an avenue for the exploration of phenomena that take place on long timescales, e.g., slow and collective relaxation in frustrated and kinetically constrained systems. Our platform makes use of the strong dipolar interactions among electronic Rydberg states and planar confinement provided by a Penning trap. We investigate how the strong electric and magnetic fields that form this trap affect the properties of the Rydberg states and show that spin-spin interaction strengths on the order of MHz are achievable under experimentally realistic conditions. As a brief illustration of the capabilities of this quantum simulator, we study the entanglement in a frustrated spin system realized by three ions.

Trapped-ion quantum computers based on segmented traps rely on shuttling operations to establish long-range connectivity between sub-registers. Qubit routing dynamically reconfigures qubit positions so that all qubits involved in a gate operation are co-located within the same segment, a task whose complexity increases with system size. To address this challenge, we propose a layout-independent compilation strategy based on large language models (LLMs). Specifically, we fine-tune pretrained LLMs to generate the required shuttling operations. We evaluate this approach on linear and branched one-dimensional architectures using quantum circuits of up to $16$ qubits. Our results show that the fine-tuned LLMs generate valid shuttling schedules and, in some cases, outperform previous shuttling compilers by requiring approximately $15\,\%$ less shuttle overhead. However, results degrade as the algorithms increase in width and depth. In future, we plan to improve LLM-based shuttle compilation by enhancing our training pipeline using Direct Preference Optimization (DPO) and Gradient Regularized Policy Optimization (GRPO).

We apply the recently proposed Sample-based Krylov Quantum Diagonalization (SKQD) method to lattice gauge theories, using the Schwinger model with a $θ$-term as a benchmark. SKQD approximates the ground state of a Hamiltonian, employing a hybrid quantum-classical approach: (i) constructing a Krylov space from bitstrings sampled from time-evolved quantum states, and (ii) classically diagonalizing the Hamiltonian within this subspace. We study the dependence of the ground-state energy and particle number on the value of the $θ$-term, accurately capturing the model's phase structure. The algorithm is implemented on trapped-ion and superconducting quantum processors, demonstrating consistent performance across platforms. We show that SKQD substantially reduces the effective Hilbert space, and although the Krylov space dimension still scales exponentially, the slower growth underscores its promise for simulating lattice gauge theories in larger volumes.

We demonstrate the operation of a dual-frequency Paul trap and characterize its performance by storing either electrons or calcium ions while applying two quadrupole fields simultaneously which oscillate at $Ω_\textrm{fast} = 2π\times 1.6$ GHz and $Ω_\textrm{slow} = 2π\times 2$ MHz. The particles are loaded and stored in the trap under various conditions followed by detection employing an electron multiplier tube. We find that tens of electrons or ions can be trapped for up to ten milliseconds and a small fraction remains trapped even after hundreds of milliseconds. During dual-frequency operation we find that while the number of trapped electrons rapidly decreases with increase of the $Ω_\textrm{slow}$ field amplitude, the number of trapped ions shows no dependence on the $Ω_\textrm{fast}$ field amplitude as supported by our extensive numerical simulations. We aim to use a similar trap for synthesising antihydrogen from antiprotons and positrons. Accordingly, we discuss open challenges such as the co-trapping of oppositely charged species and particle trap duration.

Quantum Internet in a Nutshell (QI-Nutshell) connects the fields of quantum communication and quantum computing by emulating quantum communication protocols on currently available ion-trap quantum computers. We demonstrate emulations of QKD protocols where the individual steps are mapped to physical operations within our hardware platform. This allows us to not only practically execute established protocols such as BB84 or BBM92, but also include cloning attacks by an eavesdropping party, noise sources and side-channel attacks that are generally hard to include in theoretical QKD security proofs. We deliberately inject noise and investigate its effect on quantum communication protocols. We employ numerical simulations in order to study the incorporation of small quantum error correction (QEC) codes into QKD protocols. We find that these codes can help to suppress the noise level and to monitor the noise profile of the channel. This may enable the communicating parties to detect suspicious deviations from expected noise characteristics as a result of potential eavesdropping. This suggests that QEC may serve as a means of privacy authentication for quantum communication without altering the transmitted quantum information.

We are using optical- and microwave-fields to excite Rydberg states in trapped cold 40Ca+ ions. We employ a single ion and observe spectroscopically in the manifold of a principal quantum number n=49 the dressing of Rydberg states of angular momentum states S and P. We compare our experimental spectra with a multi-level calculation of dressed states and find good agreement. The results are important for controlling the interaction of single ions in Rydberg states with electric fields of the ion trap, and for tailoring the interactions in an ion crystal in Rydberg states.

Simulations in high-energy physics are currently emerging as an application of noisy intermediate-scale quantum (NISQ) computers. In this work, we explore the multi-flavor lattice Schwinger model - a toy model inspired by quantum chromodynamics - in one spatial dimension and with nonzero chemical potential by means of variational quantum simulation on a shuttling-based trapped-ion quantum processor. This fermionic problem becomes intractable for classical numerical methods even for small system sizes due to the notorious sign problem. We employ a parametric quantum circuit executed on our quantum processor to identify ground states in different parameter regimes of the model, mapping out a quantum phase transition which is the hallmark feature of the model. The resulting states are analyzed via quantum state tomography, to reveal how characteristic properties such as correlations in the output state change across the phase transition. Moreover, we use the results to determine the phase boundaries of the model.

Planar rotors can be realized by confining molecular ions or charged nanoparticles together with atomic ions in a Paul trap. We study the case of molecular ions or charged nanoparticles that have an electric dipole moment which couples to modes of the common vibrational motion in the trap. We calculate the strength of the coupling with specific vibrational modes for rotor masses ranging from $10^2$ atomic units, as typical for diatomic molecules, to $10^{6}\,$ atomic units, corresponding to nanoclusters. Either, the coupling manifests as a resonant energy exchange between rotational states and one of ion crystal vibrational modes. Or, in the off-resonant case, the dipole-phonon coupling results in energy shifts. In both cases we discuss how the effect may be experimentally detected using sideband-resolved laser spectroscopy and measurements of decoherence.

Thorium ions and molecules, recognized for their distinctive nuclear and atomic attributes, are central to numerous trapped-ion experiments globally. Our study introduces an effective, compact source of thorium ions produced via laser ablation of microgram-scale, salt-based samples. We thoroughly analyze the variety of ion species and charge states generated at varying laser fluences. Utilizing 10$μ$g of thorium fluoride crystals and laser fluences between $1.00 - 7.00$ J$\cdot$cm$^{-2}$ we produce thorium molecular ions $^{232}$ThF$_x$$^{n+}$ (with $x= 0 - 3$ and charge states up to $n = 3+$), including ThF$^{2+}$ and ThF$^{3+}$. These species are particularly relevant for spectroscopy; ThF$^{3+}$ is valuable for its stable closed-shell configuration, while ThF$^{2+}$, which is isoelectronic to RaF, offers a unique probe for studying nuclear structure and fundamental symmetries due to its simple electronic structure with a single unpaired electron. Density functional theory calculations of the distribution of positive charge in the produced molecular cations and the simplicity of this setup indicate that this method is easily transferable to other actinide systems.

Fast entangling gate operations are a fundamental prerequisite for quantum simulation and computation. We propose an entangling scheme for arbitrary pairs of ions in a linear crystal, harnessing the high electric polarizability of highly excited Rydberg states. An all-to-all quantum gate connectivity is based on an initialization of a pair of ions to a superposition of ground- and Rydberg-states by laser excitation, followed by the entangling gate operation which relies on a state-dependent frequency shift of collective vibrational modes of the crystal. This gate operation requires applying an electric waveform to trap electrodes. Employing transverse collective modes of oscillation, we reveal order of $μs$ operation times within any of the qubit pairs in a small crystal. In our calculation, we are taking into account realistic experimental conditions and feasible electric field ramps. The proposed gate operation is ready to be combined with a scalable processor architecture to reconfigure the qubit register, either by shuttling ions or by dynamically controlling optical tweezer potentials.

Within the framework of microscopic thermodynamics, correlations can play a crucial role for energy extraction. Our work sheds light on this connection by demonstrating that entanglement governs the amount of extractable energy in a controllable setting. We experimentally investigate a fundamental link between information, encoded in tunable non-classical correlations and quantified by quantum state tomography, and its utility as fuel for energy extraction. We realize an agent-demon protocol involving two trapped-ion qubits, and show that by implementing an appropriate feedback policy, the demon can optimize the energy extraction process, capitalizing on the correlations between the system's constituents. By quantifying both the concurrence of the two-qubit resource state and the energy extraction gain from applying the feedback policy, we corroborate the connection between information and energy, solidifying the role of non-classical correlations as a resource for thermodynamic processes.

The ability for real-time control of topological defects can open up prospects for dynamical manipulation of macroscopic properties of solids. A sub-category of these defects, formed by particle dislocations, can be effectively described using the Frenkel-Kontorova chain, which characterizes the dynamics of these particles in a periodic lattice potential. This model is known to host solitons, which are the topological defects of the system and are linked to structural transitions in the chain. This work addresses three key questions: Firstly, we investigate how imperfections present in concrete implementations of the model affect the properties of topological defects. Secondly, we explore how solitons can be injected after the rapid change in lattice potential or nucleated due to quantum fluctuations. Finally, we analyze the propagation and scattering of solitons, examining the role of quantum fluctuations and imperfections in influencing these processes. Furthermore, we address the experimental implementation of the Frenkel-Kontorova model. Focusing on the trapped ion quantum simulator, we set the stage for controllable dynamics of topological excitations and their observation in this platform.

Hybrid quantum systems integrate laser-cooled trapped ions and ultracold quantum gases within a single experimental configuration, offering vast potential for applications in quantum chemistry, polaron physics, quantum information processing, and quantum simulations. In this study, we introduce the development and experimental validation of an ion trap chip that incorporates a flat atomic chip trap directly beneath it. This innovative design addresses specific challenges associated with hybrid atom-ion traps by providing precisely aligned and stable components, facilitating independent adjustments of the depth of the atomic trapping potential and the positioning of trapped ions. Our findings include successful loading of the ion trap with linear Yb$^{+}$ ion crystals and the loading of neutral $^{87}$Rb atoms into a mirror magneto-optical trap (mMOT)

Spatially-structured laser beams, eventually carrying orbital angular momentum, affect electronic transitions of atoms and their motional states in a complex way. We present a general framework, based on the spherical tensor decomposition of the interaction Hamiltonian, for computing atomic transition matrix elements for light fields of arbitrary spatial mode and polarization structures. We study both the bare electronic matrix elements, corresponding to transitions with no coupling to the atomic center-of-mass motion, as well as the matrix elements describing the coupling to the quantized atomic motion in the resolved side-band regime. We calculate the spatial dependence of electronic and motional matrix elements for tightly focused Hermite-Gaussian, Laguerre-Gaussian and for radially and azimuthally polarized beams. We show that near the diffraction limit, all these beams exhibit longitudinal fields and field gradients, which strongly affect the selection rules and could be used to tailor the light-matter interaction. The presented framework is useful for describing trapped atoms or ions in spatially-structured light fields and therefore for designing new protocols and setups in quantum optics, -sensing and -information processing.

Hybrid quantum systems that unite laser-cooled trapped ions and ultracold quantum gases in a single experimental setup have opened a rapidly advancing field of study, including Quantum chemistry, polaron physics, quantum information processing and quantum simulations. We present a fully developed and tested ion trap chip and propose a flat chip trap that can be placed beneath the ion trap. This design substantially addresses the difficulties specific to hybrid traps and features well-aligned chips that allow for independent adjustment of the depth of the atomic trap and the confinement and positioning of ions. The ion trap has been successfully tested with linear ion crystals of Yb$^{+}$ and neutral $^{87}$Rb were also loaded into a mMOT a few millimeters under the ion trapping region.

Topological defects have strong impact on both elastic and inelastic properties of materials. In this article, we investigate the possibility to controllably inject topological defects in quantum simulators of solid state lattice structures. We investigate the quench dynamics of a Frenkel-Kontorova chain, which is used to model discommensurations of particles in cold atoms and trapped ionic crystals. The interplay between an external periodic potential and the inter-particle interaction makes lattice discommensurations, the topological defects of the model, energetically favorable and can tune a commensurate-incommensurate structural transition. Our key finding is that a quench from the commensurate to incommensurate phase causes a controllable injection of topological defects at periodic time intervals. We employ this mechanism to generate quantum states which are a superposition of lattice structures with and without topological defects. We conclude by presenting concrete perspectives for the observation and control of topological defects in trapped ion experiments.

We present a method for measuring the polarizability of Rydberg ions confined in the harmonic potential of a Paul trap. For a highly excited electronic state, the coupling between the electronic wave function and the trapping field modifies the excitation probability depending on the motional state of the ion. This interaction strongly depends on the polarizability of the excited state and manifests itself in the state-dependent secular frequencies of the ion. We initialize a single trapped $^{40}$Ca$^+$ ion from the motional ground state into coherent states with $|α|$ up to 12 using electric voltages on the trap segments. The internal state, firstly initialised in the long-lived 3D$_{5/2}$ state, is excited to a Rydberg S$_{1/2}$-state via the 5P$_{3/2}$ state in a two-photon process. We probe the depletion of the 3D$_{5/2}$ state owing to the Rydberg excitation followed by a decay into the internal ground 4S$_{1/2}$ state. By analysing the obtained spectra we extract the polarizability of Rydberg states which agree with numerical calculations. The method is easy-to-implement and applicable to different Rydberg states regardless of their principal or angular quantum numbers. An accurate value of the state polarizability is needed for quantum gate operations with Rydberg ion crystals.

A promising approach for scaling-up trapped-ion quantum computer platforms is by storing multiple trapped-ion qubit sets ('ion crystals') in segmented microchip traps and to interconnect these via physical movement of the ions ('shuttling'). Already for realizing quantum circuits with moderate complexity, the design of suitable qubit assignments and shuttling schedules require automation. Here, we describe and test algorithms which address exactly these tasks. We describe an algorithm for fully automated generation of shuttling schedules, complying to constraints imposed by a given trap structure. Furthermore, we introduce different methods for initial qubit assignment and compare these for random circuit (of up to 20 qubits) and quantum Fourier transform-like circuits, and generalized Toffoli gates of up to 40 qubits each. We find that for quantum circuits which contain a fixed structure, advanced assignment algorithms can serve to reduce the shuttling overhead.

The increasing capabilities of quantum computing hardware and the challenge of realizing deep quantum circuits require fully automated and efficient tools for compiling quantum circuits. To express arbitrary circuits in a sequence of native gates specific to the quantum computer architecture, it is necessary to make algorithms portable across the landscape of quantum hardware providers. In this work, we present a compiler capable of transforming and optimizing a quantum circuit targeting a shuttling-based trapped-ion quantum processor. It consists of custom algorithms set on top of the quantum circuit framework Pytket. The performance was evaluated for a wide range of quantum circuits and the results show that the gate counts can be reduced by factors up to 5.1 compared to standard Pytket and up to 2.2 compared to standard Qiskit compilation.

We demonstrate the excitation, using a structured light beam carrying orbital angular momentum, of the center of mass motion of a single atom in the transverse direction to the beam's propagation. This interaction is achieved with a vortex beam carrying one unit of orbital angular momentum and one unit of spin/polarization angular momentum. Using a singly charged $^{40}$Ca$^+$ ion, cooled near the ground state of motion in the 3D harmonic potential of a Paul trap, we probe the narrow S$_{1/2}$ to D$_{5/2}$ transition near $729\,\mathrm{nm}$ on its motional sidebands to quantify the momentum transfer. Exchange of quanta in the perpendicular direction to the beam's wave vector $\mathbf{k}$ is observed in case of the vortex shaped beam, in strong contrast to the absence of this spin-motion coupling for the case of a Gaussian beam. We characterize the coherent interaction by an effective transverse Lamb-Dicke factor $η^\mathrm{exp}_{\perp}=0.0062(5)$ which is in agreement with our theoretical prediction $η^\mathrm{theo}_{\perp}=0.0057(1)$

Modification of spontaneous decay in space and time is a central topic of quantum physics. It has been predominantly investigated in the context of cavity quantum electrodynamics (QED), gaining new interest recently in the domain of nano-optics. Beyond cavity-QED, spontaneous emission may be modified also in free space due to correlations among the photon emitters, a phenomenon known as super- and sub-radiance. Correlations may stem either from direct interactions between the particles, from long-range exchange of photons, or by measuring single photons in a common mode. Yet, the genuine spatial spontaneous emission pattern of an atomic ensemble in an entangled quantum state has not been observed so far, due to the lack of ultra-fast cameras with high spatial resolution suited for recording single photons from single atoms. Preparing two trapped ions in free space in entangled Dicke states via photon detection, we study the resulting collective spontaneous emission patterns. Depending on the symmetry of the Dicke states, associated with the direction of detection of the first state-determining photon, we observe fundamentally different emission patterns for the subsequently scattered photon, including super- and sub-radiance. Our results demonstrate that the detection of a single photon can profoundly modify the collective emission of an atomic array, here represented by its most elementary building block of two atoms in free space.

Recently, the principle of \textit{passivity} has been used to set bounds on the evolution of a microscopic quantum system with a thermal initial state. In this work, we experimentally demonstrate the utility of two passivity based frameworks: global passivity and passivity deformation, for the detection of a "hidden" or unaccounted environment. We employ two trapped-ion qubits undergoing unitary evolution, which may optionally be coupled to an unobserved environment qubit. Evaluating the measurement data from the system qubits, we show that global passivity can verify the presence of a coupling to an unobserved environment - a heat leak - in a case where the second law of thermodynamics fails. We also show that passivity deformation is even more sensitive, detecting a heat leak where global passivity fails.

We implement three-dimensional polarization gradient cooling of trapped ions. Counter-propagating laser beams near $393\,$nm impinge in lin$\,\perp\,$lin configuration, at a frequency below the S$_{1/2}$ to P$_{3/2}$ resonance in $^{40}$Ca$^+$. We demonstrate mean phonon numbers of $5.4(4)$ at a trap frequency of $2π\times 285\,$kHz and $3.3(4)$ at $2π\times480\,$kHz, in the axial and radial directions, respectively. Our measurements demonstrate that cooling with laser beams detuned to lower frequencies from the resonance is robust against an elevated phonon occupation number, and thus works well for an initial ion motion far out of the Lamb-Dicke regime, for up to four ions, and for a micromotion modulation index $β\leq 0.1$. Still, we find that the spectral impurity of the laser field influences both, cooling rates and cooling limits. Thus, a Fabry-Pérot cavity filter is employed for efficiently suppressing amplified spontaneous emission of the diode laser.

We present an alternative approach for interconnecting trapped ion processor nodes by a deterministic single ion transfer out of the trap, into a free space trajectory, followed by recapture in the trapping potential. Our experimental realization yields a success probability of 95.1%, namely 715 out of 752 extracted ions are retrapped, cooled and observed after a transport distance of 110mm and a time of flight of 7 $μ$s. Based on the near-unity operation success, we discuss its application for scalable ion trap quantum computing and advanced quantum sensing.

We propose the use of trapped ions for detection of millicharged dark matter. Millicharged particles will scatter off the ions, giving a signal either in individual events or in the overall heating rate of the ions. Ion traps have several properties which make them ideal detectors for such a signal. First, ion traps have demonstrated significant isolation of the ions from the environment, greatly reducing the background heating and event rates. Second, ion traps can have low thresholds for detection of energy deposition, down to $\sim \text{neV}$. Third, since the ions are charged, they naturally have large cross sections for scattering with the millicharged particles, further enhanced by the low velocities of the thermalized millicharges. Despite ion-trap setups being optimized for other goals, we find that existing measurements put new constraints on millicharged dark matter which are many orders of magnitude beyond previous bounds. For example, for a millicharge dark matter mass $m_Q=10~\textrm{GeV}$ and charge $10^{-3}$ of the electron charge, ion traps limit the local density to be $n_Q \lesssim 1 \, \textrm{cm}^{-3}$, a factor $\sim 10^8$ better than current constraints. Future dedicated ion trap experiments could reach even further into unexplored parameter space.

Quantum error correction requires the detection of errors by reliable measurements of suitable multi-qubit correlation operators. Here, we experimentally demonstrate a fault-tolerant weight-4 parity check measurement scheme. An additional 'flag' qubit serves to detect errors occurring throughout the parity measurement, which would otherwise proliferate into uncorrectable weight-2 errors on the qubit register. We achieve a flag-conditioned parity measurement single-shot fidelity of 93.2(2)\%. Deliberately injecting bit and phase-flip errors, we show that the fault-tolerant protocol is capable of reliably intercepting such faults. For holistic benchmarking of the parity measurement scheme, we use entanglement witnessing to show that the implemented circuit generates genuine six-qubit multi-partite entanglement. The fault-tolerant parity measurement scheme is an essential building block in a broad class of stabilizer quantum error correction protocols, including topological color codes. Our hardware platform is based on atomic ions stored in a segmented microchip ion trap. The qubit register is dynamically reconfigured via shuttling operations, enabling effective full connectivity without operational cross-talk, which provides key capabilities for scalable fault-tolerant quantum computing.

We demonstrate super-resolution optical sensing of the size of the wave packet of a single trapped ion. Our method extends the well known ground state depletion (GSD) technique to the coherent regime. Here, we use a hollow beam to strongly saturate a coherently driven dipole-forbidden transition around a sub-diffraction limited area at its center and observe state dependent fluorescence. By spatially scanning this laser beam over a single trapped $^{40}\mathrm{Ca}^+$ ion, we are able to measure the wave packet sizes of cooled ions. Using a depletion beam waist of $4.2(1)\,μ$m we reach a spatial resolution which allows us to determine a wave packet size of $39(9)\,$nm for a near ground state cooled ion. This value matches an independently deduced value of $32(2)\,$nm, calculated from resolved sideband spectroscopy measurements. Finally, we discuss the ultimate resolution limits of our adapted GSD imaging technique in view of applications to direct quantum wave packet imaging.

A single $^{40}$Ca ion is confine in the harmonic potential of a Paul trap and cooled to a temperature of a few mK, with a wave packet of sub-m spatial and sub-m/s velocity uncertainty. Deterministically extracted from the Paul trap, the single ion is propagating over a distance of 0.27 m and detected. By engineering the ion extraction process on the initial wave packet, theoretically modeling the ion trajectories, and studying experimentally the time-of-flight distribution, we directly infer the state of the previously trapped ion. This analysis allows for accurate remote sensing of the previous motional excitation in the trap potential, both coherently or incoherently. Our method paves a way to extract, manipulate and design quantum wave packets also outside of the Paul trap.

Cross-correlation signals are recorded from fluorescence photons scattered in free space off a trapped ion structure. The analysis of the signal allows for unambiguously revealing the spatial frequency, thus the distance, as well as the spatial alignment of the ions. For the case of two ions we obtain from the cross-correlations a spatial frequency $f_\text{spatial}=1490 \pm 2_{stat.}\pm 8_{syst.}\,\text{rad}^{-1}$, where the statistical uncertainty improves with the integrated number of correlation events as $N^{-0.51\pm0.06}$. We independently determine the spatial frequency to be $1494\pm 11\,\text{rad}^{-1}$, proving excellent agreement. Expanding our method to the case of three ions, we demonstrate its functionality for two-dimensional arrays of emitters of indistinguishable photons, serving as a model system to yield structural information where direct imaging techniques fail.

A complete set of spectroscopic data is indispensable when using Rydberg states of trapped ions for quantum information processing. We carried out Rydberg series spectroscopy for $nS_{1/2}$ states with $38 \leq n \leq 65$ and for $nD_{5/2}$ states with $37\leq n \leq 50$ on a single trapped $^{40}$Ca$^+$ ion. From a nonlinear regression to resonance frequencies, we determined the ionization energy of 2 870 575.582(15) GHz, measured 60 times more accurately as compared to the accepted value and contradicting it by 7.5 standard deviations. We confirm quantum defect values of $δ_{S_{1/2}}=1.802995(5)$ and $δ_{D_{5/2}}=0.626888(9)$ for $nS_{1/2}$ and $nD_{5/2}$ states respectively, which allow for unambiguous addressing of Rydberg levels of Ca$^+$ ions.

In this chapter, we present an overview of experiments with trapped Rydberg ions and outline the advantages and challenges of developing applications of this new platform for quantum computing, sensing and simulation. Trapped Rydberg ions feature several important properties, unique in their combination: they are tightly bound in a harmonic potential of a Paul trap, in which their internal and external degrees of freedom can be controlled in a precise fashion. High fidelity state preparation of both internal and motional states of the ions has been demonstrated, and the internal states have been employed to store and manipulate qubit information. Furthermore, strong dipolar interactions can be realised between ions in Rydberg states and be explored for investigating correlated many-body systems. By laser coupling to Rydberg states, the polarisability of the ions can be both enhanced and tuned. This can be used to control the interactions with the trapping fields in a Paul trap as well as dipolar interactions between the ions. Thus, trapped Rydberg ions present an attractive alternative for fast entangling operations as compared to those mediated by normal modes of trapped ions, which are advantageous for a future quantum computer or a quantum simulator.

Ions of the isomer $^{229m}$Th are a topic of high interest for the construction of a "nuclear clock" and in the field of fundamental physics for testing symmetries of nature. They can be efficiently captured in Paul traps which are ideal for performing high precision quantum logic spectroscopy. Trapping and identification of long-lived $^{232}$Th$^{+}$ ions from a laser ablation source was already demonstrated by the TACTICa collaboration on Trapping And Cooling of Thorium Ions with Calcium. The $^{229m}$Th is most easily accessible as $α$-decay daughter of the decay of $^{233}$U. We report on the development of a source for slow Th ions, including $^{229(m)}$Th for the TACTICa experiment. The $^{229(m)}$Th source is currently under construction and comprises a $^{233}$U monolayer, from which $^{229(m)}$Th ions recoil. These are decelerated in an electric field. Conservation of the full initial charge state distribution of the $^{229(m)}$Th recoil ions is one of the unique features of this source. We present ion-flight simulations for our adopted layout and give a final design. This source will provide Th ions in their original charge state at energies suitable for capture in a linear Paul trap for spectroscopy investigations.

Photon statistics divides light sources into three different categories, characterized by bunched, antibunched or uncorrelated photon arrival times. Single atoms, ions, molecules, or solid state emitters display antibunching of photons, while classical thermal sources exhibit photon bunching. Here we demonstrate a light source in free space, where the photon statistics depends on the direction of observation, undergoing a continuous crossover between photon bunching and antibunching. We employ two trapped ions, observe their fluorescence under continuous laser light excitation, and record the spatially resolved autocorrelation function $g^{(2)}(τ)$ with a movable Hanbury Brown and Twiss detector. Varying the detector position we find a minimum value for antibunching, $g^{(2)}(0) = 0.60(5)$ and a maximum of $g^{(2)}(0)=1.46(8)$ for bunching, demonstrating that this source radiates fundamentally different types of light alike. The observed variation of the autocorrelation function is understood in the Dicke model of heralded entangled states and the observed maximum and minimum values are modeled, taking independently measured experimental parameters into account.

Thorium ions exhibit unique nuclear properties with high relevance for testing symmetries of nature, and Paul traps feature an ideal experimental platform for performing high precision quantum logic spectroscopy. Loading of stable or long-lived isotopes is well-established and relies on ionization from an atomic beam. A different approach allows trapping short-lived isotopes available as alpha-decay daughters, which recoil from a thin sample of the precursor nuclide. A prominent example is the short-lived $^{229\text{m}}$Th, populated in a decay of long-lived $^{233}$U. Here, ions are provided by an external source and are decelerated to be available for trapping. Such setups offer the option to trap various isotopes and charge states of thorium. Investigating this complex procedure, we demonstrate the observation of single $^{232}$Th$^+$ ions trapped, embedded into and sympathetically cooled via Coulomb interactions by co-trapped $^{40}$Ca$^+$ ions. Furthermore, we discuss different options for a non-destructive identification of the sympathetically cooled thorium ions in the trap, and describe in detail our chosen experimental method, identifying mass and charge of thorium ions from the positions of calcium ions, as their fluorescence is imaged on a CCD camera. These findings are verified by means of a time-of-flight signal when extracting ions of different mass-to-charge ratio from the Paul trap and steering them into a detector.

We present a feasibility study with several magnetic field configurations for creating spin-dependent forces that can split a low-energy ion beam by the Stern-Gerlach effect. To the best of our knowledge, coherent spin-splittings of charged particles have yet to be realised. Our proposal is based on ion source parameters taken from a recent experiment that demonstrated single-ion implantation from a high-brightness ion source combined with a radio-frequency Paul trap. The inhomogeneous magnetic fields can be created by permanently magnetised microstructures or from current-carrying wires with sizes in the micron range, such as those recently used in a successful implementation of the Stern-Gerlach effect with neutral atoms. All relevant forces (Lorentz force and image charges) are taken into account, and measurable splittings are found by analytical and numerical calculations.

Single dopant atoms or dopant-related defect centers in a solid state matrix provide an attractive platform for quantum simulation of topological states, for quantum computing and communication, due to their potential to realize a scalable architecture compatible with electronic and photonic integrated circuits. The production of such quantum devices calls for deterministic single atom doping techniques because conventional stochastic doping techniques are cannot deliver appropriate architectures. Here, we present the fabrication of arrays of praseodymium color centers in YAG substrates, using a deterministic source of single laser-cooled Pr$^+$ ions. The beam of single Pr$^+$ ions is extracted from a Paul trap and focused down to 30(9) nm. Using a confocal microscope we determine a conversion yield into active color centers up to 50% and realizing a placement accuracy of better than 50 nm.

We present an experimental investigation of the Rydberg 23 P$_{1/2}$ state of laser-cooled $^{40}$Ca$^+$ ions in a radiofrequency ion trap. Using micromotion sideband spectroscopy on a narrow quadrupole transition, the oscillating electric field at the ion position was precisely characterised, and the modulation of the Rydberg transition due to this field was minimised. From a correlated fit to this P line and previously measured P and F level energies of Ca II, we have determined the ionization energy of 95 751.916(32) $\rm {cm}^{-1}$, in agreement with the accepted value, and the quantum defect for the $n$ P$_{1/2}$ states.

In order to study interactions of atomic ions with ultracold neutral atoms, it is important to have sub-$μ$m control over positioning ion crystals. Serving for this purpose, we introduce a microfabricated planar ion trap featuring 21 DC electrodes. The ion trap is controlled by a home-made FPGA voltage source providing independently variable voltages to each of the DC electrodes. To assure stable positioning of ion crystals with respect to trapped neutral atoms, we integrate into the overall design a compact mirror magneto optical chip trap (mMOT) for cooling and confining neutral $^{87}$Rb atoms. The trapped atoms will be transferred into an also integrated chipbased Ioffe-Pritchard trap potential formed by a Z-shaped wire and an external bias magnetic field.We introduce the hybrid atom-ion chip, the microfabricated planar ion trap and use trapped ion crystals to determine ion lifetimes, trap frequencies, positioning ions and the accuracy of the compensation of micromotion.

Photons carrying a well-defined orbital angular momentum have been proven to modify spectroscopic selection rules in atomic matter. Excitation profiles of electric quadrupole transitions have been measured with single trapped $^{40}$Ca$^+$ ions for varying polarizations. We further develop the photo-absorption formalism to study the case of arbitrary alignment of the beam's optical axis with respect to the ion's quantization axis and mixed multipolarity. Thus, predictions for M1-dominated $^{40}Ar^{13+}$, E3-driven $^{171}Yb^+$ and $^{172}Yb^+$, and B-like $^{20}Ne^{5+}$ are presented. The latter case displays novel effects, coming from the presence of a strong photon -- magnetic dipole coupling.

We realize a heat engine using a single electron spin as a working medium. The spin pertains to the valence electron of a trapped $^{40}$Ca$^+$ ion, and heat reservoirs are emulated by controlling the spin polarization via optical pumping. The engine is coupled to the ion's harmonic-oscillator degree of freedom via spin-dependent optical forces. The oscillator stores the work produced by the heat engine and therefore acts as a flywheel. We characterize the state of the flywheel by reconstructing the Husimi $\mathcal{Q}$ function of the oscillator after different engine runtimes. This allows us to infer both the deposited energy and the corresponding fluctuations throughout the onset of operation, starting in the oscillator ground state. In order to understand the energetics of the flywheel, we determine its ergotropy, i.e. the maximum amount of work which can be further extracted from it. Our results demonstrate how the intrinsic fluctuations of a microscopic heat engine fundamentally limit performance.

Precision optical spectroscopy of exotic ions reveals accurate information about nuclear properties such as charge radii and magnetic and quadrupole moments. Thorium ions exhibit unique nuclear properties with high relevance for testing symmetries of nature. We report loading and trapping of single $^{232}$Th$^+$ ions in a linear Paul trap, embedded into and sympathetically cooled by small crystals of trapped $^{40}$Ca$^+$ ions. Trapped Th ions are identified in a non-destructive manner from the voids in the laser-induced Ca fluorescence pattern emitted by the crystal, and alternatively, by means of a time-of-flight signal when extracting ions from the Paul trap and steering them into an external detector. We have loaded and handled a total of 231 individual Th ions. We reach a time-of-flight detection efficiency of $\gtrsim 95\, \%$, consistent with the quantum efficiency of the detector. The sympathetic cooling technique is expected to be applicable for other isotopes and various charge states of Th e.g., for future studies of $^{229m}$Th.

We demonstrate the efficient generation of Be$^+$ ions with a 60 ns and 150 nJ laser pulse near 235 nm for two-step photo-ionization, proven by subsequent counting the number of ions loaded into a linear Paul trap. The bandwidth and power of the laser pulse are chosen in such a way that a first, resonant step fully saturates the entire velocity distribution of beryllium atoms effusing from a thermal oven. The second excitation step is driven by the same light field causing efficient non-resonant ionization. Our ion-loading scheme is more than 15 times more efficient as compared to former pathways using two-photon continuous wave laser excitation.

We analyze the multipole excitation of atoms with twisted light, i.e., by a vortex light field that carries orbital angular momentum. A single trapped $^{40}$Ca$^+$ ion serves as a localized and positioned probe of the exciting field. We drive the $S_{1/2} \to D_{5/2}$ transition and observe the relative strengths of different transitions, depending on the ion's transversal position with respect to the center of the vortex light field. On the other hand, transition amplitudes are calculated for a twisted light field in form of a Bessel beam, a Bessel-Gauss and a Gauss-Laguerre mode. Analyzing experimental obtained transition amplitudes we find agreement with the theoretical predictions at a level of better than 3\%. Finally, we propose measurement schemes with two-ion crystals to enhance the sensing accuracy of vortex modes in future experiments.

The propagation of light beams is well described using the paraxial approximation, where field components along the propagation direction are usually neglected. For strongly inhomogeneous or shaped light fields, however, this approximation may fail, leading to intriguing variations of the light-matter interaction. This is the case of twisted light having opposite orbital and spin angular momenta. We compare experimental data for the excitation of a quadrupole transition in a single trapped $^{40}$Ca$^+$ ion by Schmiegelow et al, Nat.\ Comm.\ 7, 12998 (2016), with a complete model where longitudinal components of the electric field are taken into account. Our model matches the experimental data and excludes by 11 standard deviations the approximation of complete transverse field. This demonstrates the importance of all field components in the interaction of twisted light with matter.

We demonstrate a coherence time of 2.1(1)~s for electron spin superposition states of a single trapped $^{40}$Ca$^+$ ion. The coherence time, measured with a spin-echo experiment, corresponds to residual rms magnetic field fluctuations $\leq$~2.7$\times$10$^{-12}$~T. The suppression of decoherence induced by fluctuating magnetic fields is achieved by combining a two-layer $μ$-metal shield, which reduces external magnetic noise by 20 to 30~dB for frequencies of 50~Hz to 100~kHz, with Sm$_2$Co$_{17}$ permanent magnets for generating a quantizing magnetic field of 0.37~mT. Our results extend the coherence time of the simple-to-operate spin qubit to ultralong coherence times which so far have been observed only for magnetic insensitive transitions in atomic qubits with hyperfine structure.

We show nanoscopic transmission microscopy, using a deterministic single particle source and compare the resulting images in terms of signal-to-noise ratio, with those of conventional Poissonian sources. Our source is realized by deterministic extraction of laser-cooled calcium ions from a Paul trap. Gating by the extraction event allows for the suppression of detector dark counts by six orders of magnitude. Using the Bayes experimental design method, the deterministic characteristics of this source are harnessed to maximize information gain, when imaging structures with a parametrizable transmission function. We demonstrate such optimized imaging by determining parameter values of one and two dimensional transmissive structures.

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 with a resolution of 5.8$\;\pm\;$1.0$\,$nm and a minimum two-sample deviation of the beam position of 1.5$\,$nm in the focal plane. The deterministic source, even when used in combination with an imperfect detector, gives rise to much higher signal to noise ratios as compared with conventional Poissonian sources. Gating of the detector signal by the extraction event suppresses dark counts by 6 orders of magnitude. We implement a Bayes experimental design approach to microscopy in order to maximize the gain in spatial information. We demonstrate this method by determining the position of a 1$\,μ$m circular hole structure to an accuracy of 2.7$\,$nm using only 579 probe particles.

We observe interference in the light scattered from trapped $^{40}$Ca$^+$ ion crystals. By varying the intensity of the excitation laser, we study the influence of elastic and inelastic scattering on the visibility of the fringe pattern and discriminate its effect from that of the ion temperature and wave-packet localization. In this way we determine the complex degree of coherence and the mutual coherence of light fields produced by individual atoms. We obtain interference fringes from crystals consisting of two, three and four ions in a harmonic trap. Control of the trapping potential allows for the adjustment of the interatomic distances and thus the formation of linear arrays of atoms serving as a regular grating of microscopic scatterers.

Photons carry one unit of angular momentum associated with their spin~\cite{Beth1936}. Structured vortex beams carry additional orbital angular momentum which can also be transferred to matter~\cite{Allen1992}. This extra twist has been used for example to drive motion of microscopic particles in optical tweezers as well as to create vortices in degenerate quantum gases~\cite{He1995,Andersen2006}. Here we demonstrate the transfer of optical orbital angular momentum from the transverse spatial structure of the beam to the internal (electronic) degrees of freedom of an atom. Probing a quadrupole transition of a single trapped $^{40}$Ca$^+$ ion localized at the center of the vortex, we observe strongly modified selection rules, accounting for both the photon spin and the vorticity of the field. In particular, we show that an atom can absorb two quanta of angular momentum from a single photon even when rotational symmetry is conserved. In contrast to previous findings~\cite{Araoka2005,Loeffler2011a,Mathevet2013}, our experiment allows for conditions where the vorticity of the laser beam determines the optical excitation, contributing to the long-standing discussion on whether the orbital angular momentum of photons can be transferred to atomic internal degrees of freedom~\cite{VanEnk1994,Babiker2002,Jauregui2004, Schmiegelow2012, Mondal2014, Scholz-Marggraf2014} and paves the way for its use to tailor light-matter interactions.

We report the experimental realization of a single-atom heat engine. An ion is confined in a linear Paul trap with tapered geometry and driven thermally by coupling it alternately to hot and cold reservoirs. The output power of the engine is used to drive a harmonic oscillation. From direct measurements of the ion dynamics, we determine the thermodynamic cycles for various temperature differences of the reservoirs. We use these cycles to evaluate power $P$ and efficiency $η$ of the engine, obtaining up to $P=342\,$yJ and $η=0.28 \,\%$, consistent with analytical estimations. Our results demonstrate that thermal machines can be reduced to the ultimate limit of single atoms.

We evaluate the feasibility of the implementation of two quantum repeater protocols with an existing experimental platform based on a $^{40}$Ca$^+$-ion in a segmented micro trap, and a third one that requires small changes to the platform. A fiber cavity serves as an ion-light interface. Its small mode volume allows for a large coupling strength of $g_c = 2 π20$ MHz despite comparatively large losses $κ= 2 π36.6$ MHz. With a fiber diameter of 125 mu m, the cavity is integrated into the microstructured ion trap, which in turn is used to transport single ions in and out of the interaction zone in the fiber cavity. We evaluate the entanglement generation rate for a given fidelity using parameters from the experimental setup. The DLCZ protocol (Duan et al, Nature, 2001, 414, 413-418) and the hybrid protocol (van Loock et al, Phys. Rev. Lett., 2006, 96, 240501) outperform the EPR protocol (Sanguard et al, New J. Phys., 2013, 15, 085004). We calculate rates of more than than 35 s$^{-1}$ for non-local Bell state fidelities larger than 0.9 with the existing platform. We identify parameters which mainly limit the attainable rates, and conclude that entanglement generation rates of 740 s$^{-1}$ at fidelities of 0.9 are within reach with current technology.

We demonstrate control of the absolute phase of an optical lattice with respect to a single trapped ion. The lattice is generated by off-resonant free-space laser beams, we actively stabilize its phase by measuring its ac-Stark shift on a trapped ion. The ion is localized within the standing wave to better than 2\% of its period. The locked lattice allows us to apply displacement operations via resonant optical forces with a controlled direction in phase space. Moreover, we observe the lattice-induced phase evolution of spin superposition states in order to analyze the relevant decoherence mechanisms. Finally, we employ lattice-induced phase shifts for inferring the variation of the ion position over 157~$μ$m range along the trap axis at accuracies of better than 6~nm.

We propose a protocol for measurement of the phonon number distribution of a harmonic oscillator based on selective mapping to a discrete spin-1/2 degree of freedom. We consider a system of a harmonically trapped ion, where a transition between two long lived states can be driven with resolved motional sidebands. The required unitary transforms are generated by amplitude-modulated polychromatic radiation fields, where the time-domain ramps are obtained from numerical optimization by application of the Chopped RAndom Basis (CRAB) algorithm. We provide a detailed analysis of the scaling behavior of the attainable fidelities and required times for the mapping transform with respect to the size of the Hilbert space. As one application we show how the mapping can be employed as a building block for experiments which require measurement of the work distribution of a quantum process.

We propose a trapped ion scheme en route to realize spin Hamiltonians on a Kagome lattice which, at low energies, are described by emergent Z2 gauge fields, and support a topological quantum spin liquid ground state. The enabling element in our scheme is the hexagonal plaquette spin-spin interactions in a 2D ion crystal. For this, the phonon-mode spectrum of the crystal is engineered by standing-wave optical potentials or by using Rydberg excited ions, thus generating localized phonon-modes around a hexagon of ions selected out of the entire two-dimensional crystal. These tailored modes can mediate spin-spin interactions between ion-qubits on a hexagonal plaquette when subject to state-dependent optical dipole forces. We discuss how these interactions can be employed to emulate a generalized Balents-Fisher-Girvin model in minimal instances of one and two plaquettes. This model is an archetypical Hamiltonian in which gauge fields are the emergent degrees of freedom on top of the classical ground state manifold. Under realistic situations, we show the emergence of a discrete Gauss's law as well as the dynamics of a deconfined charge excitation on a gauge-invariant background using the two-plaquettes trapped ions spin-system. The proposed scheme in principle allows further scaling in a future trapped ion quantum simulator, and we conclude that our work will pave the way towards the simulation of emergent gauge theories and quantum spin liquids in trapped ion systems.

We experimentally demonstrate a method to determine the temperature of trapped ions which is suitable for monitoring fast thermalization processes. We show that observing and analyzing the lineshape of dark resonances in the fluorescence spectrum provides a temperature measurement which accurate over a large dynamic range, applied to single ions and small ion crystals. Laser induced fluorescence is detected over a time of only $20\,μ$s allowing for rapid determination of the ion temperature. In the measurement range of $10^{-1}-10^{+2}\,$mK we reach better than $15\,\%$ accuracy. Tuning the cooling laser to selected resonance features allows for controlling the ion temperatures between $0.7\,$mK and more than $10\,$mK. Experimental work is supported by a solution of the 8-level optical Bloch equations when including the ions classical motion. This technique paves the way for many experiments comprising heat transport in ion strings, heat engines, non-equilibrium thermodynamics or thermometry of large ion crystals.

We experimentally demonstrate nanoscopic transmission microscopy relying on a deterministic single particle source. This increases the signal-to-noise ratio with respect to conventional microscopy methods, which employ Poissonian particle sources. We use laser-cooled ions extracted from a Paul trap, and demonstrate remote imaging of transmissive objects with a resolution of 8.6 $\pm$ 2.0nm and a minimum two-sample deviation of the beam position of 1.5nm. Detector dark counts can be suppressed by 6 orders of magnitudes through gating by the extraction event. The deterministic nature of our source enables an information-gain driven approach to imaging. We demonstrate this by performing efficient beam characterization based on a Bayes experiment design method.

We experimentally demonstrate fast separation of a two-ion crystal in a microstructured segmented Paul trap. By the use of spectroscopic calibration routines for the electrostatic trap potentials, we achieve the required precise control of the ion trajectories near the \textit{critical point}, where the harmonic confinement by the external potential vanishes. The separation procedure can be controlled by three parameters: A static potential tilt, a voltage offset at the critical point, and the total duration of the process. We show how to optimize the control parameters by measurements of ion distances, trap frequencies and the final motional excitation. At a separation duration of $80 μ$s, we achieve a minimum mean excitation of $\bar{n} = 4.16(0.16)$ vibrational quanta per ion, which is consistent with the adiabatic limit given by our particular trap. We show that for fast separation times, oscillatory motion is excited, while a predominantly thermal state is obtained for long times. The presented technique does not rely on specific trap geometry parameters and can therefore be adopted for different segmented traps.

We discuss an experimental approach allowing to prepare antihydrogen atoms for the GBAR experiment. We study the feasibility of all necessary experimental steps: The capture of incoming $\bar{\rm H}^+$ ions at keV energies in a deep linear RF trap, sympathetic cooling by laser cooled Be$^+$ ions, transfer to a miniaturized trap and Raman sideband cooling of an ion pair to the motional ground state, and further reducing the momentum of the wavepacket by adiabatic opening of the trap. For each step, we point out the experimental challenges and discuss the efficiency and characteristic times, showing that capture and cooling are possible within a few seconds.

Heat engines extract work by running cyclically between two heat reservoirs. When the two reservoirs are thermal and at different temperatures, the maximum efficiency of the engine is given by the Carnot limit. Here we consider a quantum Otto cycle for a time-dependent harmonic oscillator coupled to an engineered squeezed thermal reservoir. We show that the efficiency at maximum power increases with the degree of squeezing, exponentially approaching unity for large squeezing parameters $r$. Furthermore, we propose an experimental scheme to implement such a system by using a single trapped ion in a linear Paul trap with special geometry and coupled to engineered reservoirs. Our analytical investigations are supported with Monte Carlo simulations that demonstrate the feasibility of our proposal. For realistic trap parameters, an increase of up to a factor of four is reached, largely exceeding the classical limit.

We study theoretically the collective E$\otimes$e Jahn-Teller-Dicke distortion in a system of trapped ions. We focus in the limit of infinite range interactions in which an ensemble of effective spins interacts with two collective vibrational modes with U(1) symmetric couplings. Our model is exactly solvable in the thermodynamical limit and it is amenable to be solved by exact numerical diagonalization for a moderate number of ions. We show that trapped ions are ideally suited to study the emergence of spontaneous symmetry breaking of a continuous symmetry and magnetic structural phase transition in a mesoscopic system.

We realize fast transport of ions in a segmented micro-structured Paul trap. The ion is shuttled over a distance of more than 10^4 times its groundstate wavefunction size during only 5 motional cycles of the trap (280 micro meter in 3.6 micro seconds). Starting from a ground-state-cooled ion, we find an optimized transport such that the energy increase is as low as 0.10 $\pm$ 0.01 motional quanta. In addition, we demonstrate that quantum information stored in a spin-motion entangled state is preserved throughout the transport. Shuttling operations are concatenated, as a proof-of-principle for the shuttling-based architecture to scalable ion trap quantum computing.

We propose an experimental scheme to realize a nano heat engine with a single ion. An Otto cycle may be implemented by confining the ion in a linear Paul trap with tapered geometry and coupling it to engineered laser reservoirs. The quantum efficiency at maximum power is analytically determined in various regimes. Moreover, Monte Carlo simulations of the engine are performed that demonstrate its feasibility and its ability to operate at maximum efficiency of 30% under realistic conditions.

We study the interaction of a light beams carrying angular momentum with a single, trapped and well localized ion. We provide a detailed calculation of selection rules and excitation probabilities for quadrupole transitions. The results show the dependencies on the angular momentum and polarization of the laser beam as well as the direction of the quantization magnetic field. In order to observe optimally the specific effects, focusing the angular momentum beam close to the diffraction limit is required. We discuss a protocol for examining experimentally the effects on the S$_{1/2}$ to D$_{5/2}$ transition using a $^{40}$Ca$^+$ ion. Various applications and advantages are expected when using light carrying angular momentum: In quantum information processing, where qubit states of ion crystals are controlled, parasitic light shifts could be avoided as the ion is excited in the dark zone of the beam at zero electric field amplitude. Such interactions also open the door to high dimensional entanglement between light and matter. In spectroscopy one might access transitions which have escaped excitation so far due to vanishing transition dipole moments.

We use a single ion as an movable electric field sensor with accuracies on the order of a few V/m. For this, we compensate undesired static electric fields in a planar RF trap and characterize the static fields over an extended region along the trap axis. We observe a strong buildup of stray charges around the loading region on the trap resulting in an electric field of up to 1.3 kV/m at the ion position. We also find that the profile of the stray field remains constant over a time span of a few months.

It is expected that ion trap quantum computing can be made scalable through protocols that make use of transport of ion qubits between sub-regions within the ion trap. In this scenario, any magnetic field inhomogeneity the ion experiences during the transport, may lead to dephasing and loss of fidelity. Here we demonstrate how to measure, and compensate for, magnetic field gradients inside a segmented ion trap, by transporting a single ion over variable distances. We attain a relative magnetic field sensitivity of ΔB/B_0 ~ 5*10^{-7} over a test distance of 140 \micro m, which can be extended to the mm range, still with sub \micro m resolution. A fast experimental sequence is presented, facilitating its use as a magnetic field gradient calibration routine, and it is demonstrated that the main limitation is the quantum shot noise.

We observe the phase space trajectory of an entangled wave packet of a trapped ion with high precision. The application of a spin dependent light force on a superposition of spin states allows for coherent splitting of the matter wave packet such that two distinct components in phase space emerge. We observe such motion with a precision of better than 9% of the wave packet extension in both momentum and position, corresponding to a 0.8 nm position resolution. We accurately study the effect of the initial ion temperature on the quantum entanglement dynamics. Furthermore, we map out the phonon distributions throughout the action of the displacement force. Our investigation shows corrections to simplified models of the system evolution. The precise knowledge of these dynamics may improve quantum gates for ion crystals and lead to entangled matter wave states with large displacements.

We introduce a measurement scheme that utilizes a single ion as a local field probe. The ion is confined in a segmented Paul trap and shuttled around to reach different probing sites. By the use of a single atom probe, it becomes possible characterizing fields with spatial resolution of a few nm within an extensive region of millimeters. We demonstrate the scheme by accurately investigating the electric fields providing the confinement for the ion. For this we present all theoretical and practical methods necessary to generate these potentials. We find sub-percent agreement between measured and calculated electric field values.

We focus down an ion beam consisting of single 40Ca+ ions to a spot size of a few mum using an einzel-lens. Starting from a segmented linear Paul trap, we have implemented a procedure which allows us to deterministically load a predetermined number of ions by using the potential shaping capabilities of our segmented ion trap. For single-ion loading, an efficiency of 96.7(7)% has been achieved. These ions are then deterministically extracted out of the trap and focused down to a 1sigma-spot radius of (4.6 \pm 1.3)mum at a distance of 257mm from the trap center. Compared to former measurements without ion optics, the einzel-lens is focusing down the single-ion beam by a factor of 12. Due to the small beam divergence and narrow velocity distribution of our ion source, chromatic and spherical aberration at the einzel-lens is vastly reduced, presenting a promising starting point for focusing single ions on their way to a substrate.

Trapped, laser-cooled atoms and ions are quantum systems which can be experimentally controlled with an as yet unmatched degree of precision. Due to the control of the motion and the internal degrees of freedom, these quantum systems can be adequately described by a well known Hamiltonian. In this colloquium, we present powerful numerical tools for the optimization of the external control of the motional and internal states of trapped neutral atoms, explicitly applied to the case of trapped laser-cooled ions in a segmented ion-trap. We then delve into solving inverse problems, when optimizing trapping potentials for ions. Our presentation is complemented by a quantum mechanical treatment of the wavepacket dynamics of a trapped ion. Efficient numerical solvers for both time-independent and time-dependent problems are provided. Shaping the motional wavefunctions and optimizing a quantum gate is realized by the application of quantum optimal control techniques. The numerical methods presented can also be used to gain an intuitive understanding of quantum experiments with trapped ions by performing virtual simulated experiments on a personal computer. Code and executables are supplied as supplementary online material (http://kilian-singer.de/ent).

We propose a geometric phase gate in a decoherence-free subspace with trapped ions. The quantum information is encoded in the Zeeman sublevels of the ground-state and two physical qubits to make up one logical qubit with ultra long coherence time. Single- and two-qubit operations together with the transport and splitting of linear ion crystals allow for a robust and decoherence-free scalable quantum processor. For the ease of the phase gate realization we employ one Raman laser field on four ions simultaneously, i.e. no tight focus for addressing. The decoherence-free subspace is left neither during gate operations nor during the transport of quantum information.

Using a segmented ion trap with mK laser-cooled ions we have realised a novel single ion source which can deterministically deliver a wide range of ion species, isotopes or ionic molecules [Schnitzler et al., Phys. Rev. Lett. 102, 070501 (2009)]. Experimental data is discussed in detail and compared with numerical simulations of ion trajectories. For the novel ion source we investigate numerically the influence of various extraction parameters on fluctuations in velocity and position of the beam. We present specialized ion optics and show from numerical simulations that nm resolution is achievable. The Paul trap, which is used as a single ion source, together with the presented ion optics, constitutes a promising candidate for a deterministic ion implantation method for applications in solid state quantum computing or classical nano-electronic devices.

We present schemes to prepare two-dimensional cluster states [H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. {\bf 86}, 910 (2001)] with atomic ions confined in a micro-structured linear ion trap and coupled by an engineered spin-spin interaction. In particular, we show how to prepare a $n \times 2$ cluster state by creating a linear cluster state and adding third-neighbor entanglement using selective recoupling techniques. The scheme is based on the capabilities provided by segmented linear Paul traps to confine ions in local potential wells and to separate and transport ions between these wells. Furthermore, we consider creating 3 and 4-qubit cluster states by engineering the coupling matrix such that through the periodicity of the time evolution unwanted couplings are canceled. All entangling operations are achieved by switching of voltages and currents, and do not require interaction with laser light.

We propose a scheme to investigate the nonequilibrium work distribution of a quantum particle under well controlled transformations of the external potential, exploiting the versatility of a single ion in a segmented linear Paul trap. We describe in detail how the motional quantum state of a single ion can be prepared, manipulated and finally read out to fully determine the free energy difference in both harmonic and anharmonic potentials. Uniquely to our system, we show how an ion may be immersed in an engineered laser--field reservoir. Trapped ions therefore represent an ideal tool for investigating the Jarzynski equality in open and closed quantum systems.

Miniaturized ion trap arrays with many trap segments present a promising architecture for scalable quantum information processing. The miniaturization of segmented linear Paul traps allows partitioning the microtrap in different storage and processing zones. The individual position control of many ions - each of them carrying qubit information in its long-lived electronic levels - by the external trap control voltages is important for the implementation of next generation large-scale quantum algorithms. We present a novel scalable microchip multi-segmented ion trap with two different adjacent zones, one for the storage and another dedicated for the processing of quantum information using single ions and linear ion crystals: A pair of radio-frequency driven electrodes and 62 independently controlled DC electrodes allows shuttling of single ions or linear ion crystals with numerically designed axial potentials at axial and radial trap frequencies of a few MHz. We characterize and optimize the microtrap using sideband spectroscopy on the narrow S1/2 <-> D5/2 qubit transition of the 40Ca+ ion, demonstrate coherent single qubit Rabi rotations and optical cooling methods. We determine the heating rate using sideband cooling measurements to the vibrational ground state which is necessary for subsequent two-qubit quantum logic operations. The applicability for scalable quantum information processing is proven.

Single ions held in linear Paul traps are promising candidates for a future quantum computer. Here, we discuss a two-layer microstructured segmented linear ion trap. The radial and axial potentials are obtained from numeric field simulations and the geometry of the trap is optimized. As the trap electrodes are segmented in the axial direction, the trap allows the transport of ions between different spatial regions. Starting with realistic numerically obtained axial potentials, we optimize the transport of an ion such that the motional degrees of freedom are not excited, even though the transport speed far exceeds the adiabatic regime. In our optimization we achieve a transport within roughly two oscillation periods in the axial trap potential compared to typical adiabatic transports that take of the order 100 oscillations. Furthermore heating due to quantum mechanical effects is estimated and suppression strategies are proposed.

An excited-state atom whose emitted light is back-reflected by a distant mirror can experience trapping forces, because the presence of the mirror modifies both the electromagnetic vacuum field and the atom's own radiation reaction field. We demonstrate this mechanical action using a single trapped barium ion. We observe the trapping conditions to be notably altered when the distant mirror is shifted by an optical wavelength. The well-localised barium ion enables the spatial dependence of the forces to be measured explicitly. The experiment has implications for quantum information processing and may be regarded as the most elementary optical tweezers.

We examine in detail the theory of the intrinsic non-linearities in the dynamics of trapped ions due to the Coulomb interaction. In particular the possibility of mode-mode coupling, which can be a source of decoherence in trapped ion quantum computation, or, alternatively, can be exploited for parametric down-conversion of phonons, is discussed and conditions under which such coupling is possible are derived.