R. Blatt

Symmetry-protected topological phases have fundamentally changed our understanding of quantum matter. An archetypal example of such a quantum phase of matter is the Haldane phase, containing the spin-1 Heisenberg chain. The intrinsic quantum nature of such phases, however, often makes it challenging to study them using classical means. Here, we use trapped-ion qutrits to natively engineer spin-1 chains within the Haldane phase. Using a scalable, deterministic procedure to prepare the Affleck-Kennedy-Lieb-Tasaki (AKLT) state within the Haldane phase, we study the topological features of this system on a qudit quantum processor. Notably, we verify the long-range string order of the state, despite its short-range correlations, and observe spin fractionalization of the physical spin-1 particles into effective qubits at the chain edges, a defining feature of this system. The native realization of Haldane physics on a qudit quantum processor and the scalable preparation procedures open the door to the efficient exploration of a wide range of systems beyond spin-1/2

Well isolated quantum systems are exquisite sensors of electromagnetic fields. In this work, we use a single trapped ion for characterizing chirped ultraviolet (UV) picosecond laser pulses. The frequency swept pulses resonantly drive a strong dipole transition via rapid adiabatic passage, resulting in near deterministic population exchange caused by absorption or stimulated emission of photons. When subjecting an ion to counterpropagating pulse pairs, we observe the loss and revival of atomic coherence as a function of the pulse pair spatial overlap enabling quantification of the temporal pulse broadening caused by a frequency chirp in shaped UV pulses with a very low peak power. We find good agreement between measured and applied chirp. The ultrafast population exchange imparts an impulsive force where the estimated change in the mean phonon numbers of 0.5 is measured for two pairs of pulses. The resonant ultrafast kicks could be applied to matter wave interferometry experiments and present a step towards ultrafast entanglement operations in trapped ions.

Scaling trapped-ion quantum computing will require robust trapping of at least hundreds of ions over long periods, while increasing the complexity and functionality of the trap itself. Symmetric 3D structures enable high trap depth, but microfabrication techniques are generally better suited to planar structures that produce less ideal conditions for trapping. We present an ion trap fabricated on stacked 8-inch wafers in a large-scale MEMS microfabrication process that provides reproducible traps at a large volume. Electrodes are patterned on the surfaces of two opposing wafers bonded to a spacer, forming a 3D structure with 2.5 micrometer standard deviation in alignment across the stack. We implement a design achieving a trap depth of 1 eV for a calcium-40 ion held at 200 micrometers from either electrode plane. We characterize traps, achieving measurement agreement with simulations to within +/-5% for mode frequencies spanning 0.6--3.8 MHz, and evaluate stray electric field across multiple trapping sites. We measure motional heating rates over an extensive range of trap frequencies, and temperatures, observing 40 phonons/s at 1 MHz and 185 K. This fabrication method provides a highly scalable approach for producing a new generation of 3D ion traps.

We present an optical scheme to detect the oscillations of a two-ion string confined in a linear Paul trap. The motion is detected by analyzing the intensity correlations in the fluorescence light emitted by one or two ions in the string. We present measurements performed under continuous Doppler cooling and under pulsed illumination. We foresee several direct applications of this detection method, including motional analysis of multi-ion species or coupled mechanical oscillators, and sensing of mechanical correlations.

We describe a technique to measure the position of a dipolar scatterer based on self-homodyne detection of the scattered light. The method can theoretically reach the Heisenberg limit, at which information gained about the position is constrained only by the back-action of the scattered light. The technique has applications in the fields of levitated optomechanics and trapped ions and is generally applicable to the position determination of confined light scatterers.

We demonstrate an optical method for detecting the mechanical oscillations of an atom with single-phonon sensitivity. The measurement signal results from the interference between the light scattered by a single trapped atomic ion and that of its mirror image. The motion of the atom modulates the interference path length and hence the photon detection rate. We detect the oscillations of the atom in the Doppler cooling limit and reconstruct average trajectories in phase space. We demonstrate single-phonon sensitivity near the ground state of motion after EIT cooling. These results could be applied for motion detection of other light scatterers of fundamental interest, such as trapped nanoparticles.

The conventional approach to perform two-qubit gate operations in trapped ions relies on exciting the ions on motional sidebands with laser light, which is an inherently slow process. One way to implement a fast entangling gate protocol requires a suitable pulsed laser to increase the gate speed by orders of magnitude. However, the realization of such a fast entangling gate operation presents a big technical challenge, as such the required laser source is not available off-the-shelf. For this, we have engineered an ultrafast entangling gate source based on a frequency comb. The source generates bursts of several hundred mode-locked pulses with pulse energy $\sim$800 pJ at 5 GHz repetition rate at 393.3 nm and complies with all requirements for implementing a fast two-qubit gate operation. Using a single, chirped ultraviolet pulse, we demonstrate a rapid adiabatic passage in a Ca$^+$ ion. To verify the applicability and projected performance of the laser system for inducing entangling gates we run simulations based on our source parameters. The gate time can be faster than a trap period with an error approaching $10^{-4}$.

We propose a new protocol to implement ultra-fast two-qubit phase gates with trapped ions using spin-dependent kicks induced by resonant transitions. By only optimizing the allocation of the arrival times in a pulse train sequence the gate is implemented in times faster than the trapping oscillation period $T<2π/ω$. Such gates allow us to increase the number of gate operations that can be completed within the coherence time of the ion-qubits favoring the development of scalable quantum computers.

We present experiments on polarization gradient cooling of Ca$^+$ multi-ion Coulomb crystals in a linear Paul trap. Polarization gradient cooling of the collective modes of motion whose eigenvectors have overlap with the symmetry axis of the trap is achieved by two counter-propagating laser beams with mutually orthogonal linear polarizations that are blue-detuned from the S$_{1/2}$ to P$_{1/2}$ transition. We demonstrate cooling of linear chains of up to 51 ions and 2D-crystals in zig-zag configuration with 22 ions. The cooling results are compared with numerical simulations and the predictions of a simple model of cooling in a moving polarization gradient.

Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented - a classical simulation approach - applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed.

Quantum computers are on the brink of surpassing the capabilities of even the most powerful classical computers. This naturally raises the question of how one can trust the results of a quantum computer when they cannot be compared to classical simulation. Here we present a verification technique that exploits the principles of measurement-based quantum computation to link quantum circuits of different input size, depth, and structure. Our approach enables consistency checks of quantum computations within a device, as well as between independent devices. We showcase our protocol by applying it to five state-of-the-art quantum processors, based on four distinct physical architectures: nuclear magnetic resonance, superconducting circuits, trapped ions, and photonics, with up to 6 qubits and 200 distinct circuits.

Trapped ions are a well-studied and promising system for the realization of a scalable quantum computer. Faster quantum gates would greatly improve the applicability of such a system and allow for greater flexibility in the number of calculation steps. In this paper we present a pulsed laser system, delivering picosecond pulses at a repetition rate of 5 GHz and resonant to the S$_{1/2}$ to P$_{3/2}$ transition in Ca$^+$ for coherent population transfer to implement fast phase gate operations. The optical pulse train is derived from a mode-locked, stabilized optical frequency comb and inherits its frequency stability. Using a single trapped ion, we implement three different techniques for measuring the ion-laser coupling strength and characterizing the pulse train emitted by the laser, and show how all requirements can be met for an implementation of a fast phase gate operation.

Measuring and understanding electric field noise from bulk material and surfaces is important for many areas of physics. In this work, we introduce a method to detect in situ different sources of electric field noise using a single trapped ion as a sensor. We demonstrate the probing of electric field noise as small as $S_E = 5.2(11)\times 10^{-16}\,\text{V}^2\text{m}^{-2}\text{Hz}^{-1}$, the lowest noise level observed with a trapped ion to our knowledge. Our setup incorporates a controllable noise source utilizing a high-temperature superconductor. This element allows us, first, to benchmark and validate the sensitivity of our probe. Second, to probe non-invasively bulk properties of the superconductor, observing for the first time a superconducting transition with an ion. For temperatures below the transition, we use our setup to assess different surface noise processes. The measured noise shows a crossover regime in the frequency domain, which cannot be explained by existing surface noise models. Our results open perspectives for new models in surface science and pave the way to test them experimentally.

A quantitative assessment of the progress of small prototype quantum processors towards fault-tolerant quantum computation is a problem of current interest in experimental and theoretical quantum information science. We introduce a necessary and fair criterion for quantum error correction (QEC), which must be achieved in the development of these quantum processors before their sizes are sufficiently big to consider the well-known QEC threshold. We apply this criterion to benchmark the ongoing effort in implementing QEC with topological color codes using trapped-ion quantum processors and, more importantly, to guide the future hardware developments that shall be required in order to demonstrate beneficial QEC with small topological quantum codes. In doing so, we present a thorough description of a realistic trapped-ion toolbox for QEC, and a physically-motivated error model that goes beyond standard simplifications in the QEC literature. Our large-scale numerical analysis shows that two-species trapped-ion crystals in high-optical aperture segmented traps, with the improvements hereby described, are a very promising candidate for fault-tolerant quantum computation.

Quantum state tomography (QST) is the gold standard technique for obtaining an estimate for the state of small quantum systems in the laboratory. Its application to systems with more than a few constituents (e.g. particles) soon becomes impractical as the effort required grows exponentially in the number of constituents. Developing more efficient techniques is particularly pressing as precisely-controllable quantum systems that are well beyond the reach of QST are emerging in laboratories. Motivated by this, there is a considerable ongoing effort to develop new characterisation tools for quantum many-body systems. Here we demonstrate Matrix Product State (MPS) tomography, which is theoretically proven to allow the states of a broad class of quantum systems to be accurately estimated with an effort that increases efficiently with constituent number. We first prove that this broad class includes the out-of-equilbrium states produced by 1D systems with finite-range interactions, up to any fixed point in time. We then use the technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled spins (qubits): a size far beyond the reach of QST. Our results reveal the dynamical growth of entanglement and description complexity as correlations spread out during a quench: a necessary condition for future beyond-classical performance. MPS tomography should find widespread use to study large quantum many-body systems and to benchmark and verify quantum simulators and computers.

Well-controlled quantum devices with their increasing system size face a new roadblock hindering further development of quantum technologies: The effort of quantum tomography---the characterization of processes and states within a quantum device---scales unfavorably to the point that state-of-the-art systems can no longer be treated. Quantum compressed sensing mitigates this problem by reconstructing the state from an incomplete set of observables. In this work, we present an experimental implementation of compressed tomography of a seven qubit system---the largest-scale realization to date---and we introduce new numerical methods in order to scale the reconstruction to this dimension. Originally, compressed sensing has been advocated for density matrices with few non-zero eigenvalues. Here, we argue that the low-rank estimates provided by compressed sensing can be appropriate even in the general case. The reason is that statistical noise often allows only for the leading eigenvectors to be reliably reconstructed: We find that the remaining eigenvectors behave in a way consistent with a random matrix model that carries no information about the true state. We report a reconstruction of quantum states from a topological color code of seven qubits, prepared in a trapped ion architecture, based on tomographically incomplete data involving 127 Pauli basis measurement settings only, repeated 100 times each.

We report on the design of a cryogenic setup for trapped ion quantum computing containing a segmented surface electrode trap. The heat shield of our cryostat is designed to attenuate alternating magnetic field noise, resulting in 120~dB reduction of 50~Hz noise along the magnetic field axis. We combine this efficient magnetic shielding with high optical access required for single ion addressing as well as for efficient state detection by placing two lenses each with numerical aperture 0.23 inside the inner heat shield. The cryostat design incorporates vibration isolation to avoid decoherence of optical qubits due to the motion of the cryostat. We measure vibrations of the cryostat of less than $\pm$20~nm over 2~s. In addition to the cryogenic apparatus, we describe the setup required for an operation with $^{\mathrm{40}}$Ca$^{\mathrm{+}}$ and $^{\mathrm{88}}$Sr$^{\mathrm{+}}$ ions. The instability of the laser manipulating the optical qubits in $^{\mathrm{40}}$Ca$^{\mathrm{+}}$ is characterized yielding a minimum of its Allan deviation of 2.4$\cdot$10$^{\mathrm{-15}}$ at 0.33~s. To evaluate the performance of the apparatus, we trapped $^{\mathrm{40}}$Ca$^{\mathrm{+}}$ ions, obtaining a heating rate of 2.14(16)~phonons/s and a Gaussian decay of the Ramsey contrast with a 1/e-time of 18.2(8)~ms.

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator, this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report the first experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields in favour of exotic long-range interactions, which have a direct and efficient implementation on an ion trap architecture. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulating high-energy theories with atomic physics experiments, the long-term vision being the extension to real-time quantum simulations of non-Abelian lattice gauge theories.

Electromagnetically-induced-transparency (EIT) cooling is a ground-state cooling technique for trapped particles. EIT offers a broader cooling range in frequency space compared to more established methods. In this work, we experimentally investigate EIT cooling in strings of trapped atomic ions. In strings of up to 18 ions, we demonstrate simultaneous ground state cooling of all radial modes in under 1 ms. This is a particularly important capability in view of emerging quantum simulation experiments with large numbers of trapped ions. Our analysis of the EIT cooling dynamics is based on a novel technique enabling single-shot measurements of phonon numbers, by rapid adiabatic passage on a vibrational sideband of a narrow transition.

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits as errors can be fully characterized. For multi-qubit operations, though, this is no longer the case as in the most general case analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g.~a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum information processing.

We investigate the dynamics of an ion sympathetically cooled by another laser-cooled ion or small ion crystal. To this end, we develop simple models of the cooling dynamics in the limit of weak Coulomb interactions. Experimentally, we create a two-ion crystal of Ca$^+$ and Al$^+$ by photo-ionization of neutral atoms produced by laser ablation. We characterize the velocity distribution of the laser-ablated atoms crossing the trap by time-resolved fluorescence spectroscopy. We observe neutral atom velocities much higher than the ones of thermally heated samples and find as a consequence long sympathethic cooling times before crystallization occurs. Our key result is a new technique for detecting the loading of an initially hot ion with energy in the eV range by monitoring the motional state of a Doppler-cooled ion already present in the trap. This technique not only detects the ion but also provides information about dynamics of the sympathetic cooling process.

The static and dynamic properties of many-body quantum systems are often well described by collective excitations, known as quasiparticles. Engineered quantum systems offer the opportunity to study such emergent phenomena in a precisely controlled and otherwise inaccessible way. We present a spectroscopic technique to study artificial quantum matter and use it for characterizing quasiparticles in a many-body system of trapped atomic ions. Our approach is to excite combinations of the system's fundamental quasiparticle eigenmodes, given by delocalised spin waves. By observing the dynamical response to superpositions of such eigenmodes, we extract the system dispersion relation, magnetic order, and even detect signatures of quasiparticle interactions. Our technique is not limited to trapped ions, and it is suitable for verifying quantum simulators by tuning them into regimes where the collective excitations have a simple form.

Electric-field noise near surfaces is a common problem in diverse areas of physics, and a limiting factor for many precision measurements. There are multiple mechanisms by which such noise is generated, many of which are poorly understood. Laser-cooled, trapped ions provide one of the most sensitive systems to probe electric-field noise at MHz frequencies and over a distance range 30 - 3000 $μ$m from the surface. Over recent years numerous experiments have reported spectral densities of electric-field noise inferred from ion heating-rate measurements and several different theoretical explanations for the observed noise characteristics have been proposed. This paper provides an extensive summary and critical review of electric-field noise measurements in ion traps, and compares these experimental findings with known and conjectured mechanisms for the origin of this noise. This reveals that the presence of multiple noise sources, as well as the different scalings added by geometrical considerations, complicate the interpretation of these results. It is thus the purpose of this review to assess which conclusions can be reasonably drawn from the existing data, and which important questions are still open. In so doing it provides a framework for future investigations of surface-noise processes.

The key to explaining a wide range of quantum phenomena is understanding how entanglement propagates around many-body systems. Furthermore, the controlled distribution of entanglement is of fundamental importance for quantum communication and computation. In many situations, quasiparticles are the carriers of information around a quantum system and are expected to distribute entanglement in a fashion determined by the system interactions. Here we report on the observation of magnon quasiparticle dynamics in a one-dimensional many-body quantum system of trapped ions representing an Ising spin model. Using the ability to tune the effective interaction range, and to prepare and measure the quantum state at the individual particle level, we observe new quasiparticle phenomena. For the first time, we reveal the entanglement distributed by quasiparticles around a many-body system. Second, for long-range interactions we observe the divergence of quasiparticle velocity and breakdown of the light-cone picture that is valid for short-range interactions. Our results will allow experimental studies of a wide range of phenomena, such as quantum transport, thermalisation, localisation and entanglement growth, and represent a first step towards a new quantum-optical regime with on-demand quasiparticles with tunable non-linear interactions.

We report on the experimental violation of multipartite Bell inequalities by entangled states of trapped ions. First we consider resource states for measurement-based quantum computation of between 3 and 7 ions and show that all strongly violate a Bell-type inequality for graph states, where the criterion for violation is a sufficiently high fidelity. Second we analyze GHZ states of up to 14 ions generated in a previous experiment using stronger Mermin-Klyshko inequalities, and show that in this case the violation of local realism increases exponentially with system size. These experiments represent a violation of multipartite Bell-type inequalities of deterministically prepared entangled states. In addition, the detection loophole is closed.

Measurement-based quantum computation (MBQC) represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of MBQC using deterministically generated graph states of up to 7 qubits, in a system of trapped atomic ions. Firstly we implement a universal set of operations for quantum computing. Secondly we demonstrate a family of measurement-based quantum error correction codes, and show their improved performance as the code length is increased. We show that all our graph states violate a multipartite Bell inequality and are therefore capable of information processing tasks that cannot be described by a local hidden variable model. The methods presented can directly be scaled up to generate graph states of several tens of qubits.

We demonstrate precise control of the coupling of each of two trapped ions to the mode of an optical resonator. When both ions are coupled with near-maximum strength, we generate ion--ion entanglement heralded by the detection of two orthogonally polarized cavity photons. The entanglement fidelity with respect to the Bell state $Ψ^+$ reaches $F \geq (91.9\pm2.5)%$. This result represents an important step toward distributed quantum computing with cavities linking remote atom-based registers.

Quantum systems in mixed states can be unentangled and yet still correlated in a way that is not possible for classical systems. These correlations can be quantified by the quantum discord and might provide a resource for certain mixed-state quantum information processing tasks. Here we report on the generation of discordant states of two trapped atomic ions via Markovian decoherence processes. While entanglement is strictly non-increasing under such operations, discord can be generated in various forms. Firstly we show that, starting from two classically correlated qubits, it is possible to generate discord by applying decoherence to just one of them. Secondly, even when starting with completely uncorrelated systems, we show that discord can be generated via classically correlated decoherence processes. Finally, the Werner states are created. The generated states can be used as a resource state for quantum information transmission and could be readily extended to more ions.

The ability to detect the interaction of light and matter at the single-particle level is becoming increasingly important for many areas of science and technology. The absorption or emission of a photon on a narrow transition of a trapped ion can be detected with near unit probability, thereby enabling the realization of ultra-precise ion clocks and quantum information processing applications. Extending this sensitivity to broad transitions is challenging due to the difficulty of detecting the rapid photon scattering events in this case. Here, we demonstrate a technique to detect the scattering of a single photon on a broad optical transition with high sensitivity. Our approach is to use an entangled state to amplify the tiny momentum kick an ion receives upon scattering a photon. The method should find applications in spectroscopy of atomic and molecular ions and quantum information processing.

Proposed quantum networks require both a quantum interface between light and matter and the coherent control of quantum states. A quantum interface can be realized by entangling the state of a single photon with the state of an atomic or solid-state quantum memory, as demonstrated in recent experiments with trapped ions, neutral atoms, atomic ensembles, and nitrogen-vacancy spins. The entangling interaction couples an initial quantum memory state to two possible light-matter states, and the atomic level structure of the memory determines the available coupling paths. In previous work, these paths' transition parameters determine the phase and amplitude of the final entangled state, unless the memory is initially prepared in a superposition state, a step that requires coherent control. Here we report the fully tunable entanglement of a single 40Ca+ ion and the polarization state of a single photon within an optical resonator. Our method, based on a bichromatic, cavity-mediated Raman transition, allows us to select two coupling paths and adjust their relative phase and amplitude. The cavity setting enables intrinsically deterministic, high-fidelity generation of any two-qubit entangled state. This approach is applicable to a broad range of candidate systems and thus presents itself as a promising method for distributing information within quantum networks.

Dynamical maps describe general transformations of the state of a physical system, and their iteration can be interpreted as generating a discrete time evolution. Prime examples include classical nonlinear systems undergoing transitions to chaos. Quantum mechanical counterparts show intriguing phenomena such as dynamical localization on the single particle level. Here we extend the concept of dynamical maps to an open-system, many-particle context: We experimentally explore the stroboscopic dynamics of a complex many-body spin model by means of a universal quantum simulator using up to five ions. In particular, we generate long-range phase coherence of spin by an iteration of purely dissipative quantum maps. We also demonstrate the characteristics of competition between combined coherent and dissipative non-equilibrium evolution. This opens the door for studying many-particle non-equilibrium physics and associated dynamical phase transitions with no immediate counterpart in equilibrium condensed matter systems. An error detection and reduction toolbox that facilitates the faithful quantum simulation of larger systems is developed as a first step in this direction.

Faraday rotation of a laser field induced by a single atom is demonstrated by tightly focussing a linearly polarized laser beam onto a laser-cooled ion held in a harmonic Paul trap. The polarization rotation signal is further used to measure the phase-shift associated with electromagnetically-induced-transparency and to demonstrate read-out of the internal state on the qubit transition with a detection fidelity of 98 $\pm$ 1%. These results have direct implications for single atom magnetometery and dispersive read-out of atomic superpositions.

A scheme for entangling distant atoms is realized, as proposed in the seminal paper by Cabrillo et al. [Phys. Rev. A 59, 1025 (1999)]. The protocol is based on quantum interference and detection of a single photon scattered from two effectively one meter distant laser-cooled and trapped atomic ions. The detection of a single photon heralds entanglement of two internal states of the trapped ions with high rate and with a fidelity limited mostly by atomic motion. Control of the entangled state phase is demonstrated by changing the path length of the single-photon interferometer.

We report on the investigation and implementation of a lumped-component, radio-frequency resonator used in a cryogenic vacuum environment to drive an ion trap. The resonator was required to achieve the voltages necessary to trap (about 100 V), while dissipating as little power as possible (< 250 mW). Ultimately a voltage gain of 100 was measured at 5.7 K. Single calcium ions were confined in a trap driven by this device, providing proof of successful resonator operation at low temperature.

We perform high-resolution real-time read-out of the motion of a single trapped and laser-cooled Ba ion. By using an interferometric setup we demonstrate shot-noise limited measurement of thermal oscillations with resolution of 4 times the standard quantum limit. We apply the real-time monitoring for phase control of the ion motion through a feedback loop, suppressing the photon recoil-induced phase diffusion. Due to the spectral narrowing in phase-locked mode, the coherent ion oscillation is measured with resolution of about 0.3 times the standard quantum limit.

A digital quantum simulator is an envisioned quantum device that can be pro- grammed to efficiently simulate any other local system. We demonstrate and investigate the digital approach to quantum simulation in a system of trapped ions. Using sequences of up to 100 gates and 6 qubits, the full time dynamics of a range of spin systems are digitally simulated. Interactions beyond those naturally present in our simulator are accurately reproduced and quantitative bounds are provided for the overall simulation quality. Our results demon- strate the key principles of digital quantum simulation and provide evidence that the level of control required for a full-scale device is within reach.

By tightly focussing a laser field onto a single cold ion trapped in front of a far-distant dielectric mirror, we could observe a quantum electrodynamic effect whereby the ion behaves as the optical mirror of a Fabry-Pérot cavity. We show that the amplitude of the laser field is significantly altered due to a modification of the electromagnetic mode structure around the atom in a novel regime in which the laser intensity is already changed by the atom alone. e propose a direct application of this system as a quantum memory for single photons.

We demonstrate several building blocks for an ion-photon interface based on a trapped Ca ion in an optical cavity. We identify a favorable experimental configuration and measure system parameters, including relative motion of the trapped ion and the resonator mode. A complete spectrum of cavity-assisted Raman transitions between the $4^{2}S_{1/2}$ and $3^{2}D_{5/2}$ manifolds is obtained. On two of these transitions, we generate orthogonally polarized cavity photons, and we demonstrate coherent manipulation of the corresponding pair of atomic states. Possible implementations of atom-photon entanglement and state mapping within the ion-cavity system are discussed.

We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator. We describe extensions to solve problems that are beyond current classical computing capabilities.

Multiparticle entanglement leads to richer correlations than two-particle entanglement and gives rise to striking contradictions with local realism, inequivalent classes of entanglement, and applications such as one-way or topological quantum computing. When exposed to decohering or dissipative environments, multiparticle entanglement yields subtle dynamical features and access to new classes of states and applications. Here, using a string of trapped ions, we experimentally characterize the dynamics of entanglement of a multiparticle state under the influence of decoherence. By embedding an entangled state of four qubits in a decohering environment (via spontaneous decay), we observe a rich dynamics crossing distinctive domains: Bell-inequality violation, entanglement superactivation, bound entanglement, and full separability. We also develop new theoretical tools for characterizing entanglement in quantum states. Our techniques to control the environment can be used to enable novel quantum-computation, state-engineering, and simulation paradigms based on dissipation and decoherence.

We experimentally demonstrate a quantum walk on a line in phase space using one and two trapped ion. A walk with up to 23 steps is realized by subjecting an ion to state-dependent displacement operations interleaved with quantum coin tossing operations. To analyze the ion's motional state after each step we apply a technique that directly maps the probability density distribution onto the ion's internal state. The measured probability distributions and the position's second moment clearly show the non-classical character of the quantum walk. To further highlight the difference between the classical (random) and the quantum walk, we demonstrate the reversibility of the latter. Finally, we extend the quantum walk by using two ions, giving the walker the additional possibility to stay instead of taking a step.

Any residual coupling of a quantum computer to the environment results in computational errors. Encoding quantum information in a so-called decoherence-free subspace provides means to avoid these errors. Despite tremendous progress in employing this technique to extend memory storage times by orders of magnitude, computation within such subspaces has been scarce. Here, we demonstrate the realization of a universal set of quantum gates acting on decoherence-free ion qubits. We combine these gates to realize the first controlled-NOT gate within a decoherence-free, scalable quantum computer.

The Dirac equation is a cornerstone in the history of physics, merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and predicting the existence of anti-matter. Furthermore, it is able to reproduce accurately the spectrum of the hydrogen atom and its realm, relativistic quantum mechanics, is considered as the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an unexpected quivering motion of a free relativistic quantum particle first examined by Schrödinger. These and other predictions would be difficult to observe in real particles, while constituting key fundamental examples to understand relativistic quantum effects. Recent years have seen an increased interest in simulations of relativistic quantum effects in different physical setups, where parameter tunability allows accessibility to different physical regimes. Here, we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion, which is set to behave as a free relativistic quantum particle. We measure as a function of time the particle position and study Zitterbewegung for different initial superpositions of positive and negative energy spinor states, as well as the cross-over from relativistic to nonrelativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate elegant textbook examples of relativistic quantum physics.

In the endeavour to scale up the number of qubits in an ion-based quantum computer several groups have started to develop miniaturized ion traps for extended spatial control and manipulation of the ions. Shuttling and separation of ion strings have been the foremost issues in linear-trap arrangements and some prototypes of junctions have been demonstrated for the extension of ion motion to two dimensions (2D). While junctions require complex trap structures, small extensions to the 1D motion can be accomplished in simple linear trap arrangements. Here, control of the extended field in a planar, linear chip trap is used to shuttle ions in 2D. With this approach, the order of ions in a string is deterministically reversed. Optimized potentials are theoretically derived and simulations show that the reordering can be carried out adiabatically. The control over individual ion positions in a linear trap presents a new tool for ion-trap quantum computing. The method is also expected to work with mixed crystals of different ion species and as such could have applications for sympathetic cooling of an ion string.

The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles. Along this line, some types of hidden variable models have been experimentally ruled out. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the results of Kochen, Specker, and Bell show that non-contextuality is in conflict with quantum mechanics. The conflict resides in the structure of the theory and is independent of the properties of special states. It has been debated whether the Kochen-Specker theorem could be experimentally tested at all. Only recently, first tests of quantum contextuality have been proposed and undertaken with photons and neutrons. Yet these tests required the generation of special quantum states and left various loopholes open. Here, using trapped ions, we experimentally demonstrate a state-independent conflict with non-contextuality. The experiment is not subject to the detection loophole and we show that, despite imperfections and possible measurement disturbances, our results cannot be explained in non-contextual terms.

We report measurements of an intensity-field correlation function of the resonance fluorescence of a single trapped Ba+ ion. Detection of a photon prepares the atom in its ground state and we observe its subsequent evolution under interaction with a laser field of well defined phase. We record the regression of the resonance fluorescence source field. This provides a direct measurement of the field of the radiating dipole of a single atom and exhibits its strong non-classical behavior. In the experimental setup an interference measurement is conditioned on a fluorescence photon detection. The third-order correlation function thus recorded demonstrates an aspect of wave-particle duality at the single-atom, single-photon level.

We observe quantum interference of photons emitted by two continuously laser-excited single ions, independently trapped in distinct vacuum vessels. High contrast two-photon interference is observed in two experiments with different ion species, calcium and barium. Our experimental findings are quantitatively reproduced by Bloch equation calculations. In particular, we show that the coherence of the individual resonance fluorescence light field is determined from the observed interference.

Quantum computers hold the promise to solve certain computational task much more efficiently than classical computers. We review the recent experimental advancements towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments are discussed. Furthermore, we review some implementations of quantum algorithms such as a deterministic teleportation of quantum information and an error correction scheme.

We perform precision measurements of the branching ratios of the 4P3/2 level decay of a single 40Ca+ ion suspended in a linear Paul trap. High precision is achieved by a novel technique based on monitoring the population transfer when repeatedly pumping the ion between different internal states. The branching fractions into the 4S1/2, 3D5/2 and 3D3/2 levels are found to be 0.9347(3), 0.0587(2) and 0.00661(4), respectively. For the branching ratio A(P3/2-S1/2)/\sum_J A(P3/2-D_J)=14.31(5), we find a forty-fold improvement in accuracy as compared to the best previous measurement.

We report on the first absolute transition frequency measurement at the 10^{-15} level with a single, laser-cooled 40Ca+ ion in a linear Paul trap. For this measurement, a frequency comb is referenced to the transportable Cs atomic fountain clock of LNE-SYRTE and is used to measure the S1/2-D5/2 electric-quadrupole transition frequency. After the correction of systematic shifts, the clock transition frequency f_Ca+ = 411 042 129 776 393.2 (1.0) Hz is obtained, which corresponds to a fractional uncertainty within a factor of three of the Cs standard. Future improvements are expected to lead to an uncertainty surpassing the best Cs fountain clocks. In addition, we determine the Lande g-factor of the D5/2 level to be gD5/2=1.2003340(3).

For quantum information processing (QIP) with trapped ions, the isotope 43Ca+ offers the combined advantages of a quantum memory with long coherence time, a high fidelity read out and the possibility of performing two qubit gates on a quadrupole transition with a narrow-band laser. Compared to other ions used for quantum computing, 43Ca+ has a relatively complicated level structure. In this paper we discuss how to meet the basic requirements for QIP and demonstrate ground state cooling, robust state initialization and efficient read out for the hyperfine qubit with a single 43Ca+ ion. A microwave field and a Raman light field are used to drive qubit transitions, and the coherence times for both fields are compared. Phase errors due to interferometric instabilities in the Raman field generation do not limit the experiments on a time scale of 100 ms. We find a quantum information storage time of many seconds for the hyperfine qubit.

Algorithms for quantum information processing are usually decomposed into sequences of quantum gate operations, most often realized with single- and two- qubit gates[1]. While such operations constitute a universal set for quantum computation, gates acting on more than two qubits can simplify the implementation of complex quantum algorithms[2]. Thus, a single three-qubit operation can replace a complex sequence of two-qubit gates, which in turn promises faster execution with potentially higher Fidelity. One important three-qubit operation is the quantum Toffoli gate which performs a NOT operation on a target qubit depending on the state of two control qubits. Here we present the first experimental realization of the quantum Toffoli gate in an ion trap quantum computer. Our implementation is particular effcient as we directly encode the relevant logic information in the motion of the ion string. [1] DiVincenzo, D. P. Two-bit gates are universal for quantum computation. cond-mat/9407022, Phys.Rev. A 51, 1015-1022 (1995). [2] Chiaverini, J. et al. Realization of quantum error correction. Nature 432, 602-605 (2004).

Today ion traps are among the most promising physical systems for constructing a quantum device harnessing the computing power inherent in the laws of quantum physics. The standard circuit model of quantum computing requires a universal set of quantum logic gates for the implementation of arbitrary quantum operations. As in classical models of computation, quantum error correction techniques enable rectification of small imperfections in gate operations, thus allowing for perfect computation in the presence of noise. For fault-tolerant computation, it is commonly believed that error thresholds ranging between 10^-4 and 10^-2 will be required depending on the noise model and the computational overhead for realizing the quantum gates. Up to now, all experimental implementations have fallen short of these requirements. Here, we report on a Molmer-Sorensen type gate operation entangling ions with a fidelity of 99.3(1)% which together with single-qubit operations forms a universal set of quantum gates. The gate operation is performed on a pair of qubits encoded in two trapped calcium ions using a single amplitude-modulated laser beam interacting with both ions at the same time. A robust gate operation, mapping separable states onto maximally entangled states is achieved by adiabatically switching the laser-ion coupling on and off. We analyse the performance of a single gate and concatenations of up to 21 gate operations. The gate mechanism holds great promise not only for two-qubit but also for multi-qubit operations.

We propose a geometric phase gate of two ion qubits that are encoded in two levels linked by an optical dipole-forbidden transition. Compared to hyperfine geometric phase gates mediated by electric dipole transitions, the gate has many interesting properties, such as very low spontaneous emission rates, applicability to magnetic field insensitive states, and use of a co-propagating laser beam geometry. We estimate that current technology allows for infidelities of around 10$^{-4}$.

We discuss techniques that allow for long coherence times in laser spectroscopy experiments with two trapped ions. We show that for this purpose not only entangled ions prepared in decoherence-free subspaces can be used but also a pair of ions that are not entangled but subject to the same kind of phase noise. We apply this technique to a measurement of the electric quadrupole moment of the 3d D5/2 state of 40Ca+ and to a measurement of the linewidth of an ultrastable laser exciting a pair of 40Ca+ ions.

We experimentally investigate nonlinear couplings between vibrational modes of strings of cold ions stored in linear ion traps. The nonlinearity is caused by the ions' Coulomb interaction and gives rise to a Kerr-type interaction Hamiltonian H = n_r*n_s, where n_r,n_s are phonon number operators of two interacting vibrational modes. We precisely measure the resulting oscillation frequency shift and observe a collapse and revival of the contrast in a Ramsey experiment. Implications for ion trap experiments aiming at high-fidelity quantum gate operations are discussed.

The performance of a quantum teleportation algorithm implemented on an ion trap quantum computer is investigated. First the algorithm is analyzed in terms of the teleportation fidelity of six input states evenly distributed over the Bloch sphere. Furthermore, a quantum process tomography of the teleportation algorithm is carried out which provides almost complete knowledge about the algorithm.

Entanglement is recognized as a key resource for quantum computation and quantum cryptography. For quantum metrology, the use of entangled states has been discussed and demonstrated as a means of improving the signal-to-noise ratio. In addition, entangled states have been used in experiments for efficient quantum state detection and for the measurement of scattering lengths. In quantum information processing, manipulation of individual quantum bits allows for the tailored design of specific states that are insensitive to the detrimental influences of an environment. Such 'decoherence-free subspaces' protect quantum information and yield significantly enhanced coherence times. Here we use a decoherence-free subspace with specifically designed entangled states to demonstrate precision spectroscopy of a pair of trapped Ca+ ions; we obtain the electric quadrupole moment, which is of use for frequency standard applications. We find that entangled states are not only useful for enhancing the signal-to-noise ratio in frequency measurements - a suitably designed pair of atoms also allows clock measurements in the presence of strong technical noise. Our technique makes explicit use of non-locality as an entanglement property and provides an approach for 'designed' quantum metrology.

The hyperfine structure of the S1/2-D5/2 quadrupole transition at 729 nm in 43Ca+ has been investigated by laser spectroscopy using a single trapped 43Ca+ ion. We determine the hyperfine structure constants of the metastable level as A=-3.8931(2) MHz and B=-4.241(4) MHz. The isotope shift of the transition with respect to 40Ca+ was measured to be 4134.713(5) MHz. We demonstrate the existence of transitions that become independent of the first-order Zeeman shift at non-zero low magnetic fields. These transitions might be better suited for building a frequency standard than the well-known 'clock transitions' between m=0 levels at zero magnetic field.

A crucial building block for quantum information processing with trapped ions is a controlled-NOT quantum gate. In this paper, two different sequences of laser pulses implementing such a gate operation are analyzed using quantum process tomography. Fidelities of up to 92.6(6)% are achieved for single gate operations and up to 83.4(8)% for two concatenated gate operations. By process tomography we assess the performance of the gates for different experimental realizations and demonstrate the advantage of amplitude--shaped laser pulses over simple square pulses. We also investigate whether the performance of concatenated gates can be inferred from the analysis of the single gates.

Among the various kinds of entangled states, the 'W state' plays an important role as its entanglement is maximally persistent and robust even under particle loss. Such states are central as a resource in quantum information processing and multiparty quantum communication. Here we report the scalable and deterministic generation of four-, five-, six-, seven- and eight-particle entangled states of the W type with trapped ions. We obtain the maximum possible information on these states by performing full characterization via state tomography, using individual control and detection of the ions. A detailed analysis proves that the entanglement is genuine. The availability of such multiparticle entangled states, together with full information in the form of their density matrices, creates a test-bed for theoretical studies of multiparticle entanglement. Independently, -Greenberger-Horne-Zeilinger- entangled states with up to six ions have been created and analysed in Boulder.

We report adiabatic passage experiments with a single trapped $^{40}$Ca$^+$ ion. By applying a frequency chirped laser pulse with a Gaussian amplitude envelope we reach a transfer efficiency of 0.990(10) on an optical transition from the electronic ground state S$_{1/2}$ to the metastable state D$_{5/2}$. This transfer method is shown to be insensitive to the accurate setting of laser parameters, and therefore is suitable as a robust tool for ion based quantum computing.

It is common belief among physicists that entangled states of quantum systems loose their coherence rather quickly. The reason is that any interaction with the environment which distinguishes between the entangled sub-systems collapses the quantum state. Here we investigate entangled states of two trapped Ca$^+$ ions and observe robust entanglement lasting for more than 20 seconds.

A hybrid quantum computing scheme is studied where the hybrid qubit is made of an ion trap qubit serving as the information storage and a solid-state charge qubit serving as the quantum processor, connected by a superconducting cavity. In this paper, we extend our previous work [1] and study the decoherence, coupling and scalability of the hybrid system. We present our calculations of the decoherence of the coupled ion - charge system due to the charge fluctuations in the solid-state system and the dissipation of the superconducting cavity under laser radiation. A gate scheme that exploits rapid state flips of the charge qubit to reduce decoherence by the charge noise is designed. We also study a superconducting switch that is inserted between the cavity and the charge qubit and provides tunable coupling between the qubits. The scalability of the hybrid scheme is discussed together with several potential experimental obstacles in realizing this scheme.

We report measurements of the lifetimes of the 3d $^2$D$_{5/2}$ and 3d $^2$D$_{3/2}$ metastable states of a single laser-cooled $^{40}$Ca$^+$ ion in a linear Paul trap. We introduce a new measurement technique based on high-efficiency quantum state detection after coherent excitation to the D$_{5/2}$ state or incoherent shelving in the D$_{3/2}$ state, and subsequent free, unperturbed spontaneous decay. The result for the natural lifetime of the D$_{5/2}$ state of 1168(9) ms agrees excellently with the most precise published value. The lifetime of the D$_{3/2}$ state is measured with a single ion for the first time and yields 1176(11) ms which improves the statistical uncertainty of previous results by a factor of four. We compare these experimental lifetimes to high-precision ab initio all order calculations and find a very good agreement. These calculations represent an excellent test of high-precision atomic theory and will serve as a benchmark for the study of parity nonconservation in Ba$^+$ which has similar atomic structure.

We report the realization of an elementary quantum processor based on a linear crystal of trapped ions. Each ion serves as a quantum bit (qubit) to store the quantum information in long lived electronic states. We present the realization of single-qubit and of universal two-qubit logic gates. The qwo-qubit operation relies on the coupling of the ions through their collective quantized motion. A detailed description of the setup and the methods is included.

We present a generic model of coupling quantum optical and solid state qubits, and the corresponding transfer protocols. The example discussed is a trapped ion coupled to a charge qubit (e.g. Cooper pair box). To enhance the coupling, and achieve compatibility between the different experimental setups we introduce a superconducting cavity as the connecting element.

A single 40Ca+ ion is trapped and laser cooled to its motional ground state. Laser radiation which couples off-resonantly to a motional sideband of the ion's S1/2 to D5/2 transition causes a phase shift proportional to the ion's motional quantum state |n>. As the phase shift is conditional upon the ion's motion, we are able to demonstrate a universal 2-qubit quantum gate operation where the electronic target state {S,D} is flipped depending on the motional qubit state |n>={|0>,|1>}. Finally, we discuss scaling properties of this universal quantum gate for linear ion crystals and present numerical simulations for the generation of a maximally entangled state of five ions.

We report on tomographic means to study the stability of a qubit register based on a string of trapped ions. In our experiment, two ions are held in a linear Paul trap and are entangled deterministically by laser pulses that couple their electronic and motional states. We reconstruct the density matrix using single qubit rotations and subsequent measurements with near-unity detection efficiency. This way, we characterize the created Bell states, the states into which they subsequently decay, and we derive their entanglement, applying different entanglement measures.

A distant mirror leads to a vacuum-induced level shift in a laser-excited atom. This effect has been measured with a single mirror 25 cm away from a single, trapped barium ion. This dispersive action is the counterpart to the mirror's dissipative effect, which has been shown earlier to effect a change in the ion's spontaneous decay [J. Eschner et al., Nature 413, 495-498 (2001)]. The experimental data are well described by 8-level optical Bloch equations which are amended to take into account the presence of the mirror according to the model in [U. Dorner and P. Zoller, Phys. Rev. A 66, 023816 (2002)]. Observed deviations from simple dispersive behavior are attributed to multi-level effects.

Using optical Ramsey interferometry, we precisely measure the laser-induced AC-stark shift on the $S_{1/2}$ -- $D_{5/2}$ "quantum bit" transition near 729 nm in a single trapped $^{40}$Ca$^+$ ion. We cancel this shift using an additional laser field. This technique is of particular importance for the implementation of quantum information processing with cold trapped ions. As a simple application we measure the atomic phase evolution during a $n \times 2π$ rotation of the quantum bit.

Two-level ionic systems, where quantum information is encoded in long lived states (qubits), are discussed extensively for quantum information processing. We present a collection of measurements which characterize the stability of a qubit based on the $S_{1/2}$--$D_{5/2}$ transition of single $^{40}$Ca$^+$ ions in a linear Paul trap. We find coherence times of $\simeq$1 ms, discuss the main technical limitations and outline possible improvements.

The quadrupole S$_{1/2}$ -- D$_{5/2}$ optical transition of a single trapped Ca$^+$ ion, well suited for encoding a quantum bit of information, is coherently coupled to the standing wave field of a high finesse cavity. The coupling is verified by observing the ion's response to both spatial and temporal variations of the intracavity field. We also achieve deterministic coupling of the cavity mode to the ion's vibrational state by selectively exciting vibrational state-changing transitions and by controlling the position of the ion in the standing wave field with nanometer-precision.

A novel method of ground state laser cooling of trapped atoms utilizes the absorption profile of a three (or multi-) level system which is tailored by a quantum interference. With cooling rates comparable to conventional sideband cooling, lower final temperatures may be achieved. The method was experimentally implemented to cool a single Ca$^+$ ion to its vibrational ground state. Since a broad band of vibrational frequencies can be cooled simultaneously, the technique will be particularly useful for the cooling of larger ion strings, thereby being of great practical importance for initializing a quantum register based on trapped ions. We also discuss its application to different level schemes and for ground state cooling of neutral atoms trapped by a far detuned standing wave laser field.

Ground state cooling and coherent manipulation of ions in an rf-(Paul) trap is the prerequisite for quantum information experiments with trapped ions. With resolved sideband cooling on the optical S1/2 - D5/2 quadrupole transition we have cooled one and two 40Ca+ ions to the ground state of vibration with up to 99.9% probability. With a novel cooling scheme utilizing electromagnetically induced transparency on the S1/2 - P1/2 manifold we have achieved simultaneous ground state cooling of two motional sidebands 1.7 MHz apart. Starting from the motional ground state we have demonstrated coherent quantum state manipulation on the S1/2 - D5/2 quadrupole transition at 729 nm. Up to 30 Rabi oscillations within 1.4 ms have been observed in the motional ground state and in the n=1 Fock state. In the linear quadrupole rf-trap with 700 kHz trap frequency along the symmetry axis (2 MHz in radial direction) the minimum ion spacing is more than 5 micron for up to 4 ions. We are able to cool two ions to the ground state in the trap and individually address the ions with laser pulses through a special optical addressing channel.

Ground state laser cooling of a single trapped ion is achieved using a technique which tailors the absorption profile for the cooling laser by exploiting electromagnetically induced transparency in the Zeeman structure of a dipole transition. This new method is robust, easy to implement and proves particularly useful for cooling several motional degrees of freedom simultaneously, which is of great practical importance for the implementation of quantum logic schemes with trapped ions.

We have cooled a two-ion-crystal to the ground state of its collective modes of motion. Laser cooling, more specific resolved sideband cooling is performed sympathetically by illuminating only one of the two $^{40}$Ca$^+$ ions in the crystal. The heating rates of the motional modes of the crystal in our linear trap have been measured, and we found them considerably smaller than those previously reported by Q. Turchette {\em et. al.} Phys. Rev. A 61, 063418 (2000) in the case of trapped $^9$Be$^+$ ions. After the ground state is prepared, coherent quantum state manipulation of the atomic population can be performed. Within the coherence time, up to 12 Rabi oscillations are observed, showing that many coherent manipulations can be achieved. Coherent excitation of each ion individually and ground state cooling are important tools for the realization of quantum information processing in ion traps.

We investigate single ions of $^{40}Ca^+$ in Paul traps for quantum information processing. Superpositions of the S$_{1/2}$ electronic ground state and the metastable D$_{5/2}$ state are used to implement a qubit. Laser light on the S$_{1/2} \leftrightarrow$ D$_{5/2}$ transition is used for the manipulation of the ion's quantum state. We apply sideband cooling to the ion and reach the ground state of vibration with up to 99.9% probability. Starting from this Fock state $|n=0>$, we demonstrate coherent quantum state manipulation. A large number of Rabi oscillations and a ms-coherence time is observed. Motional heating is measured to be as low as one vibrational quantum in 190 ms. We also report on ground state cooling of two ions.

We investigate theoretically the speed limit of quantum gate operations for ion trap quantum information processors. The proposed methods use laser pulses for quantum gates which entangle the electronic and vibrational degrees of freedom of the trapped ions. Two of these methods are studied in detail and for both of them the speed is limited by a combination of the recoil frequency of the relevant electronic transition, and the vibrational frequency in the trap. We have experimentally studied the gate operations below and above this speed limit. In the latter case, the fidelity is reduced, in agreement with our theoretical findings. // Changes: a) error in equ. 24 and table III repaired b) reference Jonathan et al, quant-ph/ 0002092, added (proposes fast quantum gates using the AC-Stark effect)

Resonance fluorescence of a single trapped ion is spectrally analyzed using a heterodyne technique. Motional sidebands due to the oscillation of the ion in the harmonic trap potential are observed in the fluorescence spectrum. From the width of the sidebands the cooling rate is obtained and found to be in agreement with the theoretical prediction.

A single Ca+ ion in a Paul trap has been cooled to the ground state of vibration with up to 99.9% probability. Starting from this Fock state |n=0> we have demonstrated coherent quantum state manipulation on an optical transition. Up to 30 Rabi oscillations within 1.4 ms have been observed. We find a similar number of Rabi oscillations after preparation of the ion in the |n=1> Fock state. The coherence of optical state manipulation is only limited by laser and ambient magnetic field fluctuations. Motional heating has been measured to be as low as one vibrational quantum in 190 ms.

A method for the experimental reconstruction of the quantum state of motion for a single trapped ion is proposed. It is based on the measurement of the ground state population of the trap after a sudden change of the trapping potential. In particular, we show how the Q function and the quadrature distribution can be measured directly. In an example we demonstrate the principle and analyze the sensibility of the reconstruction process to experimental uncertainties as well as to finite grid limitations. Our method is not restricted to the Lamb-Dicke Limit and works in one or more dimensions.

The interaction of a trapped ion with a laser beam in the strong excitation regime is analyzed. In this regime, a variety of non--classical states of motion can be prepared either by using laser pulses of well defined area, or by an adiabatic passage scheme based on the variation of the laser frequency. We show how these states can be used to investigate fundamental properties of quantum mechanics. We also study possible applications of this system to build an ion interferometer.