Ion Trapping Group

Entangling interactions between distant qubits can be mediated via an additional degree of freedom. In conventional trapped-ion schemes, realizing a well-defined, coherent gate typically requires spectrally addressing a specific bus mode. As the ion number increases, the coupling to each individual motional mode becomes weaker, so gates on large ion strings mediated by a single mode are necessarily slow. Moreover, addressing a large number of modes demands complex driving schemes, and the fundamentally perturbative character of these approaches imposes constraints on achievable gate speed and fidelity. Here, we introduce a scheme for entangling trapped-ion qubits using a time-dependent magnetic-field gradient, in which all axial motional modes participate in mediating the interaction and the gate construction is nonperturbative. The framework can be used to implement both multi-qubit gates and two-qubit gates between arbitrary pairs in a linear ion string. Through several explicit examples, we highlight the advantages over existing magnetic-gradient schemes and show how gates on multiple ion pairs can be carried out simultaneously.

Accurate and low-latency qubit state measurement is critical for trapped-ion quantum computing. While deep neural networks (DNNs) have been integrated to enhance detection fidelity, their latency performance on specific hardware platforms remains underexplored. This work benchmarks the latency of DNN-based qubit detection on field-programmable gate arrays (FPGAs) and graphics processing units (GPUs). The FPGA solution directly interfaces an electron-multiplying charge-coupled device (EMCCD) with the subsequent data processing logic, eliminating buffering and interface overheads. As a baseline, the GPU-based system employs a high-speed PCIe image grabber for image input and I/O card for state output. We deploy Multilayer Perceptron (MLP) and Vision Transformer (ViT) models on hardware to evaluate measurement performance. Compared to conventional thresholding, DNNs reduce the mean measurement fidelity (MMF) error by factors of 1.8-2.5x (one-qubit case) and 4.2-7.6x (three-qubit case). FPGA-based MLP and ViT achieve nanosecond- and microsecond-scale inference latencies, while the complete single-shot measurement process achieves over 100x speedup compared to the GPU implementation. Additionally, clock-cycle-level signal analysis reveals inefficiencies in EMCCD data transmission via Cameralink, suggesting that optimizing this interface could further leverage the advantages of ultra-low-latency DNN inference, guiding the development of next-generation qubit detection systems.

Recent advances in quantum technologies have enabled quantum simulation of gauge theories -- some of the most fundamental frameworks of nature -- in regimes far from equilibrium, where classical computation is severely limited. These simulators, primarily based on neutral atoms, trapped ions, and superconducting circuits, hold the potential to address long-standing questions in nuclear, high-energy, and condensed-matter physics, and may ultimately allow first-principles studies of matter evolution in settings ranging from the early universe to high-energy collisions. Research in this rapidly growing field is also driving the convergence of concepts across disciplines and uncovering new phenomena. In this Review, we highlight recent experimental and theoretical developments, focusing on phenomena accessible in current and near-term quantum simulators, including particle production and string breaking, collision dynamics, thermalization, ergodicity breaking, and dynamical quantum phase transitions. We conclude by outlining promising directions for future research and opportunities enabled by available quantum hardware.

The mixing of scalar substances in fluid flows by stirring and diffusion is ubiquitous in natural flows, chemical engineering, and microfluidic drug delivery. Here, we present a spectral quantum algorithm for scalar mixing by solving the advection-diffusion equation in a quantum computational fluid dynamics framework. The exact gate decompositions of the advection and diffusion operators in spectral space are derived. For all but the simplest one-dimensional flows, these operators do not commute. Therefore, we use operator splitting to construct quantum circuits capable of simulating arbitrary polynomial velocity profiles in multiple dimensions, such as the Blasius profile of a laminar boundary layer. Periodic, Neumann, and Dirichlet boundary conditions can be imposed with the appropriate quantum spectral transform. We evaluate the approach in statevector simulations of a Couette flow, plane Poiseuille flow, and a polynomial Blasius profile approximation. For an advection-diffusion problem in one dimension, we compare the time evolution of an ideal quantum simulation with those of real quantum computers with superconducting and trapped-ion qubits. The required number of two-qubit gates grows with the logarithm of the number of grid points raised to one higher power than the order of the polynomial velocity profile.

Noise of a quantum processor can be an important resource for simulating open quantum dynamics. However, this requires characterizing the device noise and then transforming it into a target structure. Here we take the first step towards this goal: We analytically and numerically study reshaping the noise associated with native trapped-ion two-qubit entangling gates via quantum circuits based on repetition codes, and experimentally demonstrate our findings on the IonQ Aria-1 quantum hardware. We investigate all the building blocks, including the quantum channels describing noisy two-qubit entangling gates, the compilation of the encoding circuits into native gates, and the propagation of two-qubit errors across ideal single-qubit gates.

Modern platforms for potential qubit candidates, such as trapped ions or neutral atoms, allow long range connectivity between distant physical qubits through shuttling. This opens up an avenue for transversal logical CNOT gates between distant logical qubits, whereby physical CNOT gates are performed between each corresponding physical qubit on the control and target logical qubits. However, the transversal CNOT can propagate errors from one logical qubit to another, leading to correlated errors between logical qubits. We have developed a multi-pass iterative decoder that decodes each logical qubit separately to deal with this correlated error. We show that under circuit-level noise and only $\mathcal{O}(1)$ code cycles, a threshold can still persist, and the logical error rate will not be significantly degraded, matching the sub-threshold logical error rate scaling of $p^{\lfloor\frac{d}{2}\rfloor}$ for a distance $d$ rotated surface code.

Noise resilience of quantum information processing is a crucial precondition to reach the fault-tolerance threshold. While resilience to many types of noise can be achieved through suitable control schemes, resilience to amplitude noise seems to be elusive within the common harmonic approximation for the bus mode of trapped ions. We show that weak an-harmonicities admit control schemes that achieve amplitude noise-resilience consistent with state-of-the-art experimental requirements, and that the required an-harmonicities can be achieved with current standards of micro-structured traps or even the intrinsically an-harmonic Coulomb interaction. This approach applies broadly to any platform that employs a bosonic bus as a qubit coupler.

We present an entangling gate scheme for trapped-ion chains that achieves high-fidelity operations with excited motional states despite multiple error sources. Our approach incorporates all relevant motional modes and exhibits enhanced robustness against both motional heating effects and detuning errors, critical features for building robust and scalable trapped-ion quantum computers.

The phenomenon of confinement is well known in high-energy physics and can also be realized for low-energy domain-wall excitations in one-dimensional quantum spin chains. A bound state consisting of two domain-walls can behave like a meson, and in a recent work of Vovrosh et al. [PRX Quantum 3, 040309 (2022)] , it was demonstrated that a pair of mesons could dynamically form a meta-stable confinement-induced bound state (consisting of four domain-walls) akin to a hadronic state. However, the protocol discussed in Vovrosh et al. [PRX Quantum 3, 040309 (2022)] involving the use of interactions with characteristically non-monotonic distance dependence is not easy to come by in nature, thus, posing a challenge for its experimental realization. In this regard, Rydberg atoms can provide the required platform for simulating confinement-related physics. We exploit the flexibility offered by interacting Rydberg-dressed atoms to engineering modified spin-spin interactions for the one-dimensional transverse field Ising model. Our numerical simulations show how Rydberg-dressed interactions can give rise to a variety of effective potentials that are suitable for hadron formation, which opens the possibility of simulating confinement physics with Rydberg platforms as a viable alternative to current trapped-ion experiments.

Optimal quantum control of continuous variable systems poses a formidable computational challenge because of the high-dimensional character of the system dynamics. The framework of quantum invariants can significantly reduce the complexity of such problems, but it requires the knowledge of an invariant compatible with the Hamiltonian of the system in question. We explore the potential of a Gaussian invariant that is suitable for quadratic Hamiltonians with any given number of motional degrees of freedom for quantum optimal control problems that are inspired by current challenges in ground-state-to-ground-state shuttling of trapped-ions.

Measurement-based cooling is a method by which a quantum system, initially in a thermal state, can be prepared in its ground state through some sort of measurement. This is done by making a measurement that heralds the system being in the desired state. Here we demonstrate the application of a measurement-based cooling technique to a trapped atomic ion. The ion is pre-cooled by Doppler laser cooling to a thermal state with a mean excitation of $\bar n \approx 18$ and the measurement-based cooling technique selects those occasions when the ion happens to be in the motional ground state. The fidelity of the heralding process is greater than 95%. This technique can be applied to other systems that are not as amenable to laser cooling as trapped ions.

Controllable bosonic systems can provide post-classical computational power with sub-universal quantum computational capability. A network that consists of a number of bosons evolving through beam-splitters and phase-shifters between different modes, has been proposed and applied to demonstrate quantum advantages. While the network has been implemented mostly in optical systems with photons, recently alternative realizations have been explored, where major limitations in photonic systems such as photon loss, and probabilistic manipulation can be addressed. Phonons, the quantized excitations of vibrational modes, of trapped ions can be a promising candidate to realize the bosonic network. Here, we experimentally demonstrate a minimal-loss phononic network that can be programmed and in which any phononic states are deterministically prepared and detected. We realize the network with up to four collective-vibrational modes, which can be straightforwardly extended to reveal quantum advantage. We benchmark the performance of the network with an exemplary algorithm of tomography for arbitrary multi-mode states with a fixed total phonon number. We obtain reconstruction fidelities of 94.5 $\pm$ 1.95 % and 93.4 $\pm$ 3.15 % for single-phonon and two-phonon states, respectively. Our experiment demonstrates a clear and novel pathway to scale up a phononic network for various quantum information processing beyond the limitations of classical and other quantum systems.

Measurement-induced state disturbance is a major challenge in obtaining quantum statistics at multiple time points. We propose a method to extract dynamic information from a quantum system at intermediate time points, namely snapshotting quantum dynamics. To this end, we apply classical post-processing after performing the ancilla-assisted measurements to cancel out the impact of the measurements at each time point. Based on this, we reconstruct a multi-time quasi-probability distribution (QPD) that correctly recovers the probability distributions at the respective time points. Our approach can also be applied to simultaneously extract exponentially many correlation functions with various time-orderings. We provide a proof-of-principle experimental demonstration of the proposed protocol using a dual-species trapped-ion system by employing $^{171}\rm{Yb}^+$ and $^{138}\rm{Ba}^+$ ions as the system and the ancilla, respectively. Multi-time measurements are performed by repeated initialization and detection of the ancilla state without directly measuring the system state. The two- and three-time QPDs and correlation functions are reconstructed reliably from the experiment, negativity and complex values in the QPDs clearly indicate a contribution of the quantum coherence throughout dynamics.

A major obstacle in the way of practical quantum computing is achieving scalable and robust high-fidelity entangling gates. To this end, quantum control has become an essential tool, as it can make the entangling interaction resilient to sources of noise. Nevertheless, it may be difficult to identify an appropriate quantum control technique for a particular need given the breadth of work pertaining to robust entanglement. To this end, we attempt to consolidate the literature by providing a non-exhaustive summary and critical analysis. The quantum control methods are separated into two categories: schemes which extend the robustness to (i) spin or (ii) motional decoherence. We choose to focus on extensions of the $σ_x\otimesσ_x$ Molmer-Sorensen interaction using microwaves and a static magnetic field gradient. Nevertheless, some of the techniques discussed here can be relevant to other trapped ion architectures or physical qubit implementations. Finally, we experimentally realize a proof-of-concept interaction with simultaneous robustness to spin and motional decoherence by combining several quantum control methods presented in this manuscript.

The study of confinement in quantum spin chains has seen a large surge of interest in recent years. It is not only important for understanding a range of effective one-dimensional condensed matter realizations, but also shares some of the non-perturbative physics with quantum chromodynamics (QCD) which makes it a prime target for current quantum simulation efforts. In analogy with QCD, the confinement-induced two-particle boundstates that appear in these models are dubbed mesons. Here, we study scattering events due to meson collisions in a quantum spin chain with long-range interactions such that two mesons have an extended interaction. We show how novel hadronic boundstates, e.g. with four constituent particles akin to tetraquarks, may form dynamically in fusion events. In a natural collision their signal is weak as elastic meson scattering dominates. However, we propose two controllable protocols which allow for a clear observation of dynamical hadron formation. We discuss how this physics can be simulated in trapped ion or Rydberg atom set-ups.

Invariant-based inverse engineering is an elegant approach to quantum control with corresponding experimental implementations that perform tasks with applications in quantum information processing such as shuttling trapped ions. We build on recent work to generalise invariant-based inverse engineering to control two coupled harmonic oscillators in any number of spatial dimensions. This may be used to perform experimentally relevant tasks such as separation of trapped ions, which is demonstrated numerically, achieving transfer fidelities of over 96% as well as low motional number excitations.

Quantum coherence is one of the clearest departures from classical physics, exhibited when a system is in a superposition of different basis states. Here the coherent superposition of three motional Fock states of a single trapped ion is experimentally certified, with a procedure provably robust against imperfect operation. As the motional state cannot be directly interrogated, our scheme uses an interference pattern generated by projective measurement of the coupled qubit state. The minimum number of coherently superposed states is inferred from a series of threshold values based on analysis of the interference pattern. This demonstrates that high-level coherence can be verified and investigated with simple, nonideal control methods well-suited to noisy intermediate-scale quantum devices.

Microfabricated ion-trap devices offer a promising pathway towards scalable quantum computing. Research efforts have begun to focus on the engineering challenges associated with developing large-scale ion-trap arrays and networks. However, increasing the size of the array and integrating on-chip electronics can drastically increase the power dissipation within the ion-trap chips. This leads to an increase in the operating temperature of the ion-trap and limits the device performance. Therefore, effective thermal management is an essential consideration for any large-scale architecture. Presented here is the development of a modular cooling system designed for use with multiple ion-trapping experiments simultaneously. The system includes an extensible cryostat that permits scaling of the cooling power to meet the demands of a large network. Following experimental testing on two independent ion-trap experiments, the cooling system is expected to deliver a net cooling power of 111 W at ~70 K to up to four experiments. The cooling system is a step towards meeting the practical challenges of operating large-scale quantum computers with many qubits.

The motion of charged particles can be interfaced with electric circuitry via the current induced in nearby pick-up electrodes. Here we show how the rotational and translational dynamics of levitated objects with arbitrary charge distributions can be coupled to a circuit and how the latter acts back on the particle motion. The ensuing cooling rates in series and parallel RLC circuits are determined, demonstrating that quadrupole ion traps are well suited for implementing all-electric cooling. We derive the effective macromotion potential for general trap geometries and demonstrate numerically how consecutive rotational and translational resistive cooling of a microscale particle can be achieved in linear Paul traps.

Two-dimensional partial covariance mass spectrometry (2D-PC-MS) exploits the inherent fluctuations of fragment ion abundances across a series of tandem mass spectra, to identify correlated pairs of fragment ions produced along the same fragmentation pathway of the same parent (e.g. peptide) ion. Here, we apply 2D-PC-MS to the analysis of intact protein ions in a standard linear ion trap mass analyzer, using the fact that the fragment-fragment correlation signals are much more specific to bio-molecular sequence than 1D MS/MS signals at the same mass accuracy and resolution. We show that from the distribution of signals on a 2D-PC-MS map it is possible to extract the charge state of both parent and fragment ions without resolving the isotopic envelope. Furthermore, the 2D map of fragment-fragment correlations naturally reveals the secondary decomposition pathways of the fragment ions. We access this spectral information using an adapted version of the Hough transform. We demonstrate the successful identification of highly charged, intact protein molecules without the need for high mass resolution. Using this technique we also perform the in silico deconvolution of the overlapping fragment ion signals from two co-isolated and co-fragmented intact protein molecules, demonstrating a viable new method for the concurrent mass spectrometric identification of a mixture of intact protein ions from the same fragment ion spectrum.

Entangling operations are a necessary tool for large-scale quantum information processing, but experimental imperfections can prevent current schemes from reaching sufficient fidelities as the number of qubits is increased. Here it is shown numerically how multi-toned generalizations of standard trapped-ion entangling gates can simultaneously be made robust against noise and mis-sets of the frequencies of the individual qubits. This relaxes the degree of homogeneity required in the trapping field, making physically larger systems more practical.

Essentially all known quantum gates rely on a weak-coupling approximation resulting in linear dynamics. With the explicit example of trapped ions, we show how high-fidelity quantum gates can be achieved outside such an approximation, and we derive readily implementable driving fields to realize gates with extremely high fidelities for ions well outside the Lamb-Dicke regime with motional temperatures achievable by only Doppler cooling.

Discrete time crystals are periodically driven systems characterized by a response with periodicity $nT$, with $T$ the period of the drive and $n>1$. Typically, $n$ is an integer and bounded from above by the dimension of the local (or single particle) Hilbert space, the most prominent example being spin-$1/2$ systems with $n$ restricted to $2$. Here we show that a clean spin-$1/2$ system in the presence of long-range interactions and transverse field can sustain a huge variety of different `higher-order' discrete time crystals with integer and, surprisingly, even fractional $n > 2$. We characterize these (arguably prethermal) non-equilibrium phases of matter thoroughly using a combination of exact diagonalization, semiclassical methods, and spin-wave approximations, which enable us to establish their stability in the presence of competing long- and short-range interactions. Remarkably, these phases emerge in a model with continous driving and time-independent interactions, convenient for experimental implementations with ultracold atoms or trapped ions.

Doppler and sideband cooling are long standing techniques that have been used together to prepare trapped atomic ions in their ground state of motion. In this paper we study how these techniques can be extended to cool both radial modes of motion of a single ion in a Penning trap. We numerically explore the prerequisite experimental parameters for efficient Doppler cooling in the presence of an additional oscillating electric field to resonantly couple the radial modes. The simulations are supported by experimental data for a single $^{40}$Ca$^+$ ion Doppler cooled to $\sim$100 phonons in both modes at a magnetron frequency of 52 kHz and a modified cyclotron frequency of 677 kHz. For these frequencies, we then show that mean phonon numbers of $0.35(5)$ for the modified cyclotron and $1.7(2)$ for the magnetron motions are achieved after 68 ms of sideband cooling.

We present a technique based on high resolution imaging to measure the absolute temperature and the heating rate of a single ion trapped at the focus of a deep parabolic mirror. We collect the fluorescence light scattered by the ion during laser cooling and image it onto a camera. Accounting for the size of the point-spread function and the magnification of the imaging system, we determine the spatial extent of the ion, from which we infer the mean phonon occupation number in the trap. Repeating such measurements and varying the power or the detuning of the cooling laser, we determine the anomalous heating rate. In contrast to other established schemes for measuring the heating rate, one does not have to switch off the cooling but the ion is always maintained in a state of thermal equilibrium at temperatures close to the Doppler limit.

Proper time, ideal clocks, and boosts are well understood classically, but subtleties arise in quantum physics. We show that quantum clocks set in motion via momentum boosts do not witness classical time dilation. However, using velocity boosts we find the ideal behaviour in both cases where the quantum clock and classical observer are set in motion. Without internal state dependent forces additional effects arise. As such, we derive observed frequency shifts in ion trap atomic clocks, indicating a small additional shift, and also show the emergence of non-ideal behaviour in a theoretical clock model.

Recent studies using the quantum information theoretic approach to thermodynamics show that the presence of coherence in quantum systems generates corrections to classical fluctuation theorems. To explicate the physical origins and implications of such corrections, we here convert an abstract framework of an autonomous quantum Crooks relation into quantum Crooks equalities for well-known coherent, squeezed and cat states. We further provide a proposal for a concrete experimental scenario to test these equalities. Our scheme consists of the autonomous evolution of a trapped ion and uses a position dependent AC Stark shift.

Constructing a large scale ion trap quantum processor will require entangling gate operations that are robust in the presence of noise and experimental imperfection. We experimentally demonstrate how a new type of Mølmer-Sørensen gate protects against infidelity caused by heating of the motional mode used during the gate. Furthermore, we show how the same technique simultaneously provides significant protection against slow fluctuations and mis-sets in the secular frequency. Since this parameter sensitivity is worsened in cases where the ions are not ground state cooled, our method provides a path towards relaxing ion cooling requirements in practical realisations of quantum computing and simulation.

The text below was written during two visits that Daniel Segal made at Universit{é} Paris 13. Danny stayed at Laboratoire de Physique des Lasers the summers of 2008 and 2009 to participate in the exploration of a novel lead in the field of ultra-high resolution spectroscopy. Our idea was to probe trapped molecular ions using Quantum Logic Spectroscopy (QLS) in order to advance our understanding of a variety of fundamental processes in nature. At that time, QLS, a ground-breaking spectroscopic technique, had only been demonstrated with atomic ions. Our ultimategoals were new approaches to the observation of parity violation in chiral molecules and tests of time variations of the fundamental constants. This text is the original research proposal written eight years ago. We have added a series of notes to revisit it in the light of what has been since realized in the field.

In recent decades there has been a rapid development of methods to experimentally control individual quantum systems. A broad range of quantum control methods has been developed for two-level systems, however the complexity of multi-level quantum systems make the development of analogous control methods extremely challenging. Here, we exploit the equivalence between multi-level systems with SU(2) symmetry and spin-1/2 systems to develop a technique for generating new robust, high-fidelity, multi-level control methods. As a demonstration of this technique, we develop new adiabatic and composite multi-level quantum control methods and experimentally realise these methods using an $^{171}$Yb$^+$ ion system. We measure the average infidelity of the process in both cases to be around $10^{-4}$, demonstrating that this technique can be used to develop high-fidelity multi-level quantum control methods and can, for example, be applied to a wide range of quantum computing protocols including implementations below the fault-tolerant threshold in trapped ions.

Due to the exponential complexity of the resources required by quantum state tomography (QST), people are interested in approaches towards identifying quantum states which require less effort and time. In this paper, we provide a tailored and efficient method for reconstructing mixed quantum states up to $12$ (or even more) qubits from an incomplete set of observables subject to noises. Our method is applicable to any pure or nearly pure state $ρ$, and can be extended to many states of interest in quantum information processing, such as multi-particle entangled $W$ state, GHZ state and cluster states that are matrix product operators of low dimensions. The method applies the quantum density matrix constraints to a quantum compressive sensing optimization problem, and exploits a modified Quantum Alternating Direction Multiplier Method (Quantum-ADMM) to accelerate the convergence. Our algorithm takes $8,35$ and $226$ seconds respectively to reconstruct superposition state density matrices of $10,11,12$ qubits with acceptable fidelity, using less than $1 \%$ of measurements of expectation. To our knowledge it is the fastest realization that people can achieve using a normal desktop. We further discuss applications of this method using experimental data of mixed states obtained in an ion trap experiment of up to $8$ qubits.

Trapped ions are a promising tool for building a large-scale quantum computer. However, the number of required radiation fields for the realisation of quantum gates in any proposed ion-based architecture scales with the number of ions within the quantum computer, posing a major obstacle when imagining a device with millions of ions. Here we present a fundamentally different concept for trapped-ion quantum computing where this detrimental scaling entirely vanishes, replacing millions of radiation fields with only a handful of fields. The method is based on individually controlled voltages applied to each logic gate location to facilitate the actual gate operation analogous to a traditional transistor architecture within a classical computer processor. To demonstrate the key principle of this approach we implement a versatile quantum gate method based on long-wavelength radiation and use this method to generate a maximally entangled state of two quantum engineered clock-qubits with fidelity 0.985(12). This quantum gate also constitutes a simple-to-implement tool for quantum metrology, sensing and simulation.

An autonomous quantum thermal machine comprising a trapped atom or ion placed inside an optical cavity is proposed and analysed. Such a machine can operate as a heat engine whose working medium is the quantised atomic motion, or as an absorption refrigerator which cools without any work input. Focusing on the refrigerator mode, we predict that it is possible with state-of-the-art technology to cool a trapped ion almost to its motional ground state using a thermal light source such as sunlight. We nonetheless find that a laser or similar reference system is necessary to stabilise the cavity frequencies. Furthermore, we establish a direct and heretofore unacknowledged connection between the abstract theory of quantum absorption refrigerators and practical sideband cooling techniques. We also highlight and clarify some assumptions underlying several recent theoretical studies on self-contained quantum engines and refrigerators. Our work indicates that cavity quantum electrodynamics is a promising and versatile experimental platform for the study of autonomous thermal machines in the quantum domain.

Atomic-scale logic and the minimization of heating (dissipation) are both very high on the agenda for future computation hardware. An approach to achieve these would be to replace networks of transistors directly by classical reversible logic gates built from the coherent dynamics of a few interacting atoms. As superpositions are unnecessary before and after each such gate (inputs and outputs are bits), the dephasing time only needs to exceed a single gate operation time, while fault tolerance should be achieved with low overhead, by classical coding. Such gates could thus be a spin-off of quantum technology much before full-scale quantum computation. Thus motivated, we propose methods to realize the 3-bit Toffoli and Fredkin gates universal for classical reversible logic using a single time-independent 3-qubit Hamiltonian with realistic nearest neighbour two-body interactions. We also exemplify how these gates can be composed to make a larger circuit. We show that trapped ions may soon be scalable simulators for such architectures, and investigate the prospects with dopants in silicon.

Quantum theory is based on a mathematical structure totally different from conventional arithmetic. Due to the symmetric nature of bosonic particles, annihilation or creation of single particles translates a quantum state depending on how many bosons are already in the given quantum system. This proportionality results in a variety of non-classical features of quantum mechanics including the bosonic commutation relation. The annihilation and creation operations have recently been implemented in photonic systems. However, this feature of quantum mechanics does not preclude the possibility of realizing conventional arithmetic in quantum systems. We implement conventional addition and subtraction of single phonons for a trapped \Yb ion in a harmonic potential. In order to realize such operations, we apply the transitionless adiabatic passage scheme on the anti-Jaynes-Cummings coupling between the internal energy states and external motion states of the ion. By performing the operations on superpositions of Fock states, we realize the hybrid computation of classical arithmetic in quantum parallelism, and show that our operations are useful to engineer quantum states. Our single-phonon operations are nearly deterministic and robust against parameter changes, enabling handy repetition of the operations independently from the initial state of the atomic motion. We demonstrate the transform of a classical state to a nonclassical one of highly sub-Poissonian phonon statistics and a Gaussian state to a non-Gaussian state, by applying a sequence of the operations. The operations implemented here are the Susskind-Glogower phase operators, whose non-commutativity is also demonstrated.

We demonstrate ground-state cooling of a trapped ion using radio-frequency (RF) radiation. This is a powerful tool for the implementation of quantum operations, where RF or microwave radiation instead of lasers is used for motional quantum state engineering. We measure a mean phonon number of $\overline{n} = 0.13(4)$ after sideband cooling, corresponding to a ground-state occupation probability of 88(7)\%. After preparing in the vibrational ground state, we demonstrate motional state engineering by driving Rabi oscillations between the n=0 and n=1 Fock states. We also use the ability to ground-state cool to accurately measure the motional heating rate and report a reduction by almost two orders of magnitude compared to our previously measured result, which we attribute to carefully eliminating sources of electrical noise in the system.

Ion Coulomb crystals (ICC), formed by atomic ions at low temperatures in radiofrequency and Penning ion traps, are structures that have remarkable properties and many applications. Images of Coulomb crystals are striking and reveal the crystal structure, which arises from a balance between the trapping forces acting on the ions and their mutual Coulomb repulsion. Applications of these structures range from frequency standards and quantum simulation through to measurement of the cross sections of chemical reactions of ions.

This book chapter gives an introduction to, and an overview of, methods for cooling trapped ions. The main addressees are researchers entering the field. It is not intended as a comprehensive survey and historical account of the extensive literature on this topic. We present the physical ideas behind several cooling schemes, outline their mathematical description, and point to relevant literature useful for a more in-depth study of this topic.

We demonstrate a method for preparing and detecting all eigenstates of a three-level microwave dressed system with a single trapped ion. The method significantly reduces the experimental complexity of gate operations with dressed-state qubits, as well as allowing all three of the dressed-states to be prepared and detected, thereby providing access to a qutrit that is well protected from magnetic field noise. In addition, we demonstrate individual addressing of the clock transitions in two ions using a strong static magnetic field gradient, showing that our method can be used to prepare and detect microwave dressed-states in a string of ions when performing multi-ion quantum operations with microwave and radio frequency fields. The individual addressability of clock transitions could also allow for the control of pairwise interaction strengths between arbitrary ions in a string using lasers.

Applying a magnetic field gradient to a trapped ion allows long-wavelength microwave radiation to produce a mechanical force on the ion's motion when internal transitions are driven. We demonstrate such a coupling using a single trapped \Yb{171}~ion, and use it to produce entanglement between the spin and motional state, an essential step towards using such a field gradient to implement multi-qubit operations.

We propose an experimental scheme to verify the quantum non-equilibrium fluctuation relations using current technology. Specifically, we show that the characteristic function of the work distribution for a non-equilibrium quench of a general quantum system can be extracted from Ramsey interferometry of a single probe qubit. Our scheme paves the way for the full characterisation of non-equilibrium processes in a variety of complex quantum systems ranging from single particles to many-body atomic systems and spin chains. We demonstrate our idea using a time-dependent quench of the motional state of a trapped ion, where the internal pseudo-spin provides a convenient probe qubit.

We study a particular form of interaction Hamiltonian between qubits and quantum harmonic oscillators, whose closed system dynamics results in qubit controlled displacement operations. We show how this interaction is realizable in many setups, including nanomechanical systems, ion traps, cavity QED and circuit QED, and in each context we provide quantitative estimates for the relevant parameters. The dynamics of the system is investigated through a master equation, including typical decoherence mechanisms resulting from the coupling of the qubit and oscillator to a thermal Markovian environment. We show how to solve the master equation by adopting a phase-space representation for the oscillator, and derive analytical and approximate solutions for many special cases of interest. Finally, our techniques are applied to a relevant example by studying the dynamics of qubit-oscillator entanglement and the preparation of oscillator states with negative Wigner function.

Setting off from the classic input-output formalism, we develop a theoretical framework to characterise the Gaussian quantum channels relating the initial correlations of an open bosonic system to those of properly identified output modes. We then proceed to apply our formalism to the case of quantum harmonic oscillators, such as the motional degrees of freedom of trapped ions or nanomechanical oscillators, interacting with travelling electromagnetic modes through cavity fields and subject to external white noise. Thus, we determine the degree of squeezing that can be transferred from an intra-cavity oscillator to light, and also show that the intra-cavity squeezing can be transformed into distributed optical entanglement if one can access both output fields of a two-sided cavity.

We introduce a scheme to reconstruct arbitrary states of networks composed of quantum oscillators--e.g., the motional state of trapped ions or the radiation state of coupled cavities. The scheme uses minimal resources, in the sense that it i) requires only the interaction between one-qubit probe and one constituent of the network; ii) provides the reconstructed state directly from the data, avoiding any tomographic transformation; iii) involves the tuning of only one coupling parameter. In addition, we show that a number of quantum properties can be extracted without full reconstruction of the state. The scheme can be used for probing quantum simulations of anharmonic many-body systems and quantum computations with continuous variables. Experimental implementation with trapped ions is also discussed and shown to be within reach of current technology.

We analyze the steady state entanglement generated in a coherently coupled dimer system subject to dephasing noise as a function of the degree of Markovianity of the evolution. By keeping fixed the effective noise strength while varying the memory time of the environment, we demonstrate that non-Markovianity is an essential, quantifiable resource that may support the formation of steady state entanglement whereas purely Markovian dynamics governed by Lindblad master equations lead to separable steady states. This result illustrates possible mechanisms leading to long-lived entanglement in purely decohering, possibly local, environments. We present a feasible experimental demonstration of this noise assisted phenomenon using a system of trapped ions.

Studying mechanical resonators via radiation pressure offers a rich avenue for the exploration of quantum mechanical behavior in a macroscopic regime. However, quantum state preparation and especially quantum state reconstruction of mechanical oscillators remains a significant challenge. Here we propose a scheme to realize quantum state tomography, squeezing and state purification of a mechanical resonator using short optical pulses. The scheme presented allows observation of mechanical quantum features despite preparation from a thermal state and is shown to be experimentally feasible using optical microcavities. Our framework thus provides a promising means to explore the quantum nature of massive mechanical oscillators and can be applied to other systems such as trapped ions.

We propose a scheme of optical pumping by which a system of atoms coupled to harmonic oscillators is driven to an entangled steady state through the atomic spontaneous emission. It is shown that the optical pumping can be tailored so that the many-body atomic state asymptotically reaches an arbitrary stabilizer state regardless of the initial state. The proposed scheme can be suited to various physical systems. In particular, the ion-trap realization is well within current technology.

Structural defects in ion crystals can be formed during a linear quench of the transverse trapping frequency across the mechanical instability from a linear chain to the zigzag structure. The density of defects after the sweep can be conveniently described by the Kibble-Zurek mechanism. In particular, the number of kinks in the zigzag ordering can be derived from a time-dependent Ginzburg-Landau equation for the order parameter, here the zigzag transverse size, under the assumption that the ions are continuously laser cooled. In a linear Paul trap the transition becomes inhomogeneous, being the charge density larger in the center and more rarefied at the edges. During the linear quench the mechanical instability is first crossed in the center of the chain, and a front, at which the mechanical instability is crossed during the quench, is identified which propagates along the chain from the center to the edges. If the velocity of this front is smaller than the sound velocity, the dynamics becomes adiabatic even in the thermodynamic limit and no defect is produced. Otherwise, the nucleation of kinks is reduced with respect to the case in which the charges are homogeneously distributed, leading to a new scaling of the density of kinks with the quenching rate. The analytical predictions are verified numerically by integrating the Langevin equations of motion of the ions, in presence of a time-dependent transverse confinement. We argue that the non-equilibrium dynamics of an ion chain in a Paul trap constitutes an ideal scenario to test the inhomogeneous extension of the Kibble-Zurek mechanism, which lacks experimental evidence to date.

The non-equilibrium dynamics of an ion chain in a highly anisotropic trap is studied when the transverse trap frequency is quenched across the value at which the chain undergoes a continuous phase transition from a linear to a zigzag structure. Within Landau theory, an equation for the order parameter, corresponding to the transverse size of the zigzag structure, is determined when the vibrational motion is damped via laser cooling. The number of structural defects produced during a linear quench of the transverse trapping frequency is predicted and verified numerically. It is shown to obey the scaling predicted by the Kibble-Zurek mechanism, when extended to take into account the spatial inhomogeneities of the ion chain in a linear Paul trap.

Currently laser cooling schemes are fundamentally based on the weak coupling regime. This requirement sets the trap frequency as an upper bound to the cooling rate. In this work we present a numerical study that shows the feasibility of cooling in the strong coupling regime which then allows cooling rates that are faster than the trap frequency with state of the art experimental parameters. The scheme we present can work for trapped atoms or ions as well as mechanical oscillators. It can also cool medium size ions chains close to the ground state.

We investigate the generation of squeezing and entanglement for the motional degrees of freedom of ions in linear traps, confined by time-varying and oscillating potentials, comprised of an DC and an AC component. We show that high degrees of squeezing and entanglement can be obtained by controlling either the DC or the AC trapping component (or both), and by exploiting transient dynamics in regions where the ions' motion is unstable, without any added optical control. Furthermore, we investigate the time-scales over which the potentials should be switched in order for the manipulations to be most effective.

We demonstrate the implementation of a spin qubit with a single Ca ion in a micro ion trap. The qubit is encoded in the Zeeman ground state levels mJ=+1/2 and mJ=-1/2 of the S1/2 state of the ion. We show sideband cooling close to the vibrational ground state and demonstrate the initialization and readout of the qubit levels with 99.5% efficiency. We employ a Raman transition close to the S1/2 - P1/2 resonance for coherent manipulation of the qubit. We observe single qubit rotations with 96% fidelity and gate times below 5mus. Rabi oscillations on the blue motional sideband are used to extract the phonon number distribution. The dynamics of this distribution is analyzed to deduce the trap-induced heating rate of 0.3(1) phonons/ms.

We present a detailed study on the possibility of manipulating quantum information encoded in the "radial" modes of arrays of trapped ions (i.e., in the ions' oscillations orthogonal to the trap's main axis). In such systems, because of the tightness of transverse confinement, the radial modes pertaining to different ions can be addressed individually. In the first part of the paper we show that, if local control of the radial trapping frequencies is available, any linear optical and squeezing operation on the locally defined modes - on single as well as on many modes - can be reproduced by manipulating the frequencies. Then, we proceed to describe schemes apt to generate unprecedented degrees of bipartite and multipartite continuous variable entanglement under realistic noisy working conditions, and even restricting only to a global control of the trapping frequencies. Furthermore, we consider the transmission of the quantum information encoded in the radial modes along the array of ions, and show it to be possible to a remarkable degree of accuracy, for both finite-dimensional and continuous variable quantum states. Finally, as an application, we show that the states which can be generated in this setting allow for the violation of multipartite non-locality tests, by feasible displaced parity measurements. Such a demonstration would be a first test of quantum non-locality for "massive" degrees of freedom (i.e., for degrees of freedom describing the motion of massive particles).

A simple technique for robust generation of vibrational Fock states in a chain of trapped ions is proposed. The method is fast and easy to implement, since only a single chirped laser pulse, simultaneously addressing all of the ions, is required. Furthermore, because the approach uses collective adiabatic passage, significant fluctuations in the intensity or frequency of the laser pulse can be tolerated, and the technique performs well even on the border of the Lamb-Dicke regime. We also demonstrate how this technique may be extended in order to create non-classical superposition states of the ions' collective motion and Greenberger-Horne-Zeilinger states of their internal states. Because only a single laser pulse is required, heating effects arising under realistic experimental conditions are negligibly small.

Since Hund's work on the ammonia molecule, the double well potential has formed a key paradigm in physics. Its importance is further underlined by the central role it plays in the Landau theory of phase transitions. Recently, the study of entanglement properties of many-body systems has added a new angle to the study of quantum phase transitions of discrete and continuous degrees of freedom, i.e., spin and harmonic chains. Here we show that control of the radial degree of freedom of trapped ion chains allows for the simulation of linear and non-linear Klein-Gordon fields on a lattice, in which the parameters of the lattice, the non-linearity and mass can be controlled at will. The system may be driven through a phase transition creating a double well potential between different configurations of the ion crystal. The dynamics of the system are controllable, local properties are measurable and tunnelling in the double well potential would be observable.

An overview and status report of the new trapping facility for highly charged ions at the Gesellschaft fuer Schwerionenforschung is presented. The construction of this facility started in 2005 and is expected to be completed in 2008. Once operational, highly charged ions will be loaded from the experimental storage ring ESR into the HITRAP facility, where they are decelerated and cooled. The kinetic energy of the initially fast ions is reduced by more than fourteen orders of magnitude and their thermal energy is cooled to cryogenic temperatures. The cold ions are then delivered to a broad range of atomic physics experiments.

An overview is presented of laser spectroscopy experiments with cold, trapped, highly-charged ions, which will be performed at the HITRAP facility at GSI in Darmstadt (Germany). These high-resolution measurements of ground state hyperfine splittings will be three orders of magnitude more precise than previous measurements. Moreover, from a comparison of measurements of the hyperfine splittings in hydrogen- and lithium-like ions of the same isotope, QED effects at high electromagnetic fields can be determined within a few percent. Several candidate ions suited for these laser spectroscopy studies are presented.

We present a method of generating collective multi-qubit entanglement via global addressing of an ion chain following the guidelines of the Tavis-Cummings model, where several qubits are coupled to a collective motional mode. We show that a wide family of Dicke states and irradiant states can be generated by single global laser pulses, unitarily or helped with suitable postselection techniques.

We propose a novel type of precision laser spectrometer for trapped, highly charged ions nearly at rest. It consists of a cylindrical open-endcap Penning trap in which an externally produced bunch of highly charged ions can be confined and investigated by means of laser spectroscopy. The combination of confinement, cooling and compression of a dense ion cloud will allow the ground state hyperfine splitting in highly charged ions to be measured with an accuracy three orders of magnitude better than in any previous experiment. A systematic study of different charge states and different isotopes of the same element allows for highly sensitive tests of bound-state QED and for a precision determination of nuclear properties. Apart from stable isotopes, also radioactive species with half-lives longer than about one hour can be investigated.

We introduce an architecture for robust and scalable quantum computation using both stationary qubits (e.g. single photon sources made out of trapped atoms, molecules, ions, quantum dots, or defect centers in solids) and flying qubits (e.g. photons). Our scheme solves some of the most pressing problems in existing non-hybrid proposals, which include the difficulty of scaling conventional stationary qubit approaches, and the lack of practical means for storing single photons in linear optics setups. We combine elements of two previous proposals for distributed quantum computing, namely the efficient photon-loss tolerant build up of cluster states by Barrett and Kok [Phys. Rev. A 71, 060310(R) (2005)] with the idea of Repeat-Until-Success (RUS) quantum computing by Lim et al. [Phys. Rev. Lett. 95, 030505 (2005)]. This idea can be used to perform eventually deterministic two-qubit logic gates on spatially separated stationary qubits via photon pair measurements. Under non-ideal conditions, where photon loss is a possibility, the resulting gates can still be used to build graph states for one-way quantum computing. In this paper, we describe the RUS method, present possible experimental realizations, and analyse the generation of graph states.

Laser spectroscopy studies are being prepared to measure the 1s ground state hyperfine splitting in trapped cold highly charged ions. The purpose of such experiments is to test quantum electrodynamics in the strong electric field regime. These experiments form part of the HITRAP project at GSI. A brief review of the planned experiments is presented.

A long-lived coherent state and non-linear interaction have been experimentally demonstrated for the vibrational mode of a trapped ion. We propose an implementation of quantum computation using coherent states of the vibrational modes of trapped ions. Differently from earlier experiments, we consider a far-off resonance for the interaction between external fields and the ion in a bidimensional trap. By appropriate choices of the detunings between the external fields, the adiabatic elimination of the ionic excited level from the Hamiltonian of the system allows for beam splitting between orthogonal vibrational modes, production of coherent states and non-linear interactions of various kinds. In particular, this model enables the generation of the four coherent Bell states. Furthermore, all the necessary operations for quantum computation such as preparation of qubits, one-qubit and controlled two-qubit operations, are possible. The detection of the state of a vibrational mode in a Bell state is made possible by the combination of resonant and off-resonant interactions between the ion and some external fields. We show that our read-out scheme provides highly efficient discrimination between all the four Bell states. We extend this to a quantum register composed of many individually trapped ions. In this case, operations on two remote qubits are possible through a cavity mode. We emphasize that our remote-qubit operation scheme does not require a high quality factor resonator: the cavity field acts as a catalyst for the gate operation.

We show that bosonic fields may present anyonic behavior when interacting with a fermion in a Jaynes-Cummings-like model. The proposal is accomplished via the interaction of a two-level system with two quantized modes of a harmonic oscillator; under suitable conditions, the system acquires a fractional geometric phase. A crucial role is played by the entanglement of the system eigenstates, which provides a two-dimensional confinement in the effective evolution of the system, leading to the anyonic behavior. For a particular choice of parameters, we show that it is possible to transmute the statistics of the system continually from fermions to bosons. We also present an experimental proposal, in an ion-trap setup, in which fractional statistical features can be generated, controlled, and measured.

This paper discusses ways to implement two-qubit gate operations for quantum computing with cold trapped ions within one step. The proposed scheme is widely robust against parameter fluctuations and its simplicity might help to increase the number of qubits in present experiments. Basic idea is to use the quantum Zeno effect originating from continuous measurements on a common vibrational mode to realise gate operations with very high fidelities. The gate success rate can, in principle, be arbitrary high but operation times comparable to other schemes can only be obtained by accepting success rates below 80%.

We report a study of the axialisation and laser cooling of single ions and small clouds of ions in a Penning trap. A weak radiofrequency signal applied to a segmented ring electrode couples the magnetron motion to the cyclotron motion, which results in improved laser cooling of the magnetron motion. This allows us to approach the trapping conditions of a Paul trap, but without any micromotion. Using an ICCD camera we show that the motion of a single ion can be confined to dimensions of the order of 10 $μ$m. We have measured increased magnetron cooling rates using an rf-photon correlation technique. For certain laser cooling conditions, the magnetron motion of the centre of mass of the cloud grows and stabilises at a large value. This results in the ions orbiting the centre of the trap together in a small cloud, as confirmed by photon-photon correlation measurements.

Universal quantum cloning machines (UQCMs), sometimes called quantum cloners, generate many outputs with identical density matrices, with as close a resemblance to the input state as is allowed by the basic principles of quantum mechanics. Any experimental realization of a quantum cloner has to cope with the effects of decoherence which terminate the coherent evolution demanded by a UQCM. We examine how many clones can be generated within a decoherence time. We compare the time that a quantum cloner implemented with trapped ions requires to produce $M$ copies from $N$ identical pure state inputs and the decoherence time during which the probability of spontaneous emission becomes non-negligible. We find a method to construct an $N\to M$ cloning circuit, and estimate the number of elementary logic gates required. It turns out that our circuit is highly vulnerable to spontaneous emission as the number of gates in the circuit is exponential with respect to the number of qubits involved.

We demonstrate that quantum information processing can be implemented with ions trapped in a far detuned optical cavity. For sufficiently large detuning the system becomes insensitive to cavity decay. Following recent experimental progress, this scheme can be implemented with currently available technology.

We show that, under certain combinations of the parameters governing the interaction of a harmonically trapped ion with a laser beam, it is possible to find one or more exact eigenstates of the Hamiltonian, with no approximations except the optical rotating-wave approximation. These are related via a unitary equivalence to exact eigenstates of the full Jaynes-Cummings model (including counter-rotating terms) supplemented by a static driving term.

An effective interaction between trapped ions in thermal motion can be generated by illuminating them simultaneously with a single laser resonant with the ionic carrier frequency. The ac Stark-shift induces simultaneous `virtual' two-phonon transitions via several motional modes. Within a certain laser intensity range these transitions can interfere constructively, resulting in a relatively fast, heating-resistant two-qubit logic gate.

We propose a scheme for measuring the Berry phase in the vibrational degree of freedom of a trapped ion. Starting from the ion in a vibrational coherent state we show how to reverse the sign of the coherent state amplitude by using a purely geometric phase. This can then be detected through the internal degrees of freedom of the ion. Our method can be applied to preparation of Schrödinger cat states.

We present an alternative scheme for the generation of a 2-qubit quantum gate interaction between laser-cooled trapped ions. The scheme is based on the AC Stark shift (lightshift) induced by laser light resonant with the ionic transition frequency. At {\it specific} laser intensities, the shift of the ionic levels allows the resonant excitation of transitions involving the exchange of motional quanta. We compare the performance of this scheme with respect to that of related ion-trap proposals and find that, for an experimental realisation using travelling-wave radiation and working in the Lamb-Dicke regime, an improvement of over an order of magnitude in the gate switching rate is possible.

We investigate to what extent two trapped ions can be manipulated coherently when their coupling is mediated by a dipole-dipole interaction. We will show how the resulting level shift induced by this interaction can be used to create entanglement, while the decay of the states remains nearly negligible. This will allow us to implement conditional dynamics (a CNOT gate) and single qubit operations. We propose two different experimental realisations where a large level shift can be achieved and discuss both the strengths and weaknesses of this scheme from the point of view of a practical realization.

We examine the dynamics of a qubit stored in the motional degrees of freedom of an ultra-cold ion in an ion trap which is subject to the decoherence effects of a finite-temperature bath. We discover an encoding of the qubit, in two of the motional modes of the ion, which is stable against the occurrence of either none or one quantum jump. For the case of a zero-temperature bath we describe how to transfer only the information concerning the occurrence of quantum jumps and their types to a measuring apparatus, without affecting the ion's motional state significantly. We then describe how to generate a unitary restoration of the qubit given the jump information, through Raman processes generated by a series of laser pulses.

We propose a method for generating maximally entangled states of N two-level trapped ions. The method is deterministic and independent of the number of ions in the trap. It involves a controlled-NOT acting simultaneously on all the ions through a dispersive interaction. We explore the potential application of our scheme for high precision frequency standards.

The decoherence of nonclassical motional states of a trapped $^9 {\rm Be^+}$ ion in a recent experiment is investigated theoretically. Sources of decoherence considered here destroy the characteristic coherent quantum dynamics of the system but do not cause energy dissipation. Here they are first introduced phenomenologically and then described using a microscopic Hamiltonian formulation. Theoretical predictions are compared to experimental results.

Quantum computers require quantum logic, something fundamentally different to classical Boolean logic. This difference leads to a greater efficiency of quantum computation over its classical counter-part. In this review we explain the basic principles of quantum computation, including the construction of basic gates, and networks. We illustrate the power of quantum algorithms using the simple problem of Deutsch, and explain, again in very simple terms, the well known algorithm of Shor for factorisation of large numbers into primes. We then describe physical implementations of quantum computers, focusing on one in particular, the linear ion-trap realization. We explain that the main obstacle to building an actual quantum computer is the problem of decoherence, which we show may be circumvented using the methods of quantum error correction.

Quantum information processing rests on our ability to manipulate quantum superpositions through coherent unitary transformations. In reality the quantum information processor (a linear ion trap, or cavity qed implementation for example) exists in a dissipative environment. Dephasing, and other technical sources of noise, as well as more fundamental sources of dissipation severely restrict quantum processing capabilities. The strength of the coherent coupling needed to implement quantum logic is not always independent of dissipation. The limitations these dissipative influences present will be described and the need for efficient error correction noted. Even if long and involved quantum computations turn out to be hard to realize, one can perform interesting manipulations of entanglement involving only a few gates and qubits, of which we give examples. Quantum communication also involves manipulations of entanglement which are simpler to implement than elaborate computations. We briefly analyse the notion of the capacity of a quantum communication channel.

We describe how two vibrational degrees of freedom of a single trapped ion can be coupled through the action of suitably-chosen laser excitation. We concentrate on a two-dimensional ion trap with dissimilar vibrational frequencies in the x- and y-directions of motion, and derive from first principles a variety of quantized two-mode couplings, concentrating on a linear coupling which takes excitations from one mode to another. We demonstrate how this can result in a state rotation, in which it is possible to transfer the motional state of the ion from say the x-direction to the y-direction without prior knowledge of that motional state.

We propose a scheme for generating vibrational pair coherent states of the motion of an ion in a two-dimensional trap. In our scheme, the trapped ion is excited bichromatically by three laser beams along different directions in the X-Y plane of the ion trap. We show that if the initial vibrational state is given by a two-mode Fock state, the final steady state, indicated by the extinction of the fluorescence emitted by the ion, is a pure state. The motional state of the ion in the equilibrium realizes that of the highly-correlated pair coherent state.