M. B. Plenio

Quantum sensors typically translate external fields into a periodic response whose frequency is then determined by analyses performed in Fourier space. This allows for a linear inference of the parameters that characterize external signals. In practice, however, quantum sensors are able to detect fields only in a narrow range of amplitudes and frequencies. A departure from this range, as well as the presence of significant noise sources and short detection times, lead to a loss of the linear relationship between the response of the sensor and the target field, thus limiting the working regime of the sensor. Here we address these challenges by means of a Bayesian inference approach that is tolerant to strong deviations from desired periodic responses of the sensor and is able to provide reliable estimates even with a very limited number of measurements. We demonstrate our method for an $^{171}$Yb$^{+}$ trapped-ion quantum sensor but stress the general applicability of this approach to different systems.

We present a method that combines continuous and pulsed microwave radiation patterns to achieve robust interactions among hyperfine trapped ions placed in a magnetic field gradient. More specifically, our scheme displays continuous microwave drivings with modulated phases, phase flips, and $π$ pulses. This leads to high-fidelity entangling gates which are resilient against magnetic field fluctuations, changes on the microwave amplitudes, and crosstalk effects. Our protocol runs with arbitrary values of microwave power, which includes the technologically relevant case of low microwave intensities. We demonstrate the performance of our method with detailed numerical simulations that take into account the main sources of decoherence.

We propose a pulsed dynamical decoupling protocol as the generator of tunable, fast, and robust quantum phase gates between two microwave-driven trapped ion hyperfine qubits. The protocol consists of sequences of $π$-pulses acting on ions that are oriented along an externally applied magnetic field gradient. In contrast to existing approaches, in our design the two vibrational modes of the ion chain cooperate under the influence of the external microwave driving to achieve significantly increased gate speeds. Our scheme is robust against the dominant noise sources, which are errors on the magnetic field and microwave pulse intensities, as well as motional heating, predicting two-qubit gates with fidelities above $99.9\%$ in tens of microseconds.

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.

Sensing in the presence of environmental noise is a problem of increasing practical interest. In a master equation description, where the state of the environment is unobserved, the effect of signal and noise is described by system operators only. In this context it is well-known that noise that is orthogonal on an external signal can be corrected for without perturbing the signal, while similarly efficient strategies for non-orthogonal signal and noise operators are not known. Here we make use of the fact that system-environment interaction typically arises via local two-body interactions describing the exchange of quanta between system and environment, which are observable in principle. That two-body-interactions are usually orthogonal on system operators, allows us to develop error corrected sensing supported by the observation of the quanta that are emitted into the environment. We describe such schemes and outline a realistic proof-of-principle experiment in an ion trap set-up.

We present practical methods to measure entanglement for quantum simulators that can be realized with trapped ions, cold atoms, and superconducting qubits. Focussing on long- and short-range Ising-type Hamiltonians, we introduce schemes that are applicable under realistic experimental conditions including mixedness due to, e.g., noise or temperature. In particular, we identify a single observable whose expectation value serves as a lower bound to entanglement and which may be obtained by a simple quantum circuit. As such circuits are not (yet) available for every platform, we investigate the performance of routinely measured observables as quantitative entanglement witnesses. Possible applications include experimental studies of entanglement scaling in critical systems and the reliable benchmarking of quantum simulators.

We present a study of the performance of the trapped-ion driven geometric phase gates introduced in [New J. Phys. 15, 083001 (2013)] when realized in a stimulated Raman transition. We show that the gate can achieve errors below the fault-tolerance threshold in the presence of laser intensity fluctuations. We also find that, in order to reduce the errors due to photon scattering below the fault-tolerance threshold, very intense laser beams are required to allow for large detunings in the Raman configuration without compromising the gate speed.

Measuring heat flow through nanoscale systems poses formidable practical difficulties as there is no `ampere meter' for heat. We propose to overcome this problem by realizing heat transport through a chain of trapped ions. Laser cooling the chain edges to different temperatures induces a current of local vibrations (vibrons). We show how to efficiently control and measure this current, including fluctuations, by coupling vibrons to internal ion states. This demonstrates that ion crystals provide a suitable platform for studying quantum transport, e.g., through thermal analogues of quantum wires and quantum dots. Notably, ion crystals may give access to measurements of the elusive large fluctuations of bosonic currents and the onset of Fourier's law. These results are supported by numerical simulations for a realistic implementation with specific ions and system parameters.

We describe a hybrid laser-microwave scheme to implement two-qubit geometric phase gates in crystals of trapped ions. The proposed gates can attain errors below the fault-tolerance threshold in the presence of thermal, dephasing, laser-phase, and microwave-intensity noise. Moreover, our proposal is technically less demanding than previous schemes, since it does not require a laser arrangement with interferometric stability. The laser beams are tuned close to a single vibrational sideband to entangle the qubits, while strong microwave drivings provide the geometric character to the gate, and thus protect the qubits from these different sources of noise. A thorough analytic and numerical study of the performance of these gates in realistic noisy regimes is presented.

Traversal of a symmetry-breaking phase transition at a finite rate can lead to causallyseparated regions with incompatible symmetries and the formation of defects at their boundaries. The defect formation follows universal scaling laws prescribed by the Kibble-Zurek mechanism (KZM) important to the study of phase transitions in fields as diverse as quantum and statistical mechanics, condensed matter physics and cosmology. Here, we observe the KZM in a crystal of cold trapped ions, which is conducive to the precise control of structural phases and the detection of defects. The experiment confirms a scaling law with an exponent of 2.68 +/- 0.06, as predicted from the KZM in the finite inhomogeneous case. Such precision makes it feasible to use ion crystals for quantitative tests of classical and quantum statistical mechanics.

We introduce a scheme to perform dissipation-assisted quantum information processing in ion traps considering realistic decoherence rates, for example, due to motional heating. By means of continuous sympathetic cooling, we overcome the trap heating, and show that the damped vibrational excitations can still be exploited to mediate coherent interactions as well as collective dissipative effects. We describe how to control their relative strength experimentally, allowing for protocols of coherent or dissipative generation of entanglement. This scheme can be scaled to larger ion registers for coherent or dissipative many-body quantum simulations.

The accurate characterization of eigenmodes and eigenfrequencies of two-dimensional ion crystals provides the foundation for the use of such structures for quantum simulation purposes. We present a combined experimental and theoretical study of two-dimensional ion crystals. We demonstrate that standard pseudopotential theory accurately predicts the positions of the ions and the location of structural transitions between different crystal configurations. However, pseudopotential theory is insufficient to determine eigenfrequencies of the two-dimensional ion crystals accurately but shows significant deviations from the experimental data obtained from resolved sideband spectroscopy. Agreement at the level of 2.5 x 10^(-3) is found with the full time-dependent Coulomb theory using the Floquet-Lyapunov approach and the effect is understood from the dynamics of two-dimensional ion crystals in the Paul trap. The results represent initial steps towards an exploitation of these structures for quantum simulation schemes.

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

The spin-Peierls instability describes a structural transition of a crystal due to strong magnetic interactions. Here we demonstrate that cold Coulomb crystals of trapped ions provide an experimental testbed in which to study this complex many-body problem and to access extreme regimes where the instability is triggered by quantum fluctuations alone. We present a consistent analysis based on different analytical and numerical methods, and provide a detailed discussion of its feasibility on the basis of ion-trap experiments. Moreover, we identify regimes where this quantum simulation may exceed the power of classical computers.

The loss of coherence is one of the main obstacles for the implementation of quantum information processing. The efficiency of dynamical decoupling schemes, which have been introduced to address this problem, is limited itself by the fluctuations in the driving fields which will themselves introduce noise. We address this challenge by introducing the concept of concatenated continuous dynamical decoupling, which can overcome not only external magnetic noise but also noise due to fluctuations in driving fields. We show theoretically that this approach can achieve relaxation limited coherence times, and demonstrate experimentally that already the most basic implementation of this concept yields an order of magnitude improvement of the decoherence time for the electron spin of nitrogen vacancy centers in diamond. The proposed scheme can be applied to a wide variety of other physical systems including, trapped atoms and ions, quantum dots, and may be combined with other quantum technologies challenges such as quantum sensing and quantum information processing.

We introduce a novel scheme that combines phonon-mediated quantum logic gates in trapped ions with the benefits of continuous dynamical decoupling. We demonstrate theoretically that a strong driving of the qubit decouples it from external magnetic-field noise, enhancing the fidelity of two-qubit quantum gates. Moreover, the scheme does not require ground-state cooling, and is inherently robust to undesired ac-Stark shifts. The underlying mechanism can be extended to a variety of other systems where a strong driving protects the quantum coherence of the qubits without compromising the two-qubit couplings.

We exploit the geometry of a zig-zag cold-ion crystal in a linear trap to propose the quantum simulation of a paradigmatic model of long-ranged magnetic frustration. Such a quantum simulation would clarify the complex features of a rich phase diagram that presents ferromagnetic, dimerized antiferromagnetic, paramagnetic, and floating phases, together with previously unnoticed features that are hard to assess by numerics. We analyze in detail its experimental feasibility, and provide supporting numerical evidence on the basis of realistic parameters in current ion-trap technology.

Trapped atomic ions have been successfully used for demonstrating basic elements of universal quantum information processing (QIP). Nevertheless, scaling up of these methods and techniques to achieve large scale universal QIP, or more specialized quantum simulations remains challenging. The use of easily controllable and stable microwave sources instead of complex laser systems on the other hand promises to remove obstacles to scalability. Important remaining drawbacks in this approach are the use of magnetic field sensitive states, which shorten coherence times considerably, and the requirement to create large stable magnetic field gradients. Here, we present theoretically a novel approach based on dressing magnetic field sensitive states with microwave fields which addresses both issues and permits fast quantum logic. We experimentally demonstrate basic building blocks of this scheme to show that these dressed states are long-lived and coherence times are increased by more than two orders of magnitude compared to bare magnetic field sensitive states. This changes decisively the prospect of microwave-driven ion trap QIP and offers a new route to extend coherence times for all systems that suffer from magnetic noise such as neutral atoms, NV-centres, quantum dots, or circuit-QED systems.

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 propose to realize quantized discrete kinks with cold trapped ions. We show that long-lived solitonlike configurations are manifested as deformations of the zigzag structure in the linear Paul trap, and are topologically protected in a circular trap with an odd number of ions. We study the quantum-mechanical time evolution of a high-frequency, gap separated internal mode of a static kink and find long coherence times when the system is cooled to the Doppler limit. The spectral properties of the internal modes make them ideally suited for manipulation using current technology. This suggests that ion traps can be used to test quantum-mechanical effects with solitons and explore ideas for the utilization of the solitonic internal-modes as carriers of quantum information.

We present a robust and fast laser cooling scheme suitable for trapped atoms and ions. Based on quantum interference, generated by a special laser configuration, it is able to rapidly cool the system such that the final phonon occupation vanishes to zeroth order in the Lamb-Dicke parameter in contrast to existing cooling schemes. Furthermore, it is robust under conditions of fluctuating laser intensity and frequency, thus making it a viable candidate for experimental applications.

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.

A laser cooling scheme for trapped ions is presented which is based on the fast dynamical Stark shift gate, described in [Jonathan etal, PRA 62, 042307]. Since this cooling method does not contain an off resonant carrier transition, low final temperatures are achieved even in traveling wave light field. The proposed method may operate in either pulsed or continuous mode and is also suitable for ion traps using microwave addressing in strong magnetic field gradients.

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.

Quantum information processing rests on our ability to manipulate quantum superpositions through coherent unitary transformations, and to establish entanglement between constituent quantum components of the processor. The quantum information processor (a linear ion trap, or a cavity confining the radiation field for example) exists in a dissipative environment. We discuss ways in which entanglement can be established within such dissipative environments. We can even make use of a strong interaction of the system with its environment to produce entanglement in a controlled way.

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.

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.

Dissipation, the irreversible loss of energy and coherence, from a microsystem, is the result of coupling to a much larger macrosystem (or reservoir) which is so large that one has no chance of keeping track of all of its degrees of freedom. The microsystem evolution is then described by tracing over the reservoir states, resulting in an irreversible decay as excitation leaks out of the initially excited microsystems into the outer reservoir environment. Earlier treatments of this dissipation described an ensemble of microsystems using density matrices, either in Schroedinger picture with Master equations, or in Heisenberg picture with Langevin equations. The development of experimental techniques to study single quantum systems (for example single trapped ions, or cavity radiation field modes) has stimulated the construction of theoretical methods to describe individual realizations conditioned on a particular observation record of the decay channel, in the environment. These methods, variously described as Quantum Jump, Monte Carlo Wavefunction and Quantum Trajectory methods are the subject of this review article. We discuss their derivation, apply them to a number of current problems in quantum optics and relate them to ensemble descriptions.

We investigate the problem of factorization of large numbers on a quantum computer which we imagine to be realized within a linear ion trap. We derive upper bounds on the size of the numbers that can be factorized on such a quantum computer. These upper bounds are independent of the power of the applied laser. We investigate two possible ways to implement qubits, in metastable optical transitions and in Zeeman sublevels of a stable ground state, and show that in both cases the numbers that can be factorized are not large enough to be of practical interest. We also investigate the effect of quantum error correction on our estimates and show that in realistic systems the impact of quantum error correction is much smaller than expected. Again no number of practical interest can be factorized.