We propose a method to simulate the dynamics of spin-boson models with small crystals of trapped ions where the electronic degree of freedom of one ion is used to encode the spin while the collective vibrational degrees of freedom are employed to form an effective harmonic environment. The key idea of our approach is that a single damped mode can be used to provide a harmonic environment with Lorentzian spectral density. More complex spectral functions can be tailored by combining several individually damped modes. The protocol is especially well-suited to simulate spin-boson models with structured environments. We propose to work with mixed-species crystals such that one species serves to encode the spin while the other species is used to cool the vibrational degrees of freedom to engineer the environment. The strength of the dissipation on the spin can be controlled by tuning the coupling between spin and vibrational degrees of freedom. In this way the dynamics of spin-boson models with macroscopic and non-Markovian environments can be simulated using only a few ions. We illustrate the approach by simulating an experiment with realistic parameters and show by computing quantitative measures that the dynamics is genuinely non-Markovian.

We describe a hybrid laser-microwave scheme to implement twoqubit 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. A. Magnus expansion for the driven single-sideband Hamiltonian 28 Appendix B. Stochastic processes for the noise sources 31 References 372. Driven single-sideband geometric phase gates Two-ion crystals as the hardware for quantum logic gatesLet us start by describing the system under consideration: a two-ion (N = 2) crystal confined in a linear Paul trap [10]. Under certain conditions [11], such radio-frequency traps provide an effective quadratic confining potential, which is characterized by the so-called axial ω z , and radial {ω x , ω y } trap frequencies. Moreover, when {ω x , ω y } ω z , the ion equilibrium positions arrange in a string along the trap z-axis. As customary in these cases, when the particles only 2 During the completion of this work, we became aware of the results of [9]. In this work, the authors have also generalized the scheme described in [8] to the near-resonance regime where faster gates can be achieved. Moreover, they have presented an experimental realization of these ideas, showing that two-qubit gates with errors ε 2,q ≈ 2.6 × 10 −2 can be achieved using this hybrid laser-microwave scheme.

Ion Coulomb crystals are currently establishing themselves as a highly controllable test-bed for mesoscopic systems of statistical mechanics. The detailed experimental interrogation of the dynamics of these crystals however remains an experimental challenge. In this work, we show how to extend the concepts of multi-dimensional nonlinear spectroscopy to the study of the dynamics of ion Coulomb crystals. The scheme we present can be realized with state-of-the-art technology and gives direct access to the dynamics, revealing nonlinear couplings even in the presence of thermal excitations. We illustrate the advantages of our proposal showing how twodimensional spectroscopy can be used to detect signatures of a structural phase transition of the ion crystal, as well as resonant energy exchange between modes. Furthermore, we demonstrate in these examples how different decoherence mechanisms can be identified.Two-dimensional (2D) spectroscopy was first proposed and realized in the context of nuclear magnetic resonance (NMR) experiments and has proven to be a very valuable tool in the investigation of complex spin systems [1]. By properly designed pulse sequences complicated spectra can be unravelled by the separation of interactions originating from different physical mechanisms to different frequency axes. The method allows for the estimation of spin-spin couplings in complex spin systems and the identification of different sources of noise. 2D spectroscopy has been adapted with remarkable success to other fields, facilitating the investigation of anharmonic molecular vibrational spectra in the infrared [2], electronic dynamics in molecular aggregates [3] and photosynthetic pigment-protein complexes [4], and photochemical reactions [5].Here we propose and analyze the application of 2D spectroscopy for the precise experimental characterization of nonlinear dynamics in few-or many-body systems of interest for quantum optics, in particular, in trapped-ion Coulomb crystals. The excellent control over the internal and motional degrees of freedom makes trapped atomic ions [6] a versatile tool to study statistical mechanics of systems in and out of equilibrium [7][8][9]. A paradigmatic example is provided by the linear-to-zigzag structural transition [10, 11]. In the vicinity of the transition, the usual harmonic treatment of the motion breaks down and nonlinear terms in the potential are essential for understanding the dynamics of the Coulomb crystal. Nonlinearities added to the trap potential have also been proposed for the implementation of the Frenkel-Kontorova model [12] and the Bose-Hubbard model [13]. The scheme we present can be used for the analysis of nonlinear dynamics, and, more generally, it represents a new appproach for the interrogation of complex quantum systems constructed from ion crystals. Some features of 2D spectroscopy are especially appealing in this context: it can provide information that is not accessible in 1D Ramsey-type experiments, it can filter out the contribution from purely harmonic term...

In the last years several estimation strategies have been formulated to determine the value of an unknown parameter in the most precise way, taking into account the presence of noise. These strategies typically rely on the use of quantum entanglement between the sensing probes and they have been shown to be optimal in the asymptotic limit in the number of probes, as long as one performs measurements on shorter and shorter time scales. Here, we present a different approach to frequency estimation, which exploits quantum coherence in the state of each sensing particle in the long time limit and is obtained by properly engineering the environment. By means of a commonly used master equation, we show that our strategy can overcome the precision achievable with entanglement-based strategies for a finite number of probes. We discuss a possible implementation of the scheme in a realistic setup that uses trapped ions as quantum sensors.

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