Hermann Uys

The measurement of dynamic correlation functions of quantum systems is complicated by measurement backaction. To facilitate such measurements we introduce a protocol, based on weak ancilla--system couplings, that is applicable to arbitrary (pseudo)spin systems and arbitrary equilibrium or nonequilibrium initial states. Different choices of the coupling operator give access to the real and imaginary parts of the dynamic correlation function. This protocol reduces disturbances due to the early time measurements to a minimum, and we quantify the deviation of the measured correlation functions from the theoretical, unitarily-evolved ones. Implementations of the protocol in trapped ions and other experimental platforms are discussed. For spin-$1/2$ models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.

We have demonstrated simultaneous laser frequency stabilization of a UV and IR laser, to the same spectroscopic sample, by monitoring only the absorption of the UV laser. For trapping and cooling Yb$^{+}$ ions, a frequency stabilized laser is required at 369.95nm to drive the $^{2}S_{1/2}$ $ \rightarrow $ $ ^{2}P_{1/2}$ cooling transition. Since the cycle is not closed, a 935.18nm laser is needed to drive the $^{2}D_{3/2}$ $\rightarrow$ $^{3}D_{[3/2]1/2}$ transition which is followed by rapid decay to the $^{2}S_{1/2}$ state. Our 369nm laser is locked to Yb$^{+}$ ions generated in a hollow cathode discharge lamp using saturated absorption spectroscopy. Without pumping, the metastable $^{2}D_{3/2}$ level is only sparsely populated and direct absorption of 935nm light is difficult to detect. A resonant 369nm laser is able to significantly populate the $^{2}D_{3/2}$ state due to the coupling between the levels. Fast re-pumping to the $^{2}S_{1/2}$ state, by 935nm light, can be detected by observing the change in absorption of the 369nm laser using lock-in detection of the photodiode signal. In this way simultaneous locking of two optical frequencies in very different spectral regimes is accomplished. A rate equation model gives good qualitative agreement with the experimental results. This technique offers improved laser frequency stabilization compared to lasers locked individually to the sample and should be readily applicable to similar ion systems.

The presence of long-range quantum spin correlations underlies a variety of physical phenomena in condensed matter systems, potentially including high-temperature superconductivity. However, many properties of exotic strongly correlated spin systems (e.g., spin liquids) have proved difficult to study, in part because calculations involving N-body entanglement become intractable for as few as N~30 particles. Feynman divined that a quantum simulator - a special-purpose "analog" processor built using quantum particles (qubits) - would be inherently adept at such problems. In the context of quantum magnetism, a number of experiments have demonstrated the feasibility of this approach. However, simulations of quantum magnetism allowing controlled, tunable interactions between spins localized on 2D and 3D lattices of more than a few 10's of qubits have yet to be demonstrated, owing in part to the technical challenge of realizing large-scale qubit arrays. Here we demonstrate a variable-range Ising-type spin-spin interaction J_ij on a naturally occurring 2D triangular crystal lattice of hundreds of spin-1/2 particles (9Be+ ions stored in a Penning trap), a computationally relevant scale more than an order of magnitude larger than existing experiments. We show that a spin-dependent optical dipole force can produce an antiferromagnetic interaction J_ij ~ 1/d_ij^a, where a is tunable over 0<a<3; d_ij is the distance between spin pairs. These power-laws correspond physically to infinite-range (a=0), Coulomb-like (a=1), monopole-dipole (a=2) and dipole-dipole (a=3) couplings. Experimentally, we demonstrate excellent agreement with theory for 0.05<a<1.4. This demonstration coupled with the high spin-count, excellent quantum control and low technical complexity of the Penning trap brings within reach simulation of interesting and otherwise computationally intractable problems in quantum magnetism.

We demonstrate spectroscopy and thermometry of individual motional modes in a mesoscopic 2D ion array using entanglement-induced decoherence as a method of transduction. Our system is a $\sim$400 $μ$m-diameter planar crystal of several hundred $^9$Be$^+$ ions exhibiting complex drumhead modes in the confining potential of a Penning trap. Exploiting precise control over the $^9$Be$^+$ valence electron spins, we apply a homogeneous spin-dependent optical dipole force to excite arbitrary transverse modes with an effective wavelength approaching the interparticle spacing ($\sim$20 \nolinebreak$μ$m). Center-of-mass displacements below 1 nm are detected via entanglement of spin and motional degrees of freedom.

Building robust instruments capable of making interferometric measurements with precision beyond the standard quantum limit remains an important goal in many metrology laboratories. We describe here the basic concepts underlying spin squeezing experiments that allow one to surpass this limit. In priniciple it is possible to reach the so-called Heisenberg limit, which constitutes an improvement in precision by a factor $\sqrt{N}$, where $N$ is the number of particles on which the measurement is carried out. In particular, we focus on recent progress toward implementing spin squeezing with a cloud of beryllium ions in a Penning ion trap, via the geometric phase gate used more commonly for performing two-qubit entangling operations in quantum computing experiments.

We discuss the use of two-dimensional $^{9}$Be$^{+}$ ion crystals for experimental tests of quantum control techniques. Our primary qubit is the 124 GHz ground-state electron spin flip transition, which we drive using microwaves. An ion crystal represents a spatial ensemble of qubits, but the effects of inhomogeneities across a typical crystal are small, and as such we treat the ensemble as a single effective spin. We are able to initialize the qubits in a simple state and perform a projective measurement on the system. We demonstrate full control of the qubit Bloch vector, performing arbitrary high-fidelity rotations ($τ_π\sim$200 $μ$s). Randomized Benchmarking demonstrates an error per gate (a Pauli-randomized $π/2$ and $π$ pulse pair) of $8\pm1\times10^{-4}$. Ramsey interferometry and spin-locking measurements are used to elucidate the limits of qubit coherence in the system, yielding a typical free-induction decay coherence time of $T_{2}\sim$2 ms, and a limiting $T_{1ρ}\sim$688 ms. These experimental specifications make ion crystals in a Penning trap ideal candidates for novel experiments in quantum control. As such, we briefly describe recent efforts aimed at studying the error-suppressing capabilities of dynamical decoupling pulse sequences, demonstrating an ability to extend qubit coherence and suppress phase errors. We conclude with a discussion of future avenues for experimental exploration, including the use of additional nuclear-spin-flip transitions for effective multiqubit protocols, and the potential for Coulomb crystals to form a useful testbed for studies of large-scale entanglement.

One approach to maintaining phase coherence of qubits through dynamical decoupling consists of applying a sequence of Hahn spin-echo pulses. Recent studies have shown that, in certain noise environments, judicious choice of the delay times between these pulses can greatly improve the suppression of phase errors compared to traditional approaches. By enforcing a simple analytical condition, we obtain sets of dynamical decoupling sequences that are designed for optimized noise filtration and are spectrum-independent up to a single scaling factor set by the coherence time of the system. We demonstrate the efficacy of these sequences in suppressing phase errors through measurements on a model qubit system, $^{9}$Be$^{+}$ ions in a Penning trap. Our combined theoretical and experimental studies show that in high-frequency-dominated noise environments this approach may suppress phase errors orders of magnitude more efficiently than comparable techniques can.

We present a detailed experimental study of the Uhrig Dynamical Decoupling (UDD) sequence in a variety of noise environments. Our qubit system consists of a crystalline array of $^{9}$Be$^{+}$ ions confined in a Penning trap. We use an electron-spin-flip transition as our qubit manifold and drive qubit rotations using a 124 GHz microwave system. We study the effect of the UDD sequence in mitigating phase errors and compare against the well known CPMG-style multipulse spin echo as a function of pulse number, rotation axis, noise spectrum, and noise strength. Our results agree well with theoretical predictions for qubit decoherence in the presence of classical phase noise, accounting for the effect of finite-duration $π$ pulses. Finally, we demonstrate that the Uhrig sequence is more robust against systematic over/underrotation and detuning errors than is multipulse spin echo, despite the precise prescription for pulse-timing in UDD.