RESEARCH

Quantum Simulation with Trapped Ions:

My current research seeks to develop a well-controlled, scalable quantum simulator to understand the behavior of complex, two-dimensional (2D) quantum spin systems. Such a tunable and reconfigurable apparatus will have the capability to address important open questions in quantum many-body physics that are intractable to numerical calculation and inaccessible to any current experimental system. The approach will encode effective quantum spins within the well-isolated internal states of trapped ions, will build upon the many recent advances in one-dimensional trapped-ion quantum simulators, and will be directly scalable to more than 100 individually-controllable spins.

Feynman was the first to point out that a new paradigm of computing was needed to gain complete insight into complex quantum systems. He argued that mapping the quantum-mechanical problem of interest onto a physical, well-controlled quantum system would eliminate the need for exponential resources, allowing such problems to be solved. These ideas were experimentally realized only a few years ago, with 2-3 trapped atomic ions used to perform a quantum simulation of a simple Ising spin system. Since then, trapped ions have been a leading platform for more advanced quantum simulation experiments, since their well-isolated environment enables high-fidelity state preparation and detection, long quantum coherence times, and extraordinarily precise quantum measurements. In all cases, however, experiments have been restricted to one-dimensional ion chains emulating one-dimensional spin models.

Schematic overview of a 2D ion trap quantum simulation. (a) Ions are confined in an rf Paul trap with large axial trap frequencies. When laser-cooled to milliKelvin temperatures, they self-organize into a 2D triangular lattice crystal. Effective spins are encoded within each ion and initialized to a product state. (b) After initialization, laser-induced forces can be used to generate a variety of different spin-spin couplings between the ions (illustrated are interacting spins on a Kagome lattice). The initial state coherently evolves under the chosen spin Hamiltonian for the desired time. (c) Following the unitary time evolution, the projection of the spin states onto a chosen axis is read out by capturing the site-resolved ion fluorescence onto a CCD camera. Repeated experiments allow for measurement of any arbitrary N-body correlation function and characterization of system entanglement.

By constructing a higher-dimensional ion trap quantum simulator, one can begin to address many of the open questions in quantum many-body physics associated with geometric frustration, exotic phases of matter (such as spin glasses/liquids and many-body localized states), and the relationship between entanglement, frustration, and high-Tc superconductivity. For instance, changing the 2D lattice geometry (an off-limits concept in 1D) can give rise to contrasting, complex, and often poorly understood phases. It is well known that antiferromagnetic Ising spins on square lattices lead to simple Neel-ordered ground states, and that triangular and Kagome lattices exhibit geometric frustration, highly degenerate ground states, and massive entanglement entropy. Far less is known about the behavior of XY or Heisenberg spins on frustrated lattices, and whether their ground states display long-range Neel order, short-range correlations, broken symmetries, or some combination of the above. Less still is known about long-range interacting spin models, for which the ground state of even a simple square lattice remains an open question. Understanding the full phase diagrams and ground state behavior for these various quantum spin systems is of strong and widespread interest, but has so far proven elusive to due the complexity of numerical calculations and the dearth of clean and controllable experimental test systems.

Given these motivations, I am constructing a 2D trapped-ion quantum simulator that retains the traditional ion-trap strengths of full control at the single-particle level, site-resolved measurements and readout, and long coherence times. This well-controlled quantum system will serve as an experimental testbed for exploring many of the aforementioned questions at the center of quantum condensed matter physics. The examples given here are undoubtedly a small sampling of what is possible with such an apparatus; its reprogrammability and tunability make it adaptable for investigating nearly all types of ground state or dynamical properties of interacting quantum spin models.

(Left) Animation showing how an oscillating electric field can confine a charged particle. (Right) Animation showing the motion of a 2D crystal of ions trapped in an oscillating electric field.

Postdoctoral Work:

My postdoctoral research was in Chris Monroe's trapped ion quantum information group at the Joint Quantum Institute. Using a linear chain of 10-20 ¹⁷¹Yb⁺ ions confined in an RF Paul trap, we performed quantum simulations of the many-body physics found in interacting spin systems. A long-term goal of the project is to scale up the number of controlled spins to 30+, where calculating system properties or dynamical evolution becomes intractable using a classical computer.

Long-range spin-spin interactions between the ions are generated using phonon-mediated, spin-dependent laser forces. State-dependent fluorescence imaging of the ions onto a camera allows for readout of the individual spin states and access to all possible correlation functions. By varying laser frequencies or trap voltages, one can tune the range of spin-spin interaction and the amount of frustration in the many-body spin system - something traditionally difficult (or impossible) to accomplish in a real material.

Camera image of 16 171Yb+ ions held in the electric potential of a Paul trap. The chain is approximately 40 μm end-to-end.

Using the trapped ion system, we have performed adiabatic quantum simulations to experimentally find the ground states of a long-range Ising model with a transverse magnetic field. The spins are prepared along the transverse magnetic field, which is initially large compared to the Ising couplings and slowly reduced during the simulation. We have designed and implemented protocols to optimize the speed of these quantum simulations while remaining locally adiabatic. We have additionally proposed a technique to determine the ground state of this Hamiltonian experimentally when the ramp is non-adiabatic, and have demonstrated its success in a system of 14 interacting spins.

With our quantum simulator, we have also studied the ground state magnetic phases of a classical Ising model at zero temperature, in which phase transitions cannot be driven classically due to the absence of thermal fluctuations. Instead, we introduce controlled quantum fluctuations to drive the phase transitions, map their positions in the phase diagram, and create the multiple, classically inaccessible ground states at different longitudinal field strengths.

More recently, we have started to probe the dynamics of our many-body system. We have developed a spectroscopic method to resolve the excited state energy levels of our Hamiltonian and to coherently engineer highly entangled states. The method allows us to directly determine the strengths of all spin-spin couplings and verify that the experimentally implemented Hamiltonian closesly matches the Hamiltonian we wish to study. We expect such verification to be crucial as the system grows beyond 30+ spins, where classical computers will be unable to confirm the results of quantum simulators.

In addition, we have performed experiments to study the growth and propagation of correlations in a quantum system evolving in a far-from equilibrium state. These measurements probe the velocity with which quantum information can be transfered throughout a many-body system, which sets the minimum timescales for entanglement growth or thermalization between disparate parts of the chain. The long-range interactions are observed to give faster propagation velocities than in the short-range case studied by Lieb and Robinson, indicating that the wealth of important theoretical proofs about thermalization, entanglement growth, and classical simulability can no longer be applied to the ion system.

Measured quench dynamics in a long-range Ising model. a-c, Spatial and time-dependent correlations (a), extracted light-cone boundary (b) and correlation propagation velocity (c) following a global quench of a long-range Ising model with α=0.63. The curvature of the boundary shows an increasing propagation velocity (b), quickly exceeding the short-range Lieb-Robinson velocity bound, vLR (c). Solid lines give a power-law fit to the data, which slightly depends on the choice of fixed contour Ci,j. d-l, Complementary plots for α=0.83 (d-f), α=1.00 (g-i) and α=1.19 (j-l). As the range of the interactions decreases, correlations do not propagate as far or as quickly through the chain; the short-range velocity bound vLR is not exceeded for our shortest-range interaction. m, n, Nearest neighbour (m) and tenth-nearest-neighbour (n) correlations for our shortest and longest-range interactions show excellent agreement with the decoherence-free exact solution (with no adjustable parameters). The dashed blue curves show an improved long-range bound valid for any commuting Hamiltonian.

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