We present a detailed study of quantum simulations of coupled spin systems in surface-electrode (SE) ion-trap arrays, and illustrate our findings with a proposed implementation of the hexagonal Kitaev model (Kitaev A 2006 Ann. Phys. 321 2). The effective (pseudo)spin interactions making up such quantum simulators are found to be proportional to the dipole–dipole interaction between the trapped ions, and are mediated by motion that can be driven by state-dependent forces. The precise forms of the trapping potentials and the interactions are derived in the presence of an SE and a cover electrode. These results are the starting point to derive an optimized SE geometry for trapping ions in the desired honeycomb lattice of Kitaev's model, where we design the dipole–dipole interactions in a way that allows for coupling all three bond types of the model simultaneously, without the need for time discretization. Finally, we propose a simple wire structure that can be incorporated into a microfabricated chip to generate localized state-dependent forces which drive the couplings prescribed by this particular model; such a wire structure should be adaptable to many other situations.
We present a detailed study of quantum simulations of coupled spin systems in surface-electrode (SE) ion-trap arrays, and illustrate our findings with a proposed implementation of the hexagonal Kitaev model (Kitaev A 2006 Ann. Phys. 321 2). The effective (pseudo)spin interactions making up such quantum simulators are found to be proportional to the dipole–dipole interaction between the trapped ions, and are mediated by motion that can be driven by state-dependent forces. The precise forms of the trapping potentials and the interactions are derived in the presence of an SE and a cover electrode. These results are the starting point to derive an optimized SE geometry for trapping ions in the desired honeycomb lattice of Kitaev's model, where we design the dipole–dipole interactions in a way that allows for coupling all three bond types of the model simultaneously, without the need for time discretization. Finally, we propose a simple wire structure that can be incorporated into a microfabricated chip to generate localized state-dependent forces which drive the couplings prescribed by this particular model; such a wire structure should be adaptable to many other situations.
Trapped ions offer long internal state (spin) coherence times and strong interparticle interactions mediated by the Coulomb force. This makes them interesting candidates for quantum simulation of coupled lattices. To this end, it is desirable to be able to trap ions in arbitrary conformations with precisely controlled local potentials. We provide a general method for optimizing periodic planar radio-frequency electrodes for generating ion trapping potentials with specified trap locations and curvatures above the electrode plane. A linear-programming algorithm guarantees globally optimal electrode shapes that require only a single radio-frequency voltage source for operation. The optimization method produces final electrode shapes that are smooth and exhibit low fragmentation. Such characteristics are desirable for practical fabrication of surface-electrode trap lattices.
Current classical computing methods are unable to simulate quantum many-body interactions efficiently.Analog quantum simulators with even small numbers of quantum degrees of freedom are expected to enable the investigation of systems not fully accessible in traditional computing such as condensed-matter systems, quantum chemistry, and material science.Two-dimensional lattices of trapped ions are a good candidate to realize an analog quantum simulator for their addressability, individual control and individual detection.Here, we report on multiple unique lattice surface ion trap designs, microfabricated at Sandia National Laboratories, capable of trapping and interacting up to 4 ions.Ions are held in close proximity in individual all radio-frequency potential minima and interact via their Coulomb repulsion.The Coulomb coupling between ions allows for simulation of spin-spin interactions 1-6 .Control electrodes of the trap can be used to adjust principal axes of the ion's motion and thus enable individual control over the interaction between ions.The lattice trap array architecture tested in this work can be scaled to larger two-dimensional lattices of trapped ions.
Two-dimensional crystals of trapped ions are a promising system with which to implement quantum simulations of challenging problems such as spin frustration. Here, we present a design for a surface-electrode elliptical ion trap which produces a 2-D ion crystal and is amenable to microfabrication, which would enable higher simulated coupling rates, as well as interactions based on magnetic forces generated by currents which may be incorporated into the trap structure. Working in an 11 K cryogenic environment, we experimentally verify to within 10% a numerical model of the structure of ion crystals in the trap, given the measured trap anisotropy. We also explore the possibility of implementing quantum simulation using magnetic forces, and calculate J-coupling rates on the order of 103 s-1 for a trap scale of 10 μm, using a current of 1 A.
We propose a trapped ion scheme en route to realize spin Hamiltonians on a Kagome lattice which, at low energies, are described by emergent gauge fields, and support a topological quantum spin liquid ground state. The enabling element in our scheme is the hexagonal plaquette spin–spin interactions in a two-dimensional ion crystal. For this, the phonon-mode spectrum of the crystal is engineered by standing-wave optical potentials or by using Rydberg excited ions, thus generating localized phonon-modes around a hexagon of ions selected out of the entire two-dimensional crystal. These tailored modes can mediate spin–spin interactions between ion-qubits on a hexagonal plaquette when subject to state-dependent optical dipole forces. We discuss how these interactions can be employed to emulate a generalized Balents–Fisher–Girvin model in minimal instances of one and two plaquettes. This model is an archetypical Hamiltonian in which gauge fields are the emergent degrees of freedom on top of the classical ground state manifold. Under realistic situations, we show the emergence of a discrete Gauss's law as well as the dynamics of a deconfined charge excitation on a gauge-invariant background using the two-plaquettes trapped ions spin-system. The proposed scheme in principle allows further scaling in a future trapped ion quantum simulator, and we conclude that our work will pave the way towards the simulation of emergent gauge theories and quantum spin liquids in trapped ion systems.
The understanding of quantum mechanical systems is essential to modern physics and its applications. However, already for entangled systems of a few tens of constituents, exact numerical simulations on classical computers can become impossible. Quantum simulators based on well-controlled quantum system might provide the only effective approach to these systems. Trapped atomic ions offer a uniquely precise and controllable platform. They are used as platforms for, e.g., future quantum computers, quantum simulators, and extremely precise clocks. Quantum-simulation problems of interest and out-of-reach for modern theoretical methods include the dynamics of one-dimensional systems on long time scales and more-than-one dimensional systems with long-range interaction. For the latter, two-dimensional arrays of ions, individually trapped and controlled above microfabricated surface-electrode traps, present a promising approach. In the first realizations so far, the inter-site distance between the ions was too large to allow for sufficient inter-site coupling. We aim for a scalable approach to an analog quantum simulator based on a two-dimensional array of individually trapped ions. As a first demonstration, we operate an array of three magnesium ions in triangular arrangement with a side-length of 40µm. Therein, we previously demonstrated the individual control of internal (electronic) and external (motional) degrees of freedom of the ions. Here, we present the first realization of inter-site coupling in the two-dimensional array. We demonstrate the coupling by a transfer of large coherent states of motion between the sites. We investigate the influence of the amplitude of these motional states in the anharmonic trapping potential and extrapolate to future experiments on the single-phonon level. We apply the real-time control of the coupling and concatenate the two-site coupling to transfer motional excitation between all three sites of the array. To demonstrate the scalability of our techniques, we couple all three sites simultaneously and investigate the evolution of an initial excitation at one and two sites of the array. The latter shows interference effects of the different pathways in the triangle. Current motional heating rates on the order of the inter-site coupling, preclude working with single phonons near the motional ground-state. We present a new experimental apparatus and vacuum chamber which will allow in-situ cleaning of surface-electrode traps. This has shown to reduce motional heating rates by two orders of magnitude. In this apparatus, we additionally realize a magnetic-field insensitive qubit in 25Mg+ with a coherence time exceeding six seconds. We generate the magnetic field around 10.9mT by a hybrid magnet assembly based on solid-state magnets and fine-tuned by electric coils. Once we have reduced the heating rate in the triangle trap array using the new setup, first quantum simulation experiments, investigating spin-frustration or artificial gauge fields, will come in reach. Here our triangle represents a basic, scalable, building block of future latices designed at will.
A collection of trapped atomic ions represents one of the most attractive platforms for the quantum simulation of interacting spin networks and quantum magnetism. Spin-dependent optical dipole forces applied to an ion crystal create long-range eective spin-spin interactions and allow the simulation of spin Hamiltonians that possess nontrivial phases and dynamics. We trap linear chains of 171Yb+ ions in a Paul trap, and constrain the occupation of energy levels to the ground hyperne clock-states, creating a qubit or pseudo-spin 1/2 system. We proceed to implement spin-spin couplings between two ions using the far detuned Mlmer-Srenson scheme and perform adiabatic quantum simulations of Ising Hamiltonians with long-range couplings. We then demonstrate our ability to control the sign and relative strength of the interaction between three ions. Using this control, we simulate a frustrated triangular lattice, and for the first time establish an experimental connection between frustration and quantum entanglement. We then scale up our simulation to show phase transitions from paramagnetism to ferromagnetism for nine ions, and to anti-ferromagnetism for sixteen ions. The experimental work culminates with our most complicated Hamiltonian - a long range anti-ferromagnetic Ising interaction between 10 ions with a biasing axial field. Theoretical work presented in this thesis shows how the approach to quantum simulation utilized in this thesis can be further extended and improved. It is shown how appropriate design of laser elds can provide for arbitrary multidimensional spin-spin interaction graphs even for the case of a linear spatial array of ions. This scheme uses currently existing trap technology and is scalable to levels where classical methods of simulation are intractable.