Quantum Simulations of Antiferromagnetic Spin Chains in an Optical Lattice

Understanding exotic forms of magnetism in quantum mechanical systems is a central goal of modern condensed matter physics, with implications for systems ranging from high-temperature superconductors to spintronic devices. Simulating magnetic materials in the vicinity of a quantum phase transition is computationally intractable on classical computers, owing to the extreme complexity arising from quantum entanglement between the constituent magnetic spins. Here we use a degenerate Bose gas of rubidium atoms confined in an optical lattice to simulate a chain of interacting quantum Ising spins as they undergo a phase transition. Strong spin interactions are achieved through a site-occupation to pseudo-spin mapping. As we vary a magnetic field, quantum fluctuations drive a phase transition from a paramagnetic phase into an antiferromagnetic phase. In the paramagnetic phase, the interaction between the spins is overwhelmed by the applied field, which aligns the spins. In the antiferromagnetic phase, the interaction dominates and produces staggered magnetic ordering. Magnetic domain formation is observed through both in situ site-resolved imaging and noise correlation measurements. By demonstrating a route to quantum magnetism in an optical lattice, this work should facilitate further investigations of magnetic models using ultracold atoms, thereby improving our understanding of real magnetic materials.

Goals for this project can be distinguished from those of the field of quantum computing, as that term has come to be used. Quantum simulations build on, and use some of the tools of quantum computation, such as error correction and the concept of quantum circuits. However, the goals of quantum computation are directed toward implementing certain algorithms, such as factoring and data base search, whereas the goals we address are focused on building tools to understand quantum phenomena in systems of fundamental interest. Examples include models of high temperature superconductivity whose phase diagrams are numerically inaccessible, and the properties of superfluid condensates and their transitions, for instance the transition into a Tonks gas, or Anderson insulator phases. A variety of the CUA experiments with ultra-cold neutral atoms address these important questions from condensed matter physics. This project takes an alternative approach towards implementing these goals that is based on systems of trapped ions.

Idea of Ion Trap Quantum Simulator -- A quantum simulator is inherently a quantum computer, but there are some important distinctions. The level of complexity is much less than required for a useful quantum computer. Specifically, the physical system used for the simulator need only have a number of degrees of freedom (e.g. controllable internal quantum states) comparable to the system being simulated. For example, to simulate an N-spin Ising model it suffices to have a lattice of ~N atoms. Furthermore, the control scheme is intrinsically simpler and under the right circumstances, can be more robust (i.e. less susceptible to errors) than that required for quantum computation.

These simplicities allow for the possibility of realizing a quantum simulator with technology at hand along the following lines. The working medium is a two-dimensional array of ultracold ions whose internal hyperfine states model degrees of freedom of the system being simulated. The vibrational states of the ions are entangled via the Coulomb interaction, and mapped onto internal states of the ions with laser-driven Raman transitions. This general approach has already been applied to linear chains of four to six ions for metrology, clocks, and simple quantum algorithms, at a variety of institutions around the world.

Extending trapped-ion techniques to systems of tens to hundreds of ions presents a formidable challenge, but we believe that it is possible using a two-dimensional radio-frequency ion trap that we have implemented based on ideas advanced at NIST. The special feature of our implementation is that it is entirely planar; it is fabricated from a single layer of metal, deposited on a glass composite substrate, and lithographically patterned to produce segmented electrodes. The advantages of this geometry are that it is readily fabricated and can be extended in size because it is fully compatible with semiconductor fabrication techniques. Also, it has the intrinsic geometry of two-dimensional neutral atom traps already used at CUA, which is important in loading the cold ions. This approach opens the door to achieving the electrostatic control required to manipulate tens to hundreds of ions.

Implementation -- Although the two-dimensional ion trap offers significant advantages in its scalability, it suffers from one serious disadvantage: the perpendicular confining potential is intrinsically weak. Thus the ions must be loaded at extremely low energy, precluding the traditional technique of electron-beam ionization of thermal atoms from a hot oven. To overcome this problem we plan to photoionize neutral atoms trapped in a magneto-optical trap (MOT) above the planar surface. The recoil energy of the atom, typically ~10 µeV (0.1K), is small compared to the trap depth, typically ~0.01 eV (120K). The trapped ion is then cooled to the vibrational ground state using methods that have been demonstrated at several laboratories.

We have considered a variety of potential problems associated with our proposed approach. The most serious appears to be the unknown origin of heating in microfabricated ion traps, which show noise that scales a d-4. The probable cause of this, surface contaminants due to having a hot atom source, is avoided by our method of loading from a MOT.

We have carried out preliminary studies of this system and implemented several key components: construction and demonstration of ion crystallization in a linear trap, lithographic fabrication of a planar ion trap, and development of a MOT for loading a two-dimensional trap.

Frequency stabilized diode laser systems and linear Paul traps have been built and ion crystals have been created as a test demonstration of our ion trapping capabilities. In addition, ionized clusters of rhodamine-doped dextran clusters have been trapped in a planar ion trap, fabricated using standard lithographic techniques on a composite glass laminate substrate We can create ion chains, split and join chains, and move individual ions linearly and through four-way junctions, by applying DC bias voltages to a selection of 48 electrode segments along the trap.

A magneto-optical trap containing neutral strontium atoms has been created, and positioned in a UHV chamber above a planar substrate which will serve as an ion trap, to catch photo-ionized atoms.

The experimental program presents obvious challenges, but we believe that the following tasks can be carried out over a period of several years. The first task is to load low-energy ions into a planar trap. The next is to create two-dimensional ion crystals, cooled to their vibrational ground state. We anticipate being able to create and control crystals with about a dozen ions, in tight lateral confinement. This modest number would actually enable us to create experimental conditions under which phonon modes of the ions may be used to simulate spins interacting under a Bose-Hubbard Hamiltonian. It might also permit us to observe an (artificial) Mott-insulator to superfluid phase transition, by adiabatically modifying the hopping rate, as has been theoretically proposed. It may also be possible to simulate more sophisticated BEC behavior, such as Tonks gas dynamics, as studied by other CUA groups. Our current planar trap geometries can accommodate up to ~100 ions, and together with ~10 independently modulated, frequency stabilized laser beams incident on the ions, and photon counting imaging detectors, a quantum simulator for ~100 interacting spins should be realizable.

Implementing a single quantum simulation, even an elementary simulation with a small number of ions, would be a major advance because it could open the way to studying many different systems. This would be possible because Hamiltonian control techniques from quantum computation can enable imperfect but easily controlled systems to simulate the behavior of the idealized models that are of theoretical interest. For example, a lattice of atoms with short coherence times cannot directly be used to simulate quantum dynamics of well-isolated spins that have long coherence times; however, when quantum error correction or methods from decoherence-free subspaces are applied, remarkably, such simulations become possible. The principle condition is that the coherence time of the components of the simulator exceeds a certain, fixed threshold, which is independent of the size of the simulated system, and the desired accuracy of the simulation. Current analyses indicate that the proposed ion trap simulator could be within reach of this capability.