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arXiv:1303.0062 · AIP conference proceedings

Simulating quantum magnetism with correlated non-neutral ion plasmas

J. J. Bollinger, J. Britton, Brian C. Sawyer

Published Jan 1, 2013• 4 citations
arXivHTMLPDF

Abstract

By employing forces that depend on the internal electronic state (or spin) of an atomic ion, the Coulomb potential energy of a strongly coupled array of ions can be modified in a spin-dependent way to mimic effective quantum spin Hamiltonians. Both ferromagnetic and anti-ferromagnetic interactions can be implemented. We use simple models to explain how the effective spin interactions are engineered with trapped-ion crystals. We summarize the type of effective spin interactions that can be readily generated, and discuss an experimental implementation using single-plane ion crystals in a Penning trap.

DOI
10.1063/1.4796076

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arXiv:1102.3645•The European Physical Journal D•Jul 27, 2011•45 citations
By employing forces that depend on the internal electronic state (or spin) of an atomic ion, the Coulomb potential energy of a strongly coupled array of ions can be modified in a spin-dependent way to mimic effective quantum spin Hamiltonians. Both ferromagnetic and anti-ferromagnetic interactions can be implemented. We use simple models to explain how the effective spin interactions are engineered with trapped-ion crystals. We summarize the type of effective spin interactions that can be readily generated, and discuss an experimental implementation using single-plane ion crystals in a Penning trap.
Quantum simulations using AMO systems promise a new way to experimentally study emergent quantum phenomena, but few systems have demonstrated the capability to control ensembles in which quantum effects cannot be directly computed. The 2D array of 100s of 9Be+ ions in a Penning trap, crystallized in a triangular lattice when laser cooled, is a promising platform for intractable quantum simulations using the 9Be+ valence electron spin as a qubit [1]. Spin-dependent forces are employed to modify the strong Coulomb interaction of the ions, mimicking a quantum magnetic interaction. The range of the magnetic interaction can be tuned from infinite to a dipole-dipole like coupling. Combining the application of the spin-dependent force with a transverse magnetic field should lead to the development of quantum correlations between the spins, which can be measured through optical readout of the spin state, both globally and with site-resolved imaging. In this way, trapped atomic ions can be used to probe novel and intractable aspects of quantum magnetism, including the effects of long-range interactions and simulations of quantum non-equilibrium phenomena [2]. In addition to a general overview, I will discuss recent work from a new Penning trap set-up at NIST.
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 effective spin–spin interactions and allow the simulation of spin Hamiltonians that possess nontrivial phases and dynamics. Here we show how the appropriate design of laser fields can provide for arbitrary multidimensional spin–spin interaction graphs even for the case of a linear spatial array of ions. This scheme uses currently available trap technology and is scalable to levels where the classical methods of simulation are intractable.
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.
We derive the normal modes for a rotating Coulomb ion crystal in a Penning trap, quantize the motional degrees of freedom, and illustrate how they can be driven by a spin-dependent optical dipole force to create a quantum spin simulator on a triangular lattice with hundreds of spins. The analysis for the axial modes (oscillations perpendicular to the two-dimensional crystal plane) follow a standard normal-mode analysis, while the remaining planar modes are more complicated to analyze because they have velocity-dependent forces in the rotating frame. After quantizing the normal modes into phonons, we illustrate some of the different spin-spin interactions that can be generated by entangling the motional degrees of freedom with the spin degrees of freedom via a spin-dependent optical dipole force. In addition to the well-known power-law dependence of the spin-spin interactions when driving the axial modes blue of the phonon band, we notice certain parameter regimes in which the level of frustration between the spins can be engineered by driving the axial or planar phonon modes at different frequencies. These systems may allow for the analog simulation of quantum spin glasses with large numbers of spins.
We discuss the experimental feasibility of quantum simulation with trapped ion crystals, using magnetic field gradients. We describe a micro structured planar ion trap, which contains a central wire loop generating a strong magnetic gradient of about 20 T/m in an ion crystal held about 160 μm above the surface. On the theoretical side, we extend a proposal about spin-spin interactions via magnetic gradient induced coupling (MAGIC) [J. Phys. B At. Mol. Opt. Phys. 42, 154009 (2009)]. We describe aspects where planar ion traps promise novel physics: spin-spin coupling strengths of transversal eigenmodes exhibit significant advantages over the coupling schemes in longitudinal direction that have been previously investigated. With a chip device and a magnetic field coil with small inductance, a resonant enhancement of magnetic spin forces through the application of alternating magnetic field gradients is proposed. Such resonantly enhanced spin-spin coupling may be used, for instance, to create Schrödinger cat states. Finally we investigate magnetic gradient interactions in two-dimensional ion crystals, and discuss frustration effects in such two-dimensional arrangements.