We investigate theoretically systems of ions in segmented linear Paul traps for the quantum simulation of quantum spin models with tunable interactions. The scheme is entirely general and can be applied to the realization of arbitrary spin-spin interactions. As a specific application we discuss in detail the quantum simulation of models that exhibit long-distance entanglement in the ground state. We show how tailoring of the axial trapping potential allows for generating spin-spin coupling patterns that are suitable to create long-distance entanglement. We discuss how suitable sequences of microwave pulses can implement Trotter expansions and realize various kinds of effective spin-spin interactions. The corresponding Hamiltonians can be varied on adjustable time scales, thereby allowing the controlled adiabatic preparation of their ground states.
We investigate theoretically systems of ions in segmented linear Paul traps for the quantum simulation of quantum spin models with tunable interactions. The scheme is entirely general and can be applied to the realization of arbitrary spin-spin interactions. As a specific application we discuss in detail the quantum simulation of models that exhibit long-distance entanglement in the ground state. We show how tailoring of the axial trapping potential allows for generating spin-spin coupling patterns that are suitable to create long-distance entanglement. We discuss how suitable sequences of microwave pulses can implement Trotter expansions and realize various kinds of effective spin-spin interactions. The corresponding Hamiltonians can be varied on adjustable time scales, thereby allowing the controlled adiabatic preparation of their ground states.
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.
The quantum simulation of complex many body systems holds promise for understanding the origin of emergent properties of strongly correlated systems, such as high-Tc superconductors and spin liquids. Cold atomic systems provide an almost ideal platform for quantum simulation due to their excellent quantum coherence, initialization and readout properties, and their ability to support several forms of interactions. In this thesis, I present experiments on the quantum simulation of long range Ising models in the presence of transverse magnetic fields with a chain of up to sixteen ultracold 171-Yb+ ions trapped in a linear radio frequency Paul trap. Two hyperfine levels in each of the 171-Yb+ ions serve as the spin-1/2 systems. We detect the spin states of the individual ions by observing state-dependent fluorescence with single site resolution, and can directly measure any possible spin correlation function. The spin-spin interactions are engineered by applying dipole forces from precisely tuned lasers whose beatnotes induce stimulated Raman transitions that couple virtually to collective phonon modes of the ion motion. The Ising couplings are controlled, both in sign and strength with respect to the effective transverse field, and adiabatically manipulated to study various aspects of this spin model, such as the emergence of a quantum phase transition in the ground state and spin frustration due to competing antiferromagnetic interactions. Spin frustration often gives rise to a massive degeneracy in the ground state, which can lead to entanglement in the spin system. We detect and characterize this frustration induced entanglement in a system of three spins, demonstrating the first direct experimental connection between frustration and entanglement. With larger numbers of spins we also vary the range of the antiferromagnetic couplings through appropriate laser tunings and observe that longer range interactions reduce the excitation energy and thereby frustrate the ground state order. This system can potentially be scaled up to study a wide range of fully connected spin networks with a few dozens of spins, where the underlying theory becomes intractable on a classical computer.
While fault-tolerant quantum computation might still be years away, analog quantum simulators offer a way to leverage current quantum technologies to study classically intractable quantum systems. Cutting edge quantum simulators such as those utilizing ultracold atoms are beginning to study physics which surpass what is classically tractable. As the system sizes of these quantum simulators increase, there are also concurrent gains in the complexity and types of Hamiltonians which can be simulated. In this work, I describe advances toward the realization of an adaptable, tunable quantum simulator capable of surpassing classical computation. We simulate long-ranged Ising and XY spin models which can have global arbitrary transverse and longitudinal fields in addition to individual transverse fields using a linear chain of up to 24 Yb+ 171 ions confined in a linear rf Paul trap. Each qubit is encoded in the ground state hyperfine levels of an ion. Spin-spin interactions are engineered by the application of spin-dependent forces from laser fields, coupling spin to motion. Each spin can be read independently using state-dependent fluorescence. The results here add yet more tools to an ever growing quantum simulation toolbox. One of many challenges has been the coherent manipulation of individual qubits. By using a surprisingly large fourth-order Stark shifts in a clock-state qubit, we demonstrate an ability to individually manipulate spins and apply independent Hamiltonian terms, greatly increasing the range of quantum simulations which can be implemented. As quantum systems grow beyond the capability of classical numerics, a constant question is how to verify a quantum simulation. Here, I present measurements which may provide useful metrics for large system sizes and demonstrate them in a system of up to 24 ions during a classically intractable simulation. The observed values are consistent with extremely large entangled states, as much as ~95% of the system entangled. Finally, we use many of these techniques in order to generate a spin Hamiltonian which fails to thermalize during experimental time scales due to a meta-stable state which is often called prethermal. The observed prethermal state is a new form of prethermalization which arises due to long-range interactions and open boundary conditions, even in the thermodynamic limit. This prethermalization is observed in a system of up to 22 spins. We expect that system sizes can be extended up to 30 spins with only minor upgrades to the current apparatus. These results emphasize that as the technology improves, the techniques and tools developed here can potentially be used to perform simulations which will surpass the capability of even the most sophisticated classical techniques, enabling the study of a whole new regime of quantum many-body physics.
In a recent experiment, Barreiro et al (2011 Nature 470 486) demonstrated the fundamental building blocks of an open-system quantum simulator with trapped ions. Using up to five ions, dynamics were realized by sequences that combined single- and multi-qubit entangling gate operations with optical pumping. This enabled the implementation of both coherent many-body dynamics and dissipative processes by controlling the coupling of the system to an artificial, suitably tailored environment. This engineering was illustrated by the dissipative preparation of entangled two- and four-qubit states, the simulation of coherent four-body spin interactions and the quantum non-demolition measurement of a multi-qubit stabilizer operator. In this paper, we present the theoretical framework of this gate-based ('digital') simulation approach for open-system dynamics with trapped ions. In addition, we discuss how within this simulation approach, minimal instances of spin models of interest in the context of topological quantum computing and condensed matter physics can be realized in state-of-the-art linear ion-trap quantum computing architectures. We outline concrete simulation schemes for Kitaev's toric code Hamiltonian and a recently suggested color code model. The presented simulation protocols can be adapted to scalable and two-dimensional ion-trap architectures, which are currently under development.
Quantum emulation of the transverse field Ising model has made significant progress recently. Ions are placed in a trap and then have an optical spin-dependent force applied to them which generates an effective spin-spin interaction between the different ions. The spin-spin interaction is long range, with a power law that can be continuously adjusted from the uniform case (power law zero) to the dipole-dipole case (power law three). I will begin with a discussion of the experimental results from the Monroe group on examining the antiferromagnetic case in a linear Paul trap with up to 16 spins [1]. Next, I will discuss the progress from the Bollinger group on examining the ferromagnetic spin-spin interactions between about 300 spins in a rotating Penning trap [2]. Both of these experiments show the challenge with scaling up these systems to large sizes, namely that it is difficult to maintain the adiabaticity condition due to experimental limitations on the coherence of the system (primarily from spontaneous emission). The diabatic evolution presents a new opportunity in determining the excitation energies of the spin systems via spectroscopic techniques. Advanced signal processing techniques that employ compressive sensing are needed to efficiently process such data. I will discuss the feasibility of such analyses for future experiments.
We present schemes to prepare two-dimensional cluster states [H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001)] with atomic ions confined in a microstructured linear ion trap and coupled by an engineered spin-spin interaction. In particular, we show how to prepare a $n\ifmmode\times\else\texttimes\fi{}2$ cluster state by creating a linear cluster state and adding third-neighbor entanglement using selective recoupling techniques. The scheme is based on the capabilities provided by segmented linear Paul traps to confine ions in local potential wells and to separate and transport ions between these wells. Furthermore, we consider creating three- and four-qubit cluster states by engineering the coupling matrix such that through the periodicity of the time evolution unwanted couplings are canceled. All entangling operations are achieved by switching of voltages and currents and do not require interaction with laser light.