Universal, fault-tolerant quantum computing would require millions of physical qubits to practically implement most proposed algorithms, a target currently out of reach of experimental capability. In the near term, noisy systems on the order of tens of qubits can employ quantum simulation of particular Hamiltonians to surpass classical computational abilities and solve interesting problems. In particular, one-dimensional (1D) ion chains in radiofrequency (RF) traps have seen remarkable success in simulating 1D quantum spin systems. A comparable ability to manipulate two-dimensional (2D) ion crystals in RF traps would significantly expand the class of systems accessible to quantum simulation. Notably, 2D ion arrays are conducive to studies of many-body systems such as geometrically frustrated lattices, topological materials, and spin-liquid states.In this thesis, I present advances toward the goal of creating programmable, "radial-2D" arrays of trapped 171Yb+ ions for quantum simulation. Qubits are embedded within two hyperfine electronic energy levels, cooled to their motional ground state, and measured via spin-dependent fluorescence. A precisely controlled entangling mechanism allows for the creation of a wide variety of spin models, including Ising or Heisenberg interactions. We present an experimental study which establishes radial-2D crystals of 171Yb+ ions as a robust platform for quantum simulation, through characterization of ion positions, structural phases, normal mode frequencies, and effects from RF heating. We also design and experimentally demonstrate a new open-endcap, blade-style RF trap which can confine and resolve large numbers of ions in the radial-2D crystal phase. Finally, we examine other challenges faced by trapped ion systems: optimally cooling to the motional ground state, accurately determining ion temperature, and measuring susceptibility to the presence of ionizing radiation.

Quantum simulation is the process of using a highly controllable quantum system to simulate another less controllable system. Quantum simulation can provide insights into the properties and dynamics of complex many-body systems. Trapped ion platform is one of the leading candidates for a quantum simulator due to its properties such as ease of isolation, preparation and manipulation. Simulating high dimensional spin systems enables us to study the various physical phenomena in higher geometries. Previous proposals for simulating higher dimensions require experimental resources that don't scale favourably with the system. In this thesis, we propose a hybrid (digital-analog) approach to simulate an effective 2D lattice from a 1D chain of trapped ions. In the initial geometry, the ions interact with each other through a flip-flop kind of interactions generated using a global Molmer-Sorensen scheme. A series of single qubit gates are used to rescale and suppress the interactions in the initial chain to simulate the target geometry. These gates are applied using a laser field gradient which generates a site-dependent AC stark shift. I discuss the construction of this protocol in detail and the theoretical results for the case of 6, 9 and 16 ions. I also show that the number of gates and also the Stark gradient scale linearly with the system size. Experimental implementation of an ion trap quantum simulator has various challenges, one of the which is the laser frequency stabilization within a fraction of transition linewidth. Traditionally, this is done by locking the lasers to an atomic transition. In this thesis, I discuss two alternative schemes for locking the laser frequencies used to build a 171Yb+ ion quantum simulator. One of these solutions uses a commercial wavemeter as a measuring device for the frequency and feedbacks the lasers based on this measurement. I discuss the layout of this scheme and some results. Other solution uses a Fabry Perot (FP) cavity to transfer the stability of a stable laser to an unstable laser. In this thesis, I discuss the construction, optical layout and transmission measurements of a home-built FP cavity.

Non-equilibrium quantum many-body systems like periodically driven (Floquet) systems exhibit much richer dynamics than their equilibrium counterparts. A striking example is a time crystal, a novel non-equilibrium phase of matter which is forbidden in equilibrium. This thesis concentrates on studying Floquet many-body systems, in particularly, higher-order Floquet topology, light-matter interactions in the Floquet systems, and interesting physics of Floquet many-spin systems with a view towards quantum simulation, e.g. with trapped ions. The model Hamiltonians studied in this thesis are of two types: Floquet-Bloch Hamiltonian and Floquet many-spin Hamiltonian. We report the theoretical discovery and characterization of higher-order Floquet topological phases dynamically generated in a periodically driven system with mirror symmetries. We demonstrate numerically and analytically that these phases support lower-dimensional Floquet bound states, such as corner Floquet bound states at the intersection of edges of a two-dimensional system, protected by the nonequilibrium higher-order topology induced by the periodic drive. We characterize higher-order Floquet topologies of the bulk Floquet Hamiltonian using mirror-graded Floquet winding number. We also show that bulk vortex structures can be dynamically generated by a spatially inhomogeneous drive and can host multiple Floquet bound states. Inspired by the effectiveness of the optical probes in condensed matter systems, we study the light-matter interactions in Floquet systems in strong and weak optical field regimes. In the weak field limit, we formulate the linear response theory of a periodically driven system. We illustrate the applications of this formalism by giving general expressions for optical conductivity of Floquet systems, including its homodyne and heterodyne components and beyond. We obtain the Floquet optical conductivity of specific driven models, including two-dimensional Dirac material such as the surface of a topological insulator, graphene, and the Haldane model irradiated with circularly or linearly polarized laser, as well as a semiconductor quantum well driven by an ac potential. In the strong field limit, we formulate a theory of bulk optical current for a periodically driven system, which accounts for the mixing of external drive and laser field frequencies and, therefore, the broadening of the harmonic spectrum compared to the undriven system. We illustrate the application of this theory by studying high harmonic generation in the periodically driven Su Schrieffer-Heeger model. We find significant enhancement in higher harmonics when the system is driven, even for low field amplitudes and obtain harmonics forbidden in the undriven model. Motivated by applications on a trapped-ion quantum simulator, we study many-spin systems with periodically driven two-body interactions and external fields in the high-frequency limit. We show that a hierarchy of multi-spin interactions can be generated in powers of inverse-frequency, such as Dzyaloshinskii-Moriya interaction (DMI), three- and four-spin interactions, and additional Zeeman terms. We introduce the concept of a Floquet gauge pump where we exploit the dynamically generated interactions of a Floquet Hamiltonian to detect the topology of the ground state and edge states in interacting systems. We demonstrate this concept in 1D XY model with periodically driven couplings and a transverse field and discuss the requirements to realize the Floquet gauge pump with trapped ions.

Quantum simulations are the use of well controlled many-body quantum systems to simulate and solve other many-body quantum systems that are not understood. This thesis describes theoretical proposals and experimental progress towards simulating quantum spin Hamiltonians with trapped SSSr+ ions on microfabricated planar lattice traps. These types of quantum simulations help solve exponentially complex quantum systems, which are challenging to current classical computers. Porras and Cirac's work has shown that off-resonance laser light couples the internal states of the trapped ions with their external motional states; the external states of the ions are coupled through their Coulomb repulsion. Therefore the internal states of ions can be mapped to effective spin states and spin coupling is mediated by phonons. I propose two simulation schemes for quantum spin Hamiltonians in two dimensions: the time evolution of a spin state in a system of three ions with ferromagnetic interaction and spin frustration in a triangle. To realize these proposals, I design and microfabricate a hexagonal lattice trap and install it into an ultrahigh vacuum chamber. This thesis also presents the construction of the experimental test apparatus for the trapped ion quantum simulator, including the electronics, optics, and vacuum systems.

Abstract: The fundamental laws of quantum mechanics are far from our everyday life experiences. However, they have been extremely successful in explaining observations in microscopic physical systems and are, already, harnessed for several applications. Moreover, in combination with relativistic considerations, basic phenomena from quantum mechanics are accredited a central role in the (macroscopic) evolution of our universe. In this thesis, we employ a trapped ion quantum simulator to experimentally investigate two fundamental quantum mechanical effects that have been extensively studied for many decades. In the first investigation, we implement non-adiabatic changes of the ions' trapping potential in order to amplify quantum vacuum fluctuations. Following theoretical works by others, this mechanism can be interpreted as an analog to the creation of particles during cosmic inflation in the early universe. Such cosmological particles are considered as the seeds for the formation of the large-scale structures in our universe that we observe nowadays. In our analog quantum simulation, the creation of particles is evidenced by the detection of a squeezed state in the motion of the ions. A remarkable feature of this mechanism is the accompanying creation of quantum entanglement. In case of the cosmological particles, this entanglement spreads over large, cosmic distances. In the second investigation, we benchmark a theoretical concept in the framework of open quantum systems in a most basic system and under near-ideal conditions. In nature, any quantum system inevitably interacts with its environment and, thereby, needs ultimately to be considered an open system that can build up correlations and, even, entanglement with its environment. We implement a spin-1/2 system in the electronic degree of freedom of a single trapped ion, which we couple to a bosonic environment, formed by a motional degree of freedom of the ion. Further, we implement a tunable interaction between system and environment that, in turn, can lead to entanglement. By performing measurements on the open system only, we observe quantum non-Markovian behavior, which we quantify using a rigorously defined measure. Thereby, we reveal that the quantification of such quantum memory effects is fundamentally limited by fundamental quantum mechanical measurement uncertainties. The investigations presented in this thesis significantly expand the capabilities of our trapped ion platform to perform ana ...

Quantum mechanics, the subfield of physics that describes the behavior of very small (quantum) particles, provides the basis for a new paradigm of computing. First proposed in the 1980s as a way to improve computational modeling of quantum systems, the field of quantum computing has recently garnered significant attention due to progress in building small-scale devices. However, significant technical advances will be required before a large-scale, practical quantum computer can be achieved. Quantum Computing: Progress and Prospects provides an introduction to the field, including the unique characteristics and constraints of the technology, and assesses the feasibility and implications of creating a functional quantum computer capable of addressing real-world problems. This report considers hardware and software requirements, quantum algorithms, drivers of advances in quantum computing and quantum devices, benchmarks associated with relevant use cases, the time and resources required, and how to assess the probability of success.

The Advanced School on Quantum Foundations and Open Quantum Systems was an exceptional combination of lectures. These comprise lectures in standard physics and investigations on the foundations of quantum physics.On the one hand it included lectures on quantum information, quantum open systems, quantum transport and quantum solid state. On the other hand it included lectures on quantum measurement, models for elementary particles, sub-quantum structures and aspects on the philosophy and principles of quantum physics.The special program of this school offered a broad outlook on the current and near future fundamental research in theoretical physics.The lectures are at the level of PhD students.