The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete classification of topological phases remains elusive. In this context, experimental studies are key, both for studying the interplay between topology and complex effects and for identifying new forms of topological order. It is therefore crucial to find complementary means to measure topological properties in order to reach a fundamental understanding of topological phases. In one dimensional chiral systems, we suggest a new way to construct and identify topologically protected bound states, which are the smoking gun of these materials. In two dimensional Hofstadter strips (i.e: systems which are very short along one dimension), we suggest a new way to measure the topological invariant directly from the atomic dynamics.
The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete classification of topological phases remains elusive. In this context, experimental studies are key, both for studying the interplay between topology and complex effects and for identifying new forms of topological order. It is therefore crucial to find complementary means to measure topological properties in order to reach a fundamental understanding of topological phases. In one dimensional chiral systems, we suggest a new way to construct and identify topologically protected bound states, which are the smoking gun of these materials. In two dimensional Hofstadter strips (i.e: systems which are very short along one dimension), we suggest a new way to measure the topological invariant directly from the atomic dynamics.
The recent discovery of topological insulators has led to a tremendous interest in the exploration of topological phases of matter which do not fit into Landau's symmetry breaking paradigm. Numerous exotic topological materials are theoretically predicted. Some of them have been experimentally reported, but many remain not. In this thesis, we explore topological phases of matter from three aspects: their classification, realization and application. We first review some basic classification theories, which provide us a "big picture" and lay the foundation for the rest of the thesis. We then move on to propose a systematic method based on quaternion algebra to construct toy tight-binding Hamiltonians for all the exotic phases in a recently developed periodic table for topological insulators and superconductors. We also introduce two peculiar families of topological phases that are beyond the table---the Hopf and four-dimensional topological insulators without time reversal symmetry. Prototypical Hamiltonians are constructed and their topological properties, such as robust edge states, are numerically studied. Motivated by rapid experimental progress in engineering spin-orbit coupling and artificial gauge field for cold atoms, we continue the thesis by proposing a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice. To unambiguously probe topological phases, we also bring forth systematic and generic methods to measure the characteristic topological invariants, for both free and strongly interacting systems. Moreover, we demonstrate that a kaleidoscope of knot and link structures is encoded in the spin texture of Hopf insulators and show how to observe different knots and links in cold atoms via time-of-flight images. The last part of the thesis is about the application of topological materials. After a demonstration of how to create, braid and detect Majorana fermions with cold atoms, we put forward a proposal to construct a self-test quantum random number generator by using Majorana fermions. Majorana random number generators are able to generate certifiable true random numbers with unconditional security. They offer a new perspective to the utilization of topological materials and may have vital applications in cryptography and related areas.
This thesis is a contribution to the research field of quantum simulations of topological phases of matter, i.e. quantum states characterized by a non-local order parameter.The aim of the research on quantum simulations is to overcome the difficulties encountered in the study of quantum mechanical models with conventional numerical methods.The proposed way to accomplish this task is the direct exploitation of controllable quantum systems whose properties, for instance low-energy physics, mimic those of the model to be studied.We study several problems connected to the objective of creating a quantum simulator of topological phases of matter.In particular, we discuss how to engineer the Hamiltonian of topological models, how to measure quantities which unravel topological orders, and how to quantify the reliability of topological quantum memories.The first direction of investigation considered in this thesis regards the development of new techniques to simulate topological states of matter with cold atoms in optical lattices.We propose a scheme based on an unconventional use of a bi-chromatic optical lattice to engineer the Hamiltonians of several topological insulators and of the related non-interacting relativistic models.We also focus on the Pfaffian wavefunction, a topological state supporting non-Abelian quasi-excitations arising in the presence of three-body interactions.We present a proposal for the realization with optical lattices of systems characterized by such interactions and discuss the possibility of using such a scheme to create a topological state, such as the mentioned wavefunction.The second line of research considered in this thesis concerns the detection of a string operator in optical lattices.We report on an experiment, to which we provided theoretical support, aimed at the measurement of the non-local order parameter of a one-dimensional Mott insulator; we support the discussion with the relevant theoretical background.Although the Mott insulator is not a topological phase, the measurement of string observables is crucial to reveal the non-local order of several topological states, and the developed experimental protocol is expected to be relevant in future experiments.The third contribution of this thesis is related to the use of topological phases to build a quantum computer, which is a specific example of a quantum simulator.We focus on the important problem of understanding whether it is possible to reliably store quantum information in topological states.We consider one specific class of topological quantum memories, namely those based on a fermionic topological superconductor with zero-energy Majorana modes.At the end of the storage period, we take into account the possibility of applying an operation aimed at the recovery of the information initially encoded into the system and then diminished by the time evolution.In particular, we vii
Cold-atom experiments in optical lattices offer a versatile platform to realize various topological quantum phases. A key challenge in those experiments is to unambiguously probe the topological order. We propose a method to directly measure the characteristic topological invariants (order) based on the time-of-flight imaging of cold atoms. The method is generally applicable to detection of topological band insulators in one, two, or three dimensions characterized by integer topological invariants. Using detection of the Chern number for the two-dimensional anomalous quantum Hall states and the Chern-Simons term for the three-dimensional chiral topological insulators as examples, we show that the proposed detection method is practical, and robust to typical experimental imperfections such as limited imaging resolution, inhomogeneous trapping potential, and disorder in the system.
The detection of topological phases of matter has become a central issue in recent years. Conventionally, the realization of a specific topological phase in condensed matter physics relies on probing the underlying surface band dispersion or quantum transport signature of a real material, which may be imperfect or even absent. On the other hand, quantum simulation offers an alternative approach to directly measure the topological invariant on a universal quantum computer. However, experimentally demonstrating high-dimensional topological phases remains a challenge due to the technical limitations of current experimental platforms. Here, we investigate the three-dimensional topological insulators in the AIII (chiral unitary) symmetry class, which yet lack experimental realization. Using the nuclear magnetic resonance system, we experimentally demonstrate their topological properties, where a dynamical quenching approach is adopted and the dynamical bulk-boundary correspondence in the momentum space is observed. As a result, the topological invariants are measured with high precision on the band-inversion surface, exhibiting robustness to the decoherence effect. Our Letter paves the way toward the quantum simulation of topological phases of matter in higher dimensions and more complex systems through controllable quantum phases transitions.
We propose an experimental scheme to simulate and detect the properties of time-reversal invariant topological insulators, using cold atoms trapped in one-dimensional bichromatic optical lattices. This system is described by a one-dimensional Aubry-Andre model with an additional SU(2) gauge structure, which captures the essential properties of a two-dimensional Z2 topological insulator. We demonstrate that topologically protected edge states, with opposite spin orientations, can be pumped across the lattice by sweeping a laser phase adiabatically. This process is an elegant way to transfer topologically protected quantum states in a highly controllable environment. We discuss how density measurements could provide clear signatures of the topological phases emanating from our one-dimensional system.
The ability to precisely and coherently control atomic systems has improved dramatically in the last two decades, driving remarkable advancements in quantum computation and simulation. In recent years, atomic and atom-like systems have also been served as a platform to study topological phases of matter and non-equilibrium many-body physics. Integrated with rapid theoretical progress, the employment of these systems is expanding the realm of our understanding on a range of physical phenomena. In this dissertation, I draw on state-of-the-art experimental technology to develop several new ideas for controlling and applying atomic systems. In the first part of this dissertation, we propose several novel schemes to realize, detect, and probe topological phases in atomic and atom-like systems. We first theoretically study the intriguing properties of Hopf insulators, a peculiar type of topological insulators beyond the standard classification paradigm of topological phases. Using a solid-state quantum simulator, we report the first experimental observation of Hopf insulators. We demonstrate the Hopf fibration with fascinating topological links in the experiment, showing clear signals of topological phase transitions for the underlying Hamiltonian. Next, we propose a feasible experimental scheme to realize the chiral topological insulator in three dimensions. They are a type of topological insulators protected by the chiral symmetry and have thus far remained unobserved in experiment. We then introduce a method to directly measure topological invariants in cold-atom experiments. This detection scheme is general and applicable to probe of different topological insulators in any spatial dimension. In another study, we theoretically discover a new type of topological gapless rings, dubbed a Weyl exceptional ring, in three-dimensional dissipative cold atomic systems. In the second part of this dissertation, we focus on the application of atomic systems in quantum computation and simulation. Trapped atomic ions are one of the leading platforms to build a scalable, universal quantum computer. The common one-dimensional setup, however, greatly limits the system's scalability. By solving the critical problem of micromotion, we propose a two-dimensional architecture for scalable trapped-ion quantum computation. Hamiltonian tomography for many-body quantum systems is essential for benchmarking quantum computation and simulation. By employing dynamical decoupling, we propose a scalable scheme for full Hamiltonian tomography. The required number of measurements increases only polynomially with the system size, in contrast to an exponential scaling in common methods. Finally, we work toward the goal of demonstrating quantum supremacy. A number of sampling tasks, such as the boson sampling problem, have been proposed to be classically intractable under mild assumptions. An intermediate quantum computer can efficiently solve the sampling problem, but the correct operation of the device is not known to be classically verifiable. Toward practical verification, we present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. In a separate study, we introduce a new model built from translation-invariant Ising-interacting spins. This model possesses several advantageous properties, catalyzing the ultimate experimental demonstration of quantum supremacy.