Chiral edge states are a hallmark of quantum Hall physics. In electronic systems, they appear as a macroscopic consequence of the cyclotron orbits induced by a magnetic field, which are naturally truncated at the physical boundary of the sample. Here we report on the experimental realization of chiral edge states in a ribbon geometry with an ultracold gas of neutral fermions subjected to an artificial gauge field. By imaging individual sites along a synthetic dimension, we detect the existence of the edge states, investigate the onset of chirality as a function of the bulk-edge coupling, and observe the edge-cyclotron orbits induced during a quench dynamics. The realization of fermionic chiral edge states is a fundamental achievement, which opens the door towards experiments including edge state interferometry and the study of non-Abelian anyons in atomic systems.Ultracold atoms in optical lattices represent an ideal platform to investigate the physics of condensed-matter problems in a fully tunable, controllable environment [1,2]. One of the remarkable achievements in recent years has been the realization of synthetic background gauge fields, akin to magnetic fields in electronic systems. Indeed, by exploiting light-matter interaction, it is possible to imprint a Peierls phase onto the atomic wavefunction, which is analogous to the Aharanov-Bohm phase experienced by a charged particle in a magnetic field [3][4][5]. These gauge fields, first synthesized in Bose-Einstein condensates [6], have recently allowed for the realization of the HarperHofstadter Hamiltonian in ultracold bosonic 2D lattice gases [7,8], paving the way towards the investigation of different forms of bulk topological matter in bosonic atomic systems [5,9]. In the present work we are instead interested in the edge properties of fermionic systems under the effects of a synthetic gauge field. Fermionic edge states are a fundamental feature of 2D topological states of matter, such as quantum Hall and chiral spin liquids [10,11]. Moreover, they are robust against changing the geometry of the system by keeping its topology, and can be observed even on Hall ribbons [12]. In addition, they offer very attractive perspectives in quantum science, such as the realization of robust quantum information buses [13], and they are ideal starting points for the realization of non-Abelian anyons akin to Majorana fermions [14,15].Here, we report the observation of chiral edge states in a system of neutral fermions subjected to a synthetic magnetic field. We exploit the high level of control in our system to investigate the emergence of chirality as a function of the Hamiltonian couplings. These results have been enabled by an innovative experimental approach, where an internal (nuclear spin) degree of freedom of the atoms is used to encode a lattice structure lying in an "extra dimension" [12], providing direct access to edge physics. In addition, we validate the chiral nature of our FIG. 1. A synthetic gauge field in a synthetic dimension. a. We confine the mot...

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons [1,2]. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator [3,4], this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented [5][6][7]. Here we report the first experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model [8,9]) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism [10,11], describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields [12] in favour of exotic longrange interactions, which have a direct and efficient implementation on an ion trap architecture [13]. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulating high-energy theories with atomic physics experiments, the long-term vision being the extension to real-time quantum simulations of non-Abelian lattice gauge theories.Small-scale quantum computers exist today in the laboratory as programmable quantum devices [14]. In particular, trapped-ion quantum computers [13] provide a platform allowing a few hundred coherent quantum gates on a few qubits, with a clear roadmap towards scaling up FIG. 1. (a)The instability of the vacuum due to quantum fluctuations is one of the most fundamental effects in gauge theories. We simulate the coherent real time dynamics of particle-antiparticle creation by realising the Schwinger model (one-dimensional quantum electrodynamics) on a lattice, as described in the main text. (b) The experimental setup for the simulation consists of a linear Paul trap, where a string of 40 Ca + ions is confined. The electronic states of each ion encode a spin |↑ or |↓ ; these can be manipulated using laser beams (see Methods for details).these devices [4,15]. This provides the tools for universal digital quantum simulation [16], where the time evolution of a quantum system is approximated as a stroboscopic sequence of quantum gates [17]. Here we show how this quantum technology can be used to simulate the real time dynamics of a minimal model of a lattice gauge theory, realising the Schwinge...

Recent experimental breakthroughs in trapping, cooling and controlling ultracold gases of polar molecules, magnetic and Rydberg atoms have paved the way toward the investigation of highly tunable quantum systems, where anisotropic, long-range dipolar interactions play a prominent role at the many-body level. In this article we review recent theoretical studies concerning the physics of such systems. Starting from a general discussion on interaction design techniques and microscopic Hamiltonians, we provide a summary of recent work focused on many-body properties of dipolar systems, including: weakly interacting Bose gases, weakly interacting Fermi gases, multilayer systems, strongly interacting dipolar gases and dipolar gases in 1D and quasi-1D geometries. Within each of these topics, purely dipolar effects and connections with experimental realizations are emphasized.Comment: Review article; submitted 09/06/2011. 158 pages, 52 figures. This document is the unedited author's version of a Submitted Work that was subsequently accepted for publication in Chemical Reviews, copyright American Chemical Society after peer review. To access the final edited and published work, a link will be provided soo

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Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented-a classical simulation approachapplied to the study of lattice gauge theories together with some results on

Quantum many-body systems can have phase transitions even at zero temperature; fluctuations arising from Heisenberg's uncertainty principle, as opposed to thermal effects, drive the system from one phase to another. Typically, during the transition the relative strength of two competing terms in the system's Hamiltonian changes across a finite critical value. A well-known example is the Mott-Hubbard quantum phase transition from a superfluid to an insulating phase, which has been observed for weakly interacting bosonic atomic gases. However, for strongly interacting quantum systems confined to lower-dimensional geometry, a novel type of quantum phase transition may be induced and driven by an arbitrarily weak perturbation to the Hamiltonian. Here we observe such an effect--the sine-Gordon quantum phase transition from a superfluid Luttinger liquid to a Mott insulator--in a one-dimensional quantum gas of bosonic caesium atoms with tunable interactions. For sufficiently strong interactions, the transition is induced by adding an arbitrarily weak optical lattice commensurate with the atomic granularity, which leads to immediate pinning of the atoms. We map out the phase diagram and find that our measurements in the strongly interacting regime agree well with a quantum field description based on the exactly solvable sine-Gordon model. We trace the phase boundary all the way to the weakly interacting regime, where we find good agreement with the predictions of the one-dimensional Bose-Hubbard model. Our results open up the experimental study of quantum phase transitions, criticality and transport phenomena beyond Hubbard-type models in the context of ultracold gases.

We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimension. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered time-dependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in exisiting AMO quantum simulators, and used to measure for instance area law scaling of entanglement in two-dimensional spin models or the entanglement growth in many-body localized systems.Atomic physics provides us with the realization of engineered quantum many-body lattice models. This includes Hubbard models for bosonic and fermionic cold atoms in optical lattices [1], and spin models with Rydberg atoms [2] and chains of trapped ions [3]. Among the noticeable recent experimental advances are quantum control, and single shot measurements in lattice systems of atoms [4][5][6][7][8][9][10][11] and ions [12,13] achieving single site resolution, as illustrated for atoms in optical lattices by the quantum gas microscope [14]. This provides us not only with a unique atomic toolbox to prepare equilibrium and non-equilibrium states of quantum matter, but also with the opportunity to access in experiments novel classes of observables, beyond the familiar low order correlation functions. An outstanding example is the measurement of Rényi entropies, defined as S (n) (ρ A ) = 1 1−n log Tr(ρ n A ) (n > 1) with ρ A = Tr S\A [ρ] the reduced density matrix of a subsystem A ⊂ S of a many-body system S, which gives us a unique signature of entanglement properties in many-body phases and dynamics [15], and is also of interest in the ongoing discussion on 'quantum supremacy' [16][17][18][19][20].Below we will describe a protocol for measuring Rényi entropies S (n) (ρ A ) based on random measurements realized as random unitary operators applied to ρ A and subsequent measurements of a fixed observable [21]. In our approach the required random unitaries are implemented using the same AMO toolbox which underlies the preparation of quantum phases and dynamics (c.f. Fig. 1). This enables a physical implementation of the protocol, applicable to generic Hubbard and spin models and in arbitrary dimension. We emphasize that in contrast to recent protocols to measure n-th order Rényi entropies, which requires preparation of n identical copies [22][23][24][25], a random measurement protocol requires only a single quantum system [21], and thus can be implemented directly with existing AMO and solid state platforms [26,27]. A central aspect in any measurement scheme for Rényi

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