We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices.The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions.We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species.Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory.This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that-while pivotal for a successful implementation-have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics.We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope 173 Yb, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one-and two-dimensional settings.
We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices.The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions.We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species.Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory.This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that-while pivotal for a successful implementation-have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics.We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope 173 Yb, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one-and two-dimensional settings.
We propose a protocol for the scalable quantum simulation of SU( N ) × U(1) lattice gauge theories with alkaline-earth like atoms in optical lattices in both one- and two-dimensional systems. The protocol exploits the combination of naturally occurring SU( N ) pseudo-spin symmetry and strong inter-orbital interactions that is unique to such atomic species. A detailed ab initio study of the microscopic dynamics shows how gauge invariance emerges in an accessible parameter regime, and allows us to identify the main challenges in the simulation of such theories. We provide quantitative results about the requirements in terms of experimental stability in relation to observing gauge invariant dynamics, a key element for a deeper analysis on the functioning of such class of theories in both quantum simulators and computers.
We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with 2+1 dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a Z3 lattice gauge theory with dynamical fermionic matter in 2+1 dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.
In this thesis we theoretically investigate the quantum simulation of lattice gauge theories for cold atom experiments. We give an explicit proposal to study a (1+1) - \ndimensional U(1) lattice gauge theory. Starting with a bosonic and a fermionic atomic gas we tune the geometry and engineer the interactions such that we end up with a \none-dimensional system obeying the principle of local gauge invariance. A further important aspect of this proposal is the explicit use of highly occupied bosonic states in \norder to make the investigation of strong field quantum electrodynamics feasible. In addition, we present an approach based on the functional integral in order to theoretically treat U(1) and SU(N) gauge fields interacting with matter out of equilibrium. We show that under specific conditions the quantum theory can be accurately mapped \nonto a classical statistical ensemble. Further, exploiting this possibility, we study the prospect to observe intricate high energy phenomena like Schwinger pair-production and string breaking in such future cold atom experiments.
Lattice gauge theory has provided a crucial non-perturbative method in studying canonical models in high-energy physics such as quantum chromodynamics. Among other models of lattice gauge theory, the lattice gauge–Higgs model is a quite important one because it describes a wide variety of phenomena/models related to the Anderson–Higgs mechanism, such as superconductivity, the standard model of particle physics, and the inflation process of the early Universe. In this paper, we first show that atomic description of the lattice gauge model allows us to explore real-time dynamics of the gauge variables by using the Gross–Pitaevskii equations. Numerical simulations of the time development of an electric flux reveal some interesting characteristics of the dynamic aspect of the model and determine its phase diagram. Next, to realize a quantum simulator of the U(1) lattice gauge–Higgs model on an optical lattice filled by cold atoms, we propose two feasible methods: (i) Wannier states in the excited bands and (ii) dipolar atoms in a multilayer optical lattice. We pay attention to the constraint of Gauss's law and avoid nonlocal gauge interactions.
In the last decade there has been an intense activity aimed at the quantum simulation of interacting many-body systems using cold atoms [1, 2]. The idea of quantum simulations traces back to Feynman [3], who argued that the ideal setting to study quantum systems would be a quantum experimental setup rather then a classical one - the latter one being fundamentally limited due to its hardware classical structure. This is a particularly important problem given the intrinsic complexity of interacting many-body problems, and the difficulties that arise when tackling them with numerical simulations - two paradigmatic examples being the sign(s) problem affecting Monte Carlo simulations of fermionic systems, and the real time dynamics in more than one spatial dimension. Ultracold atoms offer a very powerful setting for quantum simulations. Atoms can be trapped in tailored optical and magnetic potentials, also controlling their dimensionality. The inter-atomic interactions can be tuned by external knobs, such as Feshbach resonances. This gives a large freedom on model building and, with suitable mappings, they allow the implementation of desired target models. This allowed an impressive exploitation of quantum simulators on the context of condensed matter physics. The simulation of high-energy physics is an important line of research in this field and it is less direct. In particular it requires the implementation of symmetries like Lorentz and gauge invariance which are not immediately available in a cold atomic setting. Gauge fields are ubiquitous in physics ranging from condensed matter [4–6] and quantum computation [7, 8] to particle physics [9], an archetypical example being Quantum Chromodynamics (QCD) [10,11], the theory of strong nuclear forces. Currently open problems in QCD, providing a long-term goal of cold atomic simulations, include confinement/deconfinement and the structure of color superconducting phases at finite chemical potential [12]. Even though QCD is a very complicated theory (to simulate or study), it is possible to envision a path through implementation of simpler models. Furthermore, it is also expected that interesting physics is found on such “intermediate models” which may deserve attention irrespectively of the QCD study. A very relevant model in this regard is the Schwinger model (Quantum Electrodynamics in 1+1 dimensions) [13]. This theory exhibits features of QCD, such as confinement [14], and is at the very same time amenable to both theoretical studies and simpler experimental schemes. This model was the target of the first experimental realization of a gauge theory with a quantum simulator [15]. The work on this Thesis is, in part, motivated by the study of toy models which put in evidence certain aspects that can be found in QCD. Such toy models provide also intermediate steps in the path towards more complex simulations. The two main aspects of QCD which are addressed here are symmetry-locking and confinement. The other main motivation for this study is to develop a systematic framework, through dimensional mismatch, for theoretical understanding and quantum simulations of long-range theories using gauge theories. The model used to study symmetry-locking consists of a four-fermion mixture [16]. It has the basic ingredients to exhibit a non-Abelian symmetry-locked phase: the full Hamiltonian has an SU (2) × SU (2) (global) symmetry which can break to a smaller SU (2) group. Such phase is found in a extensive region of the phase diagram by using a mean-field approach and a strong coupling expansion. A possible realization of such system is provided by an Ytterbium mixture. Even without tuning interactions, it is shown that such mixture falls inside the the locked-symmetry phase pointing towards a possible realization in current day experiments. The models with dimensional mismatch investigated here have fermions in a lower dimensionality d + 1 and gauge fields in higher dimensionality D + 1. They serve two purposes: establish mappings to non-local theories by integration of fields [17] and the study of confinement [18]. In the particular case of d = 1 and D = 2 it is found that some general non-local terms can be obtained on the Lagrangian [17]. This is found in the form of power-law expansions of the Laplacian mediating either kinetic terms (for bosons) or interactions (for fermions). The fact that such expansions are not completely general is not surprising since constraints do exist, preventing unphysical features like breaking of unitarity. The non-local terms obtained are physically acceptable, in this regard, since they are derived from unitary theories. The above mapping is done exactly. In certain cases it is shown that it is possible to construct an effective long-range Hamiltonian in a perturbative expansion. In particular it is shown how this is done for non-relativistic fermions (in d = 1) and 3 + 1 gauge fields. These results are relevant in the context of state of the art experiments which implement models with long-range interactions and where theoretical results are less abundant than for the case of local theories. The above mappings establish a direct relation with local theories which allow theoretical insight onto these systems. Examples of this would consist on the application of Mermin- Wagner-Hohenberg theorem [19, 20] and Lieb-Robinson bounds [21], on the propagation of quantum correlations, to non-local models. In addition they can also provide a path towards implementation of tunable long-range interactions with cold atoms. Furthermore, in terms of quantum simulations, they are in between the full higher dimensional system and the full lower dimensional one. Such property is attractive from the point of view of a gradual increase of complexity for quantum simulations of gauge theories. Another interesting property of these models is that they allow the study of confinement beyond the simpler case of the Schwinger model. The extra dimensions are enough to attribute dynamics to the gauge field, which are no longer completely fixed by the Gauss law. The phases of the Schwinger model are shown to be robust under variation of the dimension of the gauge fields [18]. Both the screened phase, of the massless case, and the confined phase, of the massive one, are found for gauge fields in 2+1 and 3+1 dimensions. Such results are also obtained in the Schwinger- Thirring model. This shows that these phases are very robust and raises interesting questions about the nature of confinement. Robustness under Thirring interactions are relevant because it shows that errors on the experimental implementation will not spoil the phase. Even more interesting is the case of gauge fields in higher dimensions since confinement in the Schwinger model is intuitively atributed to the dimensionality of the gauge fields (creating linear potentials between particles). This Thesis is organized as follows. In Chapter 1, some essential background regarding quantum simulations of gauge theories is provided. It gives both a brief introduction to cold atomic physics and lattice gauge theories. In Chapter 2, it is presented an overview over proposals of quantum simulators of gauge potentials and gauge fields. At the end of this Chapter, in Section 2.3 it is briefly presented ongoing work on a realization of the Schwinger model that we term Half Link Schwinger model. There, it is argued, some of the generators of the gauge symmetry on the lattice can be neglected without comprimising gauge invariance. In Chapter 3, the results regarding the phase diagram of the fourfermion mixture, exhibiting symmetry-locking, is presented. In Chapter 4, the path towards controlling non-local kinetic terms and interactions is provided, after a general introduction to the formalism of dimensional mismatch. At the end the construction of effective Hamiltonians is described. Finally, Chapter 5 concerns the study of confinement and the robustness of it for 1+1 fermions. The first part regards the Schwinger-Thirring with the presence of a -term while, in the second part, models with dimensional mismatch are considered. The thesis ends with conclusions and perspectives of future work based on the results presented here.
Lattice gauge theories describe fundamental phenomena in nature, but calculating their real-time dynamics on classical computers is notoriously difficult. In a recent publication [Nature 534, 516 (2016)], we proposed and experimentally demonstrated a digital quantum simulation of the paradigmatic Schwinger model, a U(1)-Wilson lattice gauge theory describing the interplay between fermionic matter and gauge bosons. Here, we provide a detailed theoretical analysis of the performance and the potential of this protocol. Our strategy is based on analytically integrating out the gauge bosons, which preserves exact gauge invariance but results in complicated long-range interactions between the matter fields. Trapped-ion platforms are naturally suited to implementing these interactions, allowing for an efficient quantum simulation of the model, with a number of gate operations that scales only polynomially with system size. Employing numerical simulations, we illustrate that relevant phenomena can be observed in larger experimental systems, using as an example the production of particle--antiparticle pairs after a quantum quench. We investigate theoretically the robustness of the scheme towards generic error sources, and show that near-future experiments can reach regimes where finite-size effects are insignificant. We also discuss the challenges in quantum simulating the continuum limit of the theory. Using our scheme, fundamental phenomena of lattice gauge theories can be probed using a broad set of experimentally accessible observables, including the entanglement entropy and the vacuum persistence amplitude.