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.
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.
Predicting the quantum dynamics of charged matter interacting with dynamical gauge fields poses an outstanding challenge in theoretical physics. Lacking a generally applicable computational method, quantum simulators offer a promising alternative. \n \nIn this thesis, we contribute to the quantum simulation of high-energy physics, focusing on the platform of ultracold atoms. Using Wilson fermions, we propose to improve implementations of lattice gauge theories based on mixtures of cold atoms in optical lattices. Numerical benchmarks indicate that this makes the realization of Schwinger pair production feasible with current technology. Our proposal is modular and an elementary building is demonstrated experimentally. We further identify dynamical topological transitions, which we discovered in the massive Schwinger model, as a suitable target for quantum simulators. Defining a gauge-invariant order parameter, these transitions are shown to persist beyond weak coupling. \n \nIn the second part of this thesis, we develop a framework for analyzing quantum simulators in terms of experimentally accessible irreducible correlation functions at equal times. We verify this approach numerically for the sine-Gordon model in thermal equilibrium, quantum simulated by two tunnel-coupled superfluids. Finally, we apply our analysis to the non-equilibrium dynamics of a spinor Bose gas, revealing suppressed effective interactions in a strongly-correlated infrared regime.
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.
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.
In the last decade the non-equilibrium dynamics of isolated quantum systems has raised a huge interest for both its conceptual and practical implications. Simulation of quantum dynamics represents by now a very active research effort, with highly controllable quantum devices composed of hundreds of qubits already being realized in numerous setups. These offer the unprecedented possibility to access real-time quantum dynamics and strongly correlated quantum many-body states, opening numerous avenues not only for future quantum technologies, but also for long-standing theoretical problems. One of the most sought-after applications is the computation of real-time evolution in theories of fundamental interactions. Since Wilson's proposal in 1974, the equilibrium phase diagrams of lattice gauge theories have been successfully investigated with Monte Carlo simulations in a wide range of parameters. Nonequilibrium properties, on the other hand, cannot be accessed with conventional techniques. Thanks to the recent advances in numerics (e.g. tensor networks) and experiments (quantum simulation), new promising tools are now available for tackling these questions. Remarkable works have already demonstrated that applying these techniques to simple theories, such as one-dimensional quantum electrodynamics, can help to detect the signatures of confinement and pair creation. Despite these impressive achievements, investigating the non-equilibrium dynamics of gauge theories with quantum simulators turned out to be extremely challenging. This is mostly due to the complex constrained dynamics of these theories, which make their simulation with realistic atomic platforms particularly difficult. At the same time, the dynamical constraints can play a key role in determining novel interesting phenomena and peculiar non-equilibrium properties. These phenomena -- which include, for example, quantum scars, fractons, Hilbert space fragmentation -- can lead to anomalous dynamics, eventually evading the paradigm of thermalization, which predicts the relaxation of physical observable to their thermodynamic values. The aim of this thesis is twofold: first, we want to address through a combination of numerical and analytical tools some of the open questions regarding the non-equilibrium dynamics of quantum many-body systems with constrained Hilbert spaces, including lattice gauge theories. In addition, the work reported here is part of a large effort of the community towards the quantum simulation of complex phenomena of condensed matter and high-energy physics.
Scalable cold-atom quantum simulator of a 3+1D U(1) lattice gauge theory with dynamical matter
The stated overarching goal of the highly active field of quantum simulation of high-energy physics (HEP) is to achieve the capability to study \textit{ab-initio} real-time microscopic dynamics of 3+1D quantum chromodynamics (QCD). However, existing experimental realizations and theoretical proposals for future ones have remained restricted to one or two spatial dimensions. Here, we take a big step towards this goal by proposing a concrete experimentally feasible scalable cold-atom quantum simulator of a U(1) quantum link model of quantum electrodynamics (QED) in three spatial dimensions, employing \textit{linear gauge protection} to stabilize gauge invariance. Using tree tensor network simulations, we benchmark the performance of this quantum simulator through near- and far-from-equilibrium observables, showing excellent agreement with the ideal gauge theory. Additionally, we introduce a method for \textit{analog quantum error mitigation} that accounts for unwanted first-order tunneling processes, vastly improving agreement between quantum-simulator and ideal-gauge-theory results. Our findings pave the way towards realistic quantum simulators of 3+1D lattice gauge theories that can probe regimes well beyond classical simulability.
We propose an analog quantum simulator for simulating real-time dynamics of (1+1)-dimensional non-Abelian gauge theory well within the existing capacity of ultracold-atom experiments. The scheme calls for the realization of a two-state ultracold fermionic system in a one-dimensional bipartite lattice, and the observation of subsequent tunneling dynamics. Being based on the loop string hadron formalism of SU(2) lattice gauge theory, this simulation technique is completely SU(2) invariant and simulates accurate dynamics of physical phenomena such as string breaking and/or pair production. The scheme is scalable and particularly effective in simulating the theory in the weak-coupling regime, and also a bulk limit of the theory in the strong-coupling regime up to certain approximations. This paper also presents a numerical benchmark comparison of the exact spectrum and real-time dynamics of lattice gauge theory to that of the atomic Hamiltonian with an experimentally realizable range of parameters.