Periodic Reporting for period 3 - StrEnQTh (Strong Entanglement in Quantum many-body Theory) Reporting period: 2021-05-01 to 2022-10-31 Summary of the context and overall objectives of the project The microscopic constituents of matter obey the counterintuitive laws of quantum mechanics, which are fundamentally different from the classical laws that we are used to in our macroscopic world. One far-reaching consequence of quantum mechanics is the existence of “entangled states” formed by groups of quantum components (particles). In such a state, the individual components relinquish their identity, and every component "knows" the behavior of the others at all times and across any distance. In recent years, major experimental strides in ultracold atomic gases and other platforms have made it possible to manipulate and design complex materials at the single-component level. We now know that entanglement at this fundamental level plays a major role. Yet, until now, entanglement is fully understood only if very few particles are involved, and no concise theoretical framework exists for entanglement involving large groups of particles. Even more, it is very difficult, even for the best state of art experiments, to detect entanglement that is shared between many particles. These constitute paramount obstacles for further theoretical and technological progress in quantum matter and quantum technologies. The "Strong Entanglement in Quantum Many-body Theory" (StrEnQTh) project aims to overcome these hurdles. The project follows three main objectives. (i) First, we aim at designing a novel mathematical framework for one particularly important type of many-particle entanglement, topological long-range entanglement. (ii) Second, we aim at developing novel methods for studying many-body quantum systems that are strongly perturbed, in particular, for understanding the role that entanglement plays in bringing the system back to a steady state at thermal equilibrium. Our focus lies on gauge theories, which form the theoretical basis for describing fundamental particles of nature and describe emergent exotic phases of matter. We aim at studying simplified situations inspired by effects investigated in heavy-ion colliders, where the impact between heavy atomic ions generates elementary particles violently away from equilibrium with extremely rich dynamics. (iii) Third, we aim at exploiting the untapped potential of engineered quantum dynamics, such as by perturbing the quantum system in a controlled manner, in order to design protocols for accessing many-particle entanglement in state-of-the-art experiments. This project thus addresses a frontier of modern physics, at the interface of quantum computing, quantum information theory, high-energy physics, quantum chemistry, and material science. The findings can be useful for a variety of new technological applications, ranging from novel paradigms of quantum computing, over the design of nano-materials and chemical compounds, to new types of quantum sensors. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The first half of the StrEnQTh project has been extremely intense and productive. A substantial effort has been put into the team creation, the selection of the team members, and formation of a collaborative team spirit. At present, the team is formed by 5 PhD students, 2 postdoctoral fellows, one senior researcher, the principal investigator, and one administrative assistant, with team members hailing from Brazil, China, Germany, India, Italy, Lebanon, and Namibia. To date, the team has held more than 40 presentations disseminating the aims and results of the project and published 9 peer-reviewed journal articles and 19 preprints. The major milestones of the project foreseen for this period have been achieved. The mathematical basis for objective (i) has been laid. In an interesting sidetrack, proposals have been developed to exploit quantumness in order to solve difficult problems from computational biology and soft-matter theory. Regarding objective (ii), we have developed competitive numerical codes and analytical methods, performed various theoretical studies into the dynamics of quantum many-body systems, and designed implementations in cold atomic gases and quantum computing chips of superconducting qubits, which enable detailed studies in analog quantum simulators. Regarding objective (iii), we have established paradigmatically new, experimentally-friendly detection tools for quantum correlations, many-body entanglement, and particle currents, which break the ground for further tools of increased complexity and power. The project is characterized by an exceptional degree of cross- and interdisciplinarity, involving collaborations with experimentalists working on a range of quantum-device platforms as well as experts in quantum information theory, high-energy and nuclear physics, cosmology, computational biophysics, soft-matter theory, and quantum chemistry. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Currently, the role of entanglement in systems of many particles is not yet understood, be it at equilibrium with a thermal bath or far from equilibrium. Thanks to the experimental realizations of a lattice gauge theory in a cold atomic mixture [Mil et al., Science 367, 1128-1130 (2020)] and an optical superlattice [Yang et al., Nature 587, 392-396 (2020)], we now have opened the door for microscopic in-depth investigations of gauge theory dynamics, using a novel platform with pristine control. Moreover, we have made significant progress in understanding the role of perturbations on the dynamics of gauge theories and how to protect against them [e.g. Halimeh, Hauke, Phys. Rev. Lett. 125, 030503 (2020); Halimeh, Ott, McCulloch, Yang, Hauke, Phys. Rev. Research 2, 033330 (2020)]. Further, thanks to novel protocols based on engineered quantum dynamics, observables that have hitherto been unreachable now become accessible [Costa de Almeida, Hauke, arXiv:2005.03049 to appear in PRResearch Letters (2021); Geier, Martone, Hauke, Stringari, arxiv:2102.02221 (2021); Geier, Hauke, arXiv:2104.03983 (2021)]. By the end of the project, we expect to have developed a concise mathematical framework for many-body entanglement as well as a range of theoretical and experimental tools to systematically study quantum dynamics in regimes that previously were out of reach. Thanks to these, we will obtain a much clearer picture of the role of entanglement in quantum many-body systems, making it possible to concisely describe and quantify its usefulness for technological applications. Moreover, with efficient methods to extract relevant entanglement in state-of-the-art experiments, it will become possible to certify how much a given quantum device can outperform a comparable classical machine. A cloud of cold atoms can reproduce some of the features of the early expanding universe. Engineered dissipation permits the measurement of quantum correlations. Many-body entanglement (light colour) can become measurable in cold atoms (simulation)
