The unprecedented control of synthetic quantum systems allows to tackle outstanding questions from high-energy physics, such as the non-equilibrium dynamics of gauge theories, using quantum simulators. In this talk, I will first discuss dynamical topological transitions in quantum electrodynamics (QED) in one spatial dimension , which bear similarities with the physics of topological insulators. This phenomenon is accessible within our proposals to microscopically engineer the Hamiltonian of lattice QED with a mixture of ultracold gases . These efforts recently resulted in a proof-of-principle experiment demonstrating an elementary building block for U(1) lattice gauge theories . In the second part of my talk, I will discuss a complementary approach, based on large-scale analog quantum simulators probing the many-body limit described by quantum field theory (QFT). Within an equal-time formulation of QFT, we established a procedure to extract irreducible correlations, enabling the measurement of effective interaction vertices . This is verified at the example of the sine-Gordon model in thermal equilibrium, quantum simulated by two tunnel-coupled superfluids. We further applied this approach to a spinor Bose gas out-of-equilibrium , revealing universal dynamics of an effective vertex in a regime where standard kinetic descriptions fail.
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