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 [1], 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 [2]. These efforts recently resulted in a proof-of-principle experiment demonstrating an elementary building block for U(1) lattice gauge theories [3]. 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 [4]. 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 [5], revealing universal dynamics of an effective vertex in a regime where standard kinetic descriptions fail.

[1] Zache et al, Phys. Rev. Lett. 122, 050403

[2] Zache et al, Quantum Sci. Technol. 3 034010

[3] Mil et al, arXiv:1909.07641

[4] Zache et al, arXiv:1909.12815

[5] Prüfer et al, arXiv:1909.05120