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Abstract

Phase transitions are crucial phenomena across fields like cosmology, particle physics, and condensed matter systems. A full understanding of these phenomena requires insights into the dynamic processes at play during transitions. In this thesis, we move beyond traditional descriptions based on an analytic continuation to imaginary (Euclidean) time by focusing on the non-equilibrium real-time properties of phase transitions. Using functional and numerical quantum field theory methods in Minkowski spacetime, we analyze both equilibrium and non-equilibrium regimes. In the first part of this thesis, we simulate the decay of a metastable non-equilibrium state. Our results demonstrate that this decay is effectively characterized by a time- dependent effective potential and decay rate, with quantum and thermal fluctuations shaping the transition dynamics. We then apply the real-time functional renormalization group to examine relaxation timescales during a second-order transition, both at criticality and under spontaneous symmetry breaking. The second part of this study applies these theoretical insights to relaxation and decay phenomena in ultracold atom systems. Using two tunnel-coupled superfluids, we propose an analog quantum simulation of string-breaking dynamics in the massive Schwinger model. We also investigate non-equilibrium dynamics within this experimental setup, revealing the instability and breakdown of the (classically stable) π-trapped state, where nonlinear instabilities drive exponential growth of quantum fluctuations. These findings provide a deeper understanding of phase transition dynamics and open new avenues for experimental investigation in ultracold atom systems.