We propose to investigate the coherent trajectories of many-body systems undergoing symmetry-breaking transitions (SBTs) in real time, where trajectories are meant here in a mathematical sense used to describe the dynamics of nonlinear systems. The key idea which makes this project possible is the development of a specific femtosecond laser spectroscopy technique which allows us to distinguish the order parameter dynamics in complex matter systems from hot-electron energy relaxation, quasiparticle recombination processes, damping and dephasing of coherent phonon oscillations. This allows real-time high resolution investigations of the critical system trajectories through SBTs, beyond the capabilities of current state of the art time-resolved techniques. We will investigate coherent collective field oscillations and the fundamentals of topological defect creation by the Kibble-Zurek mechanism including a study of their annihilation in the aftermath of SBTs. We will aim to control the coherent trajectories at bifurcation points by laser pulses and external fields. We will address fundamental questions on the effect of symmetry and fundamental interactions of underlying microscopic vacua on global behaviour. Systems included in our study belong to a number of different universality classes and include the study of nontrivial transitions to newly discovered hidden states of matter. In the general framework of reductionism, we expect our findings to have fundamental bearing on our understanding of SBTs revealing predictive tell-tale signatures of critical events of relevance in areas beyond many-body condensed matter physics, in elementary particle physics, primordial cosmological events and tipping points in nonlinear systems. Transition trajectories to and from hidden states are of particular interest for practical applications in new femtosecond state change memory devices.
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