Non-equilibrium Dynamics in Many-body Systems: From Strongly Correlated Electrons to Coupled Bose-Einstein Condensates

The pursuit of understanding quantum phenomena such as superfluidity, non-trivial topology of metals or insulators, dynamics of quasiparticles in strongly correlated materials has been the cornerstone of condensed matter physics in recent decades, supported by cutting-edge experimental and computational methods. In this thesis, I have investigated dynamical properties in a variety of physical systems; a bosonic system with trivial band structure, fermionic system with non-trivial band topology, and finally a strongly correlated system with cooperative Jahn-Teller distortions. The physics of spontaneous symmetry breakings leads to the several exotic quantum phases such as superfluid and more recently super-solid. In the latter case, both diagonal and off-diagonal long-range order, although being mutually exclusive, are present. One of the ways to realize this super-solid phase is using the artificially spin-orbit (SO) coupled condensates of Alkali atoms in a double well optical lattice. A key requirement for a true supersolid phase is that the associated symmetry breaking be spontaneous; however, in optical lattices, the relative phase between condensates often becomes pinned due to tunneling. In this work, we explore the coherence of this relative phase and its characteristic timescales, which play a crucial role in the formation and observation of the resulting density modulation. Next, we analyze Kerr and Faraday rotations in time-reversal symmetry-broken multi-Weyl semimetals (mWSMs) without external magnetic fields. Using Kubo response theory, we show that optical conductivities and polarization rotations depend non-trivially on the topological charge $n$. In thin films, polarization rotation scales with $n$ and $n^2$, while in bulk geometries, axion electrodynamics plays a crucial role in the response. Our findings provide experimentally accessible signatures to distinguish single, double, and triple Weyl nodes, offering insights into topological light-matter interactions. Finally, we develop a scalable machine learning force-field model to study the adiabatic dynamics of cooperative Jahn-Teller (JT) systems, motivated by their role in orbital ordering in manganites. By incorporating lattice and orbital symmetries into a deep-learning framework, we accurately capture electron-mediated JT forces. This enables large-scale Langevin simulations that reveal the post-quench coarsening dynamics of orbital domains, including a freezing dynamics at late times tied to complex domain morphology. Our approach offers a powerful tool for modeling non-equilibrium dynamics in strongly correlated electron-lattice systems.