Non-equilibrium entanglement dynamics and transport in engineered quantum systems

This dissertation presents theoretical and computational investigations into non-equilibrium quantum dynamics in engineered systems, with a central focus on entanglement generation for quantum metrology. The primary contributions establish the scalable and robust generation of entanglement in the form of two-mode squeezing in bilayers of power-law interacting quantum spins and uncover a novel dynamical phase transition. We develop a Floquet protocol utilizing spatio-temporal control to engineer spin-spin interactions tailored to the initial state and desired entanglement dynamics, thereby improving the standard quantum limit of 1/√N scaling of the sensitivity in system size N to approach the ultimate Heisenberg limit of 1/N. For a wide range of power-law interactions (1/rα), this approach creates exponentially fast entangled pairs of excitations between the layers. We then reveal a dynamical phase transition in the two-mode squeezing dynamics by identifying universal scaling in terms of system parameters and a divergent time-scale. We establish two distinct phases: a fully collective phase exhibiting Heisenberg-limited squeezing and a partially collective phase with scalable but sub-Heisenberg squeezing. A full set of critical exponents is derived, and a universal scaling collapse of the dynamics is demonstrated, framing entanglement generation within the theory of non-equilibrium critical phenomena. We also apply advanced computational methods to other quantum and classical engineered systems. We investigate quantum transport that maps conductivity to the disorder-driven phase diagram of two-dimensional Dirac fermions. We also design a high-efficiency, lead-free tandem photovoltaic cell. Collectively, this research provides a framework for harnessing quantum control to probe and enhance functionality in complex systems, from fundamental quantum sensing to applied materials design.