Schrödinger's orchestra Dynamics of ordered and disordered quantum ensembles

This thesis is a study of average dynamics of abstract many-body quantum systems.
The first half investigates entanglement generation and ultimately equilibration between quantum subsystems. The driving Hamiltonian is the random object, sampled from the Gaussian Unitary Ensemble (GUE), a staple of Random Matrix Theory. Analytic expressions are obtained for the reduced density matrix and purity of a subsystem while it entangles with a bath, and these expressions turn out only to depend on the dimensionality of the Hilbert Spaces. Numerical simulations confirm these findings strikingly, and the dynamics are compared to a selection of famous models, some integrable and some chaotic, from condensed matter physics.
The second half of the thesis considers an incarnation of a tau-function. Tau-functions are famous for their role in integrable systems. In this case, it is viewed as a correlation function between shifted bases of free fermions, or a recurrence measure of fidelity in a many-body version of the Aharonov-Bohm effect. Due to Orthogonality Catastrophe, each term in the series of such a function vanishes in the Thermodynamic Limit. Nevertheless, using finely tuned mathematical identities and Cauchy theory, this tau-function can be resummed to a Fredholm Determinant. A pioneering approximation scheme, using effective form-factors, points towards late-time asymptotics of this and more exotic or realistic models. Finally, a computer algorithm is detailed and tested that can efficiently scout out important terms in free-fermionic Hilbert spaces. It is used to approximate the tau-function at finite size, as well as some observables in the Lieb-Liniger model.