This doctoral thesis presents an extensive study on the applications of generalized coherent states (GCS) for the quantum dynamics of many-body systems. The research starts with exploring the fundamental properties of generalized coherent states, which are created by generators of the SU($M$) group acting on an extreme state, and demonstrating their role in representing ideal quantum condensates. A significant feature is the relationship between generalized coherent states and the more standard Glauber coherent states (CS). Similarities in their overcomplete and non-orthogonal nature are shown, alongside crucial differences with respect to $U(1)$ symmetry and entanglement properties, which generalized coherent states solely adhere to.
Furthermore, this thesis delves into the nonequilibrium dynamics of GCS as well as Glauber CS under nonlinear interactions. Combining analytical analysis and numerical calculations, it is found that while their two-point correlation functions are equivalent in the thermodynamic limit, their autocorrelation functions exhibit distinctly different characteristics. It is proven analytically that the autocorrelation functions of the evolved GCS relate to the ones of the corresponding Glauber CS through a Fourier series relation, which arises due to the $U(1)$ symmetry of the GCS.
A substantial part of this thesis is dedicated to investigating the dynamics of the Bose-Hubbard model, incorporating both nonlinear interaction and tunneling term. This investigation introduces a novel approach which employs an Ansatz for the wave function in terms of a linear combination of GCS, where the differential equations of all the variables are determined by the time-dependent variational principle without truncation. This innovative method is adeptly applied to the nonequilibrium dynamics in various scenarios, from the bosonic Josephson Junction model where some fundamental quantum effects can be revealed by a handful of GCS basis functions, to large system size implementations of the Bose-Hubbard model, where the phenomenon of thermalization can be observed. The proposed variational approach provides an alternative way to study the time-dependent dynamics in many-body quantum systems conserving particle number.
The final focus of this thesis is on the boson sampling problem within a linear optical network framework. Again adapting a linear combination of GCS, an exact analytical formula for the output state in standard boson sampling scenarios is derived by means of Kan's formula, showcasing a computational complexity that increases less severely with particle and mode number than the super-exponential scaling of the Fock state Hilbert space. The reduced density matrix of the output state is obtained by tracing out one subsystem. This part of the study extends to examining the properties of the subsystem entanglement creation, and offering novel perspectives on entanglement entropy differences between global and local optical networks.
This thesis makes several contributions to the field of quantum many-body systems, particularly highlighting the potential applications of GCS. The presented research offers a new variational method to the nonequilibrium dynamics, and paves the way for future explorations and applications in quantum simulations, quantum computing and beyond.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:92104 |
| Date | 18 June 2024 |
| Creators | Qiao, Yulong |
| Contributors | Grossmann, Frank, Tomsovic, Steven, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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