Wu-Ming Liu – författare

This book explores the diverse types of Schrödinger equations that appear in nonlinear systems in general, with a specific focus on nonlinear transmission networks and Bose–Einstein Condensates. In the context of nonlinear transmission networks, it employs various methods to rigorously model the phenomena of modulated matter-wave propagation in the network, leading to nonlinear Schrödinger (NLS) equations. Modeling these phenomena is largely based on the reductive perturbation method, and the derived NLS equations are then used to methodically investigate the dynamics of matter-wave solitons in the network. In the context of Bose–Einstein condensates (BECs), the book analyzes the dynamical properties of NLS equations with the external potential of different types, which govern the dynamics of modulated matter-waves in BECs with either two-body interactions or both two- and three-body interatomic interactions. It also discusses the method of investigating both the well-posedness and the ill-posedness of the boundary problem for linear and nonlinear Schrödinger equations and presents new results. Using simple examples, it then illustrates the results on the boundary problems. For both nonlinear transmission networks and Bose–Einstein condensates, the results obtained are supplemented by numerical calculations and presented as figures.

This book explores the diverse types of Schrödinger equations that appear in nonlinear systems in general, with a specific focus on nonlinear transmission networks and Bose–Einstein Condensates. In the context of nonlinear transmission networks, it employs various methods to rigorously model the phenomena of modulated matter-wave propagation in the network, leading to nonlinear Schrödinger (NLS) equations. Modeling these phenomena is largely based on the reductive perturbation method, and the derived NLS equations are then used to methodically investigate the dynamics of matter-wave solitons in the network. In the context of Bose–Einstein condensates (BECs), the book analyzes the dynamical properties of NLS equations with the external potential of different types, which govern the dynamics of modulated matter-waves in BECs with either two-body interactions or both two- and three-body interatomic interactions. It also discusses the method of investigating both the well-posedness and the ill-posedness of the boundary problem for linear and nonlinear Schrödinger equations and presents new results. Using simple examples, it then illustrates the results on the boundary problems. For both nonlinear transmission networks and Bose–Einstein condensates, the results obtained are supplemented by numerical calculations and presented as figures.

This book highlights a comprehensive and self-contained account of the physics of Rydberg atoms and their integration within the broader framework of cavity quantum electrodynamics (QED) and hybrid quantum systems. Rydberg atoms in cavities constitute a uniquely powerful and versatile platform for exploring strongly interacting quantum systems, where the combination of exaggerated atomic properties and engineered light–matter coupling enables access to physical regimes that are otherwise unattainable. With their exceptionally large electric dipole moments, long radiative lifetimes, and extreme sensitivity to external electric and magnetic fields, Rydberg atoms amplify the strength of interatomic interactions by several orders of magnitude compared to ground-state atoms. When placed within optical or microwave resonators, these properties are further enhanced through cavity quantum electrodynamics (QED), which provides precise control over the interaction between matter and confined electromagnetic fields. This integration allows the realization of deterministic quantum gates, scalable entanglement generation, and photonic nonlinearities at the level of single photons—capabilities that are at the core of quantum information science and next-generation quantum technologies. Our central aim is to furnish the reader with both a foundational understanding of Rydberg physics and the advanced theoretical and technical tools required to contribute to the fast-growing research landscape. The exposition begins with the preparation and characterization of Rydberg states, including excitation techniques, radiative properties, and spectroscopic methods. It then systematically develops the key interaction mechanisms—such as dipole blockade, Förster resonances, and tunable long-range van der Waals and dipole–dipole couplings—that underpin many-body correlations and collective dynamics. These mechanisms form the basis for much of the remarkable physics observed in ensembles of Rydberg atoms. The narrative then shifts to the interplay between strong atomic interactions and the confined electromagnetic modes of resonators.

This book highlights a comprehensive and self-contained account of the physics of Rydberg atoms and their integration within the broader framework of cavity quantum electrodynamics (QED) and hybrid quantum systems. Rydberg atoms in cavities constitute a uniquely powerful and versatile platform for exploring strongly interacting quantum systems, where the combination of exaggerated atomic properties and engineered light–matter coupling enables access to physical regimes that are otherwise unattainable. With their exceptionally large electric dipole moments, long radiative lifetimes, and extreme sensitivity to external electric and magnetic fields, Rydberg atoms amplify the strength of interatomic interactions by several orders of magnitude compared to ground-state atoms. When placed within optical or microwave resonators, these properties are further enhanced through cavity quantum electrodynamics (QED), which provides precise control over the interaction between matter and confined electromagnetic fields. This integration allows the realization of deterministic quantum gates, scalable entanglement generation, and photonic nonlinearities at the level of single photons—capabilities that are at the core of quantum information science and next-generation quantum technologies. Our central aim is to furnish the reader with both a foundational understanding of Rydberg physics and the advanced theoretical and technical tools required to contribute to the fast-growing research landscape. The exposition begins with the preparation and characterization of Rydberg states, including excitation techniques, radiative properties, and spectroscopic methods. It then systematically develops the key interaction mechanisms—such as dipole blockade, Förster resonances, and tunable long-range van der Waals and dipole–dipole couplings—that underpin many-body correlations and collective dynamics. These mechanisms form the basis for much of the remarkable physics observed in ensembles of Rydberg atoms. The narrative then shifts to the interplay between strong atomic interactions and the confined electromagnetic modes of resonators.

This book presents an analytical approach to treating several topics of current interest in the field of nonlinear partial differential equations and their applications to electrical and communications engineering, the physics of nonlinear dispersive media, as well as the nonlinear wave interactions. It treats analytically Ginzburg-Landau and wave equations such as higher-order nonlinear Schrodinger equations with/without dissipative terms, Gross-Pitaevskii equations with complicated potential terms, and cubic-quintic Ginzburg-Landau equations. For solving analytically various problems of mathematical physics in nonlinear dispersive media, the book explanatorily and carefully applies several powerful methods drawn from recent leading research articles. Special attentions are paid to the modulational instability phenomenon and baseband modulational instability phenomenon in nonlinear dispersive media. The theoretical results of this book are supplemented by numerical calculations and graphical illustrations. This book is intended for scientific researchers working in the field of nonlinear waves; it will be particularly useful for applied mathematicians, theoretical physicists, as well as electrical and communications engineers.