Model Reduction of a Piecewise Linear Flexible Mechanical Oscillator

We study the reduced order modeling of a nonlinear flexible oscillator in which a Bernoulli-Euler beam with a permanent tip magnet is subjected to a position-triggered electromagnetic kick force. This results in non-smooth boundary conditions capable of exciting many degrees of freedom. The system is modeled as piecewise linear with the different boundary conditions determining different regions of a hybrid phase space. With kick strength as parameter, its bifurcation diagram is found to exhibit a range of periodic and chaotic behaviors. Proper orthogonal decomposition (POD) is used to estimate the system's intrinsic dimensionality. However, conventional POD's purely statistical analysis of spatial covariance does not guarantee accuracy of reduced order models (ROMs). We therefore augment POD by employing a previously-developed energy closure criterion that selects ROM dimension by ensuring approximate energy balance on the reduced subspace. This physics-based criterion yields accurate ROMs with 8 degrees of freedom. Remarkably, we show that ROMs formulated at particular values of the kick strength can nevertheless reconstruct the entire bifurcation structure of the original system. Our results demonstrate that energy closure analysis outperforms variance-based estimates of effective dimension for nonlinear structural systems, and is capable of providing ROMs that are robust even across stability transitions.Comment: 30 pages, 9 figure

Entanglement dynamics and chaos in long-range quantum systems

Over the past twenty years, experimental and technological progresses have motivated a renewed attention to the study of non-equilibrium isolated many-body systems, leading to a relatively well-established paradigm in the case of local Hamiltonians. In the present thesis, I have used quantum information theoretical tools to study out-of-equilibrium dynamics, with particular attention on long-range interacting many-body systems. I have explored the dynamics of bipartite and multipartite entanglement in connection to chaos and scrambling in various long-range (clean and disordered) models. The results contained in this thesis contribute to establishing semi-classical tools as powerful techniques for the description of the quantum information spreading in long-range systems. I have further considered a different, yet connected question, concerning the multipartite entanglement structure of chaotic eigenstates and its generic evolution

Quantum control of molecular rotation

The angular momentum of molecules, or, equivalently, their rotation in three-dimensional space, is ideally suited for quantum control. Molecular angular momentum is naturally quantized, time evolution is governed by a well-known Hamiltonian with only a few accurately known parameters, and transitions between rotational levels can be driven by external fields from various parts of the electromagnetic spectrum. Control over the rotational motion can be exerted in one-, two- and many-body scenarios, thereby allowing to probe Anderson localization, target stereoselectivity of bimolecular reactions, or encode quantum information, to name just a few examples. The corresponding approaches to quantum control are pursued within separate, and typically disjoint, subfields of physics, including ultrafast science, cold collisions, ultracold gases, quantum information science, and condensed matter physics. It is the purpose of this review to present the various control phenomena, which all rely on the same underlying physics, within a unified framework. To this end, we recall the Hamiltonian for free rotations, assuming the rigid rotor approximation to be valid, and summarize the different ways for a rotor to interact with external electromagnetic fields. These interactions can be exploited for control --- from achieving alignment, orientation, or laser cooling in a one-body framework, steering bimolecular collisions, or realizing a quantum computer or quantum simulator in the many-body setting.Comment: 52 pages, 11 figures, 607 reference

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...