We introduce and discuss the concept of \textit{arrangement}, traditionally found in the context of chemical reactions and few-body rearrangement collisions, in the general context of an $N$-body quantum system. We show that the ability for particles to attract and bind is a key source of complexity of an $N$-body quantum system, and \textit{arrangement} is an emergent concept necessitated by the description of this complexity. For an $N$-body system made of particles of which some or all can attract and bind, the concept of arrangement is not only necessary for a full characterization of its quantum states and processes, but it also provides a basis for the understanding and the description of the structure of its energy spectrum. We also discuss and formulate multiparticle separability in an $N$-body system, the mathematical foundation that underlies the arrangement concept in an $N$-body theory.
Arrangement and chemical complexity in an N-body quantum system
We introduce and discuss the concept of \textit{arrangement}, traditionally found in the context of chemical reactions and few-body rearrangement collisions, in the general context of an N-body quantum system. We show that the ability for particles to attract and bind is a key source of complexity of an N-body quantum system, and \textit{arrangement} is an emergent concept necessitated by the description of this complexity. For an N-body system made of particles of which some or all can attract and bind, the concept of arrangement is not only necessary for a full characterization of its quantum states and processes, but it also provides a basis for the understanding and the description of the structure of its energy spectrum. We also discuss and formulate multiparticle separability in an N-body system, the mathematical foundation that underlies the arrangement concept in an N-body theory.
This paper introduces the concept of Quantum Arrangement Entropy (QAE), a novel approach to quantifying the structural organization within complex quantum systems. Traditional entropy measures, such as Shannon entropy, primarily focus on the number of microstates – the number of distinct arrangements of particles – describing a system. However, this perspective often neglects the inherent order and relationships between these states, particularly in quantum systems where entanglement and correlations play a crucial role. QAE expands upon this by incorporating a 'structure entropy' component, denoted as *SA*, designed to specifically capture the arrangement and relationships between the possible quantum states. The paper outlines the theoretical framework for QAE, detailing the mathematical formulation and the underlying assumptions. It argues that QAE offers a more complete and nuanced understanding of quantum systems, particularly those exhibiting complex correlations and multi-body interactions, and highlights its potential significance for future research and theoretical advancements within quantum information theory and complexity science. The core idea is to move beyond simply counting states and to directly measure the degree of organized structure within a quantum system's state space. ---
What is “complex”? A question which stimulates a panoply of often quite diverse associations and answers. While in condensed matter and also (macro-) molecular physics we are used to be forced into effective descriptions of large, multicomponent, interacting quantum systems, complexity is “(re-) constructed” in modern quantum optical experiments. In learning how “complexity” emerges from strongly coupling few degrees of freedom (or particles) we start to contemplate how to control complex systems, and learn to distinguish what is just complicated from what is truly complex. This special issue of Annalen der Physik collects expertise and perspectives on complex (quantum) systems from very diverse areas of research: disordered quantum systems, molecular dynamics and interactions, mesoscopic physics, quantum many body physics, open quantum systems, network and scattering theory, quantum information and control – which all share the same interest in getting a better grasp of complexity on the quantum level. Disorder physics with its long-standing history in the condensed matter context nowadays merges with the physics of ultracold atomic gases in highly controlled potential landscapes, and finds applications in “big data”. Mesoscopic transport theory finds novel playgrounds in graphene or in coupled quantum circuits. The coupling of electronic and nuclear degrees of freedom determines chemical reactions, as well as energy flow and directionality in large molecular complexes, with an intricate interplay of coherent and incoherent effects. Today's “femtochemistry” and “femtobiology” experiments reveal these ultrafast molecular pathways in detail and make the border between quantum and classical regimes directly observable. Indeed, this research area has become a test bed for quantum transport and control theory applied to realistic, truly complex, finite quantum systems. Given the generically highly structured spectral densities which underlie molecular systems, novel strategies are called for to characterize complex open quantum systems exhibiting structured, “active” environments. Well-controlled non- equilibrium many-particle quantum dynamics becomes accessible with ultracold quantum gases loaded into engineered potential landscapes, or in photonic circuitry. This challenges theory to develop novel tools for the efficient characterization of many-particle non-equilibrium dynamics and transport, and possibly for their control, e.g. through robust design principles. Furthermore, controlled many-body quantum systems allow for enhanced sensing, and give room for the creation of highly complex many-particle states. The latter may serve, e.g., as resources for quantum computation – which, in turn, may be used to characterize such states’ complexity. This collection of various contributions either on complex molecular structures which are ever better resolved, or on novel composite quantum systems or materials which are ever better controlled, provide a snapshot of current trends in the science of complex quantum systems. Our authors’ diverse specializations highlight the research field's interdisciplinary appeal, and the broad relevance of many of the unveiled structures – and of those phenomena which remain to be understood. We thus hope that this collection will inspire future work connecting concepts of complexity across disciplines. Irene Burghardt received her Ph.D. in chemistry in 1992 from the University of Lausanne. She carried out postdoctoral research at the Center for Nonlinear Phenomena and Complex Systems in Brussels in 1992–1995, followed by research fellowships at the Universities of Bonn and Heidelberg in 1996–1998. From 1999–2011, she held a CNRS research position at ENS Paris, and was appointed CNRS Research Director in 2007. In 2011, she moved to Goethe University Frankfurt where she holds a full professorship in Theoretical Chemistry. Her research addresses quantum molecular dynamics and nonequilibrium phenomena, with a focus on materials and biological systems. Andreas Buchleitner studied Physics at the Ludwig Maximilians University of Munich and received his diploma in 1989, upon research in experimental quantum optics. In 1993, he obtained his PhD, with theoretical research at Laboratoire de Spectroscopie Hertzienne (today Laboratoire Kastler Brossel) of Universite Pierre et Marie Curie and Ecole Normale Supérieure in Paris. Afterwards, he was a PostDoc at the Max Planck Institute for Quantum Optics, Garching, and a visiting researcher at the Queen's University of Belfast and at the Weizmann Institute of Science, Revohot. In 1999, he received his Habilitation from the Ludwig Maximilians University of Munich. From 1999 to 2007, he led the research group “Nonlinear Dynamics in Quantum Systems” at the Max Planck Institute for the Physics of Complex Systems in Dresden. Since 2007, he holds the Chair for “Quantum Optics and Statistics” at the Albert Ludwigs University of Freiburg. His main research interests are Quantum Chaos, Quantum Information and Quantum Transport. From October 2014 to 2015, he was a Senior Fellow, and Speaker of the Research Focus “Designed Quantum Transport in Complex Materials”, at the Freiburg Institute for Advanced Studies. He serves in different honorary capacities, e.g., for the German Physical Society and for the German National Academic Foundation.
Over the past years, the development of experimental techniques for the coherent manipulation and control of isolated quantum systems has made impressive progress. Such 'quantum-logic' methods are also highly attractive in a chemical context in view of unravelling and controlling the quantum dynamics of molecular collisions and chemical reactions. Quantum technologies have the potential to transform the way chemical dynamics are investigated - by providing highly sensitive methods for state readout and spectroscopy, by opening up new pathways for the quantum-state preparation of molecules and by enabling an improved control of their microscopic behavior on the single-particle level. However, for complex quantum systems like molecules, these techniques are still in their infancy and their considerable potential remains to be unlocked. The aim of the present research program supported by an Advanced Grant of the Swiss National Science Foundation is to merge the fields of quantum science and chemical dynamics by advancing quantum technologies to polyatomic molecular ions and by applying them to the study of ion-molecule collisions and chemical reactions. In this article, we review the salient experimental methods as well as prospects and challenges in the development of molecular quantum technologies and their applications to chemistry.
The ability of classical thermodynamics to predict phenomena using only handful of parameters is miraculous. Moreover, this framework contrasts with other theories with its ability to describe our day to day experience, i. e. we know intuitively that an ice cube melts in a cold beverage on a hot summer day. A similarly powerful yet significantly less intuitive theory is quantum mechanics which describes the world at its tiniest length scales. Given this foundational position, we would expect to be able to derive all physical theories from quantum mechanics only. Unfortunately this would require knowing the quantum state of all particles which is costly as quantum mechanics requires resources growing exponentially in the system size to describe even relatively innocent many-body systems. Given this uncontrollable growth of parameters, it is highly desirable to apply the principles of thermodynamics to quantum many-body systems as a means to describe them ef- ficiently. Yet, the two theories are not obviously compatible. Subtle differences such as the definition of thermalisation already puzzled founding fathers like John von Neumann. While these have been understood by now, the focus shifted to gaining a thorough understanding of the process of thermalisation in the quantum regime, meaning that the state of the system can be described by a thermal ensemble locally after an initial out-of-equilibrium situation. In the recent past the field of quantum thermodynamics and specifically out-of-equilibrium dynamics has witnessed many successes notably the advent of highly controllable analogue quantum simulators which allow to probe out-of-equilibrium physics to an unprecedented precision. It has been established that thermalisation takes place in quantum many-body systems, but the underlying reasons and a comprehensive theory are still elusive. Only very recently, a class of systems exhibiting many-body localisation has been identi ed that defy this paradigm of thermalisation by localising their constituent particles despite many-body interactions. This can be captured using an effective description in terms of local constants of motion. Understanding the emergence of these physically relevant operators that hinder thermalisation hopefully sheds light also on the mechanism of thermalisation itself. In this thesis, we investigate the relation between thermodynamics and quantum mechanics in the context of out-of-equilibrium physics for many-body localisation. We give numerical evidence for the stability of localisation in large, closed onedimensional chains. Furthermore, we connect the established phenomenology of many-body localisation to experiments with ultra-cold atoms in optical lattices and give a blue print of measurements that can be performed with existing technology to show that the simulator indeed exhibits many-body localisation. We then set out to obtain a deeper understanding of the underlying theory of many-body localisation by devising an algorithm that explicitly constructs exact constants of motion allowing us to put the existing theoretical framework of many-body localisation to the test. It is our firm belief that understanding local constants of motion is indispensable to not only understand thermalisation in quantum many-body systems but is also of great use in other fields. A connection of this type is displayed by an extension of our algorithm to finding edge modes in spin chains with symmetry protected order.
In this article a survey is given of the present status of the quantum theory of many-particle systems with special emphasis on the underlying principles. Very little is said about applications. A brief introduction to some models, which are of interest for an approximate or qualitative description of actual physical systems, is followed by a discussion of two important phenomena in many-particle systems: quasi-particles and collective excitations. Next, an outline is given of mathematical methods which have been used for a precise treatment of such phenomena. A rather thorough discussion of the Bogoliubov model for interacting bosons, and the BCS model for fermions, serves as an illustration. Some new features, typical of many-particle systems, are discussed on the basis of these models. Also another approach to the many-body problem is discussed which is more logical and mathematically more rigorous. This approach differs from standard procedures in that the limit Ω-> infinity, N -> infinity is not taken at the end of the calculations but in the beginning. This quantum theory of infinite systems leads, among other things, to a better understanding of the models mentioned above.
Studying chemical reactions at very low temperatures is of importance for the understanding of fundamental physical and chemical processes. At very low energies, collisions are dominated by only a few partial waves. Thus, studies in this regime allow the characterization of quantum effects which depend on the collisional angular momentum, e.g., reactive scattering resonances and tunneling through centrifugal barriers. Additionally, the dynamics of ultralow-energy collisions is dominated by long-range interactions, i.e., “universal” chemical forces. Thus, experiments in this domain probe the details of intermolecular interactions which is of general relevance for the understanding of chemical processes. Moreover, the increasingly more precise experiments provide valuable data for benchmarking theoretical models and quantum chemical calculations. Studies of cold ion–neutral reactions in the laboratory rely on techniques for the generation of cold ions and cold neutrals. Over the last decades, the technology in this domain has made impressive progress. Many of these techniques allow not only the translational cooling of the molecules, but also a precise preparation of their internal quantum state. Thus, by an accurate preparation of both the energy and state of the reaction partners as well as by the application of external electric, magnetic or optical fields, prospects open up to study and control chemical processes at an unprecedented level of accuracy. In this chapter, we review salient theoretical concepts and recent experimental developments for studies of ion–molecule reactions at low temperatures. We first discuss the theoretical background for the description of low-energy ion–molecule reactions. Next we present key experimental methods for laboratory studies of cold ion–molecule reactions. The chapter concludes with a review of illustrative results and an outlook on future directions.