Chaos on the Quantum Scale Simulations of atomic-scale billiards reveal chaotic phenomena that expand theory and applications, especially in nanotechnology
During the 20th century, two new ways of describing physical sys tems emerged to thwart scientists' hopes of ever completely predicting nature's behavior. First, the theory of quantum mechanics uncovered ram pant uncertainty at the tiniest scales of matter?in the canonical example, an electron's precise position and momen tum cannot simultaneously be known at any particular moment. The second body of thought is what we now call chaos theory. Some phenomena, it turns out, depend intimately on a sys tem's initial conditions, so that an im perceptible change in the beginning value of a variable can make the out come of a process impossible to pre dict. Chaos appears in examples as dis tinct as the frequency of drips from a faucet and the motion of planets. Given the unpredictability that char acterizes both chaos and quantum me chanics, one might well wonder what would happen if these phenomena were merged. Utter chaos, perhaps? Probably not; in fact we have tools for approaching the modeling of quantum chaos, or chaotic behavior on the quan tum scale. Early attempts to under stand this exotic realm have produced results that are both mathematically and scientifically important. Having
During the 20th century, two new ways of describing physical sys tems emerged to thwart scientists' hopes of ever completely predicting nature's behavior. First, the theory of quantum mechanics uncovered ram pant uncertainty at the tiniest scales of matter?in the canonical example, an electron's precise position and momen tum cannot simultaneously be known at any particular moment. The second body of thought is what we now call chaos theory. Some phenomena, it turns out, depend intimately on a sys tem's initial conditions, so that an im perceptible change in the beginning value of a variable can make the out come of a process impossible to pre dict. Chaos appears in examples as dis tinct as the frequency of drips from a faucet and the motion of planets. Given the unpredictability that char acterizes both chaos and quantum me chanics, one might well wonder what would happen if these phenomena were merged. Utter chaos, perhaps? Probably not; in fact we have tools for approaching the modeling of quantum chaos, or chaotic behavior on the quan tum scale. Early attempts to under stand this exotic realm have produced results that are both mathematically and scientifically important. Having
Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetus thanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models, hence permitting a direct comparison with theoretical calculations and a precise benchmarking of their reliability.The field has now reached a maturity such that one can address difficult problems, which cannot be solved efficiently using classical algorithms.These advances provide unprecedented opportunities to explore previously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation.We discuss the challenges, define both digital and analog quantum simulators, and list the demanding conditions they require.We also provide a brief account of the contributions gathered in the dossier on Quantum simulation in the Compte rendus Physique of the French Academy of Sciences [1][2][3][4][5][6].The latter completes excellent reviews that appeared previously, see for instance . Universal models and the role of simulations in many-body physicsUnderstanding the behavior of macroscopic quantum systems is a major challenge of modern physics.The basic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale.Conversely, many fundamental questions remain open, and even debated, about the collective dynamics at the macroscopic scale.By "macroscopic scale", here we mean systems made up of a huge number of constituents, or degrees of freedom, say 10 6 , 10 23 , or even more, as relevant in condensed matter physics or in astrophysics, for instance.Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at the quantum level.Yet, it is the main outcome of the thermodynamic approach that the collective behavior of a macroscopic system can drastically differ from that of its elementary constituents.For instance, the elementary interactions between the H 2 O molecules are fundamentally unchanged when a water bucket turns from the liquid phase to the solid phase at zero degree Celsius.Similarly, the interactions between the microscopic magnetic moments do not show any brutal change when a magnetic material gets magnetized underneath the Curie temperature.Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level.This observation takes a universal character, summarized in the celebrated motto "More is Different" [15].Such so-called emerging phenomena also appear in quantum systems, where new effects that are impossible in the classical world show up below some critical temperature or at zero temperature when some interaction parameter passes through a critical value.Celebrated examples include the superfluid transition in helium or super-conducting transitions and other metal-insulator transitions in electronic systems.Strikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopic systems, their germs are contained in the mutual interactions of their elementary constituents.Hence, local two-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid and magnetic transitions mentioned above.The Boltzmann statistical approach proved particularly successful in describing this connection between the microscopic and macroscopic scales.To understand emerging phenomena on grounds as fundamental as possible, the programme is now well established: One tries to identify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbard model for metal-insulator transitions [16][17][18].Then all is left to do is to check that the phenomenon of interest indeed emerges from the dynamics of the simplified model.The realization of this programme is nothing but a simulation.It consists in building up a simplified system that mimics the main properties of a real system.
As probably the most successful scientific theory ever created, quantum theory has profoundly changed our view of the world and extended the limits of our knowledge, impacting both the theoretical interpretation of a tremendous range of phenomena and the practical development of a host of technological breakthroughs. Yet for all its success, quantum theory remains utterly baffling. Quantum Reality: Theory and Philosophy, Second Edition cuts through much of the confusion to provide readers with an exploration of quantum theory that is as authoritatively comprehensive as it is intriguingly comprehensible. The book has been fully updated throughout to include the latest results in quantum entanglement, the theory and practical applications of quantum computing, quantum cosmology and quantum gravity. Needing little more than a school level physics and mathematics background, this volume requires only an interest in understanding how quantum theory came to be and the myriad ways it both explains how our universe functions and extends the reach of human knowledge. Written by well-known physics author and teacher Dr. Jonathan Allday, this highly engaging work: • Presents a thorough grounding in the theoretical machinery of quantum physics • Offers a whistle-stop tour through the early part of the 20th century when the founding fathers of quantum theory forever altered the frontiers of human thought • Provides an example-filled interpretation of the theory, its applications, and its pinnacle in quantum field theory (QFT), so crucial in shaping ideas about the nature of reality • Separates fact from speculation regarding quantum physics’ ability to provide a starting point for philosophical queries into ultimate understanding and the limits of science The world beneath the one that we experience with our senses is profoundly mysterious, and while we may never completely unravel that mystery, quantum theory allows us to come closer than ever to understanding where the science leaves off and the mystery begins. Quantum Reality: Theory and Philosophy, Second Edition makes that understanding accessible to anyone possessing a quest for knowledge and a sense of awe.
By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.
Einstein (1917) appreciated early on that within the “old” pre-1925/26 quantum mechanics absence of integrability is a serious obstacle for the quantization of classical systems. Therefore, in retrospect not surprisingly, the quantization problem was not adequately solved until the advent of the “new” quantum mechanics by Heisenberg, Born, Jordan and Schrödinger. The new quantum mechanics did not rely at all on the notion of classical paths, and this way, unwittingly, sidestepped the chaos problem. Within the framework of the new theory, any classical system can be quantized, including classically chaotic systems. But while the quantization of integrable systems is straightforward, the quantization of classically chaotic systems, even today, presents a formidable technical challenge. This is especially true for quantization in the semiclassical regime, where the quantum numbers involved are large. In fact, efficient semiclassical quantization rules for chaotic systems were not known until Gutzwiller (1971, 1990) intoduced periodic orbit expansions. Gutzwiller's method is discussed in Section 4.1.3 below. It is important to emphasize here that the existence of chaos in certain classical systems in no way introduces conceptual problems into the framework of modern quantum theory, although, let it be emphasized again, chaos came back with a vengeance from the “old” days of quantum mechanics. Even given all the modern day computer power accessible to the “practitioner” of quantum mechanics, chaos is the ultimate reason for the slow progress in the numerical computation of even moderately excited states in such important, but chaotic problems as the helium atom.
The present collection of papers is devoted to a rather new and very controversial topic, the so-called “quantum chaos”. Some researchers see nothing essentially new at all in this phenomenon (apart from a number of various examples and models), and they have good reason to believe so. Indeed, the problems in this field all belong to the traditional, “old-fashion”, and rather “simple” quantum mechanics of finite-dimensional systems with a given interaction and no quantized fields. Nevertheless, many, including ourselves, consider quantum chaos to be a new discovery, though in an old field, of a great importance for fundamental physics. To understand this, the phenomenon of quantum chaos should be put into its proper perspective in recent developments in physics. The central point of this perspective is the conception of dynamical chaos (also a rather new topic) in classical mechanics (for a good review see, e.g., Refs.[l-3]). Thus before discussing the current understanding of quantum chaos we need briefly to describe classical dynamical chaos