This paper is an introduction to the eleven works of the special issue on Quantum and Classical Frontiers of Noise. The weather, and its butterfly effect, is the typical example that explain why many natural phenomena are, in fact, not predictable with certainty. Noise in classical or quantum phenomena, understood as the difference between the empirical output of an experiment and its statistical prediction, is a measure of such uncertainty. One of the great contributions of noise appeared in 1905 when Einstein showed how the noisy (Brownian) motion of a dust particle seen in a microscope provided uncontroversial evidences on the existence of the (unseen) atoms. The aim of this special issue is to provoke discussions on how noise can help us to better understand (the reality behind) classical and quantum theories, and the frontier between them.
This paper is an introduction to the eleven works of the special issue on Quantum and Classical Frontiers of Noise. The weather, and its butterfly effect, is the typical example that explain why many natural phenomena are, in fact, not predictable with certainty. Noise in classical or quantum phenomena, understood as the difference between the empirical output of an experiment and its statistical prediction, is a measure of such uncertainty. One of the great contributions of noise appeared in 1905 when Einstein showed how the noisy (Brownian) motion of a dust particle seen in a microscope provided uncontroversial evidences on the existence of the (unseen) atoms. The aim of this special issue is to provoke discussions on how noise can help us to better understand (the reality behind) classical and quantum theories, and the frontier between them.
This Special Issue of Journal of Optics B: Quantum and Semiclassical Optics brings together the contributions of various researchers working on theoretical and experimental aspects of fluctuational phenomena in photonics and quantum optics. The topics discussed in this issue extend from fundamental physics to applications of noise and fluctuational methods from quantum to classical systems, and include: • Quantum measurement • Quantum squeezing • Solitons and fibres • Gravitational wave inferometers • Fluorescence phenomena • Cavity QED • Photon statistics • Noise in lasers and laser systems • Quantum computing and information • Quantum lithography • Teleportation.
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.
Scientific understanding proceeds through experiments that reveal how systems behave and through abstract models constructed to account for those behaviors. Persistent conceptual and philosophical confusions arise, however, when the distinction between empirical reality and its abstract representations is left implicit. This issue is particularly evident in the study of light, where multiple, mutually non-equivalent models coexist, each successful within specific limits, yet none capable of exhausting all observed phenomena. In this work, light is analyzed as a case study using the empirical–axiomatic self framework. Light is treated first as an empirical self: a system that exists and behaves in reality and constrains observation through experimental interaction. Experimental outcomes are taken as empirical facts, understood not as explanations but as constraints imposed by reality on representation. From these constraints, multiple axiomatic selves ray, wave, classical electromagnetic, particle, quantum, and quantum field models are constructed as abstract, internally coherent structures, each defined by assumptions and restricted domains of validity. The analysis shows that domains of validity are determined directly by experimental results and indirectly by comparison with more general axiomatic selves, though experimental constraint remains the ultimate authority. More general models clarify, but do not replace, earlier ones, explaining their success within restricted regimes. No axiomatic self, regardless of generality, is identified with the empirical self itself. By maintaining a strict distinction between empirical reality and abstract representation, the framework dissolves several long-standing philosophical confusions, including realism versus instrumentalism, wave–particle duality, theory replacement, determinism versus probability, and the measurement problem. These issues are shown to arise from category mistakes rather than from deep features of nature. Although light serves as the central example, the principles extracted are not specific to optical phenomena. The empirical–axiomatic distinction applies broadly across physical systems and extends naturally to complex systems. The work thus offers a clarified account of scientific understanding: to understand a system is to know the empirical constraints it imposes, the axiomatic structures used to represent it, the domains in which those structures apply, and the limits beyond which they fail.
We briefly review the difficulties of the standard version of quantum mechanics in accounting for all kinds of physical processes (in particular both the micro and the macroscopic ones) and we argue that to overcome such difficulties one has to give up the superposition principle, i.e., to break the linear nature of the theory. We then show that the simple introduction of terms violating the linearity of the deterministic evolution of the standard theory has physically unacceptable consequences. Taking this into account we are led to modify the quantum dynamics by adding to Schrödinger’s equation, besides nonlinear terms, a coupling with an appropriate noise. We show that, with such a move, one can work out a theory which is internally consistent and it allows to account, in a quite simple and physically appealing way, for natural processes including the measurements and the classical behavior of macroscopic bodies. On the basis of our analysis, we are led to conclude that the consideration of the noise represents a crucial element for understanding the laws of nature.
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.
In 2005 we celebrate the hundredth anniversary of the publication of five seminal papers by Albert Einstein establishing the basis of three fundamental fields of physics: the theory of relativity, quantum theory and the theory of Brownian motion. This happened at a time when the reality of atoms and molecules was still far from certain. The microscopic view Einstein took of molecular motion led to the calculation of Avogadro's number and the size of molecules by studying the motion of small particles in liquids. Combining kinetic theory and classical thermodynamics finally led Einstein to the conclusion that the displacement of a microparticle under Brownian motion varies as the square root of time. The experimental demonstration of this law three years later was considered as the striking proof that atoms and molecules are physically real. Today single atoms are probed directly in experiments, and we are able to observe the individual paths of colliding or dissociating particles. These new techniques are described in some of the papers of this issue in a very impressive way. An even more exciting development was initiated with the demonstration of Bose–Einstein condensation of dilute gases of ultracold atoms, a phenomenon first predicted by Einstein in papers published between 1916 and 1924. The first paper of Einstein's famous series of 1905, 'On an heuristic point of view about the creation and conversion of light', was an explanation of the photoelectric effect, demonstrated in several experiments over the previous few years. The explanation was based on the quantum hypothesis introduced by Max Planck five years earlier, and was considered as an additional and important proof of Planck's hypothesis. Today quantum theory and photons are the basis for much of modern science and technology. We have learned to experiment with single photons, and we have demonstrated the advantages of information transfer by single photons. Photons permit new and incredibly precise time and frequency standards. These are not only technically important, but may also lead to even more stringent tests of relativity and other fundamental laws and concepts of physics. Examples of these applications and many other uses of modern optics are discussed in this issue. Special relativity, the third of Einstein's 1905 topics, has always been closely connected with atomic, molecular and optical physics. Today atomic physics provides some of the most stringent tests of special relativity. Furthermore, fast electron beams are used in advanced light sources such as synchrotron radiation generators and free electron lasers. Those aspects and related applications are also discussed here. We hope that this special issue will be of great interest to the reader by highlighting recent advances in atomic, molecular and optical physics. This field continues to provide one of the most fertile areas for research one hundred years after it first emerged from the work of Einstein published in 1905, his 'miraculous year'.