References......................................................................................................................................399 Processes involving quantum tunneling can be found in a huge variety of physical and chemical systems, ranging in length from the mesoscopic scale of a few microns to the subatomic scale of a few fermi. It has been realized already three decades ago that the interaction of the tunneling degree of freedom with environmental degrees of freedom plays a crucial role. Namely, any realistic description has to account for the fact that a purely isolated system is always an idealization, an issue which tends to become more and more relevant with the growing complexity of the system and particularly with increasing system size. A quantitative understanding of experimental observations must include the presence of, for example, electromagnetic modes in electrical circuits, vibrational modes in molecular aggregates, or phonon backgrounds in solid-state systems. Moreover, as a genuine quantum effect tunneling processes serve as paradigm to analyze the boundary between the microscopic and the macroscopic world or at least to elucidate how the latter one emerges from the former one.
References......................................................................................................................................399 Processes involving quantum tunneling can be found in a huge variety of physical and chemical systems, ranging in length from the mesoscopic scale of a few microns to the subatomic scale of a few fermi. It has been realized already three decades ago that the interaction of the tunneling degree of freedom with environmental degrees of freedom plays a crucial role. Namely, any realistic description has to account for the fact that a purely isolated system is always an idealization, an issue which tends to become more and more relevant with the growing complexity of the system and particularly with increasing system size. A quantitative understanding of experimental observations must include the presence of, for example, electromagnetic modes in electrical circuits, vibrational modes in molecular aggregates, or phonon backgrounds in solid-state systems. Moreover, as a genuine quantum effect tunneling processes serve as paradigm to analyze the boundary between the microscopic and the macroscopic world or at least to elucidate how the latter one emerges from the former one.
Tunneling is a supreme quantum effect. Every introductory text [1] on quantum mechanics gives the paradigm example of a particle tunneling through a one-dimensional potential barrier despite having a total energy less than the barrier height. Indeed, the reader typically works through a number of exercises, all involving one-dimensional potential barriers of one form or another modeling several key physical phenomena ranging from atom transfer reactions to the decay of α-particles [2]. However, one seldom encounters coupled multidimenisonal tunneling in such texts since an analytical solution of the Schro¨dinger equation in such cases is not possible. Interestingly, the richness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimensional systems [3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as one approaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedom to the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result in strong mode-specificity and fluctuations in the reaction rates [4]. In fact, a proper description of tunneling of electrons and hydrogen atoms is absolutely essential [5,6] even in molecular systems as large as enzymes and proteins. Although one usually assumes tunneling effects to be significant in molecules involving light atom transfers, it is worth pointing out that neglecting the tunneling of even a heavy atom like carbon is the difference between a reaction occurring or not occurring. In particular, one can underestimate rates by nearly hundred orders of magnitude [7]. Interestingly, and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barrier tunneling have been obtained from a phase-space perspective [8,9]. The contributions by Creagh, Shudo and Ikeda, and Takahashi in the present volume provide a detailed account of the latest advances in the phasespace-based understanding of multidimensional barrier tunneling.
Having achieved this point we hope to have accomplished, at least partly, our main aim which was to convince the reader that once appropriate systems have been found (or built) they can present a very peculiar combination of microscopic parameters in such a way that quantum mechanics should be applied to general macroscopic variables to describe the collective effects therein. Moreover, the very nature of these macroscopic variables does not allow them to be treated in an isolated fashion. They must rather be considered coupled to uncontrollable microscopic degrees of freedom which is the ultimate origin of dissipative phenomena. The latter, at least in the great majority of cases, play a very deleterious role in the dynamics of the macroscopic variables and we hope to have introduced minimal phenomenological techniques in order to quantify this. We have concentrated our discussions on questions originating from a few examples of superconducting or magnetic systems where quantum mechanics and dissipative effects coexist. In particular, superconducting devices which present the possibility of displaying several different quantum effects (quantum interference, decay by quantum tunneling, or coherent tunneling) are of special importance, as we will see below. Prior to development of the modern cryogenic techniques and/or the ability to build nanometric devices, it was unthinkable to imagine the existence of subtle quantum mechanical effects such as the entanglement of macroscopically distinct quantum states.
Summary form only given. The development of methods to manipulate and measure the quantum mechanical state of a physical system represents one of the great challenges of modern science. Examples of systems in which quantum control is sought or has been accomplished include trapped atomic ions, the electromagnetic field, Rydberg electrons, nuclear spins and vibrations in molecules, and chemical reactions. For many years a paradigm for quantum coherent evolution has been the occurrence of tunneling in macroscopic and mesoscopic double-well potentials. Much theoretical work has been done on this model system, but a series of fundamental questions remain to be addressed in well-controlled experiments, notably whether coherent evolution can be enforced in the presence of a noisy environment, by suitable perturbations of the state of the tunneling particle. The challenge is to find simple physical systems, which can be made to interact with a well characterized environment in a controlled manner.
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.
Over the preceeding twenty years, the role of underlying classical dynamics in quantum mechanical tunneling has received considerable attention. A number of new tunneling phenomena have been uncovered that have been directly linked to the set of dynamical possibilities arising in simple systems that contain at least some chaotic motion. These tunneling phenomena can be identified by their novel ℏ-dependencies and/or statistical behaviors. We summarize a sampling of these phenomena and mention some applications.
When the revolutionary conceptual structure for the description of the physical world which we know as quantum mechanics was first formulated nearly 90 years ago, and its predictions tested in the laboratory, most of the experiments in question were on systems which were both very well characterized and reasonably well isolated from their environments, such as single electrons and atoms, small molecules and near-perfect crystalline solids. While from the very start most physicists have taken it for granted that the formalism of quantum mechanics, when combined with appropriate system-specific information, can “in principle” account for all phenomena occurring in the physical world, including those usually regarded as the subject-matter of biology, until quite recently the overwhelmingly majority opinion has been that in a biological context the role of quantum theory is confined to elucidating the equilibrium structures of the relevant molecules and their reaction processes, and that subtle phenomena such as superposition and entanglement, of which we can now routinely exhibit spectacular effects at the level of a few well-isolated photons or atoms, play at most a very indirect role in any phenomena of biological interest. A major reason conventionally given for this view has been that biological systems, at least working ones, are by their very nature “warm and wet” – a phrase which in the physicist's lexicon translates to “prone to massive decoherence”; it looks as though any interesting superposition, say of different energy eigenstates of one's system, would be rapidly decohered by the ever-present, and usually microscopically very complex, environment.