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CRC Press eBooks

Tractable Problems in Multidimensional Tunneling

Stephen C. Creagh

Published Mar 9, 2011• 0 citations
Publisher

Abstract

One of the appealing features of tunneling as a theoretical problem is that it poses interesting questions regarding the complexified dynamics of the classical limit. Often, these questions are of a different nature to the ones that arise in a purely dynamical systems context and can therefore push the dynamics side of the problem in novel directions, without necessarily being technically hard. In this article we explore a number of scenarios where these underlying problems of complex dynamics can be well enough understood to allow explicit analytical approximations to be given for tunneling rates. The treatment here is not intended to be comprehensive and short shrift is given to very interesting and important tunneling regimes, such as those of resonance-and chaos-assisted tunneling [3,14,16,17,40,48,78], for example. Instead, by concentrating on relatively tractable problems where answers can be given in terms of simple geometrical characterizations of the complex dynamics, we aim to achieve an understanding of problems which are important in their own right but which may also later serve as building blocks in the solution of more difficult problems.

DOI
10.1201/b10712-9

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CRC Press eBooks•Mar 9, 2011•0 citations
One of the appealing features of tunneling as a theoretical problem is that it poses interesting questions regarding the complexified dynamics of the classical limit. Often, these questions are of a different nature to the ones that arise in a purely dynamical systems context and can therefore push the dynamics side of the problem in novel directions, without necessarily being technically hard. In this article we explore a number of scenarios where these underlying problems of complex dynamics can be well enough understood to allow explicit analytical approximations to be given for tunneling rates. The treatment here is not intended to be comprehensive and short shrift is given to very interesting and important tunneling regimes, such as those of resonance-and chaos-assisted tunneling [3,14,16,17,40,48,78], for example. Instead, by concentrating on relatively tractable problems where answers can be given in terms of simple geometrical characterizations of the complex dynamics, we aim to achieve an understanding of problems which are important in their own right but which may also later serve as building blocks in the solution of more difficult problems.
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.
In this revised and expanded edition, in addition to a comprehensible introduction to the theoretical foundations of quantum tunneling based on different methods of formulating and solving tunneling problems, different semiclassical approximations for multidimensional systems are presented. Particular attention is given to the tunneling of composite systems, with examples taken from molecular tunneling and also from nuclear reactions. The interesting and puzzling features of tunneling times are given extensive coverage, and the possibility of measurement of these times with quantum clocks are critically examined. In addition by considering the analogy between evanescent waves in waveguides and in quantum tunneling, the times related to electromagnetic wave propagation have been used to explain certain aspects of quantum tunneling times. These topics are treated in both non-relativistic as well as relativistic regimes. Finally, a large number of examples of tunneling in atomic, molecular, condensed matter and nuclear physics are presented and solved.
Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking ``clustering'' phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the ``mixed'' role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.
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.
Open quantum systems are ubiquitous in the physical sciences, with widespread applications in the areas of chemistry, condensed-matter physics, material science, optics, and many more.Not surprisingly, there is significant interest in their efficient simulation.However, direct classical simulation quickly becomes intractable with coupling to an environment with an effective dimension that grows exponentially.This raises the question: Can quantum computers help model these complex dynamics?A first step in answering this question requires an understanding of the computational complexity of this task.Here, we map the temporal complexity of a process to the spatial complexity of a many-body state using a computational model known as the process-tensor framework.With this, we are able to explore the simulation complexity of an open quantum system as a dynamic sampling problem: a system coupled to an environment can be probed at successive points in time-accessing multitime correlations.The complexity of multitime sampling, which is an important and interesting problem in its own right, contains the complexity of master equations and stochastic maps as a special case.Our results show how the complexity of the underlying quantum stochastic process corresponds to the complexity of the associated family of master equations for the dynamics.We present both analytical and numerical examples where the multitime sampling of an open quantum system is as complex as sampling from a many-body state that is classically hard.This also implies that the corresponding family of master equations is classically hard.Our results pave the way for studying open quantum systems from a complexity-theoretic perspective, highlighting the role quantum computers will play in our understanding of quantum dynamics.