The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

This paper presents the results of a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment. It is primarily an extended account of results obtained over the last four years by the authors; while they try to provide some background for orientation, it is emphatically not intended as a comprehensive review of the literature on the subject. Its contents include (1) an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of a dissipative environment to the "spin-boson" problem; (2) the derivation of an exact formula for the dynamics of the latter problem; (3) the demonstration that there exists a simple approximation to this exact formula which is controlled, in the sense that we can put explicit bounds on the errors incurred in it, and that for almost all regions of the parameter space these errors are either very small in the limit of interest to us (the "slow-tunneling" limit) or can themselves be evaluated with satisfactory accuracy; (4) use of these results to obtain quantitative expressions for the dynamics of the system as a function of the spectral density $J(\ensuremath{\omega})$ of its coupling to the environment. If $J(\ensuremath{\omega})$ behaves as ${\ensuremath{\omega}}^{s}$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For $sl1$ the system is localized at zero temperature, and at finite $T$ relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. For $sg2$ it undergoes underdamped coherent oscillations for all relevant temperatures, while for $1lsl2$ there is a crossover from coherent oscillation to overdamped relaxation as $T$ increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, $s=1$, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\ensuremath{\alpha}$ as well as the temperature $T$: over most of the ($\ensuremath{\alpha}$,$T$) plane (including the whole region $\ensuremath{\alpha}g1$) the behavior is an incoherent relaxation at a rate proportional to ${T}^{2\ensuremath{\alpha}\ensuremath{-}1}$, but for low $T$ and $0l\ensuremath{\alpha}l\frac{1}{2}$ the authors predict a combination of damped coherent oscillation and incoherent background which appears to disagree with the results of all previous approximations. The case of finite bias is also discussed.

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Dynamics of the dissipative two-state system

Quantum-state engineering, i.e., active control over the coherent dynamics of suitable quantum-mechanical systems, has become a fascinating prospect of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. Low-capacitance Josephson tunneling junctions offer a promising way to realize quantum bits (qubits) for quantum information processing. The article reviews the properties of these devices and the practical and fundamental obstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux) degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltages in one case and by magnetic fields in the other case. Both kinds of device can be fabricated with present technology. In flux qubit devices, an important milestone, the observation of superpositions of different flux states in the system eigenstates, has been achieved. The Josephson charge qubit has even demonstrated coherent superpositions of states readable in the time domain. There are two major problems that must be solved before these devices can be used for quantum information processing. One must have a long phase coherence time, which requires that external sources of dephasing be minimized. The review discusses relevant parameters and provides estimates of the decoherence time. Another problem is in the readout of the final state of the system. This issue is illustrated with a possible realization by a single-electron transistor capacitively coupled to the Josephson device, but general properties of measuring devices are also discussed. Finally, the review describes how the basic physical manipulations on an ideal device can be combined to perform useful operations.

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Quantum-state engineering with Josephson-junction devices

Quantum mechanics works exceedingly well in all practical applications. No example of conflict between its predictions and experiment is known. Without quantum physics we could not explain the behavior of solids, the structure and function of DNA, the color of the stars, the action of lasers or the properties of superfluids. Yet well over half a century after its inception, the debate about the relation of quantum mechanics to the familiar physical world continues. How can a theory that can account ith precision for everything we can measure still be deemed lacking? The environment surrounding a quantum system can, in effect, monitor some of the system's observobles. As a result, the eigenstates of those observables continuously decohere and can behave like classical states.

Decoherence and the transition from quantum to classical

Explanations for the anomalously high mobility of protons in liquid water began with Grotthuss's idea1, 2 of ‘structural diffusion’ nearly two centuries ago Subsequent explanations have refined this concept by invoking thermal hopping3, 4, proton tunnelling5, 6 or solvation effects7 More recently, two main structural models have emerged for the hydrated proton Eigen8, 9 proposed the formation of an H9O4+ complex in which an H3O+ core is strongly hydrogen-bonded to three H2O molecules Zundel10, 11, meanwhile, supported the notion of an H5O2+ complex in which the proton isshared between two H2O molecules Here we use ab initio path integral12,13,14 simulations to address this question These simulations include time-independent equilibrium thermal and quantum fluctuations of all nuclei, and determine interatomic interactions from the electronic structure We find that the hydrated proton forms a fluxional defect in the hydrogen-bonded network, with both H9O4+ and H5O2+ occurring only in thesense of ‘limiting’ or ‘ideal’ structures The defect can become delocalized over several hydrogen bonds owing to quantum fluctuations Solvent polarization induces a small barrier to proton transfer, which is washed out by zero-point motion The proton can consequently be considered part of a ‘low-barrier hydrogen bond’15, 16, in which tunnelling is negligible and the simplest concepts of transition-state theory do not apply The rate of proton diffusion is determined by thermally induced hydrogen-bond breaking in the second solvation shell

The nature of the hydrated excess proton in water

The time-dependent Ginzburg-Landau theory is used to study both transverse and longitudinal transport properties of a layered superconductor in a magnetic field near the mean-field transition temperature {ital T}{sub {ital c}2}({ital H}). We evaluate the transport coefficients in the self-consistent Hartree approximation which interpolates smoothly between the high-temperature regime, dominated by Gaussian fluctuations, and the low-temperature flux-flow regime, with no intervening divergence. This behavior is in agreement with the experimental results for the Ettingshausen coefficient, Nernst coefficient, longitudinal conductivity, and Hall conductivity in high-temperature superconductors.

Effect of fluctuations on the transport properties of type-II superconductors in a magnetic field

Vortex motion in type-II superconductors is studied starting from a variant of the time-dependent Ginzburg-Landau equations, in which the order-parameter relaxation time is taken to be complex. Using a method due to Gor'kov and Kopnin, we derive an equation of motion for a single vortex (B\ensuremath{\ll}${\mathit{H}}_{\mathit{c}2}$) in the presence of an applied transport current. The imaginary part of the relaxation time and the normal-state Hall effect both break ``particle-hole symmetry,'' and produce a component of the vortex velocity parallel to the transport current, and consequently a Hall field due to the vortex motion. Various models for the relaxation time are considered, allowing for a comparison to some phenomenological models of vortex motion in superconductors, such as the Bardeen-Stephen and Nozi\`eres-Vinen models, as well as to models of vortex motion in neutral superfluids. In addition, the transport energy, Nernst effect, and thermopower are calculated for a single vortex. Vortex bending and fluctuations can also be included within this description, resulting in a Langevin-equation description of the vortex motion. The Langevin equation is used to discuss the propagation of helicon waves and the diffusional motion of a vortex line. The results are discussed in light of the rather puzzling sign change of the Hall effect which has been observed in the mixed state of the high-temperature superconductors.

Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg-Landau theory approach

We study the fluctuation Ettingshausen effect in a type-II superconductor using the time-dependent Ginzburg-Landau equation The free-fluctuation theory predicts a divergence in the Ettingshausen coefficient at T=${\mathit{T}}_{\mathit{c}2}$(H), in conflict with the recent data of Palstra et al [Phys Rev Lett 64, 3090 (1990)] Including interactions through a self-consistent Hartree approximation eliminates the divergence and provides a quantitative explanation of the experiments

Critical fluctuations in high-temperature superconductors and the Ettingshausen effect

We study quantum tunneling in a biased double-well potential in the presence of Ohmic dissipation with a friction coefficient $\ensuremath{\alpha}g1$. The tunneling rate out of the metastable well is calculated in real time as a function of the temperature $T$ and bias energy $\ensuremath{\epsilon}$. The $T=0$ result agrees with a droplet-model calculation in imaginary time. At low temperatures we find an enhancement to the $T=0$ rate varying as ${(\frac{T}{\ensuremath{\epsilon}})}^{2}$.

Alan T. Dorsey

Papers

Dynamics of the dissipative two-state system

Effect of fluctuations on the transport properties of type-II superconductors in a magnetic field

Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg-Landau theory approach

Critical fluctuations in high-temperature superconductors and the Ettingshausen effect

Dissipative quantum tunneling in a biased double-well system at finite temperatures.