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

The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images from the natural environment (i.e., images with trees, rocks, bushes, etc). In this study, various coding schemes are compared in relation to how they represent the information in such natural images. The coefficients of such codes are represented by arrays of mechanisms that respond to local regions of space, spatial frequency, and orientation (Gabor-like transforms). For many classes of image, such codes will not be an efficient means of representing information. However, the results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy (e.g., correlation between the intensities of neighboring pixels) into first-order redundancy (i.e., the response distribution of the coefficients). Such coding produces a relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.

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Relations between the statistics of natural images and the response properties of cortical cells.

Quantum tunnelling in a dissipative 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

Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins. It is believed that the ground state of some naturally occurring spin systems can be effectively attained through a process called quantum annealing. Johnson et al. use quantum annealing to find the ground state of an artificial Ising spin system comprised of an array of eight superconducting flux qubits with programmable spin–spin couplings. With an increased number of spins, the system may provide a practical physical means to implement quantum algorithms, possibly enabling more effective approaches towards solving certain classes of hard combinatorial optimization problems. Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins; however, finding such a ground state remains computationally difficult1. It is believed that the ground state of some naturally occurring spin systems can be effectively attained through a process called quantum annealing2,3. If it could be harnessed, quantum annealing might improve on known methods for solving certain types of problem4,5. However, physical investigation of quantum annealing has been largely confined to microscopic spins in condensed-matter systems6,7,8,9,10,11,12. Here we use quantum annealing to find the ground state of an artificial Ising spin system comprising an array of eight superconducting flux quantum bits with programmable spin–spin couplings. We observe a clear signature of quantum annealing, distinguishable from classical thermal annealing through the temperature dependence of the time at which the system dynamics freezes. Our implementation can be configured in situ to realize a wide variety of different spin networks, each of which can be monitored as it moves towards a low-energy configuration13,14. This programmable artificial spin network bridges the gap between the theoretical study of ideal isolated spin networks and the experimental investigation of bulk magnetic samples. Moreover, with an increased number of spins, such a system may provide a practical physical means to implement a quantum algorithm, possibly allowing more-effective approaches to solving certain classes of hard combinatorial optimization problems.

Quantum annealing with manufactured spins

A new method of quantifying correlations in symbolic sequences is applied to DNA nucleotides. Spectral density measurements of individual base positions demonstrate the ubiquity of low-frequency 1/${\mathit{f}}^{\mathrm{\ensuremath{\beta}}}$ noise and long-range fractal correlations as well as prominent short-range periodicities. Ensemble averages over classifications in the GenBank databank (primate, invertebrate, plant, etc.) show systematic changes in spectral exponent \ensuremath{\beta} with evolutionary category.

Evolution of long-range fractal correlations and 1/f noise in DNA base sequences.

Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension, a number that agrees with our intuitive notion of dimension but need not be an integer. In section I, a series of computer generated and rendered random fractal shapes provide a visual introduction to the concepts of fractal geometry. These complex images, with details on all scales, are the result of the simplest rules of fractal geometry. Their success as forgeries of the natural world has played an important role in the rapid establishment of fractal geometry as a new scientific discipline and exciting graphic technique. Section II presents a brief mathematical characterization of these forgeries, as variations on Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statistical self-similarity and self-affinity will be reviewed. Finally, section III will discuss independent cuts, Fourier filtering, midpoint displacement, successive random additions, and the Weierstrass-Mandelbrot random function as specific generating algorithms.

Random Fractal Forgeries

The probability distributions for switching out of the superconducting state of low-cur-rent-density 1-\ensuremath{\mu}m Nb Josephson junctions with capacitance \ensuremath{\approx} 0.1 pF have been measured as a function of temperature $T$ down to 3 mK. Below 100 mK the distribution widths become independent of $T$. The results are in excellent agreement with predictions for the quantum tunneling of the (macroscopic) junction phase that include the reduction of tunneling rates due to dissipation.

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Macroscopic Quantum Tunneling in 1-μm Nb Josephson Junctions

Transmission electron micrographs of thin evaporated gold films were analyzed by computer. For length scales above 10 nm, the irregular connected clusters show a perimeter linearly proportional to area. Near the percolation threshold the large-scale power-law correlations and area distributions are consistent with the scaling theory of second-order phase transitions. Geometrically, the boundary of all clusters is a fractal of dimension $D=2$ while individual boundaries are of fractal dimension ${D}_{C}\ensuremath{\approx}1.9$.

Fractal (Scaling) Clusters in Thin Gold Films near the Percolation Threshold

Measurements of the tunneling rate $\ensuremath{\Gamma}$ out of the zero-voltage state for several Nb edge junctions with differing shunt capacitances are described. At zero temperature, increasing the shunt capacitance lowers $\ensuremath{\Gamma}$ in agreement with dissipative calculations of the macroscopic-quantum-tunneling rate. As temperature increases, $\mathrm{ln}[\frac{\ensuremath{\Gamma}(T)}{\ensuremath{\Gamma}(0)}]\ensuremath{\propto}{T}^{2}$ as recently predicted.

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Richard F. Voss

Papers

Evolution of long-range fractal correlations and 1/f noise in DNA base sequences.

Random Fractal Forgeries

Macroscopic Quantum Tunneling in 1-μm Nb Josephson Junctions

Fractal (Scaling) Clusters in Thin Gold Films near the Percolation Threshold

Effects of dissipation and temperature on macroscopic quantum tunneling.