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 gives the 2010 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use. The 2010 adjustment takes into account the data considered in the 2006 adjustment as well as the data that became available from 1 January 2007, after the closing date of that adjustment, until 31 December 2010, the closing date of the new adjustment. Further, it describes in detail the adjustment of the values of the constants, including the selection of the final set of input data based on the results of least-squares analyses. The 2010 set replaces the previously recommended 2006 CODATA set and may also be found on the World Wide Web at physics.nist.gov/constants.

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CODATA Recommended Values of the Fundamental Physical Constants: 2006

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

Recent developments concerning the rapid single-flux-quantum (RSFQ) circuit family are reviewed. Elementary cells in this circuit family can generate, pass, memorize, and reproduce picosecond voltage pulses with a nominally quantized area corresponding to transfer of a single magnetic flux quantum across a Josephson junction. Functionally, each cell can be viewed as a combination of a logic gate and an output latch (register) controlled by clock pulses, which are physically similar to the signal pulses. Hand-shaking style of local exchange by the clock pulses enables one to increase complexity of the LSI RSFQ systems without loss of operating speed. The simplest components of the RSFQ circuitry have been experimentally tested at clock frequencies exceeding 100 GHz, and an increase of the speed beyond 300 GHz is expected as a result of using an up-to-date fabrication technology. This review includes a discussion of possible future developments and applications of this novel, ultrafast digital technology. >

RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems

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

This monograph offers a detailed description of the statistics, dynamics and statics of Josephson junctions. Particular emphasis is placed on the dynamics of new circuits and analog and digital devices using single quanta of magnetic flux.

Dynamics of Josephson Junctions and Circuits

Mutual phase-locking in Josephson junction arrays

We report the first observation of resonant tunneling of a system between two macroscopically distinct states: energy levels in different fluxoid wells of a weakly damped superconducting quantum interference device which differ in mean current by approximately 6 \ensuremath{\mu}A. Near 50 mK, the tunneling rate $\ensuremath{\gamma}({\ensuremath{\Phi}}_{x})$ from the metastable well vs applied flux ${\ensuremath{\Phi}}_{x}$ is found to exhibit a series of local maxima where the levels (spaced by $\ensuremath{\approx}1.9$ K) in the two wells cross. The positions of these maxima agree well with the level crossings calculated using independently determined system parameters.

Observation of Resonant Tunneling between Macroscopically Distinct Quantum Levels.

Measurement of single electron lifetimes in a multijunction trap.

A superconductor frequency divider based on rapid single flux quantum (RSFQ) logic has been demonstrated to operate from dc to 750 GHz with an error rate less than the measurement limit (25 MHz). This high operating frequency is made possible by the use of small (0.25 μm2), high critical current density Josephson junctions along with an optimized design having parameter margins of ±30%. Simulations based on the Werthamer model are in reasonable agreement with the data and give a SFQ pulse width of 1 ps and a maximum power dissipation of 1.5 μW.

James E. Lukens

Papers

Dynamics of Josephson Junctions and Circuits

Mutual phase-locking in Josephson junction arrays

Observation of Resonant Tunneling between Macroscopically Distinct Quantum Levels.

Measurement of single electron lifetimes in a multijunction trap.

Superconductor digital frequency divider operating up to 750 GHz