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

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light fields. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.

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Quantum dynamics of single trapped ions

The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it ha...

The Jaynes-Cummings Model

Dark optical solitons: physics and applications

This Letter reports on the existence of periodic spontaneous collapse and revival of coherence in the dynamics of a simple quantum model. Also given are the first accurate expressions for the intermediate-time and long-time dynamical behavior of the model.

Periodic spontaneous collapse and revival in a simple quantum model

We describe the temporal behavior of the dynamic elements of an exactly soluble quantum model The model consists of a single two-level atom or spin interacting with a single mode of the quantized radiation field in the dipole approximation, the mode being initially in an arbitrary coherent state of excitation We give new long-time numerical and closed-form approximate analytic solutions for the expectation values of the atomic dipole moment and the difference in population of the two atomic levels in the rotating wave approximation The atomic dipole-dipole correlation function is calculated All of the results are obtained without semiclassical or decorrelation approximations Unusual features found in the temporal behavior of this lossless model problem are ''collapse,'' ie, episodic nonexponential damping of both the atomic inversion and dipole moment, and two kinds of ''revival'' or partial recorrelation, in the dynamic evolution, during which the initial state is nearly recovered We give analytic formulas for the collapse function, for both of the revival times, and for the envelope of the revival maxima Some remarks are made about the nature of irreversibility in this exactly soluble and loss-free model

Coherence versus incoherence: Collapse and revival in a simple quantum model

Theory of Spontaneous-Emission Line Shape in an Ideal Cavity

The analysis performed shows that in cubic photorefractive crystals (of Bi12SiO20 and GaAs type in particular) with drift (‘‘quasilocal’’) mechanism of optical nonlinearity spatial soliton propagation can be observed for reasonable external dc electric fields 10–15 kV/cm. These photorefractive solitons which can be excited at very low cw laser power are stable in a broad region of signal(soliton)/incoherent(spatially uniform) intensity ratio, and allow easy switching from bright to dark type. Original experimental results of self‐channeling of submicrowatt HeNe laser beam in a cubic Bi12TiO20 sample under external dc field are presented.

Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity

A novel (to the best of our knowledge) liquid level sensor based on multimode interference (MMI) effects is proposed and demonstrated. By using a multimode fiber (MMF) without cladding, known as no-core fiber, liquids around the MMF modify the self-imaging properties of the MMI device and the liquid level can be detected. We show that the sensor exhibits a highly linear response with the sensing range and multiplexed operations easily controlled by just modifying the length of the no-core fiber. At the same time, we can measure the refractive index of the liquid based on the maximum peak wavelength shift. We can also use the sensor for continuous and discrete liquid level sensing applications, thus providing a liquid level sensor that is inexpensive with a very simple fabrication process.

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Jose Javier Sanchez-Mondragon

Papers

Periodic spontaneous collapse and revival in a simple quantum model

Coherence versus incoherence: Collapse and revival in a simple quantum model

Theory of Spontaneous-Emission Line Shape in an Ideal Cavity

Spatial solitons in photorefractive Bi12TiO20 with drift mechanism of nonlinearity

Fiber-optic sensor for liquid level measurement.