Time Correlation and Decoherence in a Two-Particle Interferometer

A two-particle interferometer is theoretically analyzed to show how decoherence induced by interactions with the environment affects time correlations, a process we call time correlation decoherence. Specifically, on the basis of simple mathematical analysis, we show how the interaction between a bipartite entangled system and a photon bath representing the environment can efface the oscillations in the coincidence detection rate of the interferometer. We discuss the dependence of this kind of decoherence on the photon energy and density.     

Homopolar oscillating-disc dynamo driven by parametric resonance

We use a simple model of Bullard-type disc dynamo, in which the disc rotation rate is subject to harmonic oscillations, to analyze the generation of magnetic field by the parametric resonance mechanism. The problem is governed by a damped Mathieu equation. The Floquet exponents, which define the magnetic field growth rates, are calculated depending on the amplitude and frequency of the oscillations. Firstly, we show that the dynamo can be excited at significantly subcritical disc rotation rate when the latter is subject to harmonic oscillations with a certain frequency. Secondly, at supercritical mean rotation rates, the dynamo can also be suppressed but only in narrow frequency bands and at sufficiently large oscillation amplitudes.     

Transient spike adding in the presence of equilibria

Many models of neuronal activity exhibit complex oscillations in response to an input from other neurons in a network or to an input from a stimulus. We consider the effect of a single short stimulus on a simple model designed to mimic some features of neuronal dynamics. We focus on the transient response induced by the stimulus, particularly on the spike-adding behaviour of the response. Our main goal is to explain how the transient response is affected by the presence of unstable equilibria. We also investigate the dependence of the number of spikes on the amplitude and duration of the stimulus. In our analysis, we use numerical continuation methods and exploit the presence of different time scales in the model.     

Designer gene networks: Towards fundamental cellular control

The engineered control of cellular function through the design of synthetic genetic networks is becoming plausible. Here we show how a naturally occurring network can be used as a parts list for artificial network design, and how model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics. We first review the relevant work on synthetic gene networks, highlighting the important experimental findings with regard to genetic switches and oscillators. We then present the derivation of a deterministic model describing the temporal evolution of the concentration of protein in a single-gene network. Bistability in the steady-state protein concentration arises naturally as a consequence of autoregulatory feedback, and we focus on the hysteretic properties of the protein concentration as a function of the degradation rate. We then formulate the effect of an external noise source which interacts with the protein degradation rate. We demonstrate the utility of such a formulation by constructing a protein switch, whereby external noise pulses are used to switch the protein concentration between two values. Following the lead of earlier work, we show how the addition of a second network component can be used to construct a relaxation oscillator, whereby the system is driven around the hysteresis loop. We highlight the frequency dependence on the tunable parameter values, and discuss design plausibility. We emphasize how the model equations can be used to develop design criteria for robust oscillations, and illustrate this point with parameter plots illuminating the oscillatory regions for given parameter values. We then turn to the utilization of an intrinsic cellular process as a means of controlling the oscillations. We consider a network design which exhibits self-sustained oscillations, and discuss the driving of the oscillator in the context of synchronization. Then, as a second design, we consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In this case, we show how resonance can lead to the induction of oscillations and amplification of a cellular signal. Finally, we construct a toggle switch from positive regulatory elements, and compare the switching properties for this network with those of a network constructed using negative regulation. Our results demonstrate the utility of model analysis in the construction of synthetic gene regulatory networks. (c) 2001 American Institute of Physics.     

How effective delays shape oscillatory dynamics in neuronal networks

Synaptic, dendritic and single-cell kinetics generate significant time delays that shape the dynamics of large networks of spiking neurons. Previous work has shown that such effective delays can be taken into account with a rate model through the addition of an explicit, fixed delay (Roxin et al. (2005,2006) [29] and [30]). Here we extend this work to account for arbitrary symmetric patterns of synaptic connectivity and generic nonlinear transfer functions. Specifically, we conduct a weakly nonlinear analysis of the dynamical states arising via primary instabilities of the asynchronous state. In this way we determine analytically how the nature and stability of these states depend on the choice of transfer function and connectivity. We arrive at two general observations of physiological relevance that could not be explained in previous work. These are: 1 — fast oscillations are always supercritical for realistic transfer functions and 2 — traveling waves are preferred over standing waves given plausible patterns of local connectivity. We finally demonstrate that these results show good agreement with those obtained performing numerical simulations of a network of Hodgkin-Huxley neurons.     

Nonlinear dynamics in the CO-oxidation on Pt single crystal surfaces

The types of dynamical behaviour that can be found in nonlinear systems far from equilibrium are briefly described. Then experimental manifestations of these phenomena in the CO oxidation on Pt are presented which include bistability, oscillations and chaos as well as propagating and stationary spatial patterns. It is demonstrated how a simple model can be refined to account for the observed complex behaviour.     

The influence of the chemical potential oscillations on the de Haas-van Alphen effect in quasi-two-dimensional compounds

The de Haas-van Alphen effect in quasi-two-dimensional metals is studied at arbitrary parameters. Oscillations of the chemical potential can substantially change the temperature dependence of harmonic amplitudes that is commonly used to determine the effective electron mass. The processing of the experimental data using the standard Lifshitz-Kosevich formula can therefore lead to substantial errors even in the strong harmonic damping limit. This can explain the difference between the effective electron masses determined from the de Haas-van Alphen effect and the cyclotron resonance measurements. The oscillations of the chemical potential and the deviations from the Lifshitz-Kosevich formula depend on the reservoir density of states that exists in organic metals due to open sheets of the Fermi surface. This dependence can be used to determine the density of electron states on open sheets of the Fermi surface. We present analytical results of the calculations of harmonic amplitudes in some limiting cases that show the importance of the chemical potential oscillations. We also describe a simple algorithm for a numerical calculation of the harmonic amplitudes for arbitrary reservoir density of states, arbitrary warping, spin-splitting, temperature, and Dingle temperature.     

Synthetic gene network for entraining and amplifying cellular oscillations

We present a model for a synthetic gene oscillator and consider the coupling of the oscillator to a periodic process that is intrinsic to the cell. We investigate the synchronization properties of the coupled system, and show how the oscillator can be constructed to yield a significant amplification of cellular oscillations. We reduce the driven oscillator equations to a normal form, and analytically determine the amplification as a function of the strength of the cellular oscillations. The ability to couple naturally occurring genetic oscillations to a synthetically designed network could lead to possible strategies for entraining and/or amplifying oscillations in cellular protein levels.     

Spatial bistability: a source of complex dynamics. From spatiotemporal reaction-diffusion patterns to chemomechanical structures

We show experimentally and theoretically that reaction systems characterized by a slow induction period followed by a fast evolution to equilibrium can readily generate "spatial bistability" when operated in thin gel reactors diffusively fed from one side. This phenomenon which corresponds to the coexistence of two different stable steady states, not breaking the symmetry of the boundary conditions, can be at the origin of diverse reaction-diffusion instabilities. Using different chemical reactions, we show how stationary pulses, labyrinthine patterns or spatiotemporal oscillations can be generated. Beyond simple reaction-diffusion instabilities, we also demonstrate that the cross coupling of spatial bistability with the size responsiveness of a chemosensitive gel can give rise to autonomous spatiotemporal shape patterns, referred to as chemomechanical structures.     

Periodic mode hopping induced by thermo-optic effects in continuous-wave optical parametric oscillators

We show that thermal effects can lead to periodic mode hopping in cw optical parametric oscillators (OPOs). This mode hopping may occur as soon as two modes have different intensities at the point where they exchange their stability; this condition is easily fulfilled in OPOs that are triply resonant, or doubly resonant with a weakly resonant pump. We have observed such oscillations experimentally in a type II OPO in both configurations. A simple thermo-optic multimode model reproduces well the experimental regimes. We expect that multimode instabilities based on this mechanism can be observed with various aspects in many experimental setups at high pumping rate.     

Experiments with twisted light

The generic – that is, stable under perturbations – nodes of the field in a monochromatic light beam are optical vortices. We describe here their connection to Chladni's nodal lines in the oscillations of metal plates, as well as a few experiments that have been performed with optical vortices. We will describe how optical vortices can be generated experimentally; how it can be shown that they possess orbital angular momentum; how individual photons can be sorted according to their vortex state; and how optical vortices can be used to demonstrate higher-dimensional quantum entanglement.     

Amplified Biochemical Oscillations in Cellular Systems

We describe a mechanism for pronounced biochemical oscillations, relevant to microscopic systems, such as the intracellular environment. This mechanism operates for reaction schemes which, when modeled using deterministic rate equations, fail to exhibit oscillations for any values of rate constants. The mechanism relies on amplification of the underlying stochasticity of reaction kinetics within a narrow window of frequencies. This amplification means that fluctuations have a dominant effect, even though the number of molecules in the system is relatively large. The mechanism is quantitatively studied within simple models of self-regulatory gene expression, and glycolytic oscillations.     

Two state systems in media and “Turing's paradox”

We discuss the coherence properties of two state systems subjected to random influences in a medium. We find an expression for the “damping parameter” characterizing the loss of coherence as a kind of generalization of the optical theorem. We verify previous speculations as to how an asymmetry like optical isomerism ca be stabilized by a neutral environment and suggest an experimental test of the resulting “paradoxical” behaviour where heating will slow down the relaxation rate. This is related to a certain “paradox” of quantum measurement theory and can be verified experimentally by studying the relaxation rate for the optical activity of small molecules at low temperature. As another application of the method, neutrino oscillations in dense media are considered.     

Bragg-resonance-induced Rabi oscillations in photonic lattices

We show that propagation of optical beams in periodic lattices induces power oscillations between the Fourier spectrum peaks, with the indices related by the Bragg resonance condition. In the spatial coordinates, this is reflected in the beam position oscillations. A simple resonant theory explains the phenomenon. The effect can be used for controlled generation of the Floquet-Bloch modes in photonic lattices.     

Understanding anomalous delays in a model of intracellular calcium dynamics

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP(3)) is applied to the cell. However, more detailed mathematical investigations [M. Domijan et al., "Dynamical probing of the mechanisms underlying calcium oscillations," J. Nonlinear Sci. 16, 483-506 (2006)] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of models that make different assumptions about the mechanisms underlying the calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems with three or more slow variables [M. Wechselberger, "A propos de canards (Apropos canards)," Preprint, 2010] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP(3).     

Stückelberg-interferometry with ultra-cold atoms

Stückelberg oscillations have been observed in recent experiments with ultra-cold atoms. We show that and how precisely ultra-cold atoms in an accelerated two-band lattice are a controlled realization of Landau-Zener-Stückelberg interferometry. We derive simple formulae for the time-dependent and time-averaged occupation of the energy bands and compute interference patterns for parameters directly accessible in experiments with ultra-cold atoms.     

Quantum size effects in Friedel oscillations inside films

An analytical formula of density oscillations is found for metallic films of finite thickness. The result shows that the quantum size effect on density oscillations is surprisingly more evident in the middle of the film. As a result, the density oscillations in a finite film cannot be regarded as a simple addition of the two sets of Friedel oscillations for half-infinite metal no matter how thick the film is. This analytical result is confirmed by our numerical jellium-model computation. Such quantum size effect should exist in all the electron-mediated interactions that are driven by the Friedel oscillations. As an example, we indeed find it also exists in Ruderman–Kittel–Kasuya–Yosida interactions inside films.     

From simple to complex oscillatory behavior in metabolic and genetic control networks

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.     

Gauge covariant linear response analysis of QCD plasma oscillations

We show, using linear response theory, how plasma oscillations and screening in quark-gluon plasma can be computed in perturbation theory in a gauge covariant manner. Using this method we calculate the damping constant of color electric plasma oscillations in a Colomb gauge and show that the result agrees with an earlier calculation in the A₀ = 0 gauge.     

A note on diffraction phenomena in the inelastic scattering of complex nuclei

We illustrate how oscillations in angular distributions for inelastic scattering of heavy ions can be produced by diffraction effects arising from the variation of the excitation amplitudes.