Radiative damping and synchronization in a graphene-based terahertz emitter

We investigate the collective electron dynamics in a recently proposed graphene-based terahertz emitter under the influence of the radiative damping effect, which is included self-consistently in a molecular dynamics approach. We show that under appropriate conditions synchronization of the dynamics of single electrons takes place, leading to a rise of the oscillating component of the charge current. The synchronization time depends dramatically on the applied dc electric field and electron scattering rate, and is roughly inversely proportional to the radiative damping rate that is determined by the carrier concentration and the geometrical parameters of the device. The emission spectra in the synchronized state, determined by the oscillating current component, are analyzed. The effective generation of higher harmonics for large values of the radiative damping strength is demonstrated.Comment: 9 pages, 7 figure

Transition to complete synchronization in phase coupled oscillators with nearest neighbours coupling

We investigate synchronization in a Kuramoto-like model with nearest neighbour coupling. Upon analyzing the behaviour of individual oscillators at the onset of complete synchronization, we show that the time interval between bursts in the time dependence of the frequencies of the oscillators exhibits universal scaling and blows up at the critical coupling strength. We also bring out a key mechanism that leads to phase locking. Finally, we deduce forms for the phases and frequencies at the onset of complete synchronization.Comment: 6 pages, 4 figures, to appear in CHAO

Exploring constrained quantum control landscapes

The broad success of optimally controlling quantum systems with external fields has been attributed to the favorable topology of the underlying control landscape, where the landscape is the physical observable as a function of the controls. The control landscape can be shown to contain no suboptimal trapping extrema upon satisfaction of reasonable physical assumptions, but this topological analysis does not hold when significant constraints are placed on the control resources. This work employs simulations to explore the topology and features of the control landscape for pure-state population transfer with a constrained class of control fields. The fields are parameterized in terms of a set of uniformly spaced spectral frequencies, with the associated phases acting as the controls. Optimization results reveal that the minimum number of phase controls necessary to assure a high yield in the target state has a special dependence on the number of accessible energy levels in the quantum system, revealed from an analysis of the first- and second-order variation of the yield with respect to the controls. When an insufficient number of controls and/or a weak control fluence are employed, trapping extrema and saddle points are observed on the landscape. When the control resources are sufficiently flexible, solutions producing the globally maximal yield are found to form connected `level sets' of continuously variable control fields that preserve the yield. These optimal yield level sets are found to shrink to isolated points on the top of the landscape as the control field fluence is decreased, and further reduction of the fluence turns these points into suboptimal trapping extrema on the landscape. Although constrained control fields can come in many forms beyond the cases explored here, the behavior found in this paper is illustrative of the impacts that constraints can introduce.Comment: 10 figure

Gauge Theory for Finite-Dimensional Dynamical Systems

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This theory has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems with implications to numerical integration of differential equations. We distinguish between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry is, in essence, re-scaling of the independent variable, while descriptive gauge symmetry is a Yang-Mills-like transformation of the velocity vector field, adapted to finite-dimensional systems. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a remarkable connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse engineering and scientific fields, including quantum mechanics, chemistry, rigid-body dynamics and information theory

Many-Body Theory of Synchronization by Long-Range Interactions

Synchronization of coupled oscillators on a $d$-dimensional lattice with the power-law coupling $G(r) = g_0/r^\alpha$ and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of $\epsilon = \alpha/d-1$. For $\alpha \le d$, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For $\alpha > d$, the transition is smeared by the quenched disorder, and the macroscopic order parameter \Av\psi decays slowly with $g_0$ as |\Av\psi| \propto g_0^2.Comment: 4 pages, 2 figure

Scaling and singularities in the entrainment of globally-coupled oscillators

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling. The population is described by a Fokker-Planck equation for the distribution of phases which includes the diffusive effect of noise in the oscillator frequencies. The bifurcation from the phase-incoherent state is analyzed using amplitude equations for the unstable modes with particular attention to the dependence of the nonlinearly saturated mode $|\alpha_\infty|$ on the linear growth rate $\gamma$. In general we find $|\alpha_\infty|\sim \sqrt{\gamma(\gamma+l^2D)}$ where $D$ is the diffusion coefficient and $l$ is the mode number of the unstable mode. The unusual $(\gamma+l^2D)$ factor arises from a singularity in the cubic term of the amplitude equation.Comment: 11 pages (Revtex); paper submitted to Phys. Rev. Let

Exact Phase Solutions of Nonlinear Oscillators on Two-dimensional Lattice

We present various exact solutions of a discrete complex Ginzburg-Landau (CGL) equation on a plane lattice, which describe target patterns and spiral patterns and derive their stability criteria. We also obtain similar solutions to a system of van der Pol's oscillators.Comment: Latex 11 pages, 17 eps file

The dynamics of correlated novelties

One new thing often leads to another. Such correlated novelties are a familiar part of daily life. They are also thought to be fundamental to the evolution of biological systems, human society, and technology. By opening new possibilities, one novelty can pave the way for others in a process that Kauffman has called "expanding the adjacent possible". The dynamics of correlated novelties, however, have yet to be quantified empirically or modeled mathematically. Here we propose a simple mathematical model that mimics the process of exploring a physical, biological or conceptual space that enlarges whenever a novelty occurs. The model, a generalization of Polya's urn, predicts statistical laws for the rate at which novelties happen (analogous to Heaps' law) and for the probability distribution on the space explored (analogous to Zipf's law), as well as signatures of the hypothesized process by which one novelty sets the stage for another. We test these predictions on four data sets of human activity: the edit events of Wikipedia pages, the emergence of tags in annotation systems, the sequence of words in texts, and listening to new songs in online music catalogues. By quantifying the dynamics of correlated novelties, our results provide a starting point for a deeper understanding of the ever-expanding adjacent possible and its role in biological, linguistic, cultural, and technological evolution

Collective synchronization in populations of globally coupled phase oscillators with drifting frequencies

We generalize the Kuramoto model for coupled phase oscillators by allowing the frequencies to drift in time according to Ornstein-Uhlenbeck dynamics. Such drifting frequencies were recently measured in cellular populations of circadian oscillator and inspired our work. Linear stability analysis of the Fokker-Planck equation for an infinite population is amenable to exact solution and we show that the incoherent state is unstable passed a critical coupling strength K_c(\ga, \sigf), where \ga is the inverse characteristic drifting time and \sigf the asymptotic frequency dispersion. Expectedly $K_c$ agrees with the noisy Kuramoto model in the large \ga (Schmolukowski) limit but increases slower as \ga decreases. Asymptotic expansion of the solution for \ga\to 0 shows that the noiseless Kuramoto model with Gaussian frequency distribution is recovered in that limit. Thus varying a single parameter allows to interpolate smoothly between two regimes: one dominated by the frequency dispersion and the other by phase diffusion.Comment: 5 pages, 5 figures, accepted in Phys. Rev.