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The activities of our group are centered around the interplay of dynamics, topology, time delay, and noise in nonlinear dynamical systems and complex networks. A focus is put on the control of different space-time-patterns as well as on the influence of nonlinear stochastic processes. In particular, chimera states in networks, i.e. symmetry breaking partly coherent, partly incoherent synchronization patterns are studied. Possible applications in physics, chemistry, biology, and engineering are for example neuronal networks, communication networks and power grids as well as socio-economic systems. Our methods range from analysing generic models to numerical bifurcation analysis and simulations of complex networks.


Several of the current projects are or were embedded in the following large collaborative research projects of Deutsche Forschungsgemeinschaft DFG:


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In particular, the Schöll group deals with the following topics:

Controlling complex networks: interplay of structure, noise, and delay

Nonlinear dynamics on complex networks is a thriving interdisciplinary area of research which has recently attracted much interest. The interplay of time delay with network topology, nonlinearity, and noise leads to a plethora of complex phenomena with applications to physics, chemistry, biology, engineering, and even socio-economic systems. Time delay and stochasticity arise naturally in various systems and can be exploited for control purposes. Our objective is to establish the interconnections between the structure of complex networks, noise impact, and delay configurations in order to develop efficient control mechanisms of nonlinear dynamical systems. For this purpose we study noise-induced effects as well as complex deterministic symmetry-breaking phenomena and synchronization patterns. We model the local dynamics with simple paradigmatic models for oscillatory or excitable systems. Among the examples studied are novel coherence-incoherence patterns like chimera states, amplitude and oscillation death, partial synchronization, mesoscale structures and group synchronization in networks, stochastic bifurcations, coherence resonance and its control by delayed feedback, and applications to laser systems and neuronal dynamics.





Modelling of the dynamics of Quantum Dot Lasers and Amplifiers

The aim of this project ( Project B2 of Sfb 787) are the modelling and numerical simulation of the coupled dynamics of charge carriers and light self-organized quantum dot lasers [ , ] and amplifiers [ ]. Due to the strong variation of temporal and spatial scales, it is impossible to obtain a consistent full quantum theoretical microscopic description. Therefore, one must develop a model hierarchy. This should take into consideration the following: quantum-mechanical effects, stochastic influences, the interaction between the quantum dots and their environment in a diode structure, time-delayed feedback effects through latency times and optical delay times, as well as the nonlinear dynamics on a macroscopic scale. Our main focus is on directly electrically modulated quantum dot lasers, passively mode locked lasers, semiconductor quantum dot optical amplifiers, noise effects and chaotic synchronization of coupled quantum dot lasers.

A dynamical theory for conventional directly-modulated or mode-coupled semiconductor lasers and SOAs in the strongly nonlinear regime under pulsed current injection was already developed a long time ago on the basis of rate equations and travelling-wave approaches [IEEE J. Quantum Electr. 20, 394 (1984), Int. J. Electronics 60, 23 (1986), Appl. Phys. B 46, 69 (1988), IEEE J. Quantum. Electron. 24, 435 (1988), phys. stat. sol. (b) 150, 575 (1988), IEEE J. Quantum. Electron. 26, 1005 (1990)  IEEE J. Quantum. Electron. 27, 402 (1991)]. The nonlinear transverse spatiotemporal dynamics of multi-stripe semiconductor lasers was modelled and the complex bifurcation scenarios were analyzed [Phys. Lett. A 194, 289 (1994), Phys. Rev. A 50, 787 (1994), Physica D 70, 165 (1994), Phys. Rev. E 52, 1571 (1995)].

For the modelling of the dynamics of electrically modulated edge-emitting quantum dot lasers, a microscopic 5-variable model for quantum dot (QD) lasers, based upon the coupled dynamics of photons, and electrons and holes in the QDs and in the surrounding quantum well (QW) acting as a carrier reservoir has been developed [ Phys. Rev. B 78, 035316 (2008), IEEE J. Quantum Electron. 45, 1396 (2009), Eur. Phys. J. D. 58, 167 (2010), IEEE J. Quantum Electron. 46, 1755 (2010)]. One goal is the quantitative modelling of the turn-on dynamics and the modulation response of directly modulated edge-emitting QD lasers. To achieve this goal the microscopic rate equation model has been extended by including heating effects, pump-dependent spectral properties, and Auger recombination losses in the carrier reservoir. The inclusion of separate dynamics of holes and electrons is essential in order to explain the dynamic behavior of a QD laser with a doped carrier reservoir. The dynamics of electrons and holes becomes the more synchronized, the more similar the scattering times (given by the inverse scattering rates) are. Simulations based on the Maxwell-Semiconductor-Bloch equations show strong dependence of the turn-on delay on initial cavity detuning [ Appl. Phys. Lett. 97, 111102 (2010)].

Another goal is the investigation of a quantum dot laser subject to optical feedback. For this purpose we combine a Lang-Kobayashi like field equation with the microscopically based carrier rate equations. By tuning the phase-amplitude coupling and the optical confinement factor we are able to discuss various scenarios of the dynamics, and compare them with conventional quantum well lasers. Due to the optical feedback, multistability occurs in our model in form of external cavity modes or delay-induced intensity pulsations. External cavity modes are the basic solutions of the dynamical equations having constant carrier and photon densities and a phase that varies linearly in time. Thus they correspond to cw operation of the laser with feedback. In dependence of the feedback strength we analyzed complex bifurcation scenarios for the intensity of the emitted laser light as well as time series, power spectra and phase portraits of all dynamic variables in order to elucidate the internal dynamics of the laser [ phys. stat. sol. (b) 247, 829 (2010)]. As a result we could explain the reduced feedback sensitivity found in QD devices on the one hand by their strongly damped relaxation oscillations and on the other hand by the relatively small number of external cavity modes for a given external cavity round trip time. The small number of external cavity modes originates from a weaker phase-amplitude coupling modelled by a smaller linewidth enhancement factor (α-factor) compared to quantum well devices. For the case of fast holes and slow electrons that is important for comparison with experiments we found analytic relations using asymptotic techniques.

Our modelling of quantum dot semiconductor optical amplifiers (QD-SOAs) is aimed at understanding their ultrafast gain recovery dynamics. Due to the very short timescales (fs) found in the gain dynamics it is necessary to use a full nonlinear simulation of the coupled coherent polarization and population dynamics of carriers [ Semicond. Sci. Technol. 26, 014008 (2011), Phys. Rev. B 82, 235301 (2010)]. Thus we use the semiconductor Bloch equations including microscopically calculated carrier-carrier scattering rates between the 2D carrier reservoir and the confined QD states. The ultrashort gain depletion is sensitive against changes of the pulse area and the dephasing time of the microscopic polarization, while the injection current density mainly influences the non-coherent part of the gain recovery dynamics.


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