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# Spiral wave chimera states in large populations of coupled chemical oscillators

Jan Frederik Totz, Julian Rode, Mark R. Tinsley, Kenneth Showalter & Harald Engel

Our studies suggest that the spiral wave chimeras, core expansion and core splitting observed in the BZ system are likely to be found in a range of other systems with the common properties of immediate firing following a perturbation and long-range interactions. For example, we have found similar spiral wave chimera behaviour, with core splitting and the transition to predominantly asynchronous behaviour, in populations of nonlocally coupled FitzHugh–Nagumo oscillators. The PRC for the FitzHugh–Nagumo system resembles the PRC of the BZ system and ZBKE model, with an immediate firing region. Pulse coupled oscillator models of neuronal systems can also have immediate firing dynamics, suggesting that certain neuronal networks might exhibit spiral wave chimera behaviour similar to that described here. Other possible systems where these behaviours might be found include biological tissues and arrays of physical oscillators.

Reference: Nature Physics (2017)

## Bound Pulse Trains in Arrays of Coupled Spatially Extended Dynamical Systems

Dmitry Puzyrev, Andrei G. Vladimirov, Alexander Pimenov, Svetlana V. Gurevich & Serhiy Yanchuk

We study the dynamics of an array of nearest-neighbor coupled spatially distributed systems each generating a periodic sequence of short pulses. We demonstrate that, unlike a solitary system generating a train of equidistant pulses, an array of such systems can produce a sequence of clusters of closely packed pulses, with the distance between individual pulses depending on the coupling phase. This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in good agreement with the results of direct numerical simulations of a model system describing an array of coupled mode-locked lasers.

Reference: Phys. Rev. Lett. 119, 163901 (2017)

## Path-Controlled Time Reordering of Paired Photons in a Dressed Three-Level Cascade

Samir Bounouar, Max Strauß, Alexander Carmele, Peter Schnauber, Alexander Thoma, Manuel Gschrey, Jan-Hindrik Schulze, André Strittmatter, Sven Rodt, Andreas Knorr & Stephan Reitzenstein

The two-photon dressing of a “three-level ladder” system, here the ground state, the exciton, and the biexciton of a semiconductor quantum dot, leads to new eigenstates and allows one to manipulate the time ordering of the paired photons without unitary postprocessing. We show that, after spectral postselection of the single dressed states, the time ordering of the cascaded photons can be removed or conserved. Our joint experimental and theoretical study demonstrates the high potential of a “ladder” system to be a versatile source of orthogonally polarized, bunched or antibunched pairs of photons.

Reference: Phys. Rev. Lett. 233601 (2017)

## Coherence-Resonance Chimeras in a Network of Excitable Elements

Nadezhda Semenova, Anna Zakharova, Vadim Anishchenko & Eckehard Schöll

We demonstrate that chimera behavior can be observed in nonlocally coupled networks of excitable systems in the presence of noise. This phenomenon is distinct from classical chimeras, which occur in deterministic oscillatory systems, and it combines temporal features of coherence resonance, i.e., the constructive role of noise, and spatial properties of chimera states, i.e., the coexistence of spatially coherent and incoherent domains in a network of identical elements. Coherence-resonance chimeras are associated with alternating switching of the location of coherent and incoherent domains, which might be relevant in neuronal networks.

Reference: Phys. Rev. Lett. 117, 014102 (2016)

## Strong suppression of shot noise in a feedback-controlled single-electron transistor

Timo Wagner, Philipp Strasberg, Johannes C. Bayer, Eddy P. Rugeramigabo, Tobias Brandes & Rolf J. Haug

Feedback control of quantum mechanical systems is rapidly attracting attention not only due to fundamental questions about quantum measurements but also because of its novel applications in many fields in physics. Quantum control has been studied intensively in quantum optics but progress has recently been made in the control of solid-state qubits as well. In quantum transport only a few active band passive feedback experiments have been realized on the level of single electrons, although theoretical proposals exist. Here we demonstrate the suppression of shot noise in a single-electron transistor using an exclusively electronic closed-loop feedback to monitor and adjust the counting statistics. With increasing feedback response we observe a stronger suppression and faster freezing of charge current fluctuations. Our technique is analogous to the generation of squeezed light with in-loop photodetection as used in quantum optics. Sub-Poisson single-electron sources will pave the way for high-precision measurements in quantum transport similar to optical or optomechanical equivalents.

Reference: Nat. Nanotechnol. Lett. (2016)

The book summarizes the state-of-the-art of research on control of self-organizing nonlinear systems with contributions from leading international experts in the field. The first focus concerns recent methodological developments including control of networks and of noisy and time-delayed systems. As a second focus, the book features emerging concepts of application including control of quantum systems, soft condensed matter, and biological systems. Special topics reflecting the active research in the field are the analysis and control of chimera states in classical networks and in quantum systems, the mathematical treatment of multiscale systems, the control of colloidal and quantum transport, the control of epidemics and of neural network dynamics.

## Dynamics, control and information in delay-coupled systems

The International Conference on Delayed Complex Systems held from 4 to 8 June 2012 at the Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) in Palma de Mallorca, Spain, and co-organized with the Collaborative Research Center SFB 910 Control of Self-Organizing Nonlinear Systems—Theoretical Methods and Concepts of Application, Berlin, provided a forum for such topics. We took that opportunity to assemble a list of world leading experts, which now enables us to present perspectives of the state of the art in this field. This Theme Issue covers both applications and experiments, as well as mathematical foundations. The individual contributions summarize recent research results, but also address the broader context. Thus, the presentation is kept accessible for a large audience. The 14 articles cover various aspects of delay dynamics, control and information, ranging from fundamental mathematical aspects via delayed networks and time-delayed feedback control, to applications in neural science, optoelectronics and genetic control in cells. The articles are grouped into four parts.

http://rsta.royalsocietypublishing.org/content/371/1999.toc

## Tweezers for Chimeras in Small Networks

Iryna Omelchenko, Oleh E. Omel'chenko, Anna Zakharova, Matthias Wolfrum, and Eckehard Schöll

We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generally difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.

Reference: Phys. Rev. Lett. 116, 114101 (2016)

## Pattern Formation in Systems with Multiple Delayed Feedbacks

Serhiy Yanchuk and Giovanni Giacomelli

Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities.

In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behavior in the case of two delays is shown to “encode” two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the attainment of the longdelay regime is introduced. We also demonstrate this phenomenon for a semiconductor laser with two delayed optical feedbacks.

Reference: Phys. Rev. Lett. 112, 174103 (2014)

## Chimera Death: Symmetry Breaking in Dynamical Networks

Anna Zakharova, Marie Kapeller, and Eckehard Schöll

For a network of generic oscillators with nonlocal topology and symmetry breaking coupling we establish novel partially coherent inhomogeneous spatial patterns, which combine the features of chimera states (coexisting incongruous coherent and incoherent domains) and oscillation death (oscillation suppression), which we call “chimera death”. We show that due to the interplay of nonlocality and breaking of rotational symmetry by the coupling, two distinct scenarios from oscillatory behavior to a stationary state regime are possible: a transition from an amplitude chimera to chimera death via in-phase synchronized oscillations and a direct abrupt transition for larger coupling strength.

Reference: Phys. Rev. Lett. 112, 154101 (2014)

## Quantum Criticality and Dynamical Instability in the Kicked-Top Model

Victor Manuel Bastidas, Pedro Pérez-Fernández, Malte Vogl, and Tobias Brandes

We investigate precursors of critical behavior in the quasienergy spectrum due to the dynamical instability in the kicked top. Using a semiclassical approach, we analytically obtain a logarithmic divergence in the density of states, which is analogous to a continuous excited state quantum phase transition in undriven systems. We propose a protocol to observe the cusp behavior of the magnetization close to the critical quasienergy.

Reference: Phys. Rev. Lett. 112, 140408 (2014)

## Controlling the Position of Traveling Waves in Reaction-Diffusion Systems

Jakob Löber and Harald Engel

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a prespecified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space (x) and time (t) dependent control function f(x,t) that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.

Reference: Phys. Rev. Lett. 112, 148305 (2014)

## When Nonlocal Coupling between Oscillators Becomes Stronger: Patched Synchrony or Multichimera States

Iryna Omelchenko, Oleh E. Omel’chenko, Philipp Hövel, and Eckehard Schöll

Systems of nonlocally coupled oscillators can exhibit complex spatiotemporal patterns, called chimera states, which consist of coexisting domains of spatially coherent (synchronized) and incoherent dynamics. We report on a novel form of these states, found in a widely used model of a limit-cycle oscillator if one goes beyond the limit of weak coupling typical for phase oscillators. Then patches of synchronized dynamics appear within the incoherent domain giving rise to a multi-chimera state. We find that, depending on the coupling strength and range, different multichimera states arise in a transition from classical chimera states. The additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models.

Reference: Phys. Rev. Lett. 110, 224101 (2013)

## Control of Synchronization Patterns in Neural-like Boolean Networks

David P. Rosin, Damien Rontani, Daniel J. Gauthier, and Eckehard Schöll

We study experimentally the synchronization patterns in time-delayed directed Boolean networks of excitable systems. We observe a transition in the network dynamics when the refractory time of the individual systems is adjusted. When the refractory time is on the same order of magnitude as the mean link time delays or the heterogeneities of the link time delays, cluster synchronization patterns change, or are suppressed entirely, respectively. We also show that these transitions occur when we change the properties of only a small number of driver nodes identified by their larger in degree; hence, the synchronization patterns can be controlled locally by these nodes. Our findings have implications for synchronization in biological neural networks.

Reference: Phys. Rev. Lett. 110, 104102 (2013)

## Reentry Near the Percolation Threshold in a Heterogeneous Discrete Model for Cardiac Tissue

Sergio Alonso, and Markus Bär

Arrhythmias in cardiac tissue are related to irregular electrical wave propagation in the heart. Cardiac tissue is formed by a discrete cell network, which is often heterogeneous. A localized region with a fraction of nonconducting links surrounded by homogeneous conducting tissue can become a source of reentry and ectopic beats. Extensive simulations in a discrete model of cardiac tissue show that a wave crossing a heterogeneous region of cardiac tissue can disintegrate into irregular patterns, provided the fraction of nonconducting links is close to the percolation threshold of the cell network. The dependence of the reentry probability on this fraction, the system size, and the degree of excitability can be inferred from the size distribution of nonconducting clusters near the percolation threshold.

Reference: Phys. Rev. Lett. 110, 158101 (2013)

## Thermodynamics of a physical model implementing a Maxwell demon

Philipp Strasberg, Gernot Schaller, Tobias Brandes, and Massimiliano Esposito

We present a physical implementation of a Maxwell demon which consists of a conventional single electron transistor (SET) capacitively coupled to another quantum dot detecting its state. Altogether, the system is described by stochastic thermodynamics. We identify the regime where the energetics of the SET is not affected by the detection, but where its coarse-grained entropy production is shown to contain a new contribution compared to the isolated SET. This additional contribution can be identified as the information flow generated by the “Maxwell demon” feedback in an idealized limit.

Reference: Phys. Rev. Lett. 110, 040601 (2013)

Highlighted in Phys.org (2013)

## Single Photon Delayed Feedback: A Way to Stabilize Intrinsic Quantum Cavity Electrodynamics

Alexander Carmele, Julia Kabuss, Franz Schulze, Stephan Reitzenstein, and Andreas Knorr

Extrinsic and intrinsic control of non-classical photon states is of great importance in quantum information science. We investigate theoretically a single-emitter cavity system which operates initially in the weak-coupling limit. By applying an intrinsic control scheme, in particular quantum optical time-delayed self-feedback in the single-photon limit, we observe how this system is driven into the strong-coupling regime. This peculiar transition manifests in Rabi oscillations, observable in the coupled cavity field dynamics. This quantum optical apporach to time-delayed self-feedback opens new ways to experimentially controll features of cavity quantum electrodynamics in the single-photon limit.

Reference: Phys. Rev. Lett. 110, 013601 (2013)

## Synchronized tumbling particles

Sabine H. L. Klapp

Magnetic particles have been made that undergo synchronized oscillations when suspended in liquid in a rotating magnetic field. This discovery links the fields of nonlinear dynamics and materials science.

Reference: Nature 491, 530-531 (2012)

## Experimental Observation of Chimeras in Coupled-Map Lattices

Aaron M. Hagerstrom, Thomas E. Murphy, Rajarshi Roy, Philipp Hövel, Iryna Omelchenko and Eckehard Schöll

Networks of nonlocally coupled phase oscillators can support chimera states in which identical oscillators evolve into distinct groups that exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Similar nonlocal coupling topologies implemented in networks of chaotic iterated maps also yield dynamical states displaying coexisting spatial domains of coherence and incoherence. In these discrete-time systems, the phase is not a continuous variable, so these states are generalized chimeras with respect to a broader notion of incoherence. Chimeras continue to be the subject of intense theoretical investigation, but have yet to be realized experimentally. Here we show that these chimeras can be realized in experiments using a liquid crystal spatial light modulator to achieve optical nonlinearity in a spatially extended iterated map system. We study the coherence-incoherence transition that gives rise to these chimera states through experiment, theory, and simulation.

Reference: Nature Physics 8, 658 (2012)

Highlighted in Physics Today (2012)

Also highlighted in Physics Today, Search and Discovery (2012)

## Loss of Coherence in Dynamical Networks: Spatial Chaos and Chimera States

Iryna Omelchenko, Yuri Maistrenko, Philipp Hövel, and Eckehard Schöll

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatiotemporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous Rossler systems reveal that intermediate, partially coherent states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence.

Reference: Phys. Rev. Lett. 106, 234102 (2011)

## Pattern-generation in delay-coupled neuronal systems

Serhiy Yanchuk and Markus Kantner

We show that an arbitrary periodic 2-dimensional spiking-patterns can be generated in a 2-dimensional lattice of unidirectionally delay-coupled neurons with appropriately tuned time-delayed couplings. The video shows an example: by appropriately choosing coupling delays, voltages in the array of 14x13 Hodgkin-Huxley neurons approach a stable attractor in the form of "SFB910" logo.

## Synchronization and Complex Dynamics of Oscillators with Delayed Pulse Coupling

Markus Bär, Eckehard Schöll, Alessandro Torcini

A systematic experimental study of pulse-coupled chemical oscillators with delay has confirmed a surprisingly large number of theoretical and mathematical predictions (see the dynamics for a pair of pulse-coupled oscillators; AP=antiphase (AP) and IP=in-phase oscillations, C=complex bursting dynamics, and OS=oscillator suppression). These results have implications for neuroscience and other biological fields.

## Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons

Josef Ladenbauer, Moritz Augustin, LieJune Shiau, Klaus Obermayer

Synchronization of neuronal spiking in the brain is related to cognitive functions, such as perception, attention, and memory. It is therefore important to determine which properties of neurons influence their collective behavior in a network and to understand how. A prominent feature of many types of neurons is spike frequency adaptation, shown by a decrease in spike rate during prolonged stimulation. This behavior is typically mediated by slow transmembrane potassium currents that can be controlled by the brains neuromodulatory systems. We investigated how these adaptation currents affect the synchronization tendency of synaptically coupled model neurons by applying phase reduction based on phase response curves (PRC). Therefore, we extended the adjoint method for calculating PRCs to dynamical systems with discontinuities. Using the experimentally verified and computationally efficient adaptive exponential integrate-and-fire model as well as a biophysically detailed neuron model for validation, we analyzed synchrony and phase locking of pairs and larger networks. We found that increased adaptation currents promote synchronization of coupled excitatory neurons at lower spike frequencies, as long as the conduction delays between the neurons are negligible. Inhibitory neurons on the other hand synchronize in presence of conduction delays, with or without adaptation currents. We conclude that in local populations of excitatory neurons adaptation currents provide a mechanism to generate low frequency oscillations that could be controlled through neuromodulation, while faster rhythms seem to be caused by inhibition rather than excitation. The study provides a first step towards understanding potential mechanisms underlying the top-down control of cortical states.

Reference: PLoS Comput Biol 8(4): e1002478. doi:10.1371/journal.pcbi.1002478

## Charge Qubit Purification by an Electronic Feedback Loop

We propose the manipulation of an isolated qubit by a simple instantaneous closed-loop feedback scheme in which a time-dependent electronic detector current is directly back-coupled into qubit parameters. As specific detector model we employ a capacitively coupled single-electron transistor. We demonstrate the stabilization of pure delocalized qubit states above a critical detector-qubit coupling. This electronic purification is independent of the initial qubit state and is accomplished after few electron jumps through the detector. Our simple scheme can be used for the efficient and robust initialization of solid-state qubits in quantum computational algorithms at arbitrary temperatures.

Gerold Kiesslich, Gernot Schaller, Clive Emary, and Tobias Brandes

Physical Review Letters **107**, 050501 (2011)

arXiv:1102.3771