Temporal Dissipative Solitons in Time-Delay Feedback Systems

Regenerative vectorial breathers in a delay-coupled excitable microlaser with integrated saturable absorber

We report on the polarization dynamics of regenerative light pulses in a micropillar laser with an integrated saturable absorber (SA) coupled to an external feedback mirror. The delayed self-coupled ML is operated in the excitable regime, where it regenerates incident pulses with a supra-threshold intensity-resulting in a pulse train with an inter-pulse period approximately given by the feedback delay time, in analogy with a self-coupled biological neuron. We report the experimental observation of regenerative vectorial breathers (RVBs) in a polarization angle, manifesting themselves as a modulation of the linear polarized intensity components without a significant modulation of the total intensity. A numerical analysis of a suitable model reveals that the observed polarization mode competition is a consequence of symmetry-breaking bifurcations induced by polarization anisotropy. Our model reproduces well the observed experimental results and predicts different regimes as a function of the polarization anisotropy parameters and the pump parameter. We believe that these findings are relevant for the fabrication of flexible sources of polarized pulses, as well as for neuroinspired on-chip computing applications.

Delayed-feedback oscillators replicate the dynamics of multiplex networks: Wavefront propagation and stochastic resonance

The widespread development and use of neural networks have significantly enriched a wide range of computer algorithms and promise higher speed at lower cost. However, the imitation of neural networks by means of modern computing substrates is highly inefficient, whereas physical realization of large scale networks remains challenging. Fortunately, delayed-feedback oscillators, being much easier to realize experimentally, represent promising candidates for the empirical implementation of neural networks and next generation computing architectures. In the current research, we demonstrate that coupled bistable delayed-feedback oscillators emulate a multilayer network, where one single-layer network is connected to another single-layer network through coupling between replica nodes, i.e. the multiplex network. We show that all the aspects of the multiplexing impact on wavefront propagation and stochastic resonance identified in multilayer networks of bistable oscillators are entirely reproduced in the dynamics of time-delay oscillators. In particular, varying the coupling strength allows suppressing and enhancing the effect of stochastic resonance, as well as controlling the speed and direction of both deterministic and stochastic wavefront propagation. All the considered effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence and disclosing thereby the robustness of the observed phenomena.

Normal dispersion Kerr cavity solitons: beyond the mean-field limit

We predict the existence of a novel type of temporal localized structure in injected Kerr-Gires-Tournois interferometers (KGTI). These bright pulses exist in the normal dispersion regime, yet they do not correspond to the usual scenario of domain wall locking that induces complex shape multistability, weak stability, and a reduced domain of existence. The new states are observed beyond the mean-field limit and out of the bistable region. Their shape is uniquely defined, with peak intensities beyond that of the upper steady state, and they are stable over a broad range of the injection field, highlighting their potential for optical frequency comb (OFC) generation.

Temporal dissipative soliton with controllable morphology in a time-delayed coupled optoelectronic oscillator

A temporal dissipative soliton (TDS) with controllable morphology is demonstrated in a time-delayed coupled optoelectronic oscillator (OEO) driven by two optical carriers with different wavelengths. The morphology of the TDS is controlled by the power difference between the two optical carriers and the delay difference induced by the group-velocity dispersion (GVD) in the OEO loop. When the delay difference is small, the OEO operates in a single-soliton state. With the increase of the wavelength interval between the two optical carriers, the delay difference becomes significant so that various compound TDS structures are observed, where the TDS interval is equal to the delay difference. The morphology of the compound TDSs can be switched between a pulsating TDS packet and a stable compound TDS structure by further tuning the power difference between the two optical carriers. This discovery not only facilitates the investigation of novel soliton dynamics but also provides a method for generating customized pulse waveforms.

Square waves and Bykov T-points in a delay algebraic model for the Kerr-Gires-Tournois interferometer

We study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally, we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system.

Decoherence and Turbulence Sources in a Long Laser

We investigate the turn-on process in a laser cavity where the round-trip time is several orders of magnitude greater than the active medium timescales. In this long delay limit, we show that the universal evolution of the photon statistics from thermal to Poissonian distribution involves the emergence of power dropouts. While the largest number of these dropouts vanish after a few round-trips, some of them persist and seed coherent structures similar to dark solitons or Nozaki-Bekki holes described by the complex Ginzburg-Landau equation. These coherent structures connect stationary laser emission domains having different optical frequencies. Moreover, they emit intensity bursts which travel at a different speed, and, depending on the cavity dispersion sign, they may collide with other coherent structures, thus leading to an overall turbulent dynamics. The dynamics is well-modeled by delay differential equations from which we compute the laser coherence time evolution at each round-trip and quantify the decoherence induced by the collisions between coherent structures.

Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback

We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model.

Short- and long-range temporal cavity soliton interaction in delay models of mode-locked lasers

Interaction equations governing slow time evolution of the coordinates and phases of two interacting temporal cavity solitons in a delay differential equation model of a nonlinear mirror mode-locked laser are derived and analyzed. It is shown that long-range soliton interaction due to gain depletion and recovery can lead either to a development of a harmonic mode-locking regime or to a formation of closely packed incoherent soliton bound state with weakly oscillating intersoliton time separation. Short-range soliton interaction via electric field tails can result in an antiphase or in-phase stationary and breathing harmonic mode-locking regimes.

Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources

We study the collective dynamics in a population of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the population, whereas the feedback from the resources to the population is comprised of components acting homogeneously or inhomogeneously on individual units of the population. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the population and the resources can give rise to collective activity bursting in the population. To explain the mechanisms behind this emergent phenomenon, we combine the Ott-Antonsen reduction for the collective dynamics of the population and singular perturbation theory to obtain a reduced system describing the interaction between the population mean field and the resources.

Conservative Solitons and Reversibility in Time Delayed Systems

Time delayed dynamical systems have proven to be a fertile framework for the study of physical phenomena. In natural sciences, their uses have been limited to the study of dissipative dynamics. In this Letter, we demonstrate the existence of nonlinear reversible conservative time delayed systems. We consider the example of a dispersive microcavity containing a Kerr medium coupled to a distant external mirror. At low energies and in the long delay limit, a multiscale analysis shows the equivalence with the nonlinear Schrödinger equation. We unveil some of the symmetries and conserved quantities, as well as bright temporal solitons. While elastic collisions occur for shallow wave packets, we observe the lack of integrability at higher energies. We recover the Lugiato-Lefever equation in the weakly dissipative regime.

Chaotic Diffusion in Delay Systems: Giant Enhancement by Time Lag Modulation

We consider a typical class of systems with delayed nonlinearity, which we show to exhibit chaotic diffusion. It is demonstrated that a periodic modulation of the time lag can lead to an enhancement of the diffusion constant by several orders of magnitude. This effect is the largest if the circle map defined by the modulation shows mode locking and, more specifically, fulfills the conditions for laminar chaos. Thus, we establish for the first time a connection between Arnold tongue structures in parameter space and diffusive properties of a system. Counterintuitively, the enhancement of diffusion is accompanied by a strong reduction of the effective dimensionality of the system.

Control of single and multiple phase solitons in a ring cavity

Phase solitons are localized structures characterized by phase jumps of 2π or multiples arising in forced ring lasers. Here, we show numerically that they can be created by superimposing to the constant driving field a suitable control beam matched in frequency with a different cavity mode for a time of the order of ten cavity round trip times. If the two beams are separated in frequency by n free spectral ranges of the cavity, a train of solitons like a perfect soliton crystal consisting of n equispaced phase solitons is generated. This may represent a simple way to produce frequency combs with flexible frequency spacing and high power per line.

Robust spike timing in an excitable cell with delayed feedback

The initiation and regeneration of pulsatile activity is a ubiquitous feature observed in excitable systems with delayed feedback. Here, we demonstrate this phenomenon in a real biological cell. We establish a critical role of the delay resulting from the finite propagation speed of electrical impulses in the emergence of persistent multiple-spike patterns. We predict the coexistence of a number of such patterns in a mathematical model and use a biological cell subject to dynamic clamp to confirm our predictions in a living mammalian system. Given the general nature of our mathematical model and experimental system, we believe that our results capture key hallmarks of physiological excitability that are fundamental to information processing.

Pulse-timing symmetry breaking in an excitable optical system with delay

Excitable systems with delayed feedback are important in areas from biology to neuroscience and optics. They sustain multistable pulsing regimes with different numbers of equidistant pulses in the feedback loop. Experimentally and theoretically, we report on the pulse-timing symmetry breaking of these regimes in an optical system. A bifurcation analysis unveils that this originates in a resonance phenomenon and that symmetry-broken states are stable in large regions of the parameter space. These results have impact in photonics for, e.g., optical computing and versatile sources of optical pulses.

Spatiotemporal representation of long-delayed systems: An alternative approach

Dynamical systems with long-delay feedback can exhibit complicated temporal phenomena, which once reorganized in a two-dimensional space are reminiscent of spatiotemporal behavior. In this framework, a normal forms description has been developed to reproduce the dynamics, and the opportunity to treat the corresponding variables as true space and time has since been established. However, recently, an alternative approach has been proposed [F. Marino and G. Giacomelli, Phys. Rev. E 98, 060201(R) (2018)2470-004510.1103/PhysRevE.98.060201] with a different interpretation of the variables involved, which better takes into account their physical character and allows for an easier determination of the normal forms. In this paper, we extend such idea and apply it to a number of paradigmatic examples, paving the way to a rethinking of the concept of spatiotemporal representation of long-delayed systems.

The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback

We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop and their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states and homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains.

Dynamics of a stochastic excitable system with slowly adapting feedback

We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic bursting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance or effectively control the features of the stochastic bursting. The setup can be considered a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker-Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting.

Topological localized states in the time delayed Adler model: Bifurcation analysis and interaction law

The time-delayed Adler equation is the simplest model for an injected semiconductor laser with coherent injection and optical feedback. It is, however, able to reproduce the existence of topological localized structures (LSs) and their rich interactions. In this paper, we perform the first extended bifurcation analysis of this model and we explore the mechanisms by which LSs emerge. We also derive the effective equations governing the motion of distant LSs and we stress how the lack of parity in time-delayed systems leads to exotic, non-reciprocal, interactions between topological localized states.