Nonlocality Induces Chains of Nested Localized Structures

TL;DR

This paper demonstrates that pointwise nonlocality can create chains of nested localized structures in a laser system, revealing a new type of flexible molecular-like behavior.

Contribution

It introduces the concept of nonlocality-induced chains of nested localized structures and provides experimental evidence in a laser system.

Findings

Observation of chains of nested localized structures

Nonlocality leads to flexible, non-rigid bonds in these structures

Experimental validation in a time-delayed laser system

Abstract

Localized Structures often behave as quasi-particles and they may form molecules characterized by well-defined bond distances. In this paper we show that pointwise nonlocality may lead to a new kind of molecule where bonds are not rigid. The elements of this molecule can shift mutually one with respect to the others while remaining linked together, in a way similar to interlaced rings in a chain. We report experimental observations of these chains of nested localized structures in a time-delayed laser system.

Equations

$${\partial }_t\Phi$$

$$\frac{\dot{\Phi }}{2}$$

$$I_x$$

Full text

In a seminal contribution, Alan Turing set the bases of morphogenesis turing52 . He demonstrated that, in a dissipative environment, the interplay between local nonlinearities and differential operators was sufficient to initiate self-organization and generate an infinite variety of patterns. These emergent structures can be found in many physical, biological, and laboratory systems. Among them, localized structures (LSs) are of particular interest and have been widely observed in nature WKR-PRL-84 ; MFS-PRA-87 ; NAD-PSS-92 ; UMS-NAT-96 ; AP-PLA-01 . These states can be individually addressed by a local perturbation, without affecting their surrounding environment. They are particularly relevant for applications when implemented in optical resonators as light bits for information processing RK-OS-88 ; FS-PRL-96 ; BLP-PRL-97 ; 1172836 ; BTB-NAT-02 ; LCK-NAP-10 ; GBG-PRL-08 ; TAF-PRL-08 ; HBJ-NAP-14 . Localized Structures may form bound states, also called “molecules”, via the overlap of their oscillating tails which creates “covalent” bonds corresponding to stable equilibrium distances RK-JOSAB-90 ; AP-PLA-01 ; MFH-PRE-02 .

In this letter we disclose a different kind of molecule composed by chains of nested LSs, which are globally bounded yet locally independent, and similar to an ensemble of interlaced rings. Interesting enough, similar molecular structures exist in chemistry. They are composed by interlocked macrocycles and they are called catenanes FW-JAC-61 . While usual covalent bound states of LSs move as a rigid ensemble when subject to perturbations, the stability analysis of these interlaced LSs reveals that they exhibit several neutral modes corresponding to the individual translation of each element. In other words, small displacements between the components do not relax, yet the ensemble remains stable. We show that these molecules, which challenge the usual notion of local stability for LS compounds, can be obtained in presence of a pointwise nonlocality coupling a field $\Phi\left(x,t\right)$ to a distant point in space $\Phi\left(x+a,t\right)$.

Nonlocality has been widely explored in spatially extended systems and has been shown to induce patterns RDA-PRL-98 , convective instabilities PZ-PRL-05 ; ZP-PRL-07 or N-fold structures RNP-EPL-03 . Distributed nonlocality was recently found capable of stabilizing LSs FCE-PRL-13 and identified as an important mechanism governing the morphogenesis processes in liquid crystal TFC-PRA-15 and vegetation patterns EFC-PRE-15 . Global coupling, as an extreme case of nonlocality, is known to have a deep impact on LSs bifurcation diagrams FCS-PRL-07 .

The difference between standard covalent molecules and the nested states is illustrated in the qualitative sketches of Fig. 1. We represent in Fig. 1a) an elementary LS solution stemming from a generic partial differential equation (PDE) able to sustain LSs in one spatial dimension $\left(x\right)$. The characteristic length of the LS is denoted $l$. A small pointwise nonlocal perturbation can be modeled for instance as a linear perturbation of a PDE that reads

[TABLE]

with $\mathcal{F}$ an operator representing e.g. the Ginzburg-Landau equation as in TF-JPF-88 ; FT-PRL-90 . Without loss of generality we assume $a>0$. For sufficiently small $\varepsilon,$ the overall structure of the original LS remains preserved, but it develops a small echo of amplitude $\varepsilon$ at the distance $a$, as a consequence of the nonlocal term. Additional weaker replica of this echo also appear at distances $na$ ($n\in\mathbb{N})$ with amplitudes $\varepsilon^{n}$, but they can be neglected in this discussion. If $\varepsilon=0,$ standard covalent molecules can be generated by the interaction between the decaying oscillating tails of two nearby LSs. This leads to equilibrium distances $d\sim l$ (not shown). When $\varepsilon\neq 0$, and for moderate nonlocality $a\sim l$, the rightmost LS tail will be modified, which may lead to additional equilibrium distances that persist even when $a\gg l$. Here the binding occurs as a consequence of the interaction between the main pulse of a second LS and the echo of the first, leading to an equilibrium distance $d\sim a$ , see Fig. 1b). All these rigid molecules feature well-determined bond lengths and a single neutral mode corresponding to the translation degree of freedom of the whole ensemble. A novel kind of molecule appears in the case $a\gg l$, when the second LS is placed at a distance $d$ from the first LS such that $l<d<a$. In this case, shown in Fig. 1c), the two LSs are sufficiently far so that they can move independently but, at the same time, the second LS cannot overcome the repulsive barrier induced by the echo of the first LS. As such, the two LSs are globally linked while being locally independent. A useful analogy to picture these situations can be made using rings which are rigidly bounded in the situation represented in Fig. 1b), while they are interlaced in the case of Fig. 1c).

Pointwise nonlocality may not be easy to achieve experimentally and, in order to observe such new bounded states, we have studied their realization in a time-delayed system (TDS). In recent years, building on the strong analogies between spatially extended and time-delayed systems AGL-PRA-92 ; GP-PRL-96 ; K-CMMP-98 , the latter have been proposed for controlling spatial LSs G-PRE-13 , hosting chimera states LPM-PRL-13 , domain walls GMZ-EPL-12 ; GMZ-PRE-13 ; MGB-PRL-14 ; JAH-PRL-15 , vortices YG-PRL-14 and, in particular, temporal LSs MJB-PRL-14 ; GJT-NC-15 ; RAF-SR-16 ; MJB-NAP-15 ; J-PRL-16 , see YG-JPA-17 for a review. The idea that a time delay $\tau$ is akin to a spatial dimension is rooted in the representation developed in AGL-PRA-92 that consists in cutting a single temporal time trace generated by a TDS into $n\in\mathbb{N}$ chunks of duration $T$ (with $T\simeq\tau$) and stack them into a two dimensional map. The horizontal dimension plays the role of a pseudo-spatial variable representing the profile of the pulse within the n-th period while the vertical axis depicts the discrete time index $n$. In some situations GP-PRL-96 , this two-dimensional representation can even lead to an analytical description of a TDS as a PDE containing local operators, as e.g. Laplacian. In this formalism, the temporal profile over one period –the information in speudo-space– is mapped onto the next period. As such, one understands that the inclusion of a second delay $\tau_{2}=a$ induces a pointwise nonlocality coupling each point of the temporal profile to itself, but with a shift $a$. In addition, similar results are to be found for $\tau_{2}=pT+a$ if $p\in\mathbb{N}$ is sufficiently small so that the temporal profile does not evolve much during $p$ periods.

An example of a photonic system with a double time delay capable of hosting LSs and molecules has been recently described MJB-NAP-15 and is summarized in Fig. 2a). A single-transverse mode vertical-cavity surface-emitting laser (VCSEL) is coupled to an external cavity that selects one of the linearly polarized states of the VCSEL (Y, say) and feeds it back twice, once with a time delay $\tau_{f}$, and once with a delay $\tau_{r}$ after being rotated into the orthogonal direction. When $\tau_{f}$ is much larger than the laser timescales and for properly chosen parameters, this system hosts vectorial LSs which correspond to a full rotation of the polarization vector state on the Poincaré sphere. Along the X polarization component, they correspond to upward pulses over a low intensity background while they correspond to anti-phase downward pulses in the Y polarization component, see Fig. 2b,c). The situation depicted in Fig. 2a) is well described by the so-called Spin-Flip Model SFM-PRA-95 supplemented by the inclusion of the delayed re-injection terms. Yet, a multiple timescales analysis of this model applied in the limit of large damping of the relaxation oscillations, weak dichroism $\gamma_{a}$, moderate birefringence $\gamma_{p}$ and feedback rates $\left(\eta,\beta\right)$ leads to a decoupling between the equations for the total intensity $I_{0}$, the population inversion, the field ellipticity and the optical phase, from the equation describing the polarization vector longitude $\Phi$ along the equatorial plane of the Poincaré sphere JMG-PRA-14 . The following simpler delayed equation was found to reproduce well the dynamics,

[TABLE]

with $\Phi^{\tau_{f,r}}=\Phi\left(t-\tau_{f,r}\right)$ the delayed arguments, $z=\alpha\gamma_{p}+\gamma_{a}$, $\left(\bar{\eta},\bar{\beta}\right)=\left(\eta,\beta\right)\sqrt{1+\alpha^{2}}$ with $\alpha$ the Henry’s linewidth enhancement factor. If not otherwise stated, the parameters are $\alpha=2$, $\gamma_{a}=0$, $\gamma_{p}/\pi=4.8\,$GHz, $\eta/\pi=8.68\,$GHz, $\beta/\pi=6.75\,$GHz, $\tau_{f}=3.3\,$ns and $\tau_{r}=3.5\,$ns. The X and Y components of the intensity read

[TABLE]

Figure 2b,c) shows that, in each polarization, the main pulse is followed by a small inverted kink after a time $\Delta\tau=\tau_{r}-\tau_{f}$. This echo is the signature of the nonlocal coupling induced by the additional delay $\tau_{r}$. It bears some similarity with the interaction between temporal LSs in injected Kerr fibers mediated by sound-waves JEM-NAP-13 , although here the effect is fully controllable and, as it will be shown below, it can be used to tune the interaction between LSs. These nonlocal echos create binding forces and allow the existence of LS molecules whose separation between elements is precisely $\Delta\tau$, as illustrated in Fig. 1b) and demonstrated in MJB-NAP-15 . The structure of a typical covalent molecule is depicted in terms of the Stokes parameters and the polarization resolved intensities in Fig. 3a,c). Each localized state is composed by a large polarization kink for $\Phi$ followed by their nonlocal echo at $\Delta\tau$. The second LS is linked to the first via its echo, such that the bonding distance is $\Delta\tau$.

As illustrated in Fig. 1, when $\Delta\tau$ is sufficiently large with respect to the pulse size, a new kind of molecule of interlaced LSs can be observed where the second LS is trapped between the first LS and its echo. This* *molecule of nested LSs is represented in Fig. 3d) as a function of the polarization resolved intensities and in Fig. 3b) in terms of the Stokes parameters. After a large rotation corresponding to the primary peak of the first LS, the system performs the opposed movement in correspondence with the primary peak of the second LS, thus coming back to its original state. Similar but smaller kinks follow as the nonlocal echoes of the two LSs. Accordingly, Fig. 3b) evidences that this nested molecule has the additional property of being composed of a kink and an anti-kink, such that the total topological charge is zero.

Considering the LSs as periodic solutions of a high-dimensional dynamical system permitted us to perform the analysis of their Floquet multipliers. Floquet theory allows to study the linear stability of periodic solutions, see for instance K-TE-08 . The stability analysis performed in MJB-NAP-15 confirmed that the covalent molecule in Fig. 3a,c) possesses a single neutral mode corresponding to the translation invariance of the whole molecule. The results of a similar analysis applied to the nested molecule in Fig. 3b,d) are summarized in Fig. 4. The temporal profile of the solution over which we performed the stability analysis is depicted in Fig. 4a) and one notices in Fig. 4b) not one but two quasi-degenerate Floquet multipliers close to $\mu=1$. We also show in Fig. 4a) the eigenvectors associated to these two neutral modes. One can easily identify them with the temporal derivatives of the kink and of the anti-kink composing the nested molecule, which allows for their individual translation. The residual interactions between LSs, which are always present if their separation is finite, renormalize the Floquet multipliers and explain their small deviation with respect to unity. In the limit $\left(\tau_{f},\tau_{r}\right)\rightarrow\left(\infty,\infty\right)$ we find numerically that they converge to $\mu=1$. In other words our local analysis shows that the two LSs composing the molecule are indeed locally independent. However, it fails to show their global dependence for which the consideration of the whole temporal profile is needed. The local independence of the LSs forming the bound state shown in Fig. 3b,d) contrasts with the rigid behavior of the components of the molecule shown in Fig. 3a,c). It also justifies our heuristic description of the molecule described in Fig. 1c) as interlaced rings.

This evidence can be further supported by analyzing the motion of the structure represented in Fig. 3b,d) over many round-trips in presence of noise. As independent LSs and LS molecules coexist for the same parameter values, they can only be discriminated experimentally by observing their evolution on long timescales. While independent LSs exhibit uncorrelated random walks under the action of noise present in the system, LSs forming covalent molecules behave as a unique rigid body, see for instance Fig. 3a,b) in MJB-NAP-15 . The stochastic evolution of the molecule described in Fig. 3b,d) is shown in Fig. 4d) by using a spatio-temporal diagram where the time trace for the polarization resolved intensity $I_{y}$ is folded over itself at each period $T\simeq\tau_{f}$. Because we are plotting the intensity along the Y direction $I_{y}$, the LS main pulse corresponds to a dip (D) while the echo corresponds to small peak (P) at $\Delta\tau$ from D. In order to help visualizing the fluctuations of the distance between the molecule components, we have represented the LS evolution in the reference frame where the first LS remains static. As a consequence, while Fig. 4d) does not provide any information anymore regarding the random walk of the first LS, but it magnifies the evolution of relative distance between the two LSs. It also shows that, despite the fact that the two LSs can drift one respect to the other, the distance between them remains bounded and the main kink of the second LS is always caught between the main kink of the first LS and its echo. Also, this representation allows to distinguish between two elements catenanes and two independent LSs. The signature of the first corresponds to two dips (D) followed by two peaks (P), i.e. DDPP in Fig. 4d), while two independent LSs would correspond to DPDP.

Our theoretical predictions of the existence of catenane molecules are supported by experimental observations. A wealth of these molecule states has been observed in the experiment using the setup described in Fig. 2a), and they all coexist for the same parameter values. Some examples are illustrated in Fig. 5. Besides two independent LSs in Fig. 5a), we represent in Fig. 5b) a standard covalent molecule found for small values of $\Delta\tau$, where the binding occurs via the nonlocal echo, which corresponds to the situation depicted in Fig. 1b). For larger values of $\Delta\tau$, and besides the simplest catenane corresponding to DDPP (not shown) we depict in Fig. 5c) a situation where three LSs are interlaced, giving the structures DDPDPP. Their local independence can be deduced from their relative uncorrelated motion, similar to that of independent LSs as depicted in Fig. 5a). Counting from left to right, we note that LS2 is interlaced with LS1, while LS3 is interlaced with LS2. A more complex catenane of 6 elements is shown in Fig. 5d). Here, two elements (LS2 and LS3) are very close to each other and their echoes trap a distant element (LS6) driving its diffusion, thus evidencing the binding forces induced by the echo at the source of catenane molecules.

In conclusion, we described how the presence of a pointwise nonlocality in an extended system can give rise to a new kind of molecule of LSs whose elements are simultaneously locally independent and globally locked. Owing to the strong link between spatially extended and delayed systems, we have analyzed the implementation of pointwise nonlocality using a VCSEL enclosed in a double external cavity. The experimental signature of the optical catenane is found to closely match our predictions. We note that other TDSs capable of generating LSs as in MJB-PRL-14 ; GJT-NC-15 , would yield similar catenanes molecules.