Evidence of a Critical Phase Transition in Purely Temporal Dynamics with Long-Delayed Feedback

Marco Faggian, Francesco Ginelli, Francesco Marino, and Giovanni Giacomelli

Phys. Rev. Lett. 120, 173901 – Published 26 April 2018

Abstract

Experimental evidence of an absorbing phase transition, so far associated with spatiotemporal dynamics, is provided in a purely temporal optical system. A bistable semiconductor laser, with long-delayed optoelectronic feedback and multiplicative noise, shows the peculiar features of a critical phenomenon belonging to the directed percolation universality class. The numerical study of a simple, effective model provides accurate estimates of the transition critical exponents, in agreement with both theory and our experiment. This result pushes forward a hard equivalence of nontrivial stochastic, long-delayed systems with spatiotemporal ones and opens a new avenue for studying out-of-equilibrium universality classes in purely temporal dynamics.

Received 25 July 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.173901

Physics Subject Headings (PhySH)

Critical phenomena Percolation

Nonlinear time-delay systems Semiconductor lasers

Statistical Physics & ThermodynamicsNonlinear Dynamics

Authors & Affiliations

Marco Faggian^1,2, Francesco Ginelli¹, Francesco Marino³, and Giovanni Giacomelli⁴

¹SUPA, Physics Department and Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
²Faculty of Information Studies in Novo Mesto, 8000 Novo Mesto, Slovenia
³Consiglio Nazionale delle Ricerche, Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy
⁴Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy

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Issue

Vol. 120, Iss. 17 — 27 April 2018

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Images

Figure 1
Experimental setup (top). Center and bottom: experimental laser dynamics with an upper polarization initial condition, for a fixed feedback gain and multiplicative noise amplitude; the delay is $τ = 190 ms$ . (a)–(c) P-ST patterns increasing the laser pump current in the absorbed, critical, and active phase, respectively. Darker colors mark the upper polarized state. The horizontal range is $τ / 10$ , and the vertical one spans $2 \times 10^{3}$ delay units. (d) Delay-averaged intensity $ρ (n)$ as a function of PT for different values of the normalized control parameter $Δ$ (see text). The dashed red line is the power law decay for DP scaling theory. (e) Estimated asymptotic intensity $ρ_{\infty}$ vs $Δ$ .
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Figure 2
Experimental laser dynamics with single seed initial conditions (see text). The delay is $τ = 190 ms$ . (a)–(c) P-ST patterns of a single seed in the absorbed, critical, and active phase, respectively. (d) Delay-averaged intensity $ρ (n)$ and (e) survival probability $P (n)$ as a function of PT for different values of the normalized control parameter $Δ$ (see text). Inset: survival probability exponential decay rates $r = 1 / ξ_{∥}$ vs $| Δ |$ in the absorbing phase. The dashed red lines mark the expected DP power law behavior.
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Figure 3
Simulations of Eq. (1) for fully active initial conditions. Typical P-ST patterns in the (a) absorbed ( $a = 0.93$ ), (b) critical ( $a = 0.894 55$ ), and (c) active ( $a = 0.86$ ) phases in a blue ( $x_{t} \approx y_{0}$ ) to red ( $x_{t} \approx y_{1}$ ) color scale. (d) Delay-averaged intensity $ρ (n)$ for different values of $a$ (from top to bottom: $a = 0.88$ , 0.885, 0.89, 0.89455, 0.8985, 0.9045). The delay is $τ = 2 \times 10^{5}$ ; the corrected drift $α = 5.75$ . Data have been further averaged over 5 to 50 independent realizations. Inset: scaling of $ρ_{\infty}$ [estimated by a time-average of $ρ (n)$ over the stationary regime] vs $Δ = | a - a_{c} | / a_{c}$ . (e) Mean absorbing time vs $τ$ for different values of $a$ (from top to bottom: $a = 0.891$ , 0.8947, 0.887). Data are averaged over 400 realizations. The dashed red lines mark the DP scaling behavior (lower graphs are in log-log scale), while our best fits (not shown) estimate $δ \approx 0.16 (1)$ , $β \approx 0.28 (1)$ , $z \approx 1.58 (8)$ .
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Figure 4
Single seed simulations. (a) Characteristic P-ST pattern near the critical point $a_{c}^{s s} \approx 0.894 47 (2)$ . (b) Survival probability $P (n)$ for different values of $a$ (from the top: $a = 0.88$ , 0.885, 0.89, 0.89447, 0.8955, 0.8985, 0.9045). (c) Subcritical PT correlation length $ξ_{∥}$ [estimated through the fit of $P (n) \sim n^{- δ} \exp (- n / ξ_{∥})$ vs $Δ$ ]. Inset: growth of the delay-averaged intensity $ρ (n)$ at the critical point. Data has been averaged over around $1 0^{3} - 1 0^{4}$ different independent realizations. The dashed red lines mark the DP scaling behavior, while our best fits (not shown) estimate $δ \approx 0.159 (4)$ , $ν_{∥} \approx 1.73 (2)$ , $θ \approx 0.32 (1)$ .
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