Chimera states in small optomechanical arrays

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Chimera states in small optomechanical arrays

Karl Pelka, Vittorio Peano, and André Xuereb

Phys. Rev. Research 2, 013201 – Published 25 February 2020

Abstract

Synchronization of weakly coupled nonlinear oscillators is a ubiquitous phenomenon that has been observed across the natural sciences. We study the dynamics of optomechanical arrays—networks of mechanically compliant structures that interact with the radiation pressure force—which are driven to self-oscillation. These systems offer a convenient platform to study synchronization phenomena and have potential technological applications. We demonstrate that this system supports the existence of long-lived chimera states, where parts of the array synchronize while others do not. Through a combined numerical and analytical analysis we show that these chimera states can only emerge in the presence of mechanical frequency heterogeneity.

Received 20 June 2019
Accepted 14 January 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013201

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Chimera states Collective dynamics Optomechanics Synchronization

Nonlinear Dynamics

Authors & Affiliations

Karl Pelka ^1,*, Vittorio Peano ^2,1, and André Xuereb ¹

¹Department of Physics, University of Malta, Msida MSD 2080, Malta
²Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany

^*karl.a.pelka@um.edu.mt

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Vol. 2, Iss. 1 — February - April 2020

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Images

Figure 1
A test-bed for investigating chimera states. (a) Optomechanical microtoroids are arranged in two identical arrays. Excitation of optical modes delocalized over each array causes self-sustained, synchronized mechanical oscillation at large optical powers. A third optical mode introduces springlike coupling between both arrays allowing chimeras to emerge in the compound system. (b) Driving scheme, where each optical mode is driven coherently with a specific detuning. Further detail is given in the text.
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Figure 2
Two-dimensional parameter-space diagram of the synchronization behavior, plotted as a function of the standard deviation of the natural frequencies, $σ$ (horizontal axis), and the maximal number of photons $| α_{max} |^{2}$ (vertical). The statistical evaluation process yielding the color of the smooth diagram is detailed in the text. Upon increasing the optical input power $P_{α} \propto {| α_{max} |}^{2}$ distinct behaviors are observed depending on $σ$ , characterizing the heterogeneity: No self-sustained oscillation (region A, black); self-sustained, synchronized oscillation at one frequency (region B, green); self-sustained, unsychronized oscillation (region C, blue); self-sustained, synchronized oscillation always attained at multiple frequencies (region D, red). This plot can be seen as a two-dimensional projection of a three-dimensional parameter space, with the third axis corresponding to the mechanical coupling strength. In the hatched region, additional global mechanical coupling of two arrays leads to chimera states.
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Figure 3
Numerical analysis of the behavior of two optomechanical arrays consisting of four oscillators and light field each in the absence of (upper and middle row) and with large mechanical coupling (lower row). The chosen parameters correspond to the white diamond in Fig. 2. (a) Time evolution of the mechanical oscillator positions (orange, magenta, green, blue) and light field amplitude (black) (b) Mechanical power spectral densities. In all cases, we find synchronization to one frequency, which in the case of no mechanical coupling is one of the natural frequencies of the oscillators. These natural frequencies are indicated by the vertical lines. In the large-coupling case, both arrays are synchronized in frequency and phase as can be seen by the evolution of the arrays, which are presented on the same plot.
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Figure 4
Sample trajectories of the mechanical motion (orange, magenta, green, blue) for both arrays (white circle in Fig. 2). (a) Phase reconstruction via analytic signal representation and (b) phase velocities extracted from the reconstructed phases. The top (bottom) row refers to the first (second) array in both subfigures. In the absence of mechanical coupling (left), the two arrays synchronize, indicated by the same slope of all phases and the convergence of the phase velocities to horizontal lines. At intermediate mechanical coupling strengths (middle) the oscillators in the second array are synchronized to one frequency while in the first array the phases diverge and the phase velocities stay distinct. This is a chimera state, where the first array oscillates incoherently while the second array is synchronized. When mechanical coupling dominates (right), the two arrays synchronize to the same frequency.
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Figure 5
Stability analysis of chimera states upon variation of the mechanical decay rate $Γ$ (dashed), motional mass $m$ (dotted), and optomechanical coupling $G$ (dash dotted). We see that variations on the scale of percent in each parameter still yield a large proportion of chimera states. This indicates that the symmetry of the indistinguishable arrays is only required approximately for an experiment.
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Figure 6
Stability analysis of chimera states upon variation of the mechanical oscillation frequencies. We see that variations on the scale of a linewidth still result in a significant proportion of chimera states. Thus the symmetry of the indistinguishable arrays can also be broken slightly and still yield chimera states.
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