Nonlinear dynamics and chaos in an optomechanical beam

Optical nonlinearities, such as thermo-optic mechanisms and free-carrier dispersion, are often considered unwelcome effects in silicon-based resonators and, more specifically, optomechanical cavities, since they affect, for instance, the relative detuning between an optical resonance and the excitation laser. Here, we exploit these nonlinearities and their intercoupling with the mechanical degrees of freedom of a silicon optomechanical nanobeam to unveil a rich set of fundamentally different complex dynamics. By smoothly changing the parameters of the excitation laser we demonstrate accurate control to activate two- and four-dimensional limit cycles, a period-doubling route and a six-dimensional chaos. In addition, by scanning the laser parameters in opposite senses we demonstrate bistability and hysteresis between two- and four-dimensional limit cycles, between different coherent mechanical states and between four-dimensional limit cycles and chaos. Our findings open new routes towards exploiting silicon-based optomechanical photonic crystals as a versatile building block to be used in neurocomputational networks and for chaos-based applications.

showing the contributions to gMI while observing the structure from the top and from the bottom (panels a) and b) respectively). Although the contribution from the wings surface and the hole surface have opposite signs, the former is much larger in magnitude, thus giving rise to an overall gMI/2 of 650 kHz. Figure 8. Phonon lasing and self-pulsing. a) Experimental RF spectrum at different values of l. The black and grey curves correspond to experimental RF spectra of a SP/phonon lasing regime involving a mechanical mode at m=54 MHz in the two extremes of the M=1 plateau. The green curve corresponds to SP/phonon lasing with M=2, in which the mechanical oscillation at frequency m is superimposed on a SP trace at frequency m/2. Although the black and grey curves are obtained at two different values of l, the signal is locked at the same frequency m. Here it becomes evident that the mechanical oscillator is not only pumped resonantly, but also that the large amplitude of the coherent mechanical motion acts as a feedback that stabilizes and entrains the SP and the mechanical oscillator. Since m is much more robust than SP, the SP mechanism adapts its frequency to the mechanical one. When the resonant condition with the mechanical oscillation is not fulfilled (red curve) the RF peaks in a frequency-unlocked region are inhomogenously broadened in frequency because the integration time is much greater than the typical period of the signal. b) Simulated phase portraits calculated using the Equations 5, 6 and 7 for equivalent situations to those of panel a). Coherent mechanical oscillations of nanometric amplitude are obtained in the M=1 and M=2 cases, while the red curve is flat in the u-axis. Figure 9. Numerical simulations of phase-space trajectories of SP/phonon lasing and chaos. a) Simulated limit cycle associated with a four-dimensional SP/phonon lasing (l=1531.5 nm). b) Simulated trajectory associated with a chaotic regime (l=1532.7 nm). The trajectory tends to fill a restricted volume of the phase space. The maximum intracavity photon number in both cases is no=10 5 . Figure 10. Simulated frequency spectra in the SP/phonon lasing regime and in the chaotic regime considering two mechanical oscillators. Fast Fourier Transform of the simulated transmitted signal (panel a)) and of the deformation of the first oscillator (u, panel b)), which is the one with m=54 MHz. The black curves are associated with the four-dimensional SP/phonon lasing regime (l=1531.5 nm) while the red curved are associated with the chaotic regime (l=1532.7 nm). In the latter case, it is worth to note that the spectrum of the transmitted signal is broad band, as expected from a chaotic regime. On top of it there are peaks associated with the two mechanical modes in consideration (m=54 MHz and m ' =5 MHz), thus indicating that both of them are in a regime of high amplitude oscillations. Interestingly, the Fast Fourier Transform of the deformation of the first oscillator (red curve of panel b)) indicates that the mechanical oscillation is coherent. On the contrary, the Fast Fourier Transform of either T or N is broad band, that is, chaotic. The maximum intracavity photon number in all cases is no=10 5 . is in a coherent regime. The simulated dynamics of the second oscillator (m ' =5 MHz) displays an oscillation amplitude that is high enough to display an associated peak in the frequency spectrum (see Supplementary Figure 10), but its dynamics have a little influence on the dynamics of the other magnitudes at play. The overall dynamics of the transmitted signal is chaotic, as in the experimental case. Supplementary Reference 3 is one of the few experimental works claiming the observation of chaos in an OM integrated system. Hereafter we enlist several points that may allow comparing it with our works, including the mechanisms reported in the current manuscript and previous reports, that is, Supplementary Reference 7:

Supplementary
Comparison of geometries. Supplementary Reference 3 studies a 2D crystal by exploiting the OM interaction between a slot optical mode and a single breathing mechanical mode. On the other hand, we report on a 1D nanobeam in which several mechanical modes are at stake, enriching some of the dynamical solutions. Indeed, the chaotic states reported in Figs. 4 and 5 involve the activation of an in-plane and an out-of-plane mechanical modes, thus making the chaotic state intrinsically different from that claimed in Supplementary Reference 3. Two inplane flexural modes are also involved in the laser power bi-stability reported on Fig. 3 of the main text.
Bistability and hystheresis. One of the main points of our manuscript is the observation of bistability and hysteresis between bidimensional and four-dimensional limit cycles, between different coherent mechanical states and between four-dimensional limit cycles and chaos. Both the laser wavelength and its power have been varied in different senses to unveil those features. We have also successfully reproduced those features by solving a numerical model of non-linear differential equations. Supplementary Reference 3 does not address bi-stability of any kind.
Comparison of basic dynamics: Pure self-pulsing and phonon lasing. The basic underlying mechanism leading to the dynamics observed in Supplementary Reference 3 is Thermo-Optic/Free-Carrier-Dispersion (TO/FCD) self-pulsing (SP) driven by two-photon absorption. However, as we acknowledge in our manuscript, this dynamics is not novel. Pure SP has been reported in several exclusively-photonic systems during the last decade (see pages 4-6 of Supplementary Reference 4 for a recent review on the topic). To the best of our knowledge, it was first observed a decade ago by Johnson et al. 5. In Supplementary Reference 1 we experimentally demonstrate that SP can couple to a mechanical mode through optical forces in an OM system. As a consequence of that coupling, we also demonstrate phonon lasing. Those experimental observations were reproduced with a numerical model, reported for the first time in our Supplementary Reference 1 in 2015, which is essentially equivalent to that of Supplementary Reference 3 (Eqs. 1-4). Reference 3 omits any reference to SP, including our work from Supplementary Reference 1.
The maximum frequency reached by the pure SP regime is similar in both systems (several tens of megahertz) as we understand it is mainly limited by heat dissipation to the surrounding atmosphere.
Comparison of the interpretation of the different dynamic regimes. By increasing the laser wavelength, the system of Supplementary Reference 10 displays unstable pulses (USP, the authors do not provide further insights), which are not present in our system. Then the system of Supplementary Reference 10 passes through several flat frequency regions at integer fractions of the mechanical modes. The interpretation of the dynamics within those regions is missing in Supplementary Reference 3. In 2015, we reported similar, though much wider, flat frequency regions in Supplementary Reference 1 at comparable laser powers together with the result of a numerical modelling that reproduces the experimental findings and the following interpretation: "When the mechanics/self-pulsing resonant condition is achieved, "flat regions" appear, indicating the coherent vibration of the OM photonic crystal. Since all the dynamics is coupled together, the OM oscillations provide an active feedback that stabilizes the SP. In those specific conditions, the two oscillators are frequency-entrained (FE) in a way that the SP adapts its oscillating frequency to be a simple fraction of the mechanical eigenfrequency. Similarly to the case of synchronized oscillators, the lowest M values have the largest FE zones….".

Comparison of OM interaction strengths. The only mechanical mode at play in Supplementary
Reference 3 happens at almost twice the maximum SP frequency (that is at 112 MHz) and its OM coupling rate is stated to be go,OM /2 = 110 kHz. In our case, the main mechanical mode that is at the heart of the complex dynamics discussed along the manuscript, is an in-plane flexural mode happening at 54 MHz, which falls within the frequency range covered by the pure SP dynamics, and displaying an OM coupling rate of go,OM /2 = 300 kHz. This two combined values enable a much stronger coupling between the SP and the mechanical harmonic oscillator. Indeed, in our work it is possible to use the first harmonic of the optical force to drive the mechanical mode. The stronger coupling appears evident when comparing the much larger frequency entrained regions (frequency plateaus) present in our case (4-6 nm in the best case) with those reported in Ref 3 (less than 1 nm in the best case).

Qualitative comparison of the chaotic dynamics and route towards chaos. Supplementary
Reference 3 claims the observation of chaos at specific laser-cavity detunings. In our opinion, the authors of Supplementary Reference 3 misinterpret the transition among all the previous states as a route towards chaos. A typical route to chaos is studied, for instance, in Supplementary Reference 6, where the dynamical system starts from a limit cycle and becomes chaotic through subsequent period doubling bifurcations as an external parameter is varied. Along the route, the effective dimension of the system must be always higher than two, as stated by the Poicaré-Bendixson theorem. Contrary to what is stated by the authors of Supplementary Reference 3, a SOM state (isolated SP) cannot participate in this route because of being a solution of an effective two-dimensional system. Indeed, in our opinion, the route to chaos in the case of Supplementary Reference 3 should have been studied just in the transition between the fOMO/2 state and the chaos state. Therefore, our opinion is that Figures 3 and 4e of Supplementary Reference 3 do not report a route to chaos.
In our case, we clearly state that the transition to chaos occurs only at high enough powers, is abrupt to the best of the resolution of our tuneable laser (1 pm) and starts from the M=1 coherent state. We also observe that the transition between the M =2 and M =1 at high power follows a route along which the attractor undergoes subsequent period-doubling bifurcations (see lower part of Fig. 4a, Fig. 4b and Fig. 4c).
Comparison of intracavity optical energy thresholds to disclose chaotic dynamics. An intracavity stored optical energy (calculated at perfect laser/cavity resonance) of about 50 fJ is required for observing chaos, which is slightly lower than in Supplementary Reference 10 (60 fJ).

Quantitative comparison of the analysis of the chaotic dynamics. Authors of Supplementary
Reference 3 use the Grassberger-Procaccia (GPA) method, which is known to have been the most popular method used to quantify chaos in the 80s. However, as stated in a later reference (Supplementary Reference 2), it is sensitive to variations in its parameters, for instance, number of data points, embedding dimension, reconstruction delay, and it is usually unreliable except for long, noise-free time series. Hence, the outcomes of the GPA algorithm are questionable.