Mesoscopic chaos mediated by Drude electron-hole plasma in silicon optomechanical oscillators

Chaos has revolutionized the field of nonlinear science and stimulated foundational studies from neural networks, extreme event statistics, to physics of electron transport. Recent studies in cavity optomechanics provide a new platform to uncover quintessential architectures of chaos generation and the underlying physics. Here we report the generation of dynamical chaos in silicon-based monolithic optomechanical oscillators, enabled by the strong and coupled nonlinearities of two-photon-absorption induced Drude electron-hole plasma. Deterministic chaotic oscillation is achieved, and statistical and entropic characterization quantifies the chaos complexity at 60 fJ intracavity energies. The correlation dimension D2 is determined at 1.67 for the chaotic attractor, along with maximal Lyapunov exponent rate about 2.94 the fundamental optomechanical oscillation for fast adjacent trajectory divergence. Nonlinear dynamical maps demonstrate the subharmonics, bifurcations, and stable regimes, along with distinct transitional routes into chaos. This provides a CMOS-compatible and scalable architecture for understanding complex dynamics on the mesoscopic scale.

fractal space and time [3] to self-organization in natural sciences [4], amongst others. Chaos in optical systems has emerged and drawn much attention due to its unique features and broad applications, including chaos-based synchronized secure optical communications [5][6][7], highperformance light detection and ranging [8] and ultrafast physical random bit generation [9].
Studies of chaos generation in III-V laser components have further shown progress in harnessing the broadband carriers in both the near infrared and the mid-infrared wavelength ranges [10][11][12][13][14][15][16][17], although the challenges of monolithic integration and circumventing the seemingly universal requirement of external perturbations remains to be solved.
Driven by centrifugal radiation pressure, optomechanical chaotic quivering was experimentally observed in toroidal whispering-gallery-mode (WGM) microcavities [22]. Recently, in a toroidal WGM microcavity, stochastic resonance and chaos are transferred between two optical fields [23], with the chaotic physical basis through a strong nonlinear optical Kerr response from the nonlinear optical-mechanical modal coupling. This is complemented by recent theoretical studies on chaos including electro-optomechanical systems and potential routes into chaos [24,25]. Here, we couple the prior single optomechanical basis with a second basisthat of electron-hole plasma oscillations in the same cavity -to deterministically generate dynamical chaos in a silicon photonic crystal (PhC) cavity. Differing from the prior studies, the silicon experimental platform enables electron-hole plasma dynamical generation, destabilizing the system dynamics, and provides a route for chip-scale planar electronicphotonic integration. Our photonic crystal implementation is based on a slot-type optomechanical (OM) cavity with sub-wavelength [0.051(/nair) 3 ] modal volumes V, and high quality factor (Q) to V ratios [26,27]. This provides strong optical gradient oscillation [26,28] to achieve operating intracavity energies at approximately 60 fJ, and for near-single-mode operation. Our two-oscillator OM cavity is designed with comparable dynamical oscillation timescales between Drude electron-hole plasma and radiation pressure optomechanics --this allows the chaotic attractors and unique trajectories to be uncovered for the first time. We present the statistical and entropic characteristics of the nonlinear dynamical regimes, and illustrate the transition routes into and out of chaos. Our first-principles numerical modeling, including coupled oscillations in seemingly unrelated degrees-of-freedom -two-photoninduced free-carrier and thermal dynamics with radiation pressure dynamics -capture the experimental observations, the multi-period orbits, and the trajectory divergence into chaotic states.

Results
Experimental observation of chaos. Figures 1a shows the scanning electron micrograph (SEM) of the slot-type optomechanical photonic crystal cavity mediated by Drude electronhole plasma examined in this study. The air-bridged photonic crystal cavity is introduced with shifted-centre air holes that are shifted by 15 nm, 10 nm, and 5 nm, respectively, as shown in Figure 1b. The width-modulated line-defect photonic crystal cavity design has a total quality factor Q of 54,300 ( Figure 1c) and a sub-wavelength modal volume of 0.051(/nair) 3 ( Figure   1b inset) at the 1572.8 nm resonance wavelength (o, with the effective mode index n). The optomechanical cavity consists of two (16.0 μm × 5.5 μm × 250 nm) micromechanical photonic crystal slabs, separated by a 120 nm slot width across the photonic crystal line defect. The inplane mechanical mode has a 112 MHz fundamental resonance and, when driven into the regenerative oscillation regime, has a narrow sub-15-Hz linewidth at ambient pressure and room temperature [29]. The large optical field gradient from the tight slot cavity photon confinement enables a large coherent optomechanical coupling strength g0 of ~ 690 kHz (detailed in Supplementary Note 4 ), resulting in low-threshold optomechanical oscillation (OMO) [26][27][28][29]. Concurrently, on the same cavity, strong nonlinearities such as two-photon absorption, free-carrier and thermo-optic dynamical effects lead to modulation of the intracavity field [30]. Note the characteristic timescales of the OMO and the photonic crystal carrier dynamics are made comparable, through our designed mechanical modes and intrinsic free-carrier diffusion times, enabling the coupled equations of motion to have sufficient overlap and degrees of freedom for chaos generation. Unstable pulses (USP) occur first, before the system is driven into a series of stable subharmonic pulse states such as the fomo/4 states (oscillation period being four times the OMO period), the fomo/3 states, and the fomo/2 states respectively. For detuning  between 2.0 and 2.33 nm, the system observes a chaos region characterized by both a broadband radio frequency (RF) spectrum and an intricate phase portrait. For detuning  greater than 2.33 nm, the system is driven to exit the chaos region by evolving into a fomo/2 state ( = 2.33 to 3.2 nm) before cumulating into a self-induced optical modulation (SOM) state ( = 3.2 to 4.2 nm) [30,31]. Of note, the oscillation period of SOM (about 13 ns to 17 ns), mainly determined by the Drude plasma effect and thermal dissipation rate, is comparable with that of OMO (about 9 ns). The close oscillation frequencies of SOM and OMO facilitate their effective interaction in PhC nanocavity and the occurrence of chaos [4,18].  Figure 2b presents the phase portrait of chaos in a twodimensional plane spanned by the power of temporal waveform (P, horizontal axis) and its first derivative (σ, vertical axis) [32]. The reconstructed trajectory is useful for illustrating the complex geometrical and topological structure of the strange attractor, showing the local instability, yet global stable nature, of a chaos structure [32]. In order to reveal the topological structure of chaos attractors, a state-space procedure is implemented to average the temporal waveform points in a m-dimensional embedded space [32] (detailed in Supplementary Note 1) by removing stochastic noise from the recorded raw data. The noise removal enables a clear depiction of the topological structure of attractor and is also useful for the estimation of correlation dimension and Kolmogorov entropy, the most commonly used measures of the strangeness of chaotic attractor and the randomness of chaos [33][34][35][36]. Furthermore, Figure 2c shows the corresponding RF spectrum, where the signal distributes broadly and extends up to the cutoff frequency of measurement instrumentation, showing a hallmark spectral feature of chaos. Figure 3 illustrates the detailed properties of several different dynamical states, including RF spectra, temporal waveforms, and phase portraits. First, Figure 3a shows the frequency and temporal characteristics of the fomo/2 state. We observe three characteristic features of the fomo/2 state: distinct fomo/2 components in the RF spectrum ( Figure [32][33][34][35][36][37][38]. Details of these measures are provided in Supplementary Note 1. LEs, which describe the divergence rate of nearby attractor trajectories, are the most widely employed criteria in defining chaos [33]. In Figure 4b, we show the calculated LEs, converging to values λ1 ≈ 0.329 ns -1 , λ2 ≈ -0.087 ns -1 and λ3 ≈ -0.946 ns -1 respectively, or equivalently when expressed on the intrinsic optomechanical photonic crystal cavity timescale (τomo = fomo -1 ≈ 8.9 ns) λ1 ≈ 2.94τomo -1 , λ2 ≈ -0. 78τomo -1 and λ3 ≈ -8.45τomo -1 . The maximal LE is positive, illustrating a fast divergence rate between adjacent orbits and indicating that the system is chaotic [32,33]. We further analyze the correlation dimension D2: where CD is the correlation integral of vector size D in an r radius sphere and d is the Euclidian norm distance [36]. A conservative estimate of the attractor correlation dimension is implemented through the Grassberger-Procaccia (G-P) algorithm [36,38] as detailed in Supplementary Note 1. As shown in Figure 4c, the correlation integrals CD vary with sphere radius r. In Figure 4d, the plot of correlation integral slope versus sphere radius r is obtained by extracting the slope from Figure 4c. A clear plateau of the correlation integral slope is observed, supporting the estimated value of D2 at about 1.67 (D2 ≈ 2.0 without noise filtering). The correlation dimension D2 highlights the fractal dimensionality of the attractor and demonstrates the strangeness of the complex geometrical structure [34]. We note that this D2 value is already higher than that of several canonical chaos structures such as the Hénon map (at 1.21), the logistic map (at 0.5), and the Kaplan-Yorke map (at 1.42), and is even close to that of Lorenz chaos (at 2.05) [36].
Furthermore the waveform unpredictability can be characterized by the second-order Renyi approximation of the Kolmogorov entropy K2: where  is the time series sampling rate], a measurement of the system uncertainty and a sufficient condition for chaos [38]. A positive K2 is characteristic of a chaotic system, while a completely ordered system and a totally random system will have K2 = 0 and K2 =  respectively. With the G-P algorithm, K2 is calculated as  0.17 ns -1 or expressed equivalently as ≈ 1.52τomo -1 , representing that the mean divergence rate of the orbit section (with adjoining point pairs in the phase space) is rapid within 1.52 times the fundamental OMO period. It characterizes the gross expansion of the original adjacent states on the attractor [38] and, therefore, indicates the significant unpredictability in the dynamical process of such solid-state systems.
Theoretical simulation of chaos. To further support the physical observations, we model the dynamics of the optomechanical photonic crystal cavity system under the time-domain nonlinear coupled mode formalism, taking into account the OMO oscillation [21], two-photon absorption [31], free-carrier and thermo-optic dynamics [30,31]: where x, A, N and ΔT represent respectively the motional displacement, the intracavity E-field amplitude, the free-carrier density, and the cavity temperature variation. δω = ωL -ω0 is the detuning between injection light ωL and photonic crystal cavity resonance ω0, and Pin is the  [16,18,21].When the drive power is between the SOM and OMO thresholds, TPA-associated amplitude modulations disrupt the OMO rhythm, breaking the closed OMO limit cycles and creating the non-repeating chaotic oscillations. On the other hand, if the frequency ratio between OMO and SOM is close to a rational value, they will lock each other based on the harmonic frequency locking phenomena [39,40].
Consequently, different sub-harmonic fomo states are also observed in Figure 3.

Discussion
We demonstrate chaos generation in mesoscopic silicon optomechanics, achieved through single-cavity coupled oscillations between radiation-pressure-and two-photon-induced freecarrier dynamics. Chaos generation is observed at 60 fJ intracavity energies, with a correlation dimension D2 determined at approximately 1.67. The maximal Lyapunov exponent rate is measured at 2.94 times the fundamental OMO frequency, and the second-order Renyi estimate of the Kolmogorov entropy K2 is determined at 1.9 times the fundamental OMO period -both showing fast adjacent trajectory divergence into the chaotic states. Furthermore, we route the chaos through unstable states and fractional subharmonics, tuned deterministically through the drive-laser detuning and intracavity energies. These observations set the path towards synchronized mesoscopic chaos generators for science of nonlinear dynamics and potential applications in secure and sensing application, in light of recent works about gigahertz optomechanical oscillation [41] and synchronization of coupled optomechanical oscillators [42].     curves (b, c, e, f, h, i, k and l) are the temporal waveforms and orbital phase portraits, where the blue dots are the measured raw data and the solid red curves are the noise-reduced orbital trajectories.  The dashed white horizontal line is an example corresponding to the injected power level in the measurement.