Route to hyperchaos in quadratic optomechanics

Abstract

Hyperchaos is a qualitatively stronger form of chaos, in which several degrees of freedom contribute simultaneously to exponential divergence of small changes. A hyperchaotic dynamical system is therefore even more unpredictable than a chaotic one, and has a higher fractal dimension. While hyperchaos has been studied extensively over the last decades, only a few experimental systems are known to exhibit hyperchaotic dynamics. Here we introduce hyperchaos in the context of cavity optomechanics, in which light inside an optical resonator interacts with a suspended oscillating mass. We show that hyperchaos can arise in optomechanical systems with quadratic coupling and is well within reach of current experiments. We compute the two positive Lyapunov exponents, characteristic of hyperchaos, and independently verify the correlation dimension. We also identify a possible mechanism for the emergence of hyperchaos. As systems designed for high-precision measurements, optomechanical systems enable direct measurement of all four dynamical variables and therefore the full reconstruction of the hyperchaotic attractor. Our results may contribute to better understanding of nonlinear systems and the chaos-hyperchaos transition, and allow the study of hyperchaos in the quantum regime.

Figures (7)

• Figure 1: Quadratic optomechanical coupling. (a) Membrane-in-the-middle system. A thin dielectric membrane with natural frequency $Ω$ oscillates with displacement $x$ within the mode $a$ of a Fabry-Perot cavity, shifting the cavity resonance frequency $ω_c$. The cavity decay rate through the input/output port is $κ$ and the oscillator decay rate is $γ$. (b) Dependence of $ω_c$ on $x$. Only part of the full mode spectrum is shown, with two widely separated branches, originating from the hybridization of the modes of the two half-cavities. Quadratic coupling occurs at $x=0$.
• Figure 2: The Lyapunov exponents of a system with $κ=0.4$, $γ=10^{-3}$, and $Δ=15.5$, for varying $P$. The initial condition is $(x,p,a)=(4,4,0)$. The shaded region highlights the range with two positive LEs, indicating hyperchaos.
• Figure 3: Occurrence of hyperchaos for various system parameters. (a) Values of the maximal LE $λ_1$ and second LE $λ_2$ vs. the detuning $Δ$ and pump power $P$, with fixed cavity decay rate $κ=0.4$. The dashed white lines correspond to Fig. \ref{['fig:lyapunov']}(a). (b) Effect of the linewidth $κ$ for different pumping powers $P$, where the colormaps show, each independently, the largest values of $λ_1, λ_2$ obtained in the range $Δ \in [0,20]$ for a given $(κ,P)$. The other parameters are $γ=10^{-3}$ and initial condition $(x,p,a)=(4,4,0)$.
• Figure 4: Estimation of the fractal dimension, using the corrlation dimension. System parameters are $κ=0.4$, $γ=0.007$, and $Δ=15.5$. The data (red, left axis) is overlaid on the first two Lyapunov exponents (right axis). There is excellent correlation between the second LE (dark gray) becoming positive, indicating hyperchaos, and the dimension exceeding 3, indicated by the dashed line. The dimension is obtained using the Grassberger-Procaccia method. Error bars indicate the standard error of a linear fit to the scaling curve, see Appendix \ref{['sec:numerics']}. Note the increase of $D$ in the small hyperchaotic region near $P=1.5$.
• Figure 5: Fluctuations of the LEs, highlighting unstable dimension variability at the chaos-hyperchaos transition. Parameters are the same as in Fig. \ref{['fig:lyapunov']}. The three largest LEs are computed for finite (but large) time intervals and their distributions, at points denoted by markers, is shown on top of the asymptotic LEs. The distributions are half-transparent, revealing the overlap of $λ_2$ and $λ_3$; their vertical axis corresponds to the LE value and the horizontal is arbitrary. (a) and (b) show chaos-hyperchaos transition, while (c) shows transition from non-chaotic behavior to regular chaos. Note that at some points, particularly in (c), the distributions of $λ_2$ are narrow and hardly visible. The inset in (b) shows a wider view, similar to Fig. \ref{['fig:lyapunov']}, indicating the regions in (a), (b), and (c), left to right respectively.