Formation and manipulation of optomechanical chaos via a bichromatic driving

We propose a scheme to efficiently manipulate optomechanical systems into and out of chaotic regimes. Here the optical system is coherently driven by a continuous-wave bichromatic laser field consisting of a pump field and a probe field, where the beat frequency of the bichromatic components plays an important role in controlling the appearance of chaotic motion and the corresponding chaotic dynamics. With state-of-the-art experimental parameters, we find that a broad chaos-absent window with sharp edges can be formed by properly adjusting the powers of the bichromatic input field. Moreover, the lifetime of the transient chaos and the chaotic degree of the optomechanical system can be well tuned simply by changing the initial phases of the bichromatic input field. This investigation may be useful for harnessing the optomechanical nonlinearity to manipulate rich chaotic dynamics and find applications in chaos-based communication.

Figure 1

(a) Schematic diagram of an optomechanical system driven by a probe field with frequency ${\omega }_p$ and a pump field whose frequency is detuned by $\Delta$ from the cavity resonance frequency, in which one mirror is fixed and the other is movable. (b) The spectrum of the intracavity field for four different values of the probe-field power $P_p$ at $P_l=60$ mW, $\Omega =0.6{\omega }_m$, and ${\phi }_l={\phi }_p=0$. (c) The trajectory of mechanical motion in phase space at $P_l=60$ mW, $P_p=4.5$ mW, and $\Omega =0.6{\omega }_m$. For the simulations, we employ the following experiment parameters from Ref. [21] throughout this work: $m=20$ ng, ${\omega }_m=2\pi \times 51.8$ MHz, ${\gamma }_m=2\pi \times 41.0$ kHz, $G=-2\pi \times 12$ GHz/nm, $\kappa =2\pi \times 15.0$ MHz, and $\Delta =-{\omega }_m$, respectively. We define the initial value of $\vec{\sigma }=(q,p,c_r,c_i)$ as ${\vec{\sigma }}_0=(q_0,p_0,c_{r_0},c_{i_0})$, where $q_0,p_0,c_{r_0}$, and $c_{i_0}$ correspond to the initial values of $q,p,c_r$, and $c_i$, respectively. We set ${\vec{\sigma }}_0=(0,0,0,0)$ for this figure.

Figure 2

Evolution of $ln\left({\varepsilon }_{I_c}\right)$ and cavity fields in temporal domain and phase space for four different values of the probe laser frequency under the fixed optical powers of the two input lasers ($P_l=60$ mW and $P_p=20$ mW). The phase space here is attained from the plots of the first derivative of $I_c$ in time vs $I_c$. We use ${\phi }_l={\phi }_p=0$ and ${\vec{\sigma }}_0=(0,0,0,0)$, and set the initial value of $\vec{\varepsilon }$ as $\vec{\varepsilon }=({10}^{-20},{10}^{-20},{10}^{-13},{10}^{-13})$ throughout our work.

Figure 3

(a)–(d) Lyapunov exponent varies with $\Omega$ for four different values of $P_l$ under the condition of a fixed $P_p=10$ mW. (e)–(h) Lyapunov exponent varies with $\Omega$ for four different values of $P_p$ at a fixed $P_l=60$ mW. Here, ${\phi }_l=0$ and ${\phi }_p=0$ are used. Lyapunov exponent is taken from the slope of the curve of $ln\left({\varepsilon }_{I_c}\right)$ versus time $t$ (from $0.84$ $\mu \text{s}$ to $1.2$ $\mu \text{s}$) by using linear fitting. We set ${\vec{\sigma }}_0=(0,0,0,0)$ for all panels.

Figure 4

(a)–(d) With respect to the transient chaos ($\Omega =0.6{\omega }_m$, $P_p=20$ mW), the lifetime of chaos is modulated by different relative phases $\phi$. (e) For the chaotic oscillation of cavity field ($\Omega =1.1{\omega }_m$, $P_p=10$ mW), Lyapunov exponent can be tuned by different $\phi$. We use $P_l=60$ mW and ${\vec{\sigma }}_0=(0,0,0,0)$ for all panels.

Figure 5

(a) For the chaotic oscillation of cavity field, Lyapunov exponent does not vary with initial phases of the two input drives for identical $\phi$ when $c_{r_0}=c_{i_0}=0$; that is, Lyapunov exponent does not change under the following three different conditions: (i) ${\phi }_l=\pi /2,{\phi }_p=0$; (ii) ${\phi }_l=\pi ,{\phi }_p=\pi /2$; (iii) ${\phi }_l=3\pi /2,{\phi }_p=\pi$. (b) We set $c_{r_0}=c_{i_0}=100$; then Lyapunov exponent can be modulated by different phases of two input lasers even if $\phi$ remains unchanged. We use $P_l=60$ mW, $P_p=20$ mW, $\Omega =1.1{\omega }_m$, $p_0=0$, and $q_0=0$ for these two panels.