Mechanical oscillations frozen on discrete levels by two optical driving fields

We report a synchronization phenomenon which is simulated with an optomechanical system driven by two equally strong fields with different frequencies. Once the frequencies of the driving fields are properly matched, the amplitude and phase of the mechanical oscillator coupled to the cavity field, together with its frequency spectrum, will be frozen on one of the determined trajectories like energy levels, and the phenomenon exists for the drive intensity beyond a threshold. Interestingly, the mechanical motion can become highly sensitive to its initial condition and the perturbations during the beginning period of dynamical evolution but, unlike the aperiodicity in chaotic motion, it will nonetheless evolve to one of those fixed trajectories in the end. The scenario may find applications in detecting tiny changes in the environment.

Figure 1

(a) Setup of two drives on a cavity with a fixed mirror and a movable mirror (the mechanical oscillator) connected by a spring. This system exemplifies a general model of two oscillators with the intrinsic frequencies ${\omega }_c$ and ${\omega }_m$ and the damping rates $\kappa$ and ${\gamma }_m$ (${\gamma }_m\ll \kappa$ as in [20]), respectively. The radiation pressure of the cavity field modifies the cavity frequency ${\omega }_c$ with a mechanical displacement $X_m$ much less than the cavity length, realizing a nonlinear interaction potential $V_{\mathrm{eff}}$ between the cavity field and mechanical oscillator. (b i) Stabilized $X_m\left(t\right)$ linear responses to the increase of the drive amplitudes, when their frequencies do not match (${\mathrm{\Delta }}_1=1.002{\omega }_m$ and ${\mathrm{\Delta }}_2=0$). (b ii) $X_m\left(t\right)$ becomes frozen with the fixed $A_1$ and ${\varphi }_1$, once the condition ${\mathrm{\Delta }}_1={\omega }_m$ and ${\mathrm{\Delta }}_2=0$ is satisfied. Here we choose ${\omega }_m=50\kappa$, $g={10}^{-5}\kappa$, and ${\gamma }_m={10}^{-5}\kappa$ for the system, as well as $E_{1\left(2\right)}=2.5\times {10}^5\kappa$, and the existing noises are neglected.

Figure 2

(a) Relation of the average mechanical energy $\langle {\mathcal{E}}_m\rangle =\frac{1}{2}(\langle X_m^2\rangle +\langle P_m^2\rangle )$ to the dimensionless drive amplitude up to $E/\kappa =6.3\times {10}^7$ (in terms of the logarithmic scales). The “quantized” mechanical energy on the displayed levels satisfies a power law $\langle {\mathcal{E}}_m\rangle \left(n\right)\sim n^{2.2}$, as shown in the inset. (b i) One section viewed with a scale on the order of ${10}^3$. (b2) View of another range starting from $E/\kappa ={10}^8$ with a scale of ${10}^{-4}$ (the logarithmic scale on the vertical axis appears uneven). (c) and (d) One-to-one correspondence between the stabilized mechanical oscillations and cavity oscillations. The contours of ${\mathcal{E}}_m\left(t\right)$ oscillating at the frequency ${\omega }_m$ in (c) have their amplitudes $A_{n}d$, on the order of ${10}^{-2}$ of the level positions $\frac{1}{2}(A_n^2+d^2)$. In (d i) we use a dash line to make a single peak distinct.

Figure 3

(a) Evolutions under the fixed drives with $E=3.5\times {10}^7\kappa$, but with different initial conditions $(X_m\left(0\right),P_m\left(0\right))=(0,0)$ for the red curve, $(X_m\left(0\right),P_m\left(0\right))=(4\sqrt{2}\times {10}^{-6},0)$ for the black curve, and $(X_m\left(0\right),P_m\left(0\right))=(6\sqrt{2}\times {10}^{-6},0)$ for the indigo curve. The inset shows the period of reaching the stability. (b) Evolutions of the mechanical energy under the fixed drives with $E=({10}^8+6.01\times {10}^{-4})\kappa$, but with a tiny difference in the initial conditions: $(X_m\left(0\right),P_m\left(0\right))=(0,0)$ for the red curve and $(X_m\left(0\right),P_m\left(0\right))=(0,\sqrt{2}\times {10}^{-9})$ for the indigo curve. The mechanical oscillator may evolve to the level $n=7$ if its initial momentum reaches $P_m\left(0\right)\sim {10}^{-4}$.

Figure 4

(a) and (b) Influence on the evolution courses by the drive amplitude fluctuations in the form $H(\kappa t-\kappa t_d){\xi }_{1\left(2\right)}\left(t\right)$, where ${\xi }_{1\left(2\right)}\left(t\right)$ is a random function generated with matlab, as shown in the inset. The stochastic function changes its value for every step size of $\kappa t=1.5\times {10}^{-5}$. The red curve due to the noisy perturbation added close to the stability almost coincides with the blue one (the evolution course exactly under $E={10}^8\kappa$ without any fluctuation). The black curve due to the perturbation added earlier evolves to another level. (c) Effect of a random drive $H(\kappa t-\kappa t_d){\xi }_m\left(t\right)$ on the mechanical oscillator. The noisy perturbation starting at the beginning ($t_d=0$) changes the evolution course (black curve), but the same noise acting much later will not affect the evolution (red curve).

Figure 5

Increasing drive amplitude $E$ specifies the different dynamical regimes named after the corresponding mechanical responses. The illustrated $X_c\left(t\right)$ and $X_m\left(t\right)$ exemplify the associated dynamical behaviors. In the quasilinearly responding regime the amplitude of $X_m\left(t\right)$ is proportional to $E$; in the transitional regime both the cavity field and mechanical oscillator take a much longer time to reach stability; in the regime of mode locking (the level $n=1$), $X_m\left(t\right)$ is frozen but $X_c\left(t\right)$ becomes proportional to $E$; in the highly sensitive regime a slight variation of $E$ changes both the cavity and mechanical motion (the example shows a transition between $n=5$ and $n=6$, as indicated by the peak numbers in a half period of cavity oscillation).