Steady-state control in an unstable optomechanical system

We demonstrate the stabilization of an unstable steady state in an optomechanical system for low and high pump rates. The optomechanical system consists of a pumped cavity with one movable mirror. The stabilization is achieved by applying Pyragas control, i.e., feeding back the cavity field by an external mirror. This way, a maximal cavity field in the low pumping regime can be achieved, which corresponds to a large displacement of the movable mirror.

Figure 1

The investigated system consists of a cavity, where the cavity mode couples via optomechanical coupling (arising due to the radiation pressure force) to the mechanical mode of a movable cavity mirror. This system is pumped by an external laser. To control the setup, an additional, external mirror is placed outside the cavity introducing feedback into the system, which can be exploited for the Pyragas control scheme.

Figure 2

Stationary states of the normalized mirror displacement as a function of the pump rate $E_0/{\kappa }_{\mathrm{cav}}$, where stable parts are solid (green) curves and unstable ones are dashed (red) curves. For this calculation, we used ${\omega }_{\mathrm{cav}}-{\omega }_{\mathrm{L}}={\omega }_{\mathrm{m}}=11{\kappa }_{\mathrm{cav}}=11\times 2\pi \times 7.1\text{MHz},{\kappa }_{\mathrm{m}}=2\pi \times 2.2\text{MHz}$, and $G=2\pi \times 3.4\text{kHz}$ (from Ref. [27]).

Figure 3

Dynamics of the free system without control $\left(K=0\right)$ for (a) low $\left(E_0/{\kappa }_{\mathrm{cav}}=30000\right)$ and (b) high $\left(E_0/{\kappa }_{\mathrm{cav}}=100000\right)$ pump rates starting near the stationary state with large displacement of the mirror.

Figure 4

Evolution of the displacement of the mirror with control $\left(K=17,{\kappa }_{\mathrm{cav}}\tau =0.3\right)$. (a) For $E_0/{\kappa }_{\mathrm{cav}}=30000$ the system has to be initially near the stationary state with large displacement (solid line), otherwise it ends up in the state with small displacement (dashed line). (b) For $E_0/{\kappa }_{\mathrm{cav}}=100000$ the initial conditions are insignificant for the stabilization of the steady state. Inset: The normalized control force $F=K/{\kappa }_{\mathrm{cav}}|c\left(t\right)-c(t-\tau )|$ vanishes for successful, noninvasive stabilization.

Figure 5

Success of control in the $K-\tau$ plane at $E_0/{\kappa }_{\mathrm{cav}}=30000$: In the white area no steady state is reached, in the shaded (red) area the steady state with small displacement is reached, and in the black area the steady state with large displacement is selected.

Figure 6

Success of control in the $K-\tau$ plane at $E_0/{\kappa }_{\mathrm{cav}}=100000$: In the white areas the control is unsuccessful, whereas it succeeds in the black areas.

Figure 7

Relation between the displacement and pump rate from a dynamical calculation without (solid red line) and with (dashed black line) control: Without control the system never reaches a stationary state with large displacement corresponding to the state with high photon number; with control, the state with large displacement is stabilized and maintained, until dropping below the threshold of the pump rate, where it ceases to exist. By applying the control, the hysteresis expected from a bistable system is recovered. Inset: The time dependence of the pump rate to create the hysteresis.